Embed Size (px)
Citation preview
Some mathematical models of
tumor growth
Benoıt Perthame
Universite Pierre et Marie Curie-Paris 6,CNRS UMR 7598, Laboratoire J.-L. Lions, BC187
4, place Jussieu, F-75252 Paris cedex 5
March 13, 2017
2
Contents
1 Some aspects of tumor growth through ODEs 1
1.1 Cancer: a public health problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Impact of cancer in our society . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Three major challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.3 Complexity of the phenomena . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Birth and death . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Is a bolus the optimal therapy? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Proliferative and quiescent cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5 Proliferative and quiescent cells: therapy . . . . . . . . . . . . . . . . . . . . . . . 11
1.6 Is initial tumor size decay a significant information? . . . . . . . . . . . . . . . . 12
1.7 Proliferative and quiescent cells: global stability . . . . . . . . . . . . . . . . . . . 13
1.8 Angiogenesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.9 Cancer immunotherapy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.10 Fast-slow dynamics; a simple example . . . . . . . . . . . . . . . . . . . . . . . . 20
1.11 Competition between healthy and tumor cells . . . . . . . . . . . . . . . . . . . . 22
1.12 Cancer stem cells and the cancer paradox . . . . . . . . . . . . . . . . . . . . . . 22
1.13 Other aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.13.1 Chronotherapeutics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.13.2 Resistance to therapy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.13.3 Darwinian evolution: The Peto paradox . . . . . . . . . . . . . . . . . . . 25
2 Reaction-diffusion equations: models of invasion 27
2.1 Reaction-diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 Why is Fisher-KPP model different from the heat equation . . . . . . . . . . . . 28
2.3 Elementary properties of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.4 The invasion property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.5 Examples of traveling waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3
2.5.1 Analytical example: the Fisher-KPP equation with ignition temperature . 32
2.5.2 Analytical example: the bistable equation . . . . . . . . . . . . . . . . . . 33
2.5.3 Analytical example: the Fisher-KPP equation . . . . . . . . . . . . . . . . 33
2.6 Invasion property without the subsolution condition (heat equation) . . . . . . . 36
2.7 Invasion property without the subsolution condition (Fisher-KPP equation) . . . 37
3 Transport and advection equations 39
3.1 Transport equation; method of caracteristics . . . . . . . . . . . . . . . . . . . . . 40
3.2 Advection and volumes transformation . . . . . . . . . . . . . . . . . . . . . . . . 42
3.3 Advection (Dirac masses) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.4 Examples and related equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.4.1 Long time behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.4.2 Nonlinear advection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.4.3 The renewal equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.4.4 The Fokker-Planck equation . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.4.5 Other stochastic aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4 Spatial models of tumor growth (compressible) 49
4.1 The simplest fluid biomechanical model . . . . . . . . . . . . . . . . . . . . . . . 49
4.2 The compact support property . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.3 Theoretical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.3.1 Elementary Lq estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.3.2 Contraction property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.3.3 Other a priori estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.4 Theoretical properties (derivatives) . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.5 Theoretical properties (regularizing effect) . . . . . . . . . . . . . . . . . . . . . . 57
5 Variants and extensions of the compressible model 59
5.1 On the law-of-state, viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.2 Active motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.3 Spatial model with nutrient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.4 Necrotic core and the nutrient instability . . . . . . . . . . . . . . . . . . . . . . 61
5.5 Healthy and tumor cells: the seggregation property . . . . . . . . . . . . . . . . . 62
5.6 Proliferative and quiescent models with space . . . . . . . . . . . . . . . . . . . . 64
5.7 Effect of acidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.8 Space and age structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.9 The multiphase flow approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4
6 Hele-Shaw models and free boundary formulation 71
6.1 Tumour spheroids growth (mechanical model) . . . . . . . . . . . . . . . . . . . . 72
6.2 Tumour spheroids growth (nutrients limitation) . . . . . . . . . . . . . . . . . . . 73
6.3 Tumour growth with nutrients (general) . . . . . . . . . . . . . . . . . . . . . . . 75
6.4 Proliferating, quiescent and dead cells . . . . . . . . . . . . . . . . . . . . . . . . 76
6.5 Proliferating and healthy cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
7 From the cell scale to the free boundary problem 79
7.1 Limit in the purely mechanical model . . . . . . . . . . . . . . . . . . . . . . . . 79
7.2 Front movement (purely mechanical) . . . . . . . . . . . . . . . . . . . . . . . . . 81
7.3 Limit with nutrients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
8 Physiologically structured dynamics 85
8.1 A simple example : the renewal equation . . . . . . . . . . . . . . . . . . . . . . . 85
8.2 Renewal equation: Stable Steady Distribution . . . . . . . . . . . . . . . . . . . . 87
8.3 Renewal equation: exponential convergence to SSD . . . . . . . . . . . . . . . . . 88
8.4 Cell divion and size structured equations . . . . . . . . . . . . . . . . . . . . . . . 89
8.5 Phenotypically structured equations . . . . . . . . . . . . . . . . . . . . . . . . . 91
8.6 The Perron-Froebenius theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
8.7 General Relative Entropy for the renewal equation . . . . . . . . . . . . . . . . . 96
8.8 General Relative Entropy for parabolic equation . . . . . . . . . . . . . . . . . . 98
5
Chapter 1
Some aspects of tumor growth
through ODEs
1.1 Cancer: a public health problem
1.1.1 Impact of cancer in our society
In western societies, and despite constant progresses in therapy, cancer becomes the first cause
of death. A major step towards awareness of the problem was the signing of the National Cancer
Act of 1971 by then U.S. President Richard Nixon, generally called ‘the war on cancer’.
Some figures show the importance of the desease
• 1600 Americans die every day from cancer
• since 2004, cancer is the first cause of mortality in France (34% among men, 25% among
women)
• In France the number of cases increased of 20% between 1980 and 2000.
France (2009) USA (2010)
Cancer 30 % 23 %
Heart diseases 21 % 24 %
Chronic lower respiratory 6 % 6 %
CVA 6 % 5 %
Figure 1.1: The causes of mortality in France and in the USA
Women (Fr, 1999) Men (Fr, 1999)
Lung 8.5 % 26 %
Breast/prostate 22 % 10 %
Colon and rectal 12 % 10 %
Figure 1.2: The most dangerous cancers in France
1.1.2 Three major challenges
The paper by Ben Jacob et al [6] mentions three present major challenges in understanding
cancer devopment and treatment
• metastatic colonization,
• dormancy and relapse,
• multiple drug therapy and immune resistance.
Another challenge is to use mathematical models and simulations to predict the evolution of the
disease and the best treatment.
1.1.3 Complexity of the phenomena
Compared to viral deseases, which became major killers with large societies a few hundred years
ago, cancer is an old desease, see Figure 7.1. It seems to be inherent to the complexity of or-
ganisms produced by evolution.
Figure 1.3: Metastatic cancer in a dinosaur bone (Rotschild M., Witzke B. J. and Hershkovitz
I., The Lancet 1999)
To underline the difficulties encountered in the tpic of cancer modleing, we mention below
several elementary aspects of cancer where mathematical models have been proposed.
2
Mutations
• The human genome is of order of 3.2 109 nucleotides pairs. During cell division, one estimates
that there is a copying error for 107 pairs. Therefore such errors are rather frequent.
• Most of our tissus are renewing regularly. Cnacer occurs preferentially in fast renewal tissues
epithelium (as skin, intestine), hematopeitic system (generating blood cells).
Tumor growth
• the name solid tumor is used for tumors that develop within an organ in opposition to liquid
tumors ( blood cancers, leukemia which are not treated here),
• the size of a cell is typically ≈ 10µm, a small solid tumor of 1mm contains ≈ 106 cells, to be
visible on imaging a tumor shoulf reach 107 cells,
• when growing the tumor cells push away the tissue including vasculature; when reaching the
size of ≈ 1mm, cells in the center of the tumor miss nutrients (oxygen, glucosis) and die without
control: this is necrosis (by opposition to apoptosis, the programmed death of cells),
• angiogenesis is the formation of new blood vessels as in response to Vascular Endothelial
Growth Factors excreted by necrotic cells,
• metastasis : escape of tumor cells from the mother tumor, by active motion and through the
vasculature. Figure 1.4 gives an idea of the distribution of metastases,
• ’contact inhibition’ refers to a limited proliferation of cells when they are packed too closely
and membrane receptors induces signals which inhibit proliferation.
Therapy
• cytotoxic and cytostatic drugs act differently on the cells: cytotoxic drugs kill the proliferative
cells and cytostatic drugs just block proliferation. Not only the molecule but also the dose can
induce these two effects,
• some drugs stay in the body and act during several weeks, some drugs are absorbed/eliminated
within a few hours,
• for some drugs toxicity adds up and the total amount of drug during the life is the constraint,
this is called ‘cumulative toxicity’
• it is usual to hear that the early decay profile of tumor is a good predicitve sign for long term
survival. As we will see, this conclusion holds or does not hold, depending on the assumptions
retained for the modeling,
• 40% of cancer therapies undergo resistance to drug (adaptation, mutation, collective reaction)
• two large classes of therapeutic molecules are available
• monoclonal antibodies which bind to molecules as (Tumor Growth Factors, Vascular
Endothelial Growth Factors) and inactivate them,
• Tyrosine Kinase Inhibitors (TKI) which, by binding to cell receptors, avoid activation of
3
intracellular pathways of proliferation and migration.
• 500 molecules (proteins, enzymes) are known to be involved in cancer, for 100 of them we
know inhibitors. Even if we knew all of them, what can we do with 500 drugs?
Figure 1.4: Results of a mathematical model compared to observations for metastatic prolifer-
ation. Work performed by a team in University of Aix-Marseille (D. Barbolosi, A. Benabdallah,
S. Benzekry, F. Hubert).
Based on a study by Koscielny et al (1984) with 2684 patients treated for breast cancer ant the
IGR, the table gives the proportion of patients which develop at least one visible metastasis in
terms the initial tumor size.
1.2 Birth and death
A general feature of living systems, that will be present all along this course, concerns the
processes of birth and death. These are described by general Lotka-Volterra models
d
dtN(t) = N(t)
(b(t)− d(t)
), N(0) = N0,
where
• N(t) denotes the total population density of cells,
• b(t) denotes the birth rate and d(t) the death rate.
Because access to nutrients and space availability control the cell proliferation and death, the
coefficients b and d are usually taken as nonlinear function of N leading to a self-contained
equationd
dtN(t) = N(t) R(N(t)), (1.1)
with R the bulk growth rate. With r > 0 the intrinsic birth rate in conditions where nutrients
4
and space is freely available, R satisfies one of the two conditions
R(0) = r > 0, R′(·) < 0, R(N)→ 0 as N →∞ (unlimited growth),
R(0) = r > 0, R′(·) < 0, R(K) = 0 (with K > 0 the maximal tumor size).
Several nonlinearities are proposed in the litterature which satisfy one of these two considtions.
For instance the classical Verhulst equation (a = 1 below) can be extended to
R(N) = r
[1−
(N
K
)a], a > 0, K > 0, (logistic power). (1.2)
The Gompertz law (Benjamin Gompertz, 1779–1865) is also standard and assumes a faster
growth towards the steady state K
R(N) = b ln
(K
N
), 0 < N(0) < K. (1.3)
Whatever is the nonlinearity R(·), a common feature is the
Proposition 1.1 Solutions of (1.1) are always monotonic. In other words, for R(N0) > 0, one
has ddtN(t) > 0 for all t ≥ 0 and for R(N0) < 0, one has d
dtN(t) < 0 for all t ≥ 0.
With this simplified model the answer is ‘Yes’ to the question of section 1.1.3, ‘Is initial decay
a good predictive sign’.
Proof. We differentiate the equation (1.1) and find that u(t) := ddtN(t) satisfies
d
dtu(t) = u(t)[R(N(t)) +N(t)R′(N(t))] := u(t)g(t).
Its solution u(t) = u0eG(t) with G(t) =∫ t
0 g(s)ds, does not change sign.
Exercise. Identify the decay rate R(N) ≈ bN−a for N 1 so that the solution of (1.1) is
compatible with linear growth of the radius for the large time behaviour, i.e., N(t) ≈ r t3.
Solution. a = 1/3.
Exercise. Show that the Gompertz law (1.3) is the limit a → 0 of the logistic power law
(1.2), with b = ra.
1.3 Is a bolus the optimal therapy?
Setting the problem of optimal therapy. Within the framework of ODEs developed in
section 1.2, one can already address the question of therapy. It is usual to consider that the
5
Figure 1.5: Illustration of the effect of a bolus given therapy (black curve showing a dis-
continuity at each bolus) against the equivalent drug given uniformly (continuous red curve).
Horizontal axis is time and vertical axis is N(t). Left: the dose is not enough to reach the state
N(∞) = 0. Right: the dose is enough to reach N(∞) = 0.
‘effective concentration’ c(t) of therapy that arrives in a tumor acts through a death term, and
using (1.1), this is written
d
dtN(t) = N(t) R(N(t))− cth(t)N(t), N(t = 0) = N0 > 0. (1.4)
The dose cth(t) is limited by the side effects on healthy tissues. Here, we incorporate this
limitation with the single constraint that, within a certain degradation/elimination time T , the
total dose is limited (Maximal Tolerated Dose)∫ t+T
tcth(s)ds ≤ UM (constraints on dose). (1.5)
Notice that with our notations cth(t) scales like an inverse of time and thus UM is a number
without dimension. Because of this given duration T , we will simplify the question and set it as
follows:
What is the choice of cth(t) that minimizes N(T ) in (1.4), (1.5)?
What is a bolus. The bolus is an usual therapeutic protocole that consists in giving the
highest possible amount in a single dose and repeat injections periodically
cbolus(t) =∑k≥0
δ(t− kT )UM .
It makes no sense to put a Dirac mass in the right hand side of (1.4) because N(t) is discon-
tinous then, and we cannot multiply distributions by discontinuous finctions.
6
Lemma 1.2 Assume R(N) ≥ 0 and R ∈ C1 is bounded. If we approximate the bolus, that is
the drug given at once at time T , cth(t) = UMδ(t− T ), as the limit as ε→ 0 of UMε 1IT−ε<t<T,
the case of a bolus has to be defined byddtN(t) = N(t)R(N(t)), N(0) = N0,
N(T+) = N(T−)e−UM .(1.6)
We will use the notations U(t) = lnN(t) and G(U) = R(N). Notice that G′ and R′ have the
same signs.
Proof. Indeed, if we approximate the Dirac mass as cth(t) = UMε 1IT−ε<t<T, we see that for t
close to T , the equation writes
d
dtU(t)) = G(U(t))− cth(t), T − ε < t < T.
Therefore, we have with GM = supU G(U) (assumed to be finite)
U(T − ε)− UM ≤ U(T )) ≤ U(T − ε) +GMε− UM .
In the limit ε→ 0, we find that U(T+) = U(T−)−UM , that means N(T+) = N(T−)e−UM . We
already see that the best injection time, i.e. the maximal efficacy, is to give it at the end of the
period (0, T ) because this is when U is maximal.
Theorem 1.3 (Bolus is an optimal therapy) Assume that R′(N) ≤ 0 in (1.4), then the
bolus at time T is an optimal therapy.
Because we do not impose a limitation on ‖cth(t)‖∞, this result should be understood as the
equivalent of the usual bang-bang optimal control, see [64].
Proof. With the notations above, the equations are written as
d
dtU(t) = G(U(t))− cth(t), U(t = 0) = U0 = ln(N0),
and the equation for the bolus case is
d
dtU(t) = G(U(t)), U(t = 0) = U0, U(T+) = U(T−)− UM .
First, we notice that U(t) < U(t) for 0 ≤ t < T . Indeed, we have
d
dt[U(t)− U(t)] = G(U(t))−G(U(t))− cth(t) ≤ C[U(t)− U(t)]
d
dt[U(t)− U(t)]2+ ≤ 2C[U(t)− U(t)]2+,
7
and because [U(t = 0)− U(t = 0)]2+ = 0, we conclude that [U(t)− U(t)]2+ ≤ 0 for all times.
Second, because of the above and G′ ≤ 0, we can write
d
dt[U(t)− U(t)] = G(U(t))−G(U(t))− cth(t) ≥ −cth(t).
U(T )− U(T−) ≥ −UM ,U(T )− U(T+) = U(T )− U(T+) + U(T−)− U(T−) ≥ −UM + UM ≥ 0.
Exercise. For R′ ≥ 0, show that the optimal strategy is a bolus at time t = 0.
Exercise. [The therapeutic profile is not relevant]
1. For a constant therapy cth = c, compute the relation giving the steady state Nc obtained in
(1.4). Discuss when Nc > 0.
2. Compute the relation for a periodic profile N(t) for a bolus, that is corresponding to
N0 := N(0) = N(T−)e−UM , UM = Tc.
3. Prove that there is a unique solution N0 and N0 > 0 iff Nc > 0.
4. We assume Nc > 0. Prove directly that N0 < Nc, N(T−) > Nc.
[Hint]. 1. R(Nc) = c and Nc > 0 if c < R(0). 2. G(N0) + T = G(N0eUM ) = G(N(T−)), with
G′(N) = 1/(NR(N)). 3. G(y)−G(yUM ) is monotonic. 4. Work in the variable u = ln(N) rather
than N ,∫ T
0 G(u(s))ds = TG(Nc).
General comments 1. These models assume large number of cells. For small numbers, ran-
domness should be included and this leads to use jump processes. These random effects explain
that a tumor can vanish completely after treatment, but for ODE the number of tumor cells
can decrease exponentially fast but cannot vanish exactly.
2. The question of optimal drug infusion is complex and may take into account more physio-
logical effects. For instance, even very short, the infusion length can be optimized as well as the
infusion concentration during injection. For some therapies, circadian rhythms are important,
see [4] and the references therein.
In mathematical term, optimal therapy enters the subject of Optimal Control and many
elaborate tools are available both at the theoretical and numerical level, [64].
1.4 Proliferative and quiescent cells
Obviously not all cells are duplicating because, with a cell cycle around 24 hours, an initial
tumor of 106 cells would give 1012 cells after less than one month; that is a tumor of 10cm. In
8
! ! ! !
Figure 1.6: An example of patient data fit with a P-Q model of tumor growth in a work by the
team INRIA, CNRS, ENS-Lyon ‘numed’.
fact, observations show that most of the cells are in a quiescent state and only a proportion is in
proliferative state. Transitions between these two states are controled by various environmental
conditions as nutrients, space availability, TGF (Tumor Growth Factors).
To take into account this effect one should consider at least two states of cells: proliferative
and quiescent. This leads to write P = F (P )− bP + cQ, proliferative cells,
Q = bP − cQ− dQ, quiescent cells,(1.7)
with, for instance, a logistic power growth
F (P ) = rP
(1−
(P
K
)a ). (1.8)
The size of the tumor is defined as
N(t) = P (t) +Q(t).
The coefficients b > 0 and c > 0 represent the transfer from one compartment to the other
(controled themselves by growth factors and environmental conditions) and d ≥ 0 is the death
rate of cells.
A first observation is
Lemma 1.4 The dynamic (1.7) preserves positivity
P 0 > 0, Q0 > 0 =⇒ P (t) > 0, Q(t) > 0 ∀t > 0.
9
The model is not well defined for P < 0 and, by definition, we extend the nonlinearity(PK
)aby 0 for P ≤ 0.
A simple way to prove our statement is to observe that, at the first time where P or Q vanishes,
then this quantity is increasing (except if both quantities vansih, a case we discard because (0, 0)
is a solution and unique). However, we show another method that is easily adaptable to parabolic
PDEs and that we call ‘Stampacchia’s method’.
Proof. We define w(t) = 12 [P−(t)2 +Q−(t)2] and note that w(0) = 0. Using the chain rule, we
compute
w(t) = −P−P −Q−Q
= −P−[rP − rP
(PK
)a − bP + cQ]−Q−
[bP − cQ− dQ
]≤ rP 2
− + cP−Q− + bQ−P−
because terms as P−P = −P 2− or −P−Q+ are non-positive. Therefore, we conclude that, for
some constant C > 0,
w(t) ≤ C[P 2− +Q2
−] = Cw(t)2.
Because w(0) = 0, and w(t) ≥ 0, we conclude that w(t) ≡ 0.
A second observation is that the system is ‘cooperative’, that means that the interactions
between P and Q are with positive signs (b ≥ 0, c ≥ 0). A consequence is the ‘monotonicity’
property
Lemma 1.5 The dynamic (1.7) is ‘a monotonic operator’, which means
P (t = 0) > 0, Q(t = 0) > 0 =⇒ P (t) > 0, Q(t) > 0 ∀t > 0.
However, this property does not answer positively to the question ‘Is initial behaviour a good
prediction for all time’ because one observes usually N(t) = P (t) +Q(t) and not P (t) and Q(t)
separately. See section 1.6.
Proof. We set u(t) = P (t), v(t) = Q(t). Differentiating equations (1.7) gives u = F ′(P )u− bu+ cv,
v = bu− (c+ d)v.
Arguing according to signs of v(t) and u(t), we compute
12ddt [u(t)2
− + v(t)2−] =
(F ′(P )− b
)u(t)2
− − cvu(t)− − buv(t)− − (c+ d)v(t)2−
≤[ ∣∣F ′(P )− b
]+ c+b
2 + (c+ d)] [u(t)2
− + v(t)2−]
10
Because u(0)2− + v(0)2
− = 0, we conclude that u(t)2− + v(t)2
− ≡ 0.
A second observation is on the linear stability of the two steady states
Lemma 1.6 For d small, the non-zero steady state is linearly stable
P = p0K, p0 =
(1− bd
r(c+ d)
)1/a
, Q =b
c+ dP . (1.9)
The zero steady state (0, 0) is linearly unstable if r > b.
A nonlinear stability theorem can also be proved. See section 1.7
Proof. The linearized equation near (P ,Q) is written with the matrix
D =
−ra− b+ (1 + a) dbc+d c
b −(c+ d)
For d = 0, tr(D) = −(ra+ b+ c+ d) < 0 and det(D) = racK > 0, therefore the two eigenvalues
have negative real parts. By continuity this also holds for d small.
Because Q ≥ 0, near (0, 0) the equation on P gives an increasing value of P when r > b since
P = (r − b)P.
1.5 Proliferative and quiescent cells: therapy
Tumor cells are duplicating more than normal cells and most of the therapies target proliferative
cell populations (and this is a reason why tharapies also hit healthy tissues with fast renewal
rates). To achieve this task, chemotherapies can target specific proteins that are present in one
phase of the cell cycle, that is the different stages a cell has to complete in order to duplicate.
For instance they can target cyclins (proteins that control the transition between phases) and
this will keep the cells in their G1 phase so that they stay quiescent; these are called cytostatic
drugs. More usual is to target the phases S or G2 (synthesis and reparation) when DNA is
duplicating and this can lead to irreversible damage and subsequently apoptosis of the cells;
these are cytotoxic drugs.
11
Accordingly, therapy only acts on proliferative cells. If we denote by ctox and cstat the con-
centrations of cytotoxic and cytostatic drugs, a simple option is to model that as P = F (P )− (b+ cstat)P + cQ− ctoxP (t), proliferative cells,
Q = (b+ cstat)P − cQ− dQ, quiescent cells,(1.10)
We recall the steady states in (1.9), restricting our analysis to a = 1 in order to simplify,
P = K
(1− bd
r(c+ d)
), Q =
b
c+ dP , N = K
(1− bd
r(c+ d)
)(1 +
b
c+ d
).
From these, we observe that cytostatic drugs only increases the value of b and it decreases the
proliferative compartment but the number of quiescent cells increases. For dr 1, the total
tumor size however increases.
Cytotoxic drugs are always efficient because we find
P = K
(1− ctox
r− bd
r(c+ d)
), Q =
b
c+ dP , N = K
(1− ctox
r− bd
r(c+ d)
)(1 +
b
c+ d
).
Exercise. Compute the linearized matrix at (0, 0) and understand what happens.
Exercise. Give conditions on the coefficients so that the condition F ′(P ) > 0 holds.
1.6 Is initial tumor size decay a significant information?
Figure 1.7: Two regimes for the effect of cytotoxic therapy in the P-Q model (1.7)–(1.8) with
a = 1/2. The proliferative cell density increases after the bolus (black), the quiescent cells
(blue) and the sum of the two, the total cell population (red). Left: the total tumor size
decreases by repeated injections. Right Relapse eventhough in the early stage the tumor size
decreases.
Models with prolerative and qiescent cells are used for low grade gliomas (slow dynamics
brain cancer, see Figure 1.6) and an example of practical use in medecine is shown in Figure 1.6.
12
They observe that the coefficient b controls the dynamics after therapy. This can be explained
as follows; we assume that cytotoxic therapy only targets proliferative cells. Given as a bolus
at time T this is modeled as
P (T+) = βP (T−), 0 < β < 1, Q(T+) = Q(T−).
Lemma 1.7 Assume that the steady state in (1.7)–(1.8) satisfies F ′(P ) > 0, F ′(P ) − b < 0.
Then after a strong enough therapy (β close enough to 0) given near the steady state, P (t)
increases and the total number of cells N(t) decreases for t > T close enough to T .
The effect is that in a first stage the tumor size decreases because the quiescent compartment
decreases for a while; it is not touched by therapy itself but proliferative cells do not fill in this
quiescent compartment. However the proliferative cell number N(t) increases just after therapy;
a situation that does not contradict Lemma 1.5. See Figure 1.7.
Proof. Close to this initial state and thanks to our assumption F ′(P )− b < 0, we have
P (t) = F (P (t))− bP (t) + cQ(t) ≈ F (P (T+))− bP (T+) + cQ(T )
≈ [F (P (T+))− bP (T+)]− [F (P (T−))− bP (T+)] > 0
since P (T+) < P (T−).
Then we also write
ddtN(t) = F (P (t))− dQ(t) ≈ F (P (T+))− dQ(T )
≈ F (P (T+))− F (P (T−)) < 0
by Taylor expansion and with Pint ≈ P . For δ small enough this is negative and the result is
proved.
1.7 Proliferative and quiescent cells: global stability
A more mathematical result on the ODE system for proliferative and quiescent cells is the
Theorem 1.8 Assume d = 0. For P (0) ≥ 0, Q(0) ≥ 0 and (P (0), Q(0)) 6= (0, 0), the solution
of system (1.7)–(1.8) satisfies that
P (t) −−−−→t→∞ K, Q(t) −−−−→
t→∞b
cK.
13
Proof. We divide it in several steps.
Step 1. lim supt→∞ P (t) ≤ K, lim supt→∞Q(t) ≤ bcK.
Consider the function Φ(u) := (u−K)+, Φ′(u) = φ(u) := 1Iu>K. We have
d
dt
[Φ(P ) +
b
cΦ(c
bQ)]
= F (P )φ(P )− b[φ(P )− φ(
c
bQ)] [P − c
bQ]≤ 0,
as the sum of two nonpositive terms. Therefore Φ(P ) + bcΦ( cbQ) decreases to a limit which can
only be zero otherwise the right hand side would not vanish (left as an exercise).
Also a consequence is that
−∫ ∞
0F (P (t))1IP (t)>Kdt ≤ Φ(P 0) <∞.
Step 2. N(t) := P (t) +Q(t) −−−−→t→∞ N∞
We add up the two equations and find
d
dtN(t) = F (P (t)) = F (P (t))1IP (t)>K + F (P (t))1IP (t)<K.
We can integrate and, for all T > 0,
N(T )−N(0) =
∫ T
0F (P (t))1IP (t)<Kdt+
∫ T
0F (P (t))1IP (t)>Kdt
From the boundedness of N(t) (by the first step), and the integrability of the first term in the
right hand side (still by the first step), we conclude that∫ ∞0
F (P (t))1IP (t)>Kdt <∞,∫ ∞
0
∣∣F (P (t))∣∣ <∞.
From this we directly conclude the claim of step 2.
Step 3. Q(t) −−−−→t→∞ Q∞.
From the equation on Q, we have
Q(t) = bN(t)− (b+ c)Q(t),d
dt[Q(t)e(b+c)t] = bN(t)e(b+c)t.
Therefore, for t > s we have
Q(t)e(b+c)t −Q(s)e(b+c)s = b
∫ t
sN(σ)e(b+c)σdσ
Fix ε > 0, and choose s = t/2 large enough so that N(t) approaches its limit up to o(t), we find
Q(t) = Q(s)e−(b+c)(t−s) + b (N∞ + o(t))
∫ t
se−(b+c)(t−σ)dσ
14
and this shows that Q(t) has a limit as t→∞
Q(t)→ b N∞1
b+ c:= Q∞.
Therefore P (t) = N(t)−Q(t) also has a limit, and it is immediate to identify its value.
Exercise. Extend the proof of Theorem 1.8 to the case when d > 0.
1.8 Angiogenesis
Figure 1.8: The process of neovasculature development.
The PhD thesis [7] is an excellent presentation on this topic.
After a tumor has reached the size of approximately 1mm, that is around 106 cells, the
nutrients (oxygen, glucosis) do not arrive in the center of the tumor. Cells begin to die without
control: this is called necrosis by opposition to the programmed cell death called apoptosis. The
necrotic cells emit Vascular Endothelial Growth Factors (VEGF) which induce the development
of neovasculature. Then, nutrients can arrive again in higher quantity to the tumor consequently,
it can develop furthermore even though this neovasculature is usually of ‘lower quality’.
The idea to control tumor growth using anti-angiogenetic (AA) drugs comes back to the
70’s, see [31]. In 2004, a monoclonal antibody, bevacizumab (commercial name : Avastin) has
been recognized as having anti-tumoral efficacy, but mostly in combination with traditional
chemotherapy. Several studies arrive to variable conclusions (AA might be better on primary
tumor by accelerate metastatic tumors, scheduling of AA and cytotoxic drugs might play an
15
Figure 1.9: Tumor vascularization is irregular compared to vasculature in normal tissue.
important role). Several explanantions are plaisible: by reducing the vasculature, the chemother-
apy supply to the tumor is also reduced. Recently it has been discovered that AA rather than
reducing vasculature, normalizes it.
For optimal AA therapy see [46], [7].
The Gompertz law for tumor growth has been extended by Hahnfeldt et al. [39, 62]. It
includes a variable ‘carrying capacity’ depending upon access to nutrient. This leads the author
to write ddtN(t) = bN(t) ln
(K(t)N(t)
),
ddtK(t) = cN(t)− dN(t)2/3K(t).
(1.11)
The term cN(t) accounts for the stimulation by VEGF and the negative term takes into account
tumor surface vs tumor volume for the inhibition (it is reduced from a PDE model for the
diffusion of stimulating and inhibiting molecules).
Reducing the analysis to N > 0, > 0, the unique steady state is
K = N =( cd
)3/2. (1.12)
The following result is proved in [7]
Theorem 1.9 If N0 > 0 and K0 > 0, then for all times N(t) > 0 and K(t) > 0 and
(N(t),K(t))→ (N , K) as t→∞.
Proof. This result is a consequence of the two propositions we prove below.
Next, we are going to prove the large time behaviour in two steps. First, we prove that
16
Proposition 1.10 (Upper bound on solutions) Solutions of (1.11) satisfy
max(N(t),K(t)) ≤ max(N0,K0, N),
lim supt→∞
max(N(t),K(t)) ≤ N .
In particular, in this model, it is a wrong intuition that as long as the tumor grows, the
vascularization increases which allows N(t) to grow more, and therefore without limit.
Proof. We set u(t) = max(N(t),K(t)) and compute (because if the max is attained by N(t)
then ddtN(t) ≤ 0)
ddtu(t) = [cN(t)− dN(t)2/3K(t)]1IK>N + bN(t) ln
(K(t)
N(t)
)1IK≤N︸ ︷︷ ︸
≤0
≤ dN(t)2/3 [ cdK(t)1/3 −K(t)]1IK>N
≤ dN(t)2/3K(t)1/3 [ cd −K(t)2/3]1IK>N
≤ dN(t)2/3K(t)1/3 [ cd − u(t)2/3]1IK>N.
Therefore, as long as u(t) > N , we find that u(t) decreases (notice that one of the inequalities
is strict) and the result follows.
Similarly, the healthy state N = 0 is unstable. We have
Proposition 1.11 (Lower bound on solutions) Solutions of (1.11) with N0 > 0, K0 > 0
satisfy
min(N(t),K(t)) ≥ min(N0,K0, N),
lim inft→∞
min(N(t),K(t)) ≥ N .
Proof. Set v(t) = min(N(t),K(t)) and write
ddtv(t) = [cN(t)− dN(t)2/3K(t)]1IK<N + bN(t) ln
(K(t)N(t)
)1IN≤K
≥ dN(t)2/3[ cdK(t)1/3 −K(t)]1IK<N
≥ dN(t)2/3K(t)1/3[ cd −K(t)2/3]1IK<N.
And one concludes as in the first case.
It is immediate to see that the combination of these two propositions proves the Theorem 1.9.
17
Exercise. Compute the equation on u = K/N . Show the a priori bound lim supt→∞ u(t) ≤ α
with α lnα = c/b.
[Hint] ddtu+ bu lnu = c− d K
N1/3 − bKN ln KN + bu lnu = c− d K
N1/3 ≤ c.
Exercise. Extend the proof of Theorem 1.9 to the general systemddtN(t) = rN(t)
[1−
(N(t)K(t)
)a],
ddtK(t) = cN(t)− dN(t)2/3K(t).
The system under consideration is neither competitive, neither cooperative. A consequence is
Exercise. [Non-monotonicity.] Show that the ‘monotonicity’ property of Lemma 1.5 is not
satisfied in this system. In other words N(0) > 0, K(0) > 0 does not imply N(t) > 0, K(t) > 0
for all times. And N(0) < 0, K(0) < 0 does not imply N(t) < 0, K(t) < 0 for all times, and no
other sign combination works.
For an anti-angiogenetic therapy, the control problem is written in terms of K by changing
the last ODE tod
dtK(t) = cN(t)− dN(t)2/3K(t)− cAA(t)K(t).
For a constant therapy, the non zero steady state is
K = N =
(c− cAA
d
)3/2
.
Notice also that including the vasculature quality is also important in view of the discussion
above.
1.9 Cancer immunotherapy
An important mathematical literature has been devoted to describe and analyze the immune
response to cancer development. An interesting structured population approach is developed
in [25, 26] (but too complex to be presented in this chapter). On the other hand, several
ODE models have been studied independently since the earliest 1980 (Sepanova) to the optimal
therapeutic protocol studied in [48] (2012).
Even though tumor cells are not foreign cells to the body, they can be recognized by the im-
mune system because of their many mutations. However, antigene presenting cells (macrophages,
dentritic cells) are not efficient in their task to present these cells to T-cells (lymphocytes).
In this context, the so-called ‘gene therapy’ aims at enhancing the immune response and two
routes are pursued
18
• Increasing the immune system efficiency. For example by reducing Tumor Growth Factors as
TGF-β (that inhibit production of cytokines, these are protein hormones that mediate specific
immunity). A typical example is small interferines RNA therapy, that acts by inhibiting tumor
cells ability to produce TGF, [2].
• Inoculating to patients their own immune cells that have been cultured in laboratory and
stimulated to have specific anti-tumor activity (high cytokine concentrations).
The simplest model is due to Kuznetsov et al [44], see also [50]. In this model
• cells from the immune system, the effector cells, are denoted by E(t),
• tumor cells are denoted by T (t),
and the equations are E = s+ p ETT0+T −mEET − dEE,
T = rT (1− TK )−mTET,
(1.13)
where the different terms have the following interpretations (all parameters are positive)
• −mTET loss of tumor cells, killed by immune system,
• −mEET loss of immune cells in their interaction with cancer cells,
• p ETT0+T Michaelis-Menten law for development of effector cells in response to efficient targeting,
• s and d normal production and decay rate of effector cells,
• rT (1− TK ) logistic growth of the tumor cells.
The case when p ≥ dE is more difficult mathematically. Other wise the inequality E ≤s− (dE − p)E gives immediately an upper bound on E.
Exercise. Show that, with 0 < T (0) < K, E(0) ≥ 0, solutions satisfy
1. T (t) ≥ 0, E(t) ≥ 0,
2. T (t) ≤ K,
3. We define λ := pmTT0
. Write the equation for u(t) = E(t) + λT (t) and show that
u(t) ≤ s+ λ(r + dE)K − u(t) dE .
4. Conclude that lim supt→∞ u(t) ≤ s+λ(r+dE)KdE
and that E(t) is uniformly bounded.
Kirschner and Panetta [42] improved the above model to include the fact that the effector
cells divide in response to a presenting molecule, called cytokine. They arrive to a class of 3 ∗ 3
19
models. Denote by C(t) the cytokine concentration, the model writesE = s+ cT + pE
ECC0+C − dEE,
T = rT (1− TK )−mT
ETT1+T ,
C = pCETT2+T + sC − dCC
(1.14)
This system is studied in , Tsygvintsev et al, [70]. Another system, including healthy cells, is
also proposed by Itik and Banks [41], a model which is being analyzed by Letellier et al, [49].
The qualitative behaviour of solutions of system (1.14) has been studied, see [70]. In particular
solutions can exhibit chaotic behavior. Oscillations with long time dormant periods of illness
were described.
The gene therapy studied in [44] consists in adding effector cells. In the system (1.13), it leads
to change the equation on effectors to
E = s+ pET
T0 + T−mEET − dEE + c(t),
On the medical side, no definitive positive conclusion seems to emerge in general from the
idea of using better the immune system and it remains a major field of research and hope for
progresses in therapy.
1.10 Fast-slow dynamics; a simple example
We take the example of cancer-immune system competition to illustrate the notion of fast-slow
dynamics and give a simple example where the general Tikhonov theorem (see [32]) applies
globally in time.
We suppose that the immune response is faster than tumor growth and we rescale the system,
with a parameter ε > 0, as εEε(t) = s+ p EεTεT0+Tε
−mEEεTε − dEEε,
Tε(t) = rTε(1− TεK )−mTEεTε.
(1.15)
We are going to show that we can formally take ε = 0 in order to describe the limit ε → 0,
that is 0 = s+ p ETT0+T −mEET − dEE,
T = rT (1− TK )−mTET.
(1.16)
Of course, the first equation gives the expression of E(t) as a function of T (t) which can be used
in the second equation to write a single ODE.
20
Theorem 1.12 Assume that p < dE and that the initial data is well-prepared, that means it
does not depend on ε and satisfies
0 = s+ pE(0)T (0)
T0 + T (0)−mEE(0)T (0)− dEE(0).
Then, the solutions of (1.15) satisfy
(i) Eε(t) and Tε(t) are uniformly bounded,
(ii) Eε(t), Tε(t) and Tε(t) are uniformly bounded,
(iii) on each interval [0, t0], Eε(t), Tε(t) and Tε(t) converge uniformly to E(t), T (t) and T (t),
the solution of (1.16) with initial data T (0).
It is also possible to prove a stronger result. This limit is uniform on [0,∞), that means
supt≥0
[|Eε(t)− E(t)|+ |Tε(t)− T (t)|
]−→ε→0
0.
Proof. The point (i) follows from the argument of Section 1.9.
For (ii), we define u(t) = Eε(t), v(t) = Tε(t). We first observe that v(t) is uniformly bounded
as a consequence of step (i) and of the equation on Tε. Then, we write
εu(t) + u(t)
[−p+
pT0
T0 + Tε+mETε + dE
]= v(t)
[p
EεT0
(T0 + Tε)2−mEEε
].
Because the right hand side is bounded by a constant that we denote by R, because u(0) = 0
and −p+ pT0T0+Tε
+mETε + dE ≥ α > 0, with α = dE − p, the solution of this equation remains
uniformly bounded. To see this, notice that the solution of
εz(t) + αz(t) = R, z(0) = 0,
is given by z(t) = Rα
(1− e−αt/ε
). Finally, we can differentiate the equation on Tε to conclude
that Tε(t) is bounded and thus (ii) is proved.
To prove (iii), we use Ascoli-Arzela theorem and extract a subsequence wich converges locally
uniformly. The limit staisfies the system (1.16) with the same initial data, which has a unique
solution. Therefore, all the family converges.
Exercise. Show that the proof extends to a system of equations εuε(t) = F (uε, vε),
vε(t) = G(uε, vε),
with F , G smooth bounded and Lipschitzian functions and, for some constant α > 0,
∂
∂uF (u, v) ≤ −α < 0.
21
1.11 Competition between healthy and tumor cells
Many models have been proposed in order to take into account the competition between healthy
and tumor cells. Much of them are used in combination with spacial organisation and dynamics.
However, the case of an homogeneous system is relevant for ‘liquid tumors’.
Here we take an example from [11] for ‘erythroid leukemia’, they write H = Rin + rHH(1− H+TKH
)− dH H, healthy cells,
T = rTT (1− H+TKT
)− dT T, cancer cells.(1.17)
It is supposed that stem cells produce healthy cells with a constant rate Rin. Because this is a
competition system, we readily check the
Lemma 1.13 Assume that the inequalities hold: Rin + rHH0(1 − H0+T 0
KH) − dH H0 < 0 and
rTT0(1− H0+T 0
KT)− dT T 0 > 0, then for all times it holds
d
dtH(t) < 0,
d
dtT (t) > 0.
1.12 Cancer stem cells and the cancer paradox
The cancer paradox is that therapy may increase the tumor size. An explanation has been
proposed based on the notion of cancer stem cells [Hillen lecture ICIAM, Vancouver july 2011]
and [40].
The stems cells differentiate to cancer cells that also multiply. The assumption is that stems
cells do not react to the drug. The ODE model is U = a RS(U + V ) U, stem cells,
V = (1− a) RS(U + V )U + b RT (V ) V − (d+ c)V, cancer cells,(1.18)
The model includes that stem and cancer cells duplicate untill some homeostatic state is achieved,
stem cells are sensitive to the total tumor size while cancer cells proliferate only subject to
nutrient limitation. To represent that, the smooth growth function of stem cells RS(N) is
assumed to satisfy for some K > 0
R′S(N) < 0 for 0 ≤ N < K, RS(N) = 0 for N ≥ K,
and we assume that
R′T (V ) < 0, RT (0) < d, RT (∞) = 0.
22
The other parameters are
• 0 < a < 1 represent the proportion of dividing stem cells that do not differentiate and 1 − athe proportion that differentiate to cancer cells,
• b > 0 represents the relative growth of cancer cells compared to stems cells,
• d > 0 is the detah rate of cancer cells, and c the therapy concentration.
We make the assumption that the inital state satisfies
0 < U0 := U(0) < K.
Within this framework, the cancer paradox is that for therapy c > 0 the limiting behavior of
the total cancer cells N(t) = U(t)+V (t) can become larger than for c = 0, i.e., without therapy.
0 10 20 30 40 50 60 70 80 90 1000.7
0.8
0.9
1.0
1.1
1.2
1.3
0 10 20 30 40 50 60 70 80 90 1000.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
0 10 20 30 40 50 60 70 80 90 1000.5
1.0
1.5
2.0
2.5
3.0
3.5
Figure 1.10: The model (1.18) explains that the size of a tumor can increase after therapy.
Left: the stem cells U(t). Middle: the cancer cells V (t). Right: the total tumor size N(t).
To explain this, we first analyse the large time behavior for the system and introduce the
steady state Vc+d for U = 0 characterized by
RT (Vc+d) = (c+ d)Vc+d, (1.19)
and we notice that therapy decreases the tumor size
Vc+d < Vd for c > 0.
We have
Lemma 1.14 The solution satisfies that for all t ≥ 0, U(t) ≤ UH and there are two limits
U(t) t→∞
U∞, V (t) −→t→∞
Vc+d
with
U∞ + Vc+d ≥ K.
23
We can now restate the cancer paradox as the fact that U∞+Vc+d can be larger than U∞+Vd.
The case of a bolus is very clear if, before, we have reached the steady state (K − Vd, Vd). Be-
cause V suddently decreases, U + V passes below K, U increases and can reach a value large
enough such that U∞ + Vd.
Proof of Lemma 1.14.
(i) U(t) is clearly below K because ddtU = 0 if U(t) reaches this value. This also allows to check
that V (t) is uniformly bounded. Therefore U and V are uniformly lipschitz continuous.
(ii) Also ddtU(t) ≥ 0, and thus U(t) has a limit as t → ∞ that we call U∞ > 0. Therefore,
integrating the equation on U , we find that∫ ∞0
G(N(t))U(t)dt = U∞ − U0 <∞,∫ ∞
0G(N(t))dt <∞.
Then, because G(N(t)) is also uniformly lipschitz continuous, we conclude that
G(N(t)) t→∞
0.
(iii) Back to the equation on V , we are now reduced to the case of a single equation (with a
vanishing source), and we conclude that V (t) converges as mentioned. This is just because if
V (t) > Vc+d then V decreases for time long enough, if V (t) < Vc+d then V increases.
1.13 Other aspects
1.13.1 Chronotherapeutics
Circadian rhythm is a 24 hours period of activity of our cells that result from control of higher
level activity (food uptake, light/darkness, rest/activity).
The concept of chronotherapeutics takes advantage of this rhythm in order to maximie treat-
ment efficacy on its target while minimizing toxicity on healthy organs. It is used in clinics of
colorectal cancer treated with oxaliplatin (former INSERM team ‘Rythmes biologiques et can-
cer’ for instance). We rrefer the refer to [4, 21] for modeling and analysis of the models based
on structured population dynamics.
To settle the model requires to also consider the interactions and competition between tumor
cells and other cells of the body e.g. healthy tissue cells. This issue is also considered for
immunotherapy in [41].
1.13.2 Resistance to therapy
The same conclusion as in section 1.12, that therapy might become inefficient, can be obtained
including that resistant cells are selected by therapy. Is this a Darwinian selection process of
24
existing cells or of mutant favorized by competition? Is this a collective effect by exchange of
information between cells?
1.13.3 Darwinian evolution: The Peto paradox
Why is cancer so common among species and times? Is it that each cell has a small probabilty
to become cancerous by mutation? If this was true, then whales or elephants should have much
more cancers than mice which is obviously wrong; mice develop tumors typically after 3 years
and whales after 100 years. The risk to develop a cancer is not related to the number of cells:
this is called the Peto paradox (Peto was an epidemiologist).
Evolution has clearly selected organisms that are able to produce enough anti-tumor factors
for their ‘normal’ life time. The search for these factors is of course open.
25
26
Chapter 2
Reaction-diffusion equations: models
of invasion
The simplest models describing space invasion are reaction-diffusion equations. It is widely
used to describe, in a qualitative way, an epidemic propagation or an invasion by a new species
(without predator), but also a combustion wave or the spread of an opinion. The model only
takes into account limited growth (as described in section 1.2) and movement by diffusion.
2.1 Reaction-diffusion
In a reaction-diffusion model, one considers the population number density (or chemical concen-
tration) u(x, t) at position x ∈ Rd (or a subset) and at time t ≥ 0. It is assumed to be driven
by the partial differential equation
∂
∂tu
random motion︷ ︸︸ ︷−D∆u = f(u),︸ ︷︷ ︸
population growth
(Reaction-diffusion equation) (2.1)
where we assume that the C1 nonlinearity f(·) has at least two roots for u ∈ [0, 1],
f(0) = f(1) = 0. (2.2)
The steady state u ≡ 0 means that the invasive species is not present (healthy cells only). The
state u ≡ 1 is when the invasive species has arrived to its maximum number density, the medium
is saturated.
Definition 2.1 The equation is said to be
• Monostable if f(u) > 0 for 0 < u < 1,
• KPP type if it is monostable and f(u) ≤ f ′(0)u, for 0 < u < 1,
• Ignition type if for some 0 < θ < 1, f(u) = 0 for 0 ≤ u ≤ θ, f(u) > 0 for u > θ,
• Bistable if, for some 0 < u < 1, f(u) < 0 for 0 ≤ u ≤ u, f(u) > 0 for u < u < 1.
Ignition type is used when the invador is less efficient at lower number densities, for instance
because it changes the environment to help invasion. The bistable case arises for instance when
sexual reproduction is used, then the encouter of two individuals is necessary and at too low
number densities, reproduction rate cannot overcome the death rate.
2.2 Why is Fisher-KPP model different from the heat equation
It is interesting to see that the qualitative behaviours of solutions of the reaction-diffusion
equation (2.1) differ greatly from solutions of the heat equation which corresponds to f ≡ 0 and
is written∂
∂tu−D∆u = 0, (Heat equation). (2.3)
Solutions of the heat equation spread in space according to the heat kernel
KH(t) =1
(4πDt)d/2e−|x|
2/(4Dt)
because they are given by
u(x, t) = u(0) ∗x KH(t) :=
∫Rdu(y, 0)KH(x− y)dy.
In particular, two properties are given by∫Rdu(x, t)dx =
∫Rdu(x, 0)dx, ‖u(t)‖∞ ≤
1
(4πDt)d/2‖u(0)‖1.
The above control on norms gives a behaviour which is incompatible with traveling waves as
defined in the
Definition 2.2 In one space dimension, a traveling wave is a solution of the type u(x, t) =
v(x − ct) for some c ∈ R and v(−∞) = 1, v(+∞) = 0. The parameter c is called the speed of
the wave.
A traveling wave satisfies the equation
− cv′ −Dv′′ = rf(v), v(−∞) = 1, v(+∞) = 0. (2.4)
Because of translational invariance, that means v(x+a) is also a solution, one usually normalize
it by fixing the value v(0) = 1/2.
The construction of traveling wave is a full subject in itself, and very well treated, which we
do not consider here. Note however that
28
Lemma 2.3 For the heat equation, f ≡ 0, there are no traveling waves.
Proof. Integrating (2.4), we find for f ≡ 0,
cv +Dv′ = C0 = Cst.
The solutions are v(x) = C0c +C1e
−Dcx and none of them satisfies the conditions at infinity.
For reaction-diffusion equations there are traveling waves, and we give examples later in Sec-
tion 2.5.
2.3 Elementary properties of solutions
Existence of solutions follows from the general theory of semi-linear equations. We now recall
some properties
Theorem 2.4 (Existence, uniqueness) Assume (2.2) and that the initial data satisfies
u0 ∈ L1(Rd), 0 ≤ u0 ≤ 1.
Then, there is a unique solution u ∈ C([0,∞);L1(Rd)
)of (2.1) and
0 < u(x, t) < 1, ∀t > 0, x ∈ Rd.
We give two results, which statements are left unaccurate, to explain the a priori bounds
which are behind this theorem.
Proposition 2.5 (Positivity principle) Assume that v0 ≥ 0 and for some constant A, a(x, t) ≤A, then solutions to
∂
∂tv −D∆v ≥ a(x, t)v, v(t = 0) := v0 ≥ 0, (2.5)
are nonnegative, i.e., v(x, t) ≥ 0.
Counter-example. The function v = −√
(t− 1)+ satisfies dvdt = 1
2(t−1)+v. This shows that a
size condition on a(x, t) is needed. It can be relaxed to supx a(x, t) ∈ L1(0, T ) for all T > 0.
Proof. As we did for ODEs (method of Stampacchia), we multiply by −v−, integrate in x
and find12
ddt
∫Rd v−(x, t)2dx ≤ D
∫Rd ∇v−.∇vdx+
∫Rd a(x, t)v−(x, t)2dx
≤ −D∫Rd |∇v−|2dx+A
∫Rd v−(x, t)2dx
≤ A∫Rd v−(x, t)2dx.
With v−(0) = 0, the Gronwall lemma gives that v− = 0 and thus v ≥ 0.
29
Proposition 2.6 (Comparison principle) For two initial data u01, u0
2 as in Theorem 2.4,
associated with two Lipschitzian nonlinearities f1(·), f2(·), one has
u01(·) ≤ u0
2(·)
f1(·) ≤ f2(·)
=⇒ u1(x, t) ≤ u2(x, t), ∀t ≥ 0, x ∈ Rd.
Proof. We set v = u2 − u1, which satisfies
∂∂tv −D∆v = f2
(u2(x, t)
)− f1
(u1(x, t)
)≥ f1
(u2(x, t)
)− f1
(u1(x, t)
)≥ a(x, t)v
with
a(x, t) =f1
(u2(x, t)
)− f1
(u1(x, t)
)u2(x, t)− u1(x, t)
,
and a is bounded because f1 is Lipschitz continuous. Because v(t = 0) ≥ 0 we may apply
Proposition (2.5) and find v(x, t) ≥ 0.
2.4 The invasion property
A mathematical property explains why the reaction-diffusion equation describes an invasion
process: if initialy the population is growing, it will continue to grow for all times.
Theorem 2.7 (Invasion property) Assume u0 is a subsolution to the equation (2.1), that
means −D∆u0 ≤ f(u0), then
∂
∂tu(x, t) ≥ 0, ∀t ≥ 0, x ∈ Rd.
Proof. Set v = ∂∂tu(x, t). It satisfies v0 = ∂
∂tu0(x) = D∆u0 + f(u0) ≥ 0 and
∂
∂tv −D∆v = a(x, t)v, a(x, t) = f ′(u(x, t)).
Therefore, we may apply the Proposition 2.5 and find v(x, t) ≥ 0.
Of course such a statement is also true for the (linear) heat equation, taking f ≡ 0. But there is
no ‘nice’ subsolution to the heat equation. Indeed, in one dimension to fix idea, −D d2
dx2u0 ≤ 0,
means u0 convex. This is not compatible with the other usual assumptions 0 ≤ u0 ≤ 1 or
30
u0 ∈ L1(Rd).
Construction of subsolutions. However for the Fisher-KPP type of nonlinearity f in equa-
tion (2.1), there are bounded subsolutions. For instance, in dimension 1 for f(u) = ru(1 − u),
one can choose
u0 =
α cos(π2
xR) for |x| ≤ R,
0, otherwise.
We claim this is a subsolution if the parameters α > 0, R > 0 stafisfy
0 < α ≤ 1− D
r
( π
2R
)2
Indeed, we have
−D(u0)′′ = D( π
2R
)2u0 −D π
2Rαδ(|x| = R) ≤ D
( π
2R
)2u0 ≤ f(u0),
if we choose, for |x| ≤ R,
D
r
( π
2R
)2≤ 1− u0 = 1− α cos(
π
2
x
R),
and it is enough to take Dr
(π
2R
)2 ≤ 1− α.
This does not work for ignition temperature or bistable cases, but other constructions are
possible as we see it below.
Exercise. For a general nonlinearity which satisfies f ′(0) > 0, give the smallness condition on
R and α ensuring that u0 is a subsolution.
Solution. D(πR
)2 ≤ min0≤u≤αf(u)u .
Exercise. In higher dimension, let v be the first eigenfunction of the Laplace operator, −∆v =
λv, v ∈ H10 (BR), v ≥ 0 where BR is the ball centered at 0 and radius R. Build a subsolution of
the Fisher/KPP equation as u0 = αv. Explain why R has to be large enough.
2.5 Examples of traveling waves
In this section, we take D = 1. We come back on the traveling wave equation on R,
− cv′ − v′′ = f(v), v(−∞) = 1, v(+∞) = 0. (2.6)
We notice that this problem is translational invariant, i.e., v(x + x0) is also a solution for all
x0 ∈ R.
31
2.5.1 Analytical example: the Fisher-KPP equation with ignition tempera-
ture
For θ ∈ (0, 1), µ > 0, consider the discontinuous function
f(u) =
0 for 0 ≤ u < θ,
µ(1− u) for θ < u ≤ 1.(2.7)
We refer to Section 2.1 for the explanation of the terminology ‘ignition temperature’ for this
case.
Lemma 2.8 For f given by (2.7), there is a unique solution (c∗, v) of (2.6) such that v is
decreasing and normalized with v(0) = θ. It satisfies c∗ > 0.
Proof. Thanks to the normalization and because we look for a decreasing solution, for x < 0
we look for a solution with v > θ and the equation reads cv′ + v′′ + µ(1− v) = 0. The solutions
are all of the form v = 1 − w with cw′ + w′′ − µw = 0 and thus w is a linear combination
of two exponential functions. Hence, we simply consider the characteristic polynomial, that is,
λ2 + cλ− µ = 0. It has two roots of which only one is positive. Therefore, the solution which is
decaying to zero at −∞, is given by
v = 1− (1− θ)eλ+x, x ≤ 0, λ+ = λ+(c) :=1
2[−c+
√c2 + 4µ ] > 0.
For x > 0 we look for v < θ and the equation is cv′ + v′′ = 0. It has a solution decaying to
zero only for c > 0, given by
v = θe−cx, x ≥ 0.
It remains to check that v is differentiable at x = 0 (and v′′ has a jump at 0 because of the
discontinuity of f at θ), that is
(1− θ) λ+(c) = θc.
Because 2 ddcλ+(c) = −1 + c√
c2+4µ< 0, there is indeed a unique solution c∗ to this equation.
The explicit formulas show that v is decreasing.
The traveling speed c∗ results from a combination of the solution behind and in front of the
transition point x = 0. For that reason it is called a ‘pushed front’.
Exercise. Prove that a solution which satisfies v ∈ (0, 1) is always decreasing and unique.
32
2.5.2 Analytical example: the bistable equation
We can extend the argument above to the bistable case. For θ ∈ (0, 1), µ > 0, ν > 0, consider
the discontinuous function
f(u) =
−νu for 0 ≤ u < θ,
µ(1− u) for θ < u ≤ 1.(2.8)
Lemma 2.9 For f given by (2.8), there is a unique solution (c∗, v) of (2.6) with v decreasing
and normalized with v(0) = θ.
Proof. Again we may compute the unique solutions of the linear equations in (−∞, 0) and
(0,∞). For x > 0, the equation is v′′+cv′−νv = 0; the characteristic polynomial λ2 +cλ−ν = 0
has a unique negative root that gives us
v(x) = θe−λrx, λr =1
2
[c+
√c2 + 4ν
].
The same occurs for x < 0, the equation is v′′ + cv′ − µ(1 − v) = 0, that is, v = 1 − w with
w′′ + cw′ − µw = 0. This is a linear differential equation and the solutions are exponentials eλx
with λ2 + cλ − µ = 0. Therefore, there is a unique solution which is decaying to 0 at infinity
and it is given by
v(x) = 1− (1− θ)eλlx, λl =1
2
[−c+
√c2 + 4µ
].
To match the derivatives at x = 0, we have to impose
λr(c)θ = (1− θ)λl(c).
Observe that 2 ddcλr(c) = 1 + c√
c2+4ν> 0 and 2 d
dcλl(c) = −1 + c√c2+4µ
< 0 and that the limits
at ±∞ of λr,l are ±∞. This shows there is a unique c that makes the equality.
Again, the traveling speed c∗ results from a combination of the solution behind and in front
of the transition point x = 0 and we still have a pushed front.
Exercise. Build a similar example with u = 0, u = 1 unstable and conclude there is no traveling
wave connecting these states.
2.5.3 Analytical example: the Fisher-KPP equation
For θ ∈ (0, 1), µ > 0, consider the continuous piecewise linear function
f(u) =
µ(1− θ)u for 0 ≤ u ≤ θ,µθ(1− u) for θ ≤ u ≤ 1.
(2.9)
33
Lemma 2.10 For f given by (2.9), there is a minimal speed c∗ = 2√µ(1− θ); for all c ≥ c∗
there is a unique solution (c, v) of (2.6) normalized by v(0) = θ and with v decreasing.
The solution v decays exponentially to 0 with the ’slowest possible’ rate of decay of the corre-
sponding equation (see below, for c = c∗ it is xeλ−x not eλ−x).
Therefore, the situation is very different from the case of Lemma 2.8 where there is a unique
wave speed.
Proof. For x < 0, we want v > θ and the equation is
−cv′ − v′′ = µθ(1− v).
Therefore, we find as in the proof of Lemma 2.8 that the unique solution that tends to 1 at −∞is given by
v = 1− (1− θ)eλ+x, x ≤ 0, λ+ =1
2
[− c+
√c2 + 4µθ
].
For x > 0, the equation is written as cv′ + v′′ + µ(1− θ)v = 0. The new feature is that both
roots to the characteristic polynomial λ2 + cλ + µ(1 − θ) are negative. Thus, there is a one
parameter family of solutions which decay to 0 at infinity
v = θeµ−x + a(eµ+x − eµ−x), x ≥ 0 µ± =1
2
[− c±
√c2 − 4µ(1− θ)
]< 0.
Note that v is positive if, and only if, a ≥ 0.
It remains to check that the derivatives match at x = 0, that is
−(1− θ)λ+ = θµ− + a(µ+ − µ−).
or, making explicit the various expressions, our result is reduced to checking that
−(1− θ)[− c+
√c2 + 4µθ
]= −θ
[c+
√c2 − 4µ(1− θ)
]+ 2a
√c2 − 4µ(1− θ) ,
c− (1− θ)√c2 + 4µθ + θ
√c2 − 4µ(1− θ) = 2a
√c2 − 4µ(1− θ) .
For any c > c∗, the left hand side is a positive quantity (this is left as an exercise). Consequently,
we can compute a unique a > 0 that satisfies this equality. This corresponds to a positive and
decreasing function v.
For c = c∗ see the exercise below.
Here we observe a different situation than before. The minimal traveling speed c∗ is solely
determined by the solution in front of the transition point x = 0 (in fact by the decay at +∞).
For that reason it is called a pulled front.
34
Exercise. Consider the case c = c∗.
1. Show that for x > 0, v is given by θeµ−x + axeµ−x with a > 0.
2. Compute the compatibility relation for the derivatives.
3. Show that there is a unique solution (decaying or with values in (0, 1)).
Exercise. For 0 < c < c∗
1. Prove that there is no positive traveling wave.
2. Prove there may exist traveling waves but they change sign (oscillate) around x = +∞, v ≈ 0.
Exercise. For θ ∈ (0, 1), µ > 0, ν > 0 we define the discontinuous piecewise linear function
f(u) =
ν u for 0 ≤ u < θ,
µ (1− u) for θ < u ≤ 1.(2.10)
We consider the traveling wave problem, that is, to find for which c there is a decreasing solution
v to −v′′(x)− cv′(x) = f
(v(x)
), x ∈ R,
v(−∞) = 1, v(+∞) = 0, v(0) = θ.
(2.11)
We always assume that c > 2√ν.
1. Give the expression of v for x < 0.
2. Give the the one parameter family of decreasing solutions for x > 0 and indicate the condition
for the parameter.
3. Give the matching condition on v′ at x = 0.
4. Characterize the minimal speed c∗ which is defined such that for c > c∗ one can find a
traveling wave, for c < c∗ there is no traveling wave.
Hint. The relation which gives the free parameter is
F (c) := c− (1− θ)√c2 + 4µ + θ
√c2 − 4ν = 2a
√c2 − 4ν > 0.
The function F is increasing in c. Therefore, for ν sufficiently large, the minimal speed is de-
fined by c∗ = 2√ν; this is as long as F (2
√ν) > 0, that is, ν > µ (1−θ)2
θ(2−θ) . For ν smaller, then
F (2√ν) < 0 and c∗ is defined by F (c∗) = 0.
The interest here is to show that c∗ > 2√ν when ν is small, and thus there is an interesting
question to understand the general rule for this minimal speed.
35
2.6 Invasion property without the subsolution condition (heat
equation)
Here, and as a warm-up for more elaborate examples, we give a general result going in the
direction of a generic growth property.
We consider the heat equation with a positive initial data∂∂tu−D∆u = 0, t ≥ 0, x ∈ Rd,
u(x, t = 0) = u0(x) > 0, u0 ∈ L1 ∩ L∞(Rd).(2.12)
We are going to prove that
Theorem 2.11 The solution of the heat equation (2.12) satisfies the universal lower bounds
(independent of the initial data)
D∆ lnu ≥ − d2t,
∂
∂tlnu ≥ − d
2t.
Proof. Because u > 0, we can define v = D lnu, u = ev/D and compute
∂
∂tu =
ev/D
D
∂v
∂t∆u = ev/D
[∆v
D+|∇v|2D2
].
Therefore, v satisfies,∂
∂tv = D∆v + |∇v|2 ≥ Dw, w := ∆v. (2.13)
We compute, taking the Laplacian of this equation,
∂
∂tw = D∆w + 2∇v.∇w + 2
d∑i, j=1
| ∂2
∂xi∂xjv|2 ≥ D∆w + 2∇v.∇w +
2
dw2
because the Cauchy-Schwarz inequality gives us
d∑i, j=1
| ∂2
∂xi∂xjv|2 ≥
d∑i=1
| ∂2
(∂xi)2v|2 ≥ 1
d|∆v|2.
Consider the function W (t) = − d2t . Because it is homogeneous, it satisfies
∂
∂tW =
d
2t2= D∆W + 2∇v.∇W +
2
dW 2.
Because w is a supersolution of the same equation, and w(x, 0) ≥ −∞ = W (0), we conclude that
w ≥W . This is the first statement of Theorem 2.11. The second statement is a consequence of
the first one and of (2.13).
36
2.7 Invasion property without the subsolution condition (Fisher-
KPP equation)
We consider the particular case
f(u) = ru(1− u).
Theorem 2.12 The solutions 0 ≤ u ≤ 1 of the Fisher-KPP equation (2.1) satisfy the universal
lower bounds (independent of the initial data), with rm = rmin(1, 2/d),
D∆ lnu+r
2(1− u) ≥ −d
2
rme−rmt
1− e−rmt,
∂
∂tlnu ≥ −d
2
rm e−rmt
1− e−rmt.
Note that
• as r → 0 we recover exactly the result of Theorem 2.11,
• there is a regularizing effect because these expessions in the right hand sides are −∞ at t = 0,
• because negative exponentials go quickly to zero (in particular for r large), this means that
the growth regime, ∂∂tu ≥ 0, is reached much faster than for the heat equation.
Proof. We define again v = D lnu, u = ev/D and compute as before
∂
∂tv = D∆v + |∇v|2 + rD(1− u) = Dw + |∇v|2 +
rD
2(1− u), w := ∆v +
r
2(1− u). (2.14)
We first compute
∂
∂t∆v = D∆w + 2∇v.∇(∆v) + 2
∑i,j
| ∂2
∂xi∂xjv|2 − rD
2∆u
∂
∂t
r
2(1− u) = −rD
2∆u− r2
2u(1− u)
∂
∂tw = D∆w + 2∇v.∇(∆v) + 2
∑i,j
| ∂2
∂xi∂xjv|2 − rD∆u− r2
2u(1− u).
But we have
−rD∆u = −rD u
[ |∇v|2D2
+∆v
D
]= r[∇v.∇(1− u)− u∆v
],
and thus, using again the Cauchy-Schwarz inequality,
∂
∂tw ≥ D∆w + 2∇v.∇w +
2
d(∆v)2 − ruw.
We write it as∂
∂tw ≥ D∆w + 2∇v.∇w +
2
d
(w − r
2(1− u)
)2 − ruw37
≥ D∆w + 2∇v.∇w +2
d
(w2 − r(1− u)w
)− ruw
= D∆w + 2∇v.∇w +2
dw2 − rw[u+
2
d(1− u)].
Notice that r[u+ 2d(1− u) ≥ rm = rmin(1, 2/d). Consider the function
W (t) = −drm2
e−rmt
1− e−rmt.
It satisfies∂
∂tW = D∆W + 2∇v.∇W +
2
dW 2 − rmW.
Because W is negative, we have, for 0 ≤ u ≤ 1,
−rmW ≤ −rW [u+2
d(1− u)].
The comparison principle gives w := ∆v + r2(1− u) ≥W and the result follows.
38
Chapter 3
Transport and advection equations
We have seen how to describe the diffusion of a population under the effect of random (Brow-
nian) motion of its individuals. Another description of motion is when individuals or particles
follow an underlying fluid that transport them deterministically. We speak of transport when
one follows the positions (trajectories) of individuals/particles and of advection for the resulting
effect on density distribution. But let us mention that the word convection is also used, mostly
when the fluid motion is generated by heat transfer.
We consider a given velocity field v(x, t) ∈ Rd. We call the transport equation, that means the
equation in the strong form, which is to find u : Rd × R→ R such that∂∂tu(x, t) + v(x, t).∇u = 0, t ≥ 0, x ∈ Rd,
u(x, t = 0) = u0(x).(3.1)
We call the advection equation, the equation in divergence form, which is to find n : Rd×R→ Rsuch that
∂∂tn(x, t) + div
(n(x, t)v(x, t)
)= 0, t ≥ 0, x ∈ Rd,
n(x, t = 0) = n0(x).(3.2)
One can build solutions to these equations using the notion of forward characteristics, that
are the solutions of the system of differential equationsX(t; y) = v(X(t; y), t),
X(t = 0; y) = y ∈ Rd.(3.3)
We recall that these trajectories exist, and that y 7→ X(t; y) is one-to-one on Rd, under the
Cauchy-Lipschitz assumptions,∀R > 0, T > 0, ∃M1(R, T ), M2(T ) (two constants), such that, ∀|t| ≤ T
|v(x, t)− v(y, t)| ≤M1(R, T )|x− y|, ∀x, y with |x| ≤ R, |y| ≤ R,
|v(x, t)| ≤M2(T )(1 + |x|) ∀x ∈ Rd.
(3.4)
3.1 Transport equation; method of caracteristics
The solution of equation (3.1) is given by a simple formula using the forward characteristics
Lemma 3.1 (Method of characteristics) For C1 data v and u0, the solution of (3.1) are
C1 and given by
u(X(t; y), t) = u0(y) ∀t ∈ R, ∀y ∈ Rd. (3.5)
In other words, solutions are constant along the characteristics.
Proof. For a C1 fields v and initial data u0, using the regularity theory in the Cauchy-Lipschitz
theory, one can write
ddtu(X(t; y), t) = ∂
∂tu(X(t; y), t) + X(t; y).∇u(X(t; y), t)
= ∂∂tu(X(t; y), t) + v
(X(t; y), t
).∇u(X(t; y), t).
This derivative vanishes if, and only if, the transport equation (3.1) is satisfied.
To define u(x, t) at a given point x ∈ Rd, one needs to invert the mapping y 7→ X(t; y) := x.
This is easy and given by the backward characteristics departing at time t from the position x,dYds (s;x, t) = v(Y (s;x, t), s), s ∈ R,Y (s;x, t) = x ∈ Rd.
(3.6)
The inversion formula reads
x = X (t, Y (0;x, t)) , (3.7)
which yields the variant
u(x, t) = u0(Y (0;x, t)
)∀x ∈ Rd, t ∈ R. (3.8)
From the representation formula (3.8), and because x 7→ Y (0;x, t) is an homeomorphism, we
conclude that
Theorem 3.2 (Weak solutions of the transport equation) For u0 ∈ L∞(Rd), there is a
unique bounded distributional solution of (3.1) given by (3.8) for almost every (x, t).
40
Proof. For existence, consider a sequence of initial data u0n ∈ C1(Rd) such that ‖u0
n‖∞ ≤‖u0‖∞ + 1/n and u0
n −→n→∞ u0 almost everywhere. Then, we define un(x, t) = u0n
(Y (0;x, t)
)and,
passing to the limit n → ∞, we recover the formula (3.1). To see that this defines indeed a
distributional solution, we just pass to the limit in the definition of distributional solutions: for
any test function ϕ ∈ C1c
([0, T ]× Rd
)with ϕ(·, T ) = 0, we have
−∫ T
0
∫Rdun(x, t)
[∂tϕ+ divϕ
]dxdt =
∫Rdu0n(x)ϕ(x, t = 0)dx.
Therefore, using the Lebesgue convergence theorem, we have also
−∫ T
0
∫Rdu(x, t)
[∂tϕ+ divϕ
]dxdt =
∫Rdu0(x)ϕ(x, t = 0)dx. (3.9)
Uniqueness is proved by Hilbert’s duality method. Substracting two solutions, (3.9) holds
with u0 ≡ 0. Using the next section, it remains to build, for ψ ∈ C1c
((0,∞)×Rd
), a solution of
∂tϕ+ div(ϕv) = ψ, t ∈ (0, T ), x ∈ Rd, ϕ(·, T ) = 0.
Notice that the C1 theory uses the additional assumption divv ∈ C1 which is unnecessary
relaxing the assumption ϕ ∈ C1 to Lipschitz continuity.
We also conclude from the representation formula (3.8) several properties
• u0 ≥ 0 =⇒ u(x, t) ≥ 0,
• ‖u(t)‖∞ ≤ ‖u0‖∞,
• the solution is defined for all times t ∈ R, in particular we can choose the ‘initial data’ at any
time T ∈ R• the solution has the same regularity as its initial data (no regularizing effects).
Another extension to velocity fields v which are merely W 1,1 or BV can be developed thanks
to the DiPerna-Lions theory and its extensions, see [27, 1].
A geometric interpretation of the formula in Lemma 3.1 is to take A0 a measurable subset
of Rd. Then, we consider the transported set
A(t) := X(t, y); y ∈ A0. (3.10)
The set A(t) can be determined as the (weak) solution ot equation (3.1)
u(x, t) = 1IA(t), for u0 = 1IA0.
41
Exercise. For a ∈ C1, give the formulaalong the charateristics to solve the equation∂∂tu(x, t) + v(x, t).∇u+ a(x, t)u = 0, t ≥ 0, x ∈ Rd,
u(x, t = 0) = u0(x) ∈ C1.(3.11)
Exercise. For f ∈ C1, give the formulaalong the charateristics to solve the equation∂∂tu(x, t) + v(x, t).∇u+ a(x, t)u = f(x, t), t ≥ 0, x ∈ Rd,
u(x, t = 0) = u0(x) ∈ C1.(3.12)
3.2 Advection and volumes transformation
The geometry of transport by the differential system (3.3) yields another question. How are
densities evolved? In other words, what is the volume of the set A(t) compared to that of A0
in (3.10). This is what equation (3.2) tells us.
The general formula for the solution of equation (3.2) is not as simple than for transport, and
can be derived from the expression
∂
∂tn(x, t) + v(x, t).∇n(x, t) + n(x, t)div v = 0.
Using the proof of Lemma 3.1 one finds, along the characteristics and using the chain rule, that
d
dtn(X(t; y), t) + (div v)(t,X(t; y))n(X(t; y), t) = 0.
and thus
Proposition 3.3 The solution of equation (3.2) is
n(X(t, y), t)
[exp
∫ t
0div v (X(s; y), s)ds
]= n0(y) ∀t ≥ 0, ∀y ∈ Rd. (3.13)
To give an interpretation of this expression, we recall some facts on the differential system (3.3).
We consider the d× d matrix Dxv(X(t; y), t) and the volume transformation
J(y, t) = det
(∂X(t; y)
∂y
). (3.14)
They satisfy respectively the equations
d
dt
∂X(t; y)
∂y= Dxv(t,X(t; y)).
∂X(t; y)
∂y,
42
and (this is not an obvious calculation)ddtJ(y, t) = div v(t,X(t; y)) J(y, t),
J(0, y) = 1.(3.15)
Therefore, we have
J(y, t) = e∫ t0 (div v)(s,X(s,y)) ds = det
(∂X(t; y)
∂y
).
We can summarize these results and write
Theorem 3.4 (Strong solutions) With the Cauchy-Lipschitz assumptions (3.4) and with v ∈C1(R × Rd), div v ∈ C1(R × Rd), and n0 ∈ C1, there is a unique solution n ∈ C1(Rd × R) of
equation (3.2) given by the formula
n(X(t; y), t)J(y, t) = n0(y). (3.16)
It satisfies ∫Rdn(x, t)dx =
∫Rdn0(x)dx,
∫Rd|n(x, t)|dx =
∫Rd|n0(x)|dx. (3.17)
The absolute value (and |n|+ and other related variants) are the only nonlinearities which are
preservec by the flow in general. Compressible flows (see below) are the exception with this
respect.
Exercise. Using the solution of (3.2) along the characteritics and mass conservation, prove that
I(y, t) = exp∫ t
0 div v (X(s; y), s)ds and J(y, t) defined by (3.14) are equal.
[Hint]. Use that n(x, t) can be used as an initial data and thus all reasonable functions can be
attained at time t.
As before, the theory can be extended to distributional solutions and the natural space is L1
Theorem 3.5 (Weak solutions) With the assumptions of Theorem 3.4 for v and for n0 ∈L1(Rd), there is a unique distributional solution n ∈ C(R;L1(Rd)) of equation (3.2) and it
satisfies the equalities (3.17).
A consequence of the theorem is that equation (3.2) transport the densities; indeed coming
back to the transport of sets in (3.10), we have
Vol(A(t)) =∫n(x, t)dx =
∫n(X(t; y))det
(∂X(t;y)∂y
)dy
=∫n0(y)dy = Vol(A0).
43
In other words, the transport equation transports the values but transforms the volumes. The
advection equation rearrange the values so as to preserv the volumes (and thus the densities).
We conclude from the above the properties
• n0 ≥ 0 =⇒ n(x, t) ≥ 0,
• ‖n(t)‖1 = ‖n0‖1,
• the solution is defined for all times t ∈ R, in particular we can choose the ‘initial data’ at any
time T ∈ R,
• the solution has the same regularity as its initial data (no regularizing effects).
A flow that preserves volume is a flow for which J(y, t) ≡ 1 in (3.15), this called an incom-
pressible flow
Definition 3.6 A flow v is incompressible if
div v(x, t) = 0, ∀x ∈ Rd, t ∈ R.
That also means J(y, t) ≡ 1.
The role of compressibility can be seen when computing nonlinear quantities S(n). We have
∂
∂tS(n) + div
(S(n) v
)+ [nS′(n)− S(n)]divv = 0,
In particular, incompressible flows will preserv any Lp norm
d
dt
∫Rdnp(x, t)dx = 0.
The absolute value plays a special role among the choices of nonlinearities because for S(n) =
|n| one has S′(n) = sgn(n) and nS′(n)−S(n) = 0. Therefore, in distribution sense, for solutions
of (3.2), we also have∂
∂t|n(x, t)|+ div(|n(x, t)|v(x, t)) = 0
Exercise. For f ∈ C1, give the formula which solves the equation∂∂tn(x, t) + div(nv(x, t)) = f, t ≥ 0, x ∈ Rd,
u(x, t = 0) = u0(x) ∈ C1.(3.18)
Exercise. Determine the volume of the set A(t) defined by (3.10).
44
3.3 Advection (Dirac masses)
Another point of view gives the same conclusion that mass conservation is a fundamental prop-
erty of the advection equation. We have a simple representation formula for the solution is the
Lemma 3.7 For n0(x) :=K∑k=1
ρ0kδ(x− y0
k), the solution of (3.2) is given by the formula
n(x, t) =K∑k=1
ρ0kδ(x−X(t; y0
k)).
More generally, we may represent the initial data as n0(x) =∫Rd n
0(y)δ(x− y) (replacing the
finite sum by an integral) and we find
n(x, t) =
∫Rdn0(y)δ
(x−X(t; y)
)dy. (3.19)
For this reason we recover that the population density is transported by the flow field v ac-
cording to the equation (3.2).
Proof of Lemma 3.7. Because this is a linear equation, we only have to prove the formula for
one Dirac mass and the weight ρ0 = 1. Then, the definition of a weak solution means that for
all smooth test functions u(x, t) we have, for all T > 0,∫Rdn(x, T )u(x, T )dx−
∫ T
0
∫Rdn(x, t)
[ ∂∂tu(x, t) + v(x, t).∇u(x, t)
]=
∫Rdn0(x)u
(0, x).
We choose for u, a solution of the transport equation (3.1) and find∫Rdn(x, T )u(x, T ) dx = u
(y0, 0) = u(X(T ; y0), T )
by the method of characteristics. see Lemma 3.1. Because for the transport equation we may
choose the initial data at time T , this equality holds true for all C1 functions v(·, T ), which
means that n(x, T ) is a Dirac mass as announced.
3.4 Examples and related equations
3.4.1 Long time behaviour
Theorem 3.8 Assume that there is α > 0 such that
(x− y).(v(x, t)− v(y, t)
)≤ −α|x− y|2, ∀x, y ∈ Rd, t ≥ 0, (3.20)
45
then, there is a position X(t) such that, whatever is the initial data n0, as t→∞,
n(x, t) ≈ ρδ(x−X(t)).
Proof. We recall that n(x, t) = ρδ(x −X(t)) is a solution for the initial data n0(x) = ρδ(x −X(0)).
Consider two solutions with different initial data n01(x), n0
2(x) ≥ 0 (this not a restriction
because of one can always consider separately the positive and negative parts),
∂
∂t
(n1(x, t)n2(y, t)
)+divx
(v(x, t)n1(x, t)n2(y, t)
)+divy
(v(y, t)n1(x, t)n2(y, t)
)= 0, t ≥ 0, x, y ∈ Rd,
Therefore
d
dt
∫|x− y|2n1(x, t)n2(y, t)dxdy = 2
∫(x− y).
(v(x, t)− v(y, t)
)n1(x, t)n2(y, t)dxdy
≤ −2α
∫|x− y|2n1(x, t)n2(y, t)dxdy.
We find that ∫|x− y|2n1(x, t)n2(y, t)dxdy ≤ C0e−2αt.
This means that both ni concentrate at the same Dirac mass ρ(t)δ(x−X(t)). But the mass
conservation tells us that ρ(t) is constant.
Exercise. For v ∈ C1, show that condition (3.20) implies that the symmetric matrix DSv =(∂vi∂xj
+∂vj∂xi
)di,j=1
satisfies: DSv ≤ −2αI.
Exercise. For v ∈ C1 and div v(x, t) ≥ α > 0 show that, for all p > 1
‖n(t)‖p → 0 as t→∞.
3.4.2 Nonlinear advection
Several models in physics and biology lead to consider a the nonlinear drift. Examples are
• n(x, t) denotes the number of cells of size x and v(x, t) is the growth rate of cells which might
depend on environmental conditions (nutrients) which are changed by all the cells, whatever is
their size,
• n(x, t) denotes the number of polymers of length x which can increase or decrease by addition
of monomers, with a rate v which may depend on the number of polymers.
46
We consider here a formalism which uses a given weight ψ ∈ C1(Rr;R+) and measures the
growth rate with
v(x, t) = V (x, I(t)), I(t) =
∫Rdψ(x)n(x, t)dx.
Here, we make the assumption that there is α > 0 such that
(x− y).(V (x, I)− V (y, I)
)≤ −α|x− y|2, ∀x, y ∈ Rd, I ≥ 0.
Theorem 3.9 Assume that∫n0 := M0 and
M0‖DIv‖∞‖Dψ‖∞ < α.
Then, there is a position X(t), independant of the initial data, such that, as t→∞,
n(x, t) ≈ ρδ(x−X(t)).
Proof. For two different initial data n0i , i = 1, 2 we obtain two functions Ii(t). We get (see
before)
1
2
d
dt
∫|x− y|2n1(x, t)n2(y, t)dxdy =
∫(x− y).
(v(x, I1(t))− v(y, I2(t))
)n1(x, t)n2(y, t)dxdy
=
∫(x− y).
(v(x, I1(t))− v(y, I1(t))
)n1(x, t)n2(y, t)dxdy
+
∫(x− y).
(v(y, I1(t)))− v(y, I2(t))
)n1(x, t)n2(y, t)dxdy
≤ −α∫|x−y|2n1(x, t)n2(y, t)dxdy+M0‖DIv‖∞|I1(t)−I2(t)|
(∫|x− y|2n1(x, t)n2(y, t)dxdy
)1/2
.
We also have
M0(I1(t)− I2(t)) =
∫[ψ(x)− ψ(y)]n1(x, t)n2(y, t)dxdy
∣∣I1(t)−I2(t)∣∣ ≤ ‖Dψ‖∞
M0
∫|x−y|n1(x, t)n2(y, t)dxdy ≤ ‖Dψ‖∞
(∫|x− y|2n1(x, t)n2(y, t)dxdy
)1/2
.
Therefore, we control the above quantity as
12ddt
∫|x− y|2 n1(x, t)n2(y, t)dxdy
≤∫|x− y|2n1(x, t)n2(y, t)dxdy
[− α+M0‖DIv‖∞‖Dψ‖∞
].
And we conclude as before.
47
3.4.3 The renewal equation
In many different areas of biology, the age structured equation is used widely. It describes a
population of individuals who have age a at time t. The equation is∂∂tn(a, t) + ∂
∂an(a, t) + d(a)n(a, t) = 0, a ≥ 0, t ≥ 0,
n(a = 0, t) =∫∞
0 b(a)n(a, t)da,
where d and b denote respectively the age dependent death and birth rates.
This equation expresses that age and time evolve with the same speed.
When a characteristic enters the domain where the equation is stated, which is the case here
at a = 0, a boundary condition is needed. Here it is used to express that new born are borned
at age a = 0.
For a theory, see [54].
3.4.4 The Fokker-Planck equation
3.4.5 Other stochastic aspects
48
Chapter 4
Spatial models of tumor growth
(compressible)
Space plays an important role for the development of solid tumors and general ODE models,
as presented earlier, can be extended in various ways for a more detailed description of cell
prolifaration including competition for nutrients, several cell types, various interactions with the
environment as chemical signaling.
This chapter presents classical examples of approaches used in order to include space in con-
tinuous models, an old subject that began with [38]. The course by D. Drasdo gives more details
on ways cells interact and related Individual Based Models.
4.1 The simplest fluid biomechanical model
In order to present the simplest extension of the ODE models, we ignore the nutrients and con-
sider a tumor which is limited only by availlability of space, contact inhibition stops proliferation.
See [16, 59].
We introduce the space position x ∈ Rd (d = 2 or 3 for in vitro experiments, d = 3 for in
vivo tumors) and time t ≥ 0. We ignore the time being healthy cells and just write that cell
proliferation increases the local tissue pressure thus creating a velocity field. We introduce
• n(x, t) ≥ 0, the population density of cells (number of cells per mm3) located at x and time t,
• p(x, t) ≥ 0 represents the pressure induced by the cell number density n,
• v(x, t) ∈ Rd represents the local velocity field which itself drives the cell motion; the Darcy
law relates this velocity field to the pressure gradient,
• G(p) ∈ R represents the growth/death rate of cells which is limited by contact inhibition (and
thus by pressure as postulated by [16]).
The resulting equations are∂∂tn+ div
(nv)
= nG(p(x, t)
), x ∈ Rd, t ≥ 0,
n(x, t = 0) = n0(x) ≥ 0,
v = −∇p(x, t), p(x, t) ≡ Π(n) := κnγ , γ > 1.
(4.1)
We choose to make the following assumptions
• the growth/death rate G(·) satisfies
G(0) = GM > 0, G′(·) < 0, G(K) = 0 for some K > 0. (4.2)
The value K has been called the ‘homeostatic pressure’ ([59]).
• Π(·) denotes the pressure dependency as a function of the cell population density. For most of
our forthcoming developments, we take Π(n) = nγ for some γ > 1 but several other nonlinear-
ities have been proposed (see for instance [3]). In classical fluid mechanics, an explicit relation
p = Π(n) is called the state-of-law and the fluid is said to be compressible.
We may prefer two other ways to write the equation (4.1). For the, first we just use that
nv = −n∇p = −∇A(n) and find∂∂tn−∆A(n) = nG
(p(x, t)
), x ∈ Rd, t ≥ 0,
A(n) := A0nm, m = γ + 1, A0 = κγ
γ+1 .(4.3)
When the growth rate G is ignored, i.e., G ≡ 0, this equation is called the porous medium
equation and it arises in fluid mechanics; very much is known on properties of the solutions and
there is an enormous mathematical litterature on this model, see [71].
For the second, we multiply equation (4.1) by Π′(n) and re-write the equation on n as an
equation on the pressure
∂
∂tp− nΠ′(n)∆p = |∇p|2 + nΠ′(n)G
(p(x, t)
). (4.4)
This equation is very useful and this is because several properties, as regularity exponents, can
be different on p and n. If we specialize the state-of-law to Π(n) = nγ , we find
∂
∂tp− γp∆p = |∇p|2 + γpG
(p(x, t)
). (4.5)
4.2 The compact support property
The porous medium equation. It has a famous particular solution, the Barenblatt solution
50
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Figure 4.1: Effect of γ large. A solution of the mechanical model (4.1) in one dimension with
G(p) = 5(1 − p). Left: γ = 5. Right: γ = 40. The upper line is x 7→ n(x, t); the bottom line
is x 7→ p(x, t) (scale enlarged for visibility). Notice that the density scales are not the same in
the two figures. The initial data is taken with compact support and the solution displayed for
a time large enough. It keeps a compact support.
Lemma 4.1 For G ≡ 0, a solution of (4.3) is given by
nB(x, t) =1
tdα(C − k |x|
2
t2α)β
+, (4.6)
with
α =1
dm− d+ 2, β =
1
m− 1, k =
α(m− 1)
2mA0
Note that the scale of the term 1tdα
is related to the term |x|2t2α
for the property of mass conser-
vation when G ≡ 0 because the equation is in the divergence form. The constant C is deduced
from the initial mass.
Two consequences are that solutions with compact support exist and that, for these solutions,
the pressure is Lipschitz continuous for t > 0 because p = κ 1tdα(m−1)
(C − k |x|2
t2α
)+
.
Proof. The computation is simple, because the power in A(n) is nmm−1 > 1 and there is no
singularity to take into account for. Therefore, we just have to compute on the support of n,
using the notation Y = |x|2t2α
, Z = C − k |x|2t2α
,
∂tnB = − dα
tdα+1Zβ +
2αβk
tdα+1Y Zβ−1,
∂xinmB =
2mβk
tdαm−xit2α
Zmβ−1,
∆nmB = − 2dmβk
tα(dm+2)Zmβ−1 +
4mβk2(mβ − 1)
tα(dm+2)Y Zmβ−2.
51
It remains to equalize the two terms in each of these expressions which leads to β = mβ− 1 and
gives the numbers.
The tumor growth model. One can carry out a similar calculation to the expense of building
a supersolution (in next section we show that the comparison principle holds true). We also
work on the equation written on the pressure (4.5)
Lemma 4.2 There is a family of supersolutions under the form
PS(x, t) = B(R(t)2 − |x|2
)+,
for B ≥ GM2d , R(t) = R(0)e2Bt.
Here again, choosing B and R(0) allows a large choice of function to be put over an initial data.
Proof. We compute
∂PS(x, t)
∂t= 2BRR1I|x|≤R(t) ≥ 4B2R21I|x|≤R(t) ≥ |∇PS |2 = 4B2|x|21I|x|≤R(t)
and, for some µ > 0
−∆PS = −µδ|x|=R(t) + 2Bd1I|x|≤R(t) ≥ GM1I|x|≤R(t).
Since the Dirac mass on the sphere |x| = R(t) is killed by the vanishing pressure in the term
PS∆PS , we conclude that
−PS∆PS ≥ PSGM ≥ PSG(PS).
And we have proved that
∂PS(x, t)
∂t− γPS∆PS − |∇PS |2 ≥ γPSG(PS).
4.3 Theoretical properties
4.3.1 Elementary Lq estimate
The first a priori estimate is, for q ≥ 1
d
dt
∫Rdn(x, t)qdx ≤ qGM
∫Rdn(x, t)qdx, (4.7)
which is obtained multiplying the equation by qnq−1. Notice that mass increases but we have
(for n0 ≥ 0)
q
∫Rdn(x, t)dx ≤ eGM t
∫Rdn0(x)dx. (4.8)
52
4.3.2 Contraction property
Theorem 4.3 (Contraction property) For two solutions n1, n2, one has
∂|n2 − n1|∂t
−∆|A(n2)−A(n1)| ≤ GM |n2 − n1|, (4.9)
d
dt
∫Rd|n2(x, t)− n1(x, t)|dx ≤ eGM t
∫Rd|n0
2(x)− n01(x)|dx.
From the a priori estimates in section 4.3.1, it is classical ([71]) to conclude the following
existence result that we do not prove
Proposition 4.4 For n0 ∈ L1∩L∞(Rd), equation (4.1) has a unique solution n ∈ C(R+;L1(Rd)
),
n ∈ L∞(R+ × Rd
).
Proof of Theorem 4.3. For two initial data n0i , i = 1, 2, for n(x, t) = n2(x, t) − n1(x, t) we
have∂
∂t(n2 − n1)−∆
(A(n2)−A(n1)
)= (n2 − n1)G(p1) + n2[G(p2)−G(p1)].
By convexity of n 7→ |n| we find, because G(·) is decreasing and p = Π(n) increasing,
∂
∂t|n2 − n1| −∆
∣∣A(n2)−A(n1)∣∣ ≤ GM |n2 − n1|.
To justify that sgn(n2 − n1)∆(A(n2) − A(n1)
)≤ ∆
∣∣A(n2) − A(n1)∣∣, we argue as follows (we
omit technical details) assuming n2 and n1 are smooth (change A to have A′(·) > 0 and then
pass to the limit). Let Sδ(·) be a family of smooth convex functions such that Sδ(0) = 0 and
Sδ(n) −→δ→0|n|. We have,
S′δ(n2 − n1)∆(A(n2)−A(n1)
)= div
[S′δ(n2 − n1)∇
(A(n2)−A(n1)
)]︸ ︷︷ ︸I
−S′′δ (n2 − n1)∇(n2 − n1).∇(A(n2)−A(n1)
)︸ ︷︷ ︸II
.
We have
I = div[S′δ(A(n2)−A(n1)
)∇(A(n2)−A(n1)
)]+ divRδ
= ∆[Sδ(A(n2)−A(n1)
)]+ divRδ
and the term Rδ is written and estimated as
Rδ =[S′δ(A(n2)−A(n1)
)− S′δ(n2 − n1)
]∇(A(n2)−A(n1)
).
53
And with Φ = ∇(A(n2)−A(n1)
)a smooth function (with our regularization mentioned before),
we can use the Lebesgue Convergence Theorem to get
|Rδ| =∣∣S′δ(A(n2)−A(n1)
)− S′δ(n2 − n1)
∣∣ |Φ|−→δ→0
0
because A(·) is non-decreasing and S′δ(·) → sgn(·) and consequently sgn(A(n2) − A(n1)
)=
sgn(n2 − n1).
For the second term, we write, with a = A′
II = S′′δ (n2 − n1)∇(n2 − n1).(a(n2)∇n2 − a(n1)∇n1
)≥ S′′δ (n2 − n1)
(a(n2)− a(n1)
)∇(n2 − n1).∇n1
≥ a(n2)− a(n1)
n2 − n1∇Qδ(n2 − n1).∇n1 −→
δ→00
and
S′′δ (n2 − n1)(a(n2)− a(n1)
)−→δ→0
0
in distributional sense because (in uniform norm)
Qδ(u) =
∫ u
0S′′δ (v)vdv = O(δ).
Therefore we have completed the derivation of inequality (4.9).
After integration we find
d
dt
∫Rd|n2 − n1|(x, t)dx ≤ GM
∫Rd|n2 − n1|dx
∫Rd|n2 − n1|(x, t)dx ≤ eGM t
∫Rd|n0
2 − n01|(x)dx.
4.3.3 Other a priori estimates
Proposition 4.5 The solutions satisfy
n02 ≤ n0
1 =⇒ n2(t) ≤ n1(t), (Comparison principle) (4.10)
p0 ≤ K =⇒ p(x, t) ≤ K,
54
∫Rdp(x, t)dx ≤ κK(γ−1)/γeGM t
∫Rdn0(x)dx,
supp n0 ⊂ BR(0) =⇒ supp n(t) ⊂ BR(t) ∀t ≥ 0
with R(t) built considering a supersolution as in Lemma 4.2.
Proof. 1. Comparison principle. In the proof of Theorem 4.3, we may also use the convex
function |n2 − n1|+ instead of |n2 − n1| and conclude that∫Rd|n2 − n1|+(x, t)dx ≤ eCRt
∫Rd|n0
2 − n01|+(x)dx.
In particular we find the comparison principle (4.10).
Notice that the choice n2 ≡ 0 shows that n(x, t) is nonnegative when n0 is nonnegative.
2. L∞ bound on the pressure. Because n = NK , a constant such that p = K, is a so-
lution and using the comparison principle (4.10), we find the bound on the pressure (a slight
difficulty here is that we have been working in L1 and the comparison has to be extended to L∞).
3. Control of the pressure. We just estimate∫Rdp(x, t)dx = κ
∫Rdn(x, t)γ−1n(x, t)dx = κ
∫Rdp(x, t)
γ−1γ n(x, t)dx ≤ κKγ−1
∫Rdn(x, t)dx
and use (4.8).
4. Control of the support. See Section 4.2 and use the comparison principle.
4.4 Theoretical properties (derivatives)
We continue our analysis with estimates on derivatives that are independent of γ. We have
Theorem 4.6 (Estimates on the derivatives) For n0 ∈ L1∩L∞(Rd), solutions of equation
(4.1) satisfy ∫Rd|∇n(x, t)|dx ≤ eGM t
∫Rd|∇n0(x)|dx,
∂n(x, t = 0)
∂t≥ 0 =⇒ ∂n(x, t)
∂t≥ 0 ∀t ≥ 0,∫ T
0
∫Rd|∇p(x, t)|2dxdt ≤ γ
γ − 1GMκe
GMTK(γ−1)/γ‖n0‖1.
55
With the notation G′(·) ≤ −aG < 0, we also get
γ
γ + 1aG
∫ T
0
∫Rd|∇ p(x, t)
γ+1γ |dxdt ≤ C(T,GM )‖n0‖TV ).
Proof. 1. BV estimate on n. We may choose for instance n01 = n0, n0
2(x) = n0(x+ h) and by
translational invariance, we conclude that for h ∈ Rd, we have using Theorem 4.3,∫Rd|n(x+ h, t)− n(x, t)|dx ≤ eGM t
∫Rd|n0(x+ h)− n0(x)|dx.
Letting |h| → 0 we obtain (in fact for each directional derivate it holds true)∫Rd|∇n(x, t)|dx ≤ eGM t
∫Rd|∇n0(x)|dx.
This BV estimate is strong enough for existence when the initial data is BV . For L1 initial data
it is standard to use an approximation n0ε ∈ BV and build a Cauchy sequence in L1 thanks to
the contraction principle.
2. Sign of ∂n∂t . We set w = ∂n
∂t and compute, differentiating (4.3) in time,
∂
∂tw −∆[A′(n)w] = w[G(p) + γpG′(p)].
From the comparison principle, see Proposition 2.5, we conclude that w0 ≥ 0 implies w(t) ≥ 0
and the result follows. Notice the other consequence
d
dt
∫Rd|w(x, t)|dx ≤
∫Rd|w(x, t)|[G(p) + γpG′(p)]dx.
3. H1 estimate on p. We just integrate the form (4.5) of the equation and write∫Rdp(T )dx+ (γ − 1)
∫ T
0
∫Rd|∇p(x, t)|2dxdt ≤ γGM
∫ T
0
∫Rdp(x, t)dxdt+
∫Rdp0(x)dx.
It remains to use the second estimate in Proposition 4.5 to find the result.
4. BV estimate on p. This estimate is of different nature and depends on propertie of the right
hand side G. We set, for a given i = 1, 2... d, v = ∂n∂xi
and obtain
∂v
∂t−∆[A′(n)v] = v[G(p) + γpG′(p)].
Multiplying by sgn(v) as we explained it in section 4.3.2 we find
∂
∂t|v| −∆[A′(n)|v|] ≤ |v|[G(p) + γpG′(p)].
56
Notice that
|v|γp = γnγ | ∂n∂xi| = γ
γ + 1|∂n
γ+1
∂xi|.
It remains to integrate in space and time, and we get (using the step 1.)∫Rd |v(T )|dx+ γ
γ+1aG∫ T
0
∫Rd | ∂∂xi p
γ+1γ |dxdt ≤ GM
∫ T0
∫Rd |v|dxdt
≤ GM∫ T
0 eGM t∫Rd |
∂n0(x)∂xi|dxdt
and the last result follows.
4.5 Theoretical properties (regularizing effect)
We show a regularizing effect which has some similarity with those proved in sections 2.6 and
2.7 and which expresses that the equation at hand describes well only growth.
We assume that there is a constant cG such that
G(p)− pG′(p) ≥ cG > 0. (4.11)
We define W (t) < 0 as the negative universal solution of
W (t) = γW (t)2 − γcGW (t)
that means
W (t) = − cGe−γcGt
1− e−γcGt.
Theorem 4.7 For solution of (4.1), we have
∆p+G(p) ≥W (t),∂p
∂t≥ γpW (t).
Notice that
W (0) = −∞, γW (t) −→t→∞
0.
Therefore any initial data for p can fit the inequality of Theorem 4.7 and ∂p∂t becomes ‘very
quickly non-negative’.
Proof. We define w(x, t) = ∆p+G(p) and rewrite the form (4.5) under the form
∂p
∂t= γpw + |∇p|2.
57
Therefore
∂w∂t = ∆∂p
∂t +G′(p)∂p∂t
= ∆[γpw + |∇p|2
]+G′(p)
[γpw + |∇p|2
]= γ
[p∆w + 2∇p.∇w + w∆p
]+ 2
∑i,j
(∂2p
∂xi∂xj
)2+ 2∇p.∇(∆p) +G′(p)
[γpw + |∇p|2
]From this, we conclude that
∂w∂t ≥ γ
[p∆w + 2∇p.∇w + w2 − wG(p)
]+ 2∇p.∇w −G′(p)|∇p|2 + γpG′(p)w
≥ γp∆w + 2(γ + 1)∇p.∇w + γw2 − γ[G(p)− pG′(p)]w
But W (t) is a subsolution because the space derives vanish for W (t) < 0 and
∂W (t)
∂t= γW (t)2 − γcGW (t) ≤ γW (t)2 − γ[G(p)− pG′(p)]W (t).
Using the comparison principle, we conclude the proof.
58
Chapter 5
Variants and extensions of the
compressible model
5.1 On the law-of-state, viscosity
We have chosen a simple power law-of-state for the pressure p = Π(n) = κnγ . But for the
mathematical anaalysis, any smooth increasing function can be used. Physically, short range
attraction/long range repulsion gives a more realistic law because it represents the adhesion
forces when cells touch each other but are not packed. This leads to the assumptions
Π′(n) ≤ 0 for n ≤ nc, Π′(n) ≥ 0 for n ≥ nc,
which may induce istabilities [5] and which is ill-posed mathamtically. A way to circumvent this
issue is to use viscosity and not only friction, that is replave Darcy’s law by Brinkman’s law and
replace equation (4.1) by ∂∂tn+ div
(nv)
= nG(p(x, t)
),
−ν∆v + v = −∇p, p = Π(n).
5.2 Active motion
It is also possible to include active cell motion thanks to additional diffusion, then writing∂∂tn+ div
(nv)− ν∆n = nG
(p(x, t)
),
v = −∇pThis is physically a diffrent type of effect. The velocity v is prodiced by pressure forces resulting
from proliferation. The diffusion term is due to random motion of cells. This generates long
tails of cell far away from the tumor. See [56].
59
5.3 Spatial model with nutrient
It is usual to include nutrients (glocosis, oxygen) as a limitation for growth. This is in particular
necessary for avascularized tumors to explain appearance of necrotic core: the center of the
tumor becomes quiescent by lack of nutrients and then necrotic. In other words, the result of
Theorem ?? might be wrong when nutrients are included because cells can die in the core of the
tumor.
With a single concentration for nutrients, the equation (4.1) is extended as follows∂∂tn− div
(n∇Π(n)
)= nR
(p(x, t), c(x, t)
), x ∈ Rd, t ≥ 0,
τ ∂∂tc−∆c+ λcn+ rc = rcb,
Π(n) = nγ .
(5.1)
where
• c(x, t) represents the nutrients which is diffused though the tumor
• cb (a constant here) is the fresh nutrient provided by the vasculature with a rate r,
• R(p, n) is the growth/death rate. We assume that there is a minimal nutrient concentration
cmin needed for maintenance (if c is below the concentration cmin < cb, cells are dying whatever
p),∂R(p, c)
∂p< 0,
∂R(p, c)
∂c> 0, R(p, cmin) ≤ 0, R(K, cb) = 0. (5.2)
Following the manipulation leading to (4.4), we may establish an equation for the pressure.
We multiply by Π′(n) and write the equation on n as
∂
∂tp− nΠ′(n)∆p+ |∇p|2 = nΠ′(n)R
(p(x, t), c(x, t)
)and for the special case at hand, p = nγ , we find
∂
∂tp− γp∆p = |∇p|2 + γpG
(p(x, t), c(x, t)
).
A typical result in this direction is
Theorem 5.1 For n0 ∈ L1 ∩ L∞(Rd), 0 ≤ c0 ≤ cb and p0 ≤ K, the system (5.1) has a unique
solution n ∈ C(R+;L1(Rd)
), n, c ∈ L∞
(R+ × Rd
), and
0 ≤ c(x, t) ≤ cb, p(x, t) ≤ K,
supp n0 ⊂ BR(0) =⇒ supp n(t) ⊂ BR(t) ∀t ≥ 0
60
These results follow from an adaptation of the methods in Theorem ??. The main difference
is that it is difficult (impossible?) to state a rgeneral statement on the growth of the tumor. Let
us explain the difficulty. If one wishes to estimate the time derivatives
w =∂n
∂t, z =
∂c
∂t,
a natural statement would be that tumor is growing that is w(x, t) ≥ 0 and nutrient depleted
z(x, t) ≤ 0. The equations on these two unknowns are∂∂tw − div
(Q′(n)∇w
)= w
[R(p, c)
+ nΠ′(n)Rp(p, c)]
+ znRc(p, c),
τ ∂∂tz −∆z + λzn+ rz = −λcw.
If w ≥ 0, the term −λcw is compatible with a local depletion of nutrient (that is z ≤ 0). How-
ever the term znRc(p, c) would be nonpositive and is not compatible with a growth of the tumor
(w ≥ 0). For this reason the natural statement needs not hold. In term of medical observations,
tumors have often a necrotic core that is obtained by numerical simulations. See section 5.4.
Question. Is there a family of smallness assumptions which lead to the conclusion w ≥ 0,
z ≤ 0.
5.4 Necrotic core and the nutrient instability
Nutrients not only explain the necrotic core in the center of the tumor; they also play a role on
stability of the PDE.
The growth property ∂n∂t > 0 stated in Theorem ?? is also a type of stability condition. This
property becomes wrong when nutrients are included in the model, leading to the ’nutrients
instability’. The intuition is simple; if some cells are in front of the core of the tumor, they
have access to fresh nutrients with higher concentration and thus they can proliferate faster.
The numerical simulations in Figure 5.1 are based on a standard variant of system (5.1) where
nutrients are provided from the boundary of the tumor
∂∂tn− div
(n∇Π(n)
)= nR
(c(x, t)
), x ∈ Rd, t ≥ 0,
−∆c+ λcn = 0, c(x)→ cb for |x| → ∞,
Π(n) = nγ .
(5.3)
See also [5] for another route to instability.
61
Figure 5.1: The nutrient instability for tumor growth model (5.3). Snapshots of the tumor
invasion, with a developing necrotic core and surface instability.
5.5 Healthy and tumor cells: the seggregation property
Sanchez, Gardino and Maini (1994), Medvedev et al (2003), Sherratt (2010), Diaz and Kamin
(2012), Mimura et al 2002, Bertsch-Gurtin et al 2010, Presiozi and Galdani, [9] etc... consider
that the response to compression differs for healthy and tumor cells. We write∂∂tnT + div
(nT v
)= nTGT (p), nT (±∞) = 0,
∂∂tnH + div
(nHv
)= nHGH(p), nH(±∞) = KH ,
v = −κ(nT , nH) ∇p, p = Π(nH , nT ).
(5.4)
A typical law-of-state is
Π(nH , nT ) = (nH + nT )γ , γ > 0.
For this model, the authors mentioned above define
Definition 5.2 The system satisfies the seggregation property if, for all t ≥ 0,
nH(x, t)nT (x, t) = 0 when n0H(x)n0
T (x) = 0.
62
0
0.001
0.002
0.003
0.004
0.005
0.006
0 50 100 150 200 250 300
nucle
i densit
y [
nucle
i /
mic
rom
ete
r^2
]
distance to border [micrometer]
allproliferating
necrotic
0
0.001
0.002
0.003
0.004
0.005
0.006
0 50 100 150 200 250 300nucle
i densit
y [
nucle
i /
mic
rom
ete
r^2
]
distance to border [micrometer]
allproliferating
necrotic
Figure 5.2: Cell culture data in vitro at two different times. From N. Jagiella PhD thesis,
INRIA and UPMC (2012).
The meaning is very intuitive, the healthy and tumor zones do not overlap! Surprisingly the
system (5.4) which has a flavor of parabolic equations satisfies this ’front’ property of hyperbolic
nature
Theorem 5.3 The system (5.4) satisfies the seggregation property in definition 5.2.
Proof. To show that we multiply the first equation by nH and the second by nT . After adding
them and integration, we compute
d
dt
∫RdnH(x, t)nT (x, t)dx+
∫Rd
[∇nH(x, t) nT v +∇nT (x, t) nHv]
=
∫RdnTnH
[GT (p) +GH(p)
].
After another integration by parts, we get
d
dt
∫RdnH(x, t)nT (x, t)dx =
∫RdnTnH
[− divv +GT (p) +GH(p)
].
And assuming divv is bounded and using nonnegativity we conclude that
d
dt
∫RdnH(x, t)nT (x, t)dx ≤ C
∫RdnH(x, t)nT (x, t)dx,
and because∫Rd n
0H(x)n0
T (x)dx = 0, we find that∫Rd nH(x, t)nT (x, t)dx ≤ 0, which means that∫
Rd nH(x, t)nT (x, t)dx ≤ 0 and thus that nH(x, t)nT (x, t) = 0.
Remark 5.4 This seggregation property is false when one assumes different mobilities for healthy
and tumor cells
vT = −κT (nT , nH)∇p, vH = −κH(nT , nH)∇p p = Π(nH , nT ).
63
5.6 Proliferative and quiescent models with space
Another extension of the simplest spatial model is to include different states of the cells as done
in section 1.4. We call nP the number density of proliferative cells, nQ the quiescent cells and
nN the necrotic cells, see [67, 9, 8],
∂∂tnP + div
(nP v
)= nPR
(nP + nQ + nN
)− bnP ,
∂∂tnQ + div
(nQv
)= bnP − dnQ,
∂∂tnN = dnQ,
v = −∇p, p = Π(nP + nQ + nN ),
(5.5)
with a pressure defined through an increasing pressure function Π(n) which can be taken as
Π(n) = nγ . In the above references, it is supposed that the necrotic cells do not participate
to establish a pressure, i.e., p = Π(nP + nQ+) but are present to stop proliferation. Indeed, a
typical examples of growth rate are
R(nP + nQ + nN
)= K − nP − nQ − nN ,
or, closer to the model of Chapter 4, is the rate
R(nP + nQ + nN
)= K − (nP + nQ + nN )γ .
A possible extension is to add the nutrients as in section 5.3.
Another variant is to consider two different mobilities in the definition of velocity from pressure
and thus two different velocities. This leads to write
∂∂tnP + div
(nP vP
)= nPR
(nP + nQ + nN
)− bnP ,
∂∂tnQ + div
(nQvQ
)= bnP − dnQ,
∂∂tnN = dnQ,
vP = −κP∇p, vQ = −κQ∇p, p = Π(nP + nQ + nN ),
(5.6)
and κP,Q could depend on nP , nQ.
Exercise. Explain why the system (5.5) does not satisfy the seggregation property in Defini-
tion 5.2.
64
5.7 Effect of acidity
When oxygen is missing, tumor cells can use glucose and this produced lactic acid. This creates
an acidic environment where tumor cells can survive but not healthy cells. The mathematical
modeling of this phenomena has been proposed [36, 37] (see also [51]) and leads to a reaction-
diffusion system for the unknown
• nH the normal cells,
• nC the cancer cells,
• A the acidity, that is the ions H+ concentration produced by cancer cells,∂∂tnH = rHnH(1− nH)− anHA,∂∂tnC = rCnC(1− nC) +Ddiv
[(1− nH)∇nT
],
∂∂tA = bnC − cA.
In these equations, healthy cells are supposed unmotile and undergo a Fisher dynamics with
supplementary death term proportional to the H+ concentration. Cancer cells also undergo a
Fisher dynamics and invade space left by dying healthy cells. Finally H+ ions are produced by
cancer cells and are also degraded by the environment.
5.8 Space and age structure
In order to take into account the cell cycle and describe correctly cell division, the spatial models
should also contain a physiological variable that is able to represent the internal state of a cell.
This has been proposed and used by several authors [29, 12, 13].
We will not enter the details of the modeling here but just mention the type of equation
proposed by these authors. In a simplied setting, they read as
∂
∂tnP (x, a, t) +
∂
∂anP + div
(nP v
)+ [dP + b(a)]nP = 0,
associated with the boundary condition
nP (x, a = 0, t) = 2
∫ ∞0
b(a)nP (x, a, t)da.
The coefficient b(a) in dP + b(a) is included because, after mitosis according to the rate b(a),
the two cells come back at age a = 0 in the cell cycle. Note that a quiescent compartment is
usually added also.
The velocity v results from pressure forces and can be handled as before
v = −∇p, p = Π( ∫
n(x, a, t)da).
65
5.9 The multiphase flow approach
Continuing on the fluid mechanical view of a tissue, it has been proposed to consider a tumor
as a multiphase flow, see [17, 18, 58] and the result therein. In the simplest view,
• only two phases are considered: tumor cells and ’liquid’ (a generic name for extracellular fluid)
with local volume ratio φT (x, t) ≥ 0, φL(x, t) ≥ 0,
• the saturation regime (no void, incompressible) is written φT + φL = 1,
• the matter, tumor cells and liquid, flows with velocity vT , vL.
One writes usual Navier-Stokes equations for the different phases and their momentum, leading
to the system describing mass exchange and momemtum exchange
∂∂tφT + div
(φT vT
)= ΓT ,
∂∂tφL + div
(φLvL
)= ΓL,
ρTφT[∂∂tvT + vT .∇vT
]= div.TT +mT ,
ρLφL[∂∂tvL + vL.∇vL
]= div.TL +mL,
φT + φL = 1.
(5.7)
• The cell and liquid densities ρT and ρL (two constants) are usually assumed to be equal, which
means that the mass balance ρTΓT + ρLΓL = 0 reads
ΓT + ΓL = 0.
To fix ideas, a possible expression for this growth term is Fisher type (before necrosis occurs)
− ΓL = ΓT = G0 φT φL = G0 φT (1− φT ). (5.8)
• The interaction forces, mT , mL, are due to interfacial pressure P and friction terms (friction
between phases and for tumor cells an adhesion on the extracellular matrix)
mT = P∇φT + k1φTφL(vL − vT )− k2vT , mL = P∇φL + k1φTφL(vT − vL), (5.9)
with a dependency k2(φT )
• The stress tensors TT,L represent internal forces to each phase. They should be defined in a
closed form and there are many possible choices. It helps at this stage to neglect the acceleration
terms ∂∂tv + v.∇v, and also the viscosity terms. Then we retain only the pressure terms under
the form TT = −PT I , TL = −PL I . We arrive at
mT = ∇PT , mL = ∇PL. (5.10)
66
Various choices for the definition of pressures lead to various self-contained models and we
review some of them now.
• The one phase closure.
The simplest choice is to neglect interactions between the phases, that is k1 = 0, P = 0. To
close the system one takes PT = Π(φT ). One ends up with a very simplified system which can
be written ∂∂tφT + div
(φT vT
)= ΓT ,
k2vT = −∇Π(φT ) (Darcy’s law).(5.11)
The dynamics of the liquid phase is reduced to the equation
div(φT vT + (1− φT )vL
)= 0.
This is the simplest model that has been presented in section 4.1.
• The single pressure closure.
One may also choose to keep a sinple pressure P (x, t) and set
PT = φT P, PL = φL P.
This allows to solve for vT and vL in terms of P ; first we add the two momentum equations in
(5.10) and find mT +mL = ∇p. We find
vT = − 1
k2∇P, vL = vT −
1
k1φT∇P.
This finally leads to the system in closed form∂∂tφT + div
(φT vT
)= ΓT ,
vT = −K2(φT )∇P (x, t),
div[K1(φT )∇P ] = 0.
(5.12)
This is also a standard two phase flow problem (in particular it is widely used for oil/gas mixtures
in oil recovery). Our understanding is very little because it shares hyperbolic nature (having in
mind that the pressure is smoooth) and the parabolic nature of equations derived from Darcy’s
law.
To conclude this case, let us explain the a priori estimates 0 ≤ φT ≤ 1 directly from (5.12).
Because ΓT vanishes when φT = 0, see (5.8), we get −φT ≥ 0. For the other inequality, we have
to write the equation as∂
∂tφT + div
(φT
K2
K1K1∇P
)= ΓT ,
67
∂
∂tφT +∇
(φT
K2(φT )
K1(φT )
).K1∇P = ΓT .
Solutions satisfy φT ≤ 1 because it is in strong form and ΓT vanishes for φT = 1. These expres-
sions show that these models can be classified as nonlinear hyperbolic conservation laws.
• A detour by biofilms.
From [43, 65]. Bacterial biofilms are a usual form of slowly expanding community organized
BIOFILM
AIR
A G A R
x
z = h(x)
water flow water flow
!t "t !t "1t
"2t
! 1 ! !(p,w) (p + ", v)
"(!)
#t! + (!v) = g!
#t(1 ! !) + ((1 ! !)w) = !g!
µ#v ! !"p ! "" ! $ (v ! w) = 0
! (1 ! !)"p ! $ (w ! v) = 0
v = 0, p = " ! "! z = 0
p = p , !(p + " ! "!)n + µ#v
#n= !p n z = h
p "!n "t
!t v
V = v "t
V
Figure 5.3: The structure of a biofilm modeled by the two phase flow system (5.13).
within a polymeric structure secreted by the bacteria themselves. A biofilm usually grows from
a surface (agar in the experiment at hand) and bacteria are mostly unmotile. Once secreted,
the polymer will be at equilibrium with a certain proportion of water (related to the agar) thus
creating a flow towards newly formed biofilm.
In this framework, the biofilm replaces the tumor and we keep the index φT for the volume
fraction of biofilm, the liquid is just water with nutrients. The authors in [65] retain the osmotic
pressure Π(φT ) and the biofilm viscosity (in place of friction) for the momentum equations for
vT while they retain only interfacial pressure P for water. This leads to the system
∂∂tφT + div
(φT vT
)= ΓT ,
div(φT vT + (1− φT )vL) = 0,
∇Π(φT ) + φT∇P − ν∆vT = k1(φT ) (vL − vT ),
(1− φT )∇P = k1(φT ) (vT − vL).
(5.13)
The system can be simplified by an algebraic manipulation so as to eliminate vL. Then it is
reduced to three equations for the three unknowns (φT , P, vT )∂∂tφT + div
(φT vT
)= ΓT ,
div(vT − (1−φT )2
k1(φT ) ∇P)
= 0,
∇Π(φT ) +∇P − ν∆vT = 0,
(5.14)
68
Notice that retaining friction rather than viscosity, leads to replace the third equation by
∇Π(φT ) +∇P = −k2vT
which allows to change the system in∂∂tφT + div
(φT vT
)= ΓT ,
div(KP (φT )∇P
)= ∆ζ(φT ),
∇Π(φT ) +∇P = −k2vT ,
(5.15)
This type of system has also been proposed for cell motion with chemotaxis and congestion
effects in [53, 24] ∂∂tφT + div
(φT vT
)= ΓT ,
−∆c+ c = n, −∆S = div[φT∇c],
vT = ∇(c+ S).
(5.16)
69
70
Chapter 6
Hele-Shaw models and free
boundary formulation
A usual modeling of tumor growth relies on the observation that the tumor contours are usually
well defined, see Figure 6.1 and 6.2, see also the ’seggregation property’ in Theorem 5.3. This
leads to consider geometric models where the tumor is defined as an expanding set Ω(t), which
can be a ’spheroid’ then Ω(t) = BR(t). One assumes that the tumor has a constant density of
cells nmax (incompressibility of tissue). Then, the question is then to describe the evolution of
the dynamics of free boundary ∂Ω(t). The first free boundary problem for tumor growth goes
back to H. Greenspan [?] in 1972 (see Section 6.2), and many variants have been used. It is now
a well established subject with many surveys available [51, 33, 61].
Figure 6.1: Spheroid structure of a early stage tumour: outer rim of proliferating cells and
inner necrotic core. From Sutherland et al., Cancer Res. 46 (1986), 5320–5329.
pieces. We model the migration of cells by the Metropolisalgorithm and a proper definition of time scales. A cell inisolation performs a random-walk-like movement while inthe neighborhood of other cells it tends to move into thedirection which minimizes the free energy. We quantify themigration activity of a cell by its diffusion constant D inisolation. We perform a number of successive migration andorientation trials between two successive growth trials. Thetrials are accepted with probability min(1,exp!!V/FT.FT is a parameter that controls the cell activity: it may becompared to the thermal energy kBT in fluids (kB: Boltzmanconstant, T : temperature). Together with the choice of stepsizes for growth, orientation change, and migration, ouralgorithm mimics a multi-cellular configuration changingwith time. The step sizes are chosen in such a way, that thesimulation reflects a realistic growth scenario. (The detailsof our model are explained in [8])We recently used this single-cell based model to studytumor spheroid growth in liquid suspension [8], which hasbeen extensively studied experimentally [21], [14] (for anoverview of tumor growth models, please see Ref. [8] andreferences therein). Here, we study growing tumors in atissue-like medium composed of cells to analyze the influ-ence of an embedding medium on the tumor morphology(for a simulation example, see Fig. 1)
Fig. 1. Typical simulated tumor growth scenario. Red: embeddingcells, white: cells of the expanding clone. The embedding cells areinitially placed on the nodes of a square lattice and subsequentlyrelaxed before the growth of the embedded clone is started.
The embedding medium was modeled as non-dividingcells with the same parameters as the dividing cell clonewith the following exceptions: (1) ”motX” within the nameof the dataset denotes that D " D/X with D being theDiffusion constant mentioned above, (2) except of the dataset”id100_mot1_adh” the embedding cells do not adhere.The id-value refers to the initial distance of embedding cellswhich is l for id100 and l = 1.2 (= 120/100) for id120.For selected parameter sets, we have validated that the resultsdo not change if we replace the embedding cells by granularparticles with the same physical properties, but with only
passive movement (i.e., no capability to migrate actively).Experiments to validate our findings can thus be easilyconducted in in-vitro studies with an experimental settingsimilar to that in Ref. [10] by growing tumor spheroids in agranular embedding medium.
IV. GEOMETRIC SHAPE PROPERTIES
A. Morphological Operators
All datasets were given as binary 3D images, which aregenerally defined as the quadruple P = (Z3,m,n,B), whereevery element of Z3 is a point (voxel) in P. The set B # Z3
is the image foreground, or the object, whereas Z3 \B is thebackground. The neighborhood relation between the voxelsis given by m and n with m being the connectivity of objectvoxels and n the connectivity of the background. To avoidtopological paradoxa, only the following combinations arepossible: (6,26), (26,6), (6,18) and (18,6) [15].
Morphological operators are well-known in image proces-sing. Erosion and Dilation are in fact binary convolutionswith a mask describing the background-connectivity of avoxel [11]. The Hit-Miss-Operator extracts specific featuresof a binary image. For morphological Thinning, this operatoris used with a set of masks, where each mask is applied tothe original image, and all resulting images undergo a logicalOR-operation and will be subtracted from the original image[15].
B. Distance Transform
The distance field of a binary digital image is a discretescalar field of the same size with the property, that eachvalue of the scalar field specifies the shortest distance ofthe voxel to the boundary of the object. The signed distancetransform contains negative values for distances outside theobject. Distance transforms using the L1 or L" metrics can becomputed using Erosion for successive border generation andlabeling of the removed voxels until the object is completelyremoved [11]. The computation of the Euclidean distancetransform is described in [19].
C. Medial Axis Transform and Skeletonization
In a continuous space, the medial axis of an object is theset of points, which are the centers of maximally inscribedspheres. A sphere is maximally inscribed, if it touches theobject boundary in at least two points, if it lies completelywithin the object, and if there is no larger sphere withthe same properties. The skeleton of a binary object is acompact representation of its geometry and shape. It is asubset of the object with three properties [17]: (1) topologicalequivalence, (2) thinness, and (3) central location within theobject. Topological equivalence implies that the medial axishas the same number of connected components, enclosedbackground regions and holes as the original object.
In discrete space, the medial axis can be approximated byiterative Thinning as described in [15].
pieces. We model the migration of cells by the Metropolisalgorithm and a proper definition of time scales. A cell inisolation performs a random-walk-like movement while inthe neighborhood of other cells it tends to move into thedirection which minimizes the free energy. We quantify themigration activity of a cell by its diffusion constant D inisolation. We perform a number of successive migration andorientation trials between two successive growth trials. Thetrials are accepted with probability min(1,exp!!V/FT.FT is a parameter that controls the cell activity: it may becompared to the thermal energy kBT in fluids (kB: Boltzmanconstant, T : temperature). Together with the choice of stepsizes for growth, orientation change, and migration, ouralgorithm mimics a multi-cellular configuration changingwith time. The step sizes are chosen in such a way, that thesimulation reflects a realistic growth scenario. (The detailsof our model are explained in [8])We recently used this single-cell based model to studytumor spheroid growth in liquid suspension [8], which hasbeen extensively studied experimentally [21], [14] (for anoverview of tumor growth models, please see Ref. [8] andreferences therein). Here, we study growing tumors in atissue-like medium composed of cells to analyze the influ-ence of an embedding medium on the tumor morphology(for a simulation example, see Fig. 1)
Fig. 1. Typical simulated tumor growth scenario. Red: embeddingcells, white: cells of the expanding clone. The embedding cells areinitially placed on the nodes of a square lattice and subsequentlyrelaxed before the growth of the embedded clone is started.
The embedding medium was modeled as non-dividingcells with the same parameters as the dividing cell clonewith the following exceptions: (1) ”motX” within the nameof the dataset denotes that D " D/X with D being theDiffusion constant mentioned above, (2) except of the dataset”id100_mot1_adh” the embedding cells do not adhere.The id-value refers to the initial distance of embedding cellswhich is l for id100 and l = 1.2 (= 120/100) for id120.For selected parameter sets, we have validated that the resultsdo not change if we replace the embedding cells by granularparticles with the same physical properties, but with only
passive movement (i.e., no capability to migrate actively).Experiments to validate our findings can thus be easilyconducted in in-vitro studies with an experimental settingsimilar to that in Ref. [10] by growing tumor spheroids in agranular embedding medium.
IV. GEOMETRIC SHAPE PROPERTIES
A. Morphological Operators
All datasets were given as binary 3D images, which aregenerally defined as the quadruple P = (Z3,m,n,B), whereevery element of Z3 is a point (voxel) in P. The set B # Z3
is the image foreground, or the object, whereas Z3 \B is thebackground. The neighborhood relation between the voxelsis given by m and n with m being the connectivity of objectvoxels and n the connectivity of the background. To avoidtopological paradoxa, only the following combinations arepossible: (6,26), (26,6), (6,18) and (18,6) [15].
Morphological operators are well-known in image proces-sing. Erosion and Dilation are in fact binary convolutionswith a mask describing the background-connectivity of avoxel [11]. The Hit-Miss-Operator extracts specific featuresof a binary image. For morphological Thinning, this operatoris used with a set of masks, where each mask is applied tothe original image, and all resulting images undergo a logicalOR-operation and will be subtracted from the original image[15].
B. Distance Transform
The distance field of a binary digital image is a discretescalar field of the same size with the property, that eachvalue of the scalar field specifies the shortest distance ofthe voxel to the boundary of the object. The signed distancetransform contains negative values for distances outside theobject. Distance transforms using the L1 or L" metrics can becomputed using Erosion for successive border generation andlabeling of the removed voxels until the object is completelyremoved [11]. The computation of the Euclidean distancetransform is described in [19].
C. Medial Axis Transform and Skeletonization
In a continuous space, the medial axis of an object is theset of points, which are the centers of maximally inscribedspheres. A sphere is maximally inscribed, if it touches theobject boundary in at least two points, if it lies completelywithin the object, and if there is no larger sphere withthe same properties. The skeleton of a binary object is acompact representation of its geometry and shape. It is asubset of the object with three properties [17]: (1) topologicalequivalence, (2) thinness, and (3) central location within theobject. Topological equivalence implies that the medial axishas the same number of connected components, enclosedbackground regions and holes as the original object.
In discrete space, the medial axis can be approximated byiterative Thinning as described in [15].
pieces. We model the migration of cells by the Metropolisalgorithm and a proper definition of time scales. A cell inisolation performs a random-walk-like movement while inthe neighborhood of other cells it tends to move into thedirection which minimizes the free energy. We quantify themigration activity of a cell by its diffusion constant D inisolation. We perform a number of successive migration andorientation trials between two successive growth trials. Thetrials are accepted with probability min(1,exp!!V/FT.FT is a parameter that controls the cell activity: it may becompared to the thermal energy kBT in fluids (kB: Boltzmanconstant, T : temperature). Together with the choice of stepsizes for growth, orientation change, and migration, ouralgorithm mimics a multi-cellular configuration changingwith time. The step sizes are chosen in such a way, that thesimulation reflects a realistic growth scenario. (The detailsof our model are explained in [8])We recently used this single-cell based model to studytumor spheroid growth in liquid suspension [8], which hasbeen extensively studied experimentally [21], [14] (for anoverview of tumor growth models, please see Ref. [8] andreferences therein). Here, we study growing tumors in atissue-like medium composed of cells to analyze the influ-ence of an embedding medium on the tumor morphology(for a simulation example, see Fig. 1)
Fig. 1. Typical simulated tumor growth scenario. Red: embeddingcells, white: cells of the expanding clone. The embedding cells areinitially placed on the nodes of a square lattice and subsequentlyrelaxed before the growth of the embedded clone is started.
The embedding medium was modeled as non-dividingcells with the same parameters as the dividing cell clonewith the following exceptions: (1) ”motX” within the nameof the dataset denotes that D " D/X with D being theDiffusion constant mentioned above, (2) except of the dataset”id100_mot1_adh” the embedding cells do not adhere.The id-value refers to the initial distance of embedding cellswhich is l for id100 and l = 1.2 (= 120/100) for id120.For selected parameter sets, we have validated that the resultsdo not change if we replace the embedding cells by granularparticles with the same physical properties, but with only
passive movement (i.e., no capability to migrate actively).Experiments to validate our findings can thus be easilyconducted in in-vitro studies with an experimental settingsimilar to that in Ref. [10] by growing tumor spheroids in agranular embedding medium.
IV. GEOMETRIC SHAPE PROPERTIES
A. Morphological Operators
All datasets were given as binary 3D images, which aregenerally defined as the quadruple P = (Z3,m,n,B), whereevery element of Z3 is a point (voxel) in P. The set B # Z3
is the image foreground, or the object, whereas Z3 \B is thebackground. The neighborhood relation between the voxelsis given by m and n with m being the connectivity of objectvoxels and n the connectivity of the background. To avoidtopological paradoxa, only the following combinations arepossible: (6,26), (26,6), (6,18) and (18,6) [15].
Morphological operators are well-known in image proces-sing. Erosion and Dilation are in fact binary convolutionswith a mask describing the background-connectivity of avoxel [11]. The Hit-Miss-Operator extracts specific featuresof a binary image. For morphological Thinning, this operatoris used with a set of masks, where each mask is applied tothe original image, and all resulting images undergo a logicalOR-operation and will be subtracted from the original image[15].
B. Distance Transform
The distance field of a binary digital image is a discretescalar field of the same size with the property, that eachvalue of the scalar field specifies the shortest distance ofthe voxel to the boundary of the object. The signed distancetransform contains negative values for distances outside theobject. Distance transforms using the L1 or L" metrics can becomputed using Erosion for successive border generation andlabeling of the removed voxels until the object is completelyremoved [11]. The computation of the Euclidean distancetransform is described in [19].
C. Medial Axis Transform and Skeletonization
In a continuous space, the medial axis of an object is theset of points, which are the centers of maximally inscribedspheres. A sphere is maximally inscribed, if it touches theobject boundary in at least two points, if it lies completelywithin the object, and if there is no larger sphere withthe same properties. The skeleton of a binary object is acompact representation of its geometry and shape. It is asubset of the object with three properties [17]: (1) topologicalequivalence, (2) thinness, and (3) central location within theobject. Topological equivalence implies that the medial axishas the same number of connected components, enclosedbackground regions and holes as the original object.
In discrete space, the medial axis can be approximated byiterative Thinning as described in [15].
Figure 6.2: Simulated tumor growth scenario (grey) in embedding cells (red). From [60]: Shape
characterization of extracted and simulated tumor samples using topological and geometrical
measures.
6.1 Tumour spheroids growth (mechanical model)
The simplest geometrical model of tumour growth is based on increase of pressure by cell division
without reference to a nutrient which is supposed sufficient (e.g. by angiogenesis effects). The
model we present now is taken from [16, 51, 33].
We suppose that the density of cells is a constant nmax within the tumor that occupies the
spatial domain Ω(t). Then, we compute the pressure as −∆p = G(p) x ∈ Ω(t),
p = 0 on ∂Ω(t).(6.1)
As before the pressure dependent growth term is supposed to satisfy for some K > 0,
G′(·) < 0, G(K) = 0
and Darcy’s assumptions allows us to relate the growth speed and the pressure with
v(t, x) = −∇p(x, t).
The tumor still grows with the normal velocity on its boundary (See Figure 6.3)
X(t) = v(X(t), t)
), X(t) ∈ ∂Ω(t). (6.2)
Often, surface tension is included and then the Dirichlet condition is changed to
p(x, t) = ηκ(x, t), on ∂Ω(t)
with κ the mean curvature on the boundary.
72
It is easier to see this on a tumor spheroid. Then the curvature is just given by κ(x, t) = 1/R(t)
and the set of equation becomes
−∆p = G(p) x ∈ BR(t),
p = η/R(t) on ∂BR(t),
v = − x|x| .∇p on ∂BR(t),
R(t) = v(R(t), t)
).
(6.3)
Another boundary condition is introduced in [20] in order to represent a tumor spheroid
growing in a gel
p(R(t), t) = κ[R(t)−R0],
with κ a parameter that represents the stiffness of the surrounding gel and the radius R0 that
results in an equilibrium between the tissue pressure and the force exerted by the gel.
Exercise. Consider the case G = H(K − p) the Heavyside function. Compute the radius R(t)
in this model following the calculation in Section 6.2. Show that it depends on the curvature
coefficient η, does R(t) converge to a finite value R∞ as t→∞?
Ω(t)
Figure 6.3: Motion of a boundary with normal velocity.
6.2 Tumour spheroids growth (nutrients limitation)
Following [14], the tumor is considered as a spheroid, that is a ball BR(t) of R3 centered at the
origin and of radius R(t), and the cell population density is still constant inside this ball. The
available nutrient is denoted by c(x, t) for 0 ≤ |x| ≤ R(t) and is provided from the boundary
∂BR(t) with a concentration cb (blood concentration). The system reads −∆c+ λc = 0, x ∈ BR(t),
c = cb, on ∂BR(t)
(6.4)
73
R(t) =1
Sd−1Rd−1
∫BR(t)
G(c(t, x))dx, R(0) = R0 > 0. (6.5)
The equation on c just means that the nutrient is diffused instantaneously from the boundary
where the nutrient is available at a concentration cb and consumed with rate λ. The constant
Sd−1 denotes the surface of the sphere in dimension d. The functions G : [0, cb]→ R determines
the net growth rate of cells (birth and death) depending upon the available nutrient. The
simplest example is based on consumption for maintenance at a certain level c and gives
G1(c) = c− c.
We make the general assumption on G ∈ C1 that there is c > 0 such that it holds
G′(·) > 0, G(c) = 0. (6.6)
Theorem 6.1 With assumption (6.6), there is a unique solution of (6.4)–(6.5) with radial
symmetry, R(t) is monotonic and satisfies
(i) for cb ≤ c, then R(t)→ 0 as t→∞,
(ii) for cb > c, there is a R∞ > 0 such that
R(t)→ R∞ <∞ as t→∞.
The model is irrealistic in supposing that the dead cells disappear immediately and thus are
replaced by other cells moving back from a proliferating rim to the center to maintain a constant
density in the ball of radius R(t). This creates an equilibrium between proliferation and death.
The extension to the case where nutrients do not diffuse immediately is treated in [35] and is
more complicated, that is∂
∂tc−∆c+ λc = 0,
Proof. We consider dimension d = 3 only to simplify notations. The problem can be rewritten,
taking into account radial symmetry and rescalling, as
c(t, x) = C(R(t),
|x|R(t)
), (6.7)
where C(ρ, u), 0 ≤ u ≤ 1 is the solution of −1u2
∂∂u [u2 ∂C(ρ,u)
∂u ] + λρ2C = 0, 0 < u < 1,
C(ρ, 1) = cb, C ′(ρ, 0) = 0.
74
Notice that the monotonicity property of Laplace equation implies that
C(ρ, ·) is decreasing in ρ, C(0, u) = cb, C(∞, u) = 0. (6.8)
We compute, after changing variable |x| 7→ u = |x|/R(t),
∫BR(t)
G(c(t, x))dx = 4π R(t)2
∫ R(t)
0G(c(t, |x|)
)|x|2d|x|
= 4π R(t)3
∫ 1
0G(C(R(t), u
)u2du.
All together, we can obtain the equation on R(t); we define
W (ρ) := 4π
∫ 1
0G(C(ρ, u))u2du
and arrive to
R(t) = R(t)W(R(t)
). (6.9)
Recall the elementary properties (6.8) and thus, thanks to assumption (6.6)
d
dρW (ρ) < 0, W (0) = G(cb), W (∞) = G(0) < 0.
Therefore, by continuity we get the
Lemma 6.2 For cb > c, there is a unique radius R∞ > 0 such that
W(R∞)
= 0, W (R) > 0 for R < R∞, W (R) < 0 for R > R∞.
For cb < c, G(cb) < G(c) = 0 and W < 0.
The result follows directly from the sign properties of W in the differential equation (6.9).
6.3 Tumour growth with nutrients (general)
In a general domain, the Hele-Shaw setting is as follows. A pressure field equation is defined in
the domain Ω(t) −∆p = G(c(x, t)
), x ∈ Ω(t),
p = 0 on ∂Ω(t).(6.10)
The equation for the nutrient is unchanged −∆c+ λc = 0, x ∈ Ω(t),
c = cb, on ∂Ω(t)(6.11)
75
And the domain moves with the velocity −∇p(x, t).ν(x, t) with ν the outward normal to the
boundary of Ω(t)
X(t) = −∇p(X(t), t).ν(X(t), t).
An analysis can be found in [23], Sominet-Walker. Nonradial solutions and stability questions
are surveyed by A. Friedman [34].
Here we would like to explain the
Proposition 6.3 When set in a ball Ω(t) = BR(t), the model (6.10)–(6.11) is equivalent to that
in section 6.2.
Proof. To do so, we integrate the equation (6.10) and find
−∫
Ω(t)∆pdx = −
∫∂Ω(t)
∇p(x, t).ν(x, t)dx =
∫Ω(t)
G(c(x, t)
)dx.
In radial symmetry the integrand is constant and thus we find
R(t) = −∇p(X(t), t).ν(X(t), t) =1
Sd−1R(t)d−1
∫Ω(t)
G(c(x, t)
)dx
which is the model (6.4)–(6.5).
6.4 Proliferating, quiescent and dead cells
This geometric description can be extended to include more biological ingredients in the model
as quiescent cells. To keep simplicity, we ignore the nutrients and we introduce cell densities
nP (x, t), nQ(x, t) and nD(x, t) for proliferating, quiescent and dead cells. The ingredients are
those of the corresponding model introduced for ODEs in section 1.4. Following [33], the cell
movement is included as∂nP∂t + div[v nP ] = G
(nP , nQ
)− anP + bnQ, x ∈ Ω(t),
∂nQ∂t + div[v nQ] = anP − bnQ − dnD,∂nD∂t + div[v nD] = dnD − µnD,
(6.12)
This is coupled with the Darcy law for velocity
v(x, t) = −∇p(x, t)
and the pressure is implicitely determined by the incompressibility condition
nP + nQ + nD = nmax, ∀x ∈ Ω(t), t ≥ 0.
76
In order to get a self-contained model we have to find explicitely the pressure p. We add the
equations on the cell densities and find the elliptic equation
−∆p = div v = G(nP , nQ
)− µnD,
which we complete with the usual boundary conditions, that is in the simplest case
p(x, t) = 0 for x ∈ ∂Ω(t).
The free boundary ∂Ω(t) moves with the normal velocity −∇p
X(t) = −∇p(X(t), t)
and this makes that there is no need of boundary conditions in (6.12).
6.5 Proliferating and healthy cells
One can also include healthy cells. Again for the sake of simplicity we take simply proliferating
and healthy cells and ignore quiescent cells and nutrients. Following [22], the system reads∂nP∂t + div[v nP ] = G
(nP , nH
),
∂nH∂t + div[v nH ] = 0,
(6.13)
This is coupled with the Darcy law for velocity (with non-constant permeability)
v(x, t) = −κ(nP , nH)∇p(x, t)
and the pressure is again implicitely determined by the incompressibility condition
nP + nH = nmax, ∀x ∈ Rd, t ≥ 0.
In practice, the variable permeability is taken as
κ(nP , nH) = κH + (κP − κH)nPnmax
.
with κH the permeability of healthy cells and κP the permeability of proliferating cells.
As before, in order to compute the pressure p, we add the equations on the cell densities and
find
−div(κ(nP , nH)∇p(x, t)
)= div v = G
(nP , nH
).
When using healthy cells, the main difference is that there is not a precisely determined tumor
Ω(t) and thus no free boundary to evolve.
77
78
Chapter 7
From the cell scale to the free
boundary problem
This chapter aims at showing how the simple behaviors described previously can be used, com-
bined or generalized in order to give really intriguing patterns related to experimental observa-
tions. It is impossible to give an exhaustive presentation of these issues because the parabolic
formalism covers too many subjects where an enormous amount of models have been used.
7.1 Limit in the purely mechanical model
We consider the number density of tumor cells n(x, t) with the simplest model when growth is
limited by pressure as already presented in Section ?. We now introduce a parameter γ in the
model at cell scale ∂∂tn− div
(n∇p(n)
)= nG
(p(n)
), x ∈ Rd, t ≥ 0,
p(n) = nγ .(7.1)
We assume that the initial data satisfies
p(n0) ≤ PM , n0,∂
∂xin0 ∈ L1, div
(n0∇p(n0)
)+ n0G
(p(n0)
)≥ 0, (7.2)
and that the nonlinear function G(·) satisfies
G′(·) < 0, G(pM ) = 0, GM := G(0) = maxG(·). (7.3)
We recall the uniform estimates (independent of γ), for all t ≥ 0, p(n(t)) ≤ pM , eGM tn(t), eGM t ∂∂xin(t) ∈ L1,
∂∂tn ≥ 0, ∂
∂tp(n) ≥ 0,(7.4)
We now denote nγ , pγ the solution. We are going to prove the
Theorem 7.1 With the assumptions (7.2), (7.3), as γ →∞, we have strong convergence
nγ → n∞ ≤ 1, pγ → p∞ ≤ pM .
Let us define Ω(t) = x s. t. p∞(x, t) > 0. Then, we have
n∞(x, t) = 1 ∀x ∈ Ω(t),
p∞[∆p∞ +G(p∞)
]= 0, (7.5)
∂∂tn∞ − div
(n∞∇p∞
)= n∞G
(p∞),
p∞ = 0 for n∞(x, t) < 1.(7.6)
Remark 7.2 1. One can prove that, together with an initial data, (7.6) has a unique solution.
2. However the equation (7.5) is bad and does not predict what is the set Ω(t). Furthermore, for
any set Ω we can sole the elliptic equation −∆p = G(p), p ∈ H10 (Ω) and find a solution!
3. In section 7.2 we explain why this formulation is equivalent to the free boundary problem
(6.1), (6.2).
4. However, for general initial data there is more information contained in (7.5); in the region
0 < n∞ < 1 one has indeed p∞ = 0 and
∂
∂tn∞ = n∞G(0).
Proof. The proof is rather long because techinical functional analytic issues arise for compact-
ness. We refer to [55] for a complete proof and only give a general idea here. In particular we
take for granted the strong convergence of nγ and nγ which can be derived from uniform (in γ)
a priori estimates in BV .
(i) Because pγ = (nγ)γ ≤ pM , we conclude that nγ ≤ p1/γM and thus n∞ ≤ 1. Also p∞ = 0
when n∞ < 1.
(ii) We multiply the equation by p′(nγ) and use the chain rule div(n∇p(n)
)= ∇n∇p(n) + n∆p
to write the equation on pγ as
∂
∂tpγ − nγp′(nγ)∆pγ − |∇pγ |2 = nγp
′γ(n)G
(pγ(x, t)
)and for the special case p(n) = nγ at hand we find
∂
∂tpγ − γpγ∆pγ + |∇pγ |2 = γpγG
(pγ(x, t)
).
80
As γ →∞ we find (formally, there are nonlinearities!)
−p∞∆p∞ = p∞G(p∞)
that is (7.5). The main difficulty in the proof is to justify this equality which is equivalent to
prove strong convergence of ∇pγ to ∇p∞.
(iii) Passing to the limit in (7.1), we directly find (in the weak sense) the equation (7.6).
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Figure 7.1: A solution to (7.1) with γ = 40 that gives a good idea of the Hele-Shaw free
boundary. The density nγ is close to an indicator function. The pressure p∞ has a corner at the
two discontinuity points of n∞.
7.2 Front movement (purely mechanical)
For a general class of initial data, the formulation (7.5), (7.6) is equivalent to the free boundary
problem (6.1), (6.2). This is when
n0∞(x) = 1IΩ0, Ω0 = p0
∞ > 0,
then from equation (7.6) we conclude that for all times there is a set Ω(t) such that
n∞(x, t) = 1IΩ(t), Ω(t) = p∞(t) > 0.
Then one can check that (7.6) is equivalent to say that Ω(t) is moving with the normal velocity
v = −∇p∞, that is the free boundary problem in Section 6.2. This conclusion is based on the
following observations
(i) From (7.5) and the definition Ω(t) = p∞(t) > 0, we conclude −∆p∞ = G(p∞)
x ∈ Ω(t),
p∞ = 0 on ∂Ω(t).(7.7)
81
(ii) A second observation is that equation (7.6) can be written also as∂∂tn∞ −∆p∞ = n∞G
(p∞),
p∞ = 0 for n∞(x, t) < 1.(7.8)
To see this, in equation (7.1), we write the term
nγ∇p(nγ) = ∇Q(nγ), Q(nγ) =γ
γ + 1
(nγ)γ+1
and, using the second statement in Theorem 7.6,(nγ)γ+1
= nγpγ −→γ→∞
n∞p∞ = p∞.
(iii) We can now analyze the singularities in (7.8) (in the distributional sense). We only treat
dimension 1and radial symmetry for simplicity, that is n∞ = 1I(−R(t),R(t). Then, we compute
∂
∂tn∞ = R(t)
[δ(x+R(t))− δ(x+R(t))
],
∆p∞ =d
dxp′∞(x, t)
and p′∞(x, t) has a corner at x = ±R(t) because p∞ vanishes outside (−R(t), R(t). Therefore
∆p∞ = p′′∞(x, t)1I(−R(t),R(t) + p′∞(R(t), t)[− δ(x+R(t)) + δ(x+R(t))
],
where p′∞(R(t), t) means the value from inside.
Therefore equation (7.8) reduces to (7.7) for the smooth part and for Dirac part to
R(t) = p′∞(R(t), t).
That is the velocity of the free boundary predicited by Hele-Shaw model.
7.3 Limit with nutrients
Next, we consider the number density of tumor cells n(x, t) and we assume growth is directly
related to the available nutrients (oxygen, glucosis) c(x, t) through a nonlinear function G(·)that satisfies
G′(·) > 0, G(c) = 0, for some c < cb (7.9)
where c is the minimal concentration for cell survival and cb the maximal concentration brought
by blood vessels.
82
The model from section?? is written∂∂tnγ − div
(nγ∇p(nγ)
)= nγG
(cγ(x, t)
), x ∈ Rd, t ≥ 0,
α ∂∂tcγ −∆cγ + λcγnγ = 0, cγ(∞) = cb,
p(n) = nγ .
(7.10)
The limiting system for γ →∞ is now (formally, not everything is proved at the moment)
p∞[∆p∞ +G(c∞)
]= 0, (7.11)
n∞(x, t) = 1 ∀x ∈ Ω(t) := x, s. t. p∞(x, t) > 0,∂∂tn∞ − div
(n∞∇p∞
)= n∞G
(c∞),
p∞ = 0 for n∞(x, t) < 1.(7.12)
α∂
∂tc∞ −∆c∞ + λn∞c∞ = 0, c∞(|x| =∞) = cb.
Proof. As before, we multiply by p′(n) and write the equation on p
∂
∂tpγ − nγp′(nγ)∆pγ − |∇pγ |2 = nγp
′(nγ)G(cγ(x, t)
)and for the special case p = nγ at hand we find
∂
∂tpγ − γpγ∆pγ + |∇pγ |2 = γpγG
(cγ(x, t)
).
As γ →∞ we find formally
−p∞∆p∞ = p∞G(c∞)
that is the Hele-Shaw model
n∞(x, t) = 1 for x ∈ Ω(t) = p∞(t) > 0,−∆p∞ = G
(c∞(x, t)
)in Ω(t),
p∞ = 0 on ∂Ω(t),
and the other statements are as in section 7.1.
Is this the system in section (6.3)? The answer is NO because (7.12) can sustain a necrotic
core where c < c and n∞ < 1 as we saw it. In this region p∞ = 0 and the equation (7.11) by
itself is unable to predict what is the correct set Ω(t).
83
84
Chapter 8
Physiologically structured dynamics
As far as growth is concerned, all cells in a population do not necessarily behave exactly in the
same way and several ’physiological’ of ’phenotypical’ characters may modulate the growth rate
as
• Advancement in the cell cycle, size of cells, number of prior divisions,
• Variability (genetic, epigenetic)
8.1 A simple example : the renewal equation
The simplest model to understand how to model the role of age in a population is certainly
the renewal equation. Consider a ’closed’ population with no immigration, neither emigration.
Neglect also for the time being death and consider only aging and birth
• n(x, t) the number density of individuals of age x ≥ 0 at time t,
• d(x) ≥ 0 the death rate of individuals of age x,
• b(x) ≥ 0 the birth (fertility) rate of individuals of age x,
• B(t) =∫∞
0 b(x)n(x, t)dx, the the total number of newborn
• N(t) =∫∞
0 n(x, t)dx, the the total number of individuals.
At first imagine there is no death. The population density n(x, t) of individuals of age x > 0
at time t > 0 then satisfies
n(x+ s, t+ s) = n(x, t), ∀s ≥ 0.
As a consequence, differentiating in s, and taking s = 0, we find
∂
∂tn(x, t) +
∂
∂xn(x, t) = 0 t ≥ 0, x ≥ 0, . (8.1)
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.40
0.42
0.44
0.46
0.48
0.50
0.52
0.54
0.56
0.58
Figure 8.1: For two different birth rates b(x) (upper pictures), the corresponding total popu-
lation distribution N(t) is drawn below. The red curve is the steady state, drawn for visibility,
which corresponds to the solution of the ODE with the corresponding average death and birth
rates. The constant death rate d is computed to keep a bounded population.
This equation has to be complemented by the ’boundary condition’ at x = 0, i.e. the number
of newborns at time t; this is given by the quantity
n(x = 0, t) =
∫ ∞0
b(y)n(y, t)dy, (8.2)
In order to include death, we arrive to the so-called renewal equation∂∂tn(x, t) + ∂
∂xn(x, t) + d(x)n(x, t) = 0, t ≥ 0, x ≥ 0,
n(x = 0, t) =
∫ ∞0
b(y)n(y, t)dy.(8.3)
If b and d are independent of x, we may integrate in x and find (assuming the intuitive
condition n(∞, t) = 0) a simple ODE on N(t)
d
dtN(t)− n(0, t) + dN(t) = 0,
d
dtN(t) = (b− d) N(t).
If b = b0 + b1e−µx and d(x) = d (constant) one can also find a closure of (8.3) as a system on
the quantities
N(t) =
∫ ∞0
n(x, t)dx, I(t) =
∫ ∞0
n(x, t)e−µxdx
d
dtN(t) = (b0 − d) N(t) + b1 I(t),
d
dtI(t) + µI(t) = b0N(t) + (b1 − d) I(t).
86
But in general the information carried in (8.3) is much richer than id any of these systems as
depicted in Figure 8.1.
Exercise. Close the PDE (8.3) as a system for d constant and b(x) = b0 +∑
k bke−µkx.
8.2 Renewal equation: Stable Steady Distribution
Because the equation is linear, it is natural to look for a ’first eigenvalue’ λ0 that is called
Malthus parameter in this field. We introduce the notation
D(x) =
∫ x
0d(y) dy.
Theorem 8.1 Assume 1 <∫b(y)e−D(y)dy < ∞ (birth are strong enough) then there exists
unique λ0 > 0 and Φ(x) ≥ 0 (up to a multiplicaton constant) such that eλ0tΦ(x) solves the
renewal equation (8.3), that is equivalent to
∂
∂xΦ(x) + d(x)Φ(x) + λ0Φ(x) = 0, Φ(0) =
∫ ∞0
b(y)Φ(y)dy,
and it can be normalized by ∫ ∞0
Φ(x)dx = 1.
The function Φ is called the Stable Steady Distribution (SSD in short) because if represents
of the experimentally observed ’frequency’ (that is a distribution after renormalization to a
probability). As we will see in Section 8.3, any solution indeed ressembles to eλ0tΦ(x).
Proof. The one parameter family of solutions is written
Φ(x) = Φ(0)e−D(x)−λ0x.
It remains to find the parameter λ0 such that
Φ(0) =
∫ ∞0
b(y)Φ(0)e−D(y)−λ0y)dy ⇐⇒ 1 =
∫ ∞0
b(y)e−D(y)−λ0y)dy.
To do so we just notice that I : λ 7→∫∞
0 b(y)e−D(y)−λydy is a continuous function from [0,∞)
into (0,∫b(y)e−D(y)dy > 1] by the Lebesgue Theorem. It is increasing and I(∞) = 0. Therefore
the existence of (λ0,Φ) follows.
The dual equation. It is useful to introduce the solution to the dual problem
− ∂
∂xΨ(x) + d(x)Ψ(x) + λ0Ψ(x) = Ψ(0)b(x),
∫Ψ(x)N(x)dx = 1. (8.4)
87
Its solution is also explicitely built in writing
∂
∂x[Ψ(x)Φ(x)] = −Ψ(0)b(x)Φ(x), (8.5)
Ψ(x)Φ(x) = Ψ(0)Φ(0)−Ψ(0)
∫ x
0b(y)Φ(y)dy.
For integrability, we have no choice but to impose the value at x = 0 as Ψ(0)N(0) = Ψ(0)∫∞
0 b(y)N(y)dy
arriving to
Ψ(x)Φ(x) = Ψ(0)
∫ ∞x
b(y)Φ(y)dy. (8.6)
It remains to choose Ψ(0) so that
1 = Ψ(0)
∫ ∞x=0
∫ ∞y=x
b(y)Φ(y)dy = Ψ(0)
∫ ∞0
yb(y)Φ(y)dy
(an integrable weight because λ0 > 0)
Exponential growth. We may now see that all solutions of the renewal equation (8.3) increase
with the rate λ0. Indeed, we multiply it by Ψ and find
∂
∂t[Ψ(x)n(x, t)] +
∂
∂x[Ψ(x)n(x, t)] = λ0[Ψ(x)n(x, t)].
Then, we integrate (assuming still that the decay at infinity is enough) and find
d
dt
∫ ∞0
Ψ(x)n(x, t)dx = λ0
∫ ∞0
Ψ(x)n(x, t)dx
that means ∫ ∞0
Ψ(x)n(x, t)dx = eλ0t∫ ∞
0Ψ(x)n0(x)dx. (8.7)
Exercise. For d a constant and b(x) = b1I(x0,x1) compute the dual eigenfunction Ψ. For x0 = 0
show there is a ν > 0 such that b ≥ νΨ(x)/Ψ(0).
8.3 Renewal equation: exponential convergence to SSD
In view of (8.7), the next step is to show that n(x, t)e−λ0t is close to Φ (the SSD) as we can
expect from the Perron-Froebenius Theorem for matrices.
Theorem 8.2 Assume that
∃ν > 0, s.t. b(x) ≥ νΨ(x)
Ψ(0), (8.8)
88
the solutions of (8.3) satisfy∫|n(x, t)e−λ0t − ρΦ(x)|Ψ(x)dx ≤ e−νt
∫|n0(x)− ρΦ(x)|Ψ(x)dx (8.9)
with ρ =∫∞
0 n0(x)Ψ(x)dx, see (8.7).
Proof. We define
h(x, t) = n(x, t)e−λ0t − ρΦ(x).
By linearity it satisfies the equation∂∂th(x, t) + ∂
∂xh(x, t) + [d(x) + λ0]h(x, t) = 0, t ≥ 0, x ≥ 0,
h(x = 0, t) =∫b(y)h(y, t)dy.
Using the dual equation (8.4), we have again by a simple combination of these two equations,∂∂t
[h(x, t)Ψ(x)
]+ ∂
∂x
[h(x, t)Ψ(x)
]= −Ψ(0)b(x)h(x, t), t ≥ 0, x ≥ 0,
Ψ(0)h(x, t = 0) = Ψ(0)∫b(y)h(y, t)dy.
Therefore∂∂t
[|h(x, t)|Ψ(x)
]+ ∂
∂x
[|h(x, t)|Ψ(x)
]= −Ψ(0)b(x)|h(x, t)|, t ≥ 0, x ≥ 0,
Ψ(0)|h(x = 0, t)| = Ψ(0)|∫b(y)h(t, y)dy|.
We notice that by definition of ρ that∫
Ψ(x)h(x, t)dx = 0. Therefore, after integration in x, we
find
ddt
∫|h(x, t)|Ψ(x)dx = −Ψ(0)
∫b(x)|h(x, t)|dx+ Ψ(0)|
∫b(x)h(x, t)dx|
= −Ψ(0)∫B(x)|h(x, t)|dx+ |
∫[Ψ(0)b(x)− νΨ(x)]h(x, t)dx|
≤ −Ψ(0)∫b(x)|h(x, t)|dx+
∫[Ψ(0)b(x)− νΨ(x)]|h(x, t)|dx
= −ν∫|h(x, t)|Ψ(x)dx.
We conclude using the Gronwall lemma.
8.4 Cell divion and size structured equations
Closer to applications in biology is a model for cell divisions which is widely used for bacteria as
E. Coli. We assume that their growth and division rates are well described by a size parameter
(length or volume). A cell of size X(t) will grow with a rule
X(t) = g(X(t)),
89
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Figure 8.2: Comparison between an experimental cell size distribution (left) and the solution
to equation (8.10) (right).
with a growth rate which we assume is positive g(x) > 0. With certain rate b(X), a cell of size
X will divide and this produces two cells of size X2 . This gives rise to an equation on the number
n(x, t) of cells of size x∂∂tn(x, t) + ∂
∂x [g(x)n(x, t)] + b(x)n(x, t) = 4b(2x)n(2x, t), t ≥ 0, x ≥ 0,
n(x = 0, t) = 0.(8.10)
To explain the factor 4 we may compute
• N(t) =∫∞
0 n(x, t)dx the total number of cells
• M(t) =∫∞
0 xn(x, t)dx the total biomass (with x the mass of a cell)
Integration gives
d
dtN(t) +
∫b(x)n(x, t)dx = 4
∫b(2x)n(2x, t)dx = 2
∫b(x)n(x, t)dx,
that is the correct interpretation for B(x) as a division rate
d
dtN(t) =
∫b(x)n(x, t)dx.
Integration with the weight x gives
d
dtM(t)−
∫g(x)n(x, t)dx+
∫xb(x)n(x, t)dx = 4
∫xb(2x)n(2x, t)dx =
∫xb(x)n(x, t)dx,
that is the correct interpretation for g(x) as a growth rate
d
dtM(t) =
∫g(x)n(x, t)dx.
90
An example of typical solution (SSD), compared to experimental data is given in Figure 8.2
Again the dynamics can be understood with the help of eigenelements (λ0,Φ ≥ 0,Ψ) defined
by ∂∂x [g(x)Φ(x)] + [b(x) + λ0]Φ(x) = 4b(2x)Φ(2x), x ≥ 0,
Φ(x = 0, t) = 0,∫∞
0 Φ(x)dx = 1,
−g(x) ∂∂xΨ(x) + [b(x) + λ0]Ψ(x) = 2b(x)Ψ(x2 ),
∫∞0 Φ(x)Ψ(x)dx = 1.
(8.11)
One can prove existence of these eigenelements and that∫|n(x, t)e−λ0t − ρΦ(x)|Ψ(x)dx −→
t→∞0
with ρ =∫∞
0 n0(x)Ψ(x)dx.
8.5 Phenotypically structured equations
We may also consider parameters which distinguish the behaviour of adult cells and not their
state along the dynamics of interest. Then we enter a different modeling issue for which we
might go back to the simplest equation of Chapter 1. We consider that in the simplest model
d
dtN(t) = rN(t)
(1− N
K
)(8.12)
cells have different intrinsic growth rates r(x) (x here is a phenotypic parameter describing
growth it could be r(x) = x) and are all in competition for the nutrient. We arrive to the
generalisation
d
dtn(x, t) = r(x)n(x, t)
(1−
∫∞0 n(x, t)dx
K
). (8.13)
This equation is intrinsically nonlinear and leads to a completely different approach with different
types of result (selection of the fittest). See Chapter [?] on resistance to therapy.
8.6 The Perron-Froebenius theory
Motivation. We come back to the renewal equation and consider the steady state problem∂∂xn(x) + [d(x) + λ]n(x) = f(x), x ≥ 0,
n(x = 0) =
∫ ∞0
b(y)n(y)dy.(8.14)
91
For f ∈ L1 ∩L∞(0,∞) and λ large enough (to ensure µ < 1 in the expression below), there is a
unique solution built by the Duhamel formula
n(x) = n(0)e−D(x)−λx +
∫ x
0f(y)eD(y)−D(x)+λ(y−x)dy
with the value n(0) built thanks to the boundary condition
n(0) = n(0)
∫ ∞0
b(x)e−D(x)−λxdx+
∫ ∞x=0
b(x)
∫ x
0f(y)eD(y)−D(x)+λ(y−x)dydx,
n(0) =1
1− µ
∫ ∞x=0
b(x)
∫ x
0f(y)eD(y)−D(x)+λ(y−x)dydx, µ =
∫ ∞0
b(x)e−D(x)−λxdx.
We can write this as
n = A(f),
and A is a positive operator in the sense that
f ≥ 0, f 6= 0 =⇒ n = A(f) > 0.
One can check that (integrate the equation formally)
‖u‖1 ≤‖f‖1
λ− ‖b‖∞.
For a matrix A the condition: f ≥ 0, f 6= 0 =⇒ n = A(f) > 0 is equivalent to say that A has
positive coefficients. The Krein-Rutman theorem extends to infinite dimension (in the correct
setting) the Perron-Froebenius in finite dimension.
Many (but small) difficulties arise with the Krein-Rutman theory in particular that Lp are
Banach spaces where the positive cone has empty interior. Much of the time, the most efficient
method is to work by approximation and pass to the limit in a finite dimension problem.
Setting in terms of matrices. Let aij > 0, 1 ≤ i, j ≤ d, be the coefficients of a matrix
A ∈ Md×d(R) (there are interesting issues with the case aij ≥ 0 but we try to keep simplicity
here). The Perron-Frobenius theorem (see [66] for instance) tells us that A has a first eigen-
value λ0 > 0 associated with a positive right eigenvector Φ ∈ Rd, and a positive left eigenvector
Ψ ∈ Rd A.Φ = λ0Φ, Φi > 0 for i = 1, . . . , d,
Ψ.A = λ0Ψ, Ψi > 0 for i = 1, . . . , d.
For later purposes, it is convenient to normalize these vectors, so that they are now uniquely
defined. We choosed∑i=1
Φi = 1,d∑i=1
Φi Ψi = 1.
92
We set A = A− λ0Id and consider the evolution equation
d
dtn(t) = A.n(t), n(0) = n0. (8.15)
The solutions to this system converge as t→∞ with an exponential rate. Indeed, the following
result is classical
Proposition 8.3 For positive matrices A and solutions to the differential system (8.15), we
have
ρ :=d∑i=1
Ψini(t) =d∑i=1
Ψin0i , (8.16)
d∑i=1
Ψi|ni(t)| ≤d∑i=1
Ψi|n0i |, (8.17)
CΦi ≤ ni(t) ≤ CΦi with constants given by CΦi ≤ n0i ≤ CΦi, (8.18)
and there is a constant ν > 0 such that, with ρ given in (8.16), we have
d∑i=1
ΨiΦi
(ni(t)− ρΦi
Φi
)2 ≤ d∑i=1
ΨiΦi
(n0i − ρΦi
Φi
)2e−νt. (8.19)
General Relative Entropy. All the statements of this Proposition can be justified thanks to
the entropy inequality that follows
Proposition 8.4 Let H(·) be a convex function on R, then the solution to (8.15) satisfies
d
dt
d∑i=1
ΨiΦiH(ni(t)
Φi
)=
d∑i,j=1
ΨiaijΦj
[H ′(ni(t)
Φi
)[nj(t)
Φj− ni(t)
Φi]−H
(nj(t)Φj
)+H
(ni(t)Φi
)]≤ 0.
Definition 8.5 We call General Relative Entropy, the quantity
d∑i=1
ΨiΦiH(ni(t)
Φi
). The entropy
dissipation is the right hand side in the formula of Proposition 8.4.
Proof of Proposition 8.4. We denote by aij the coefficients of the matrix A and compute
d
dt
∑i
ΨiΦiH(ni(t)
Φi
)=∑i,j
ΨiH′(ni(t)
Φi
)aijnj(t)
=∑i,j
ΨiaijΦjH′(ni(t)
Φi
)[nj(t)Φj− ni(t)
Φi
],
93
because the additional term ni(t)Φi
vanishes since A.Φ = 0. But we also have, again thanks to the
equation on Φ and Ψ, that ∑i,j
ΨiaijΦj
[H(nj(t)
Φj
)−H
(ni(t)Φi
)]= 0.
Combining these two identities, we arrive to the equality in Proposition 8.4. The inequality
follows because only the coefficients out of the diagonal, that satisfy aij = aij ≥ 0, enters here.
Proof of Proposition 8.3. Notice that, as a special case of H in Proposition 8.4, we can choose
H(u) = u, which being convex together with −H gives the equality
d
dt
d∑i=1
Ψini(t) = 0.
And (8.16) follows. In particular this identifies the value ρ mentioned in (8.16).
The second statement (8.17) follows immediately by choosing the (convex) entropy function
H(u) = |u|.
As for the third statement (8.18), let us consider for instance the upper bound. It follows
choosing the (convex) entropy function H(u) = (u−C)2+ because for this nonnegative function
we haved∑i=1
ΨiΦiH(n0
i
Φi
)= 0.
Therefore, because the General Relative Entropy decays, it remains zero for all times,
d∑i=1
ΨiΦiH(ni(t)
Φi
)= 0,
which proves the result.
It remains to prove the exponential time decay statement (8.19). To do that, we work on the
vector
h(t) = n(t)− ρΦ,
which satisfies the same linear equation as n and∑i
Ψihi(t) =∑i
Ψi[ni(t)− ρΦ]dx = 0.
94
Then, we use the quadratic entropy function H(u) = u2 and the General Entropy Inequality
gives
d
dt
d∑i=1
ΨiΦi
(hi(t)
Φi
)2
= −d∑
i,j=1
ΨiaijΦj
(hj(t)
Φj− hi(t)
Φi
)2
≤ 0.
Then, we need a discrete Poincare inequality
Lemma 8.6 Being given Ψi > 0, Φi > 0, aij > 0 for i = 1, . . . , d, j = 1, . . . , d, i 6= j, there is
a constant ν > 0 such that for all vector m of components mi, 1 ≤ i ≤ d satisfying
d∑i=1
Ψimi = 0,
we have (Poincare-Wirtinger type inequality)
d∑i,j=1
ΨiaijΦj
(mj
Φj− mi
Φi
)2
≥ νd∑i=1
ΨiΦi
(mi
Φi
)2
.
With this lemma, we conclude
d
dt
d∑i=1
ΨiΦi
(hi(t)
Φi
)2
≤ −νd∑i=1
Φi
(hi(t)
Φi
)2
,
and then, (8.19) follows by a simple use of the Gronwall lemma.
Proof of Lemma 8.6. After renormalizing the vector m (when it does not vanish, otherwise the
result is obvious), we may suppose that
d∑i=1
Ψimi = 0,d∑i=1
ΨiΦi
(mi
Φi
)2
= 1.
Then we argue by contradiction. If such a ν does not exist, this means that we can find a
sequence of vectors (mk)(k≥1) such that
d∑i=1
Ψimki = 0,
d∑i=1
ΨiΦi
(mki
Φi
)2
= 1,d∑
i,j=1
ΨiaijΦj
(mkj
Φj− mk
i
Φi
)2
≤ 1/k.
After extraction of a subsequence, we may pass to the limit mk → m and this vector satisfies
d∑i=1
Ψimi = 0,d∑i=1
ΨiΦi
(mi
Φi
)2
= 1,
95
d∑i,j=1
ΨiaijΦj
(mj
Φj− mi
Φi
)2
= 0.
Therefore, from this last relation, for all i and j = 1, . . . , d, we have
mi
Φi=mj
Φj:= ν.
By the zero sum condition, we have ν = 0 because
ν
d∑i=1
Ψi = 0.
In other words, m = 0 which contradicts the normalization and thus such a ν should exist.
Remark 8.7 1. The matrix with (positive) coefficients bij = Ψi aij Nj is doubly stochastic,
i.e., the sum of the lines and columns is 1 (see for instance[66]).
2. Notice that aii − λ0 < 0 because∑
j aijΦj = 0. Therefore the matrix C with coefficients
cij = 1Φiaij Φj is that of a Markov process. In other words, we set yi = xi/Φi, then it satisfies
d
dtyi(t) = cijyj(t),
and the vector (1, 1, . . . , 1) is the (positive) eigenvector associated to the eigenvalue 0 of the
matrix C, i.e., cii =∑
j 6=i cij and cij ≥ 0. Then, (ΦiΨi)(i=1,...,d) is the invariant measure of the
Markov process. In particular this explains the entropy property which is classical for Markov
processes, see [69].
8.7 General Relative Entropy for the renewal equation
Motivation For h > 0 a space discretization step and I ∈ N the number of discretization points,
we can use a semi-disrete version of the renewal equationddtni(t) + 1
h [ni(t)− ni−1(t)] + dini(t) = 0, 1 ≤ i ≤ I,
n0(t) = hI∑i=I
ni(t)bi.(8.20)
Written as a matrix this is alsod
dt~n(t) = A.~n(t)
96
A =
− 1h − d1 + b1 b2 b3 b4
1h − 1
h − d2 0 0
0 1h − 1
h − d3 0
0 0 1h − 1
h − d4
(for I = 4) which means that A+ λI has positive corefficients for λ large enough.
Because of this structure, one can try to find a continuous version of the dissipation inequality
in Proposition 8.4.
GRE for the renewal equation
Theorem 8.8 (i) For all convex function H : R→ R and for all t > 0,
d
dt
∫ ∞0
Ψ(x)Φ(x)H(e−λ0tn(t, x)
Φ(x)
)dx := −DH(t) ≤ 0,
(ii) for the probability measure dµ(x) = b(x)Φ(x)Φ(0)dx,
DH(t) = Ψ(0)Φ(0)
[∫ ∞0
H(e−λ0tn(t, x)
Φ(x)
)dµ(x)−H
( ∫ ∞0
e−λ0tn(t, x)
Φ(x)dµ(x)
)].
The quantity DH is a dissipation term. Its expression means that the only dissipation occurs
from the boundary term at x = 0. The fact that dµ(x) is a probability relies on the boundary
condition for the steady state Φ.
Proof. We use that∂
∂x
1
Φ− d(x)
Φ= 0
and conclude∂
∂t
e−λ0tn(t, x)
Φ(x)+
∂
∂x
e−λ0tn(t, x)
Φ(x)= 0.
The chain rule gives∂
∂tH(e−λ0tn(t, x)
Φ(x)
)+
∂
∂xH(e−λ0tn(t, x)
Φ(x)
)= 0.
Combined with (8.5), we get
∂
∂t[Ψ(x)Φ(x)H
(e−λ0tn(t, x)
Φ(x)
)]+
∂
∂x[Ψ(x)Φ(x)H
(e−λ0tn(t, x)
Φ(x)
)] = −Ψ(0)b(x)Φ(x)H
(e−λ0tn(t, x)
Φ(x)
).
97
After integration in x ∈ R+ we find, using the Jensen inequality
ddt
∫Ψ(x)Φ(x)H
( e−λ0tn(t,x)Φ(x)
)dx = −Ψ(0)Φ(0)
∫H( e−λ0tn(t,x)
Φ(x)
)dµ(x) + Ψ(0)Φ(0)H
( e−λ0tn(t,0)Φ(0)
)= Ψ(0)Φ(0)
[−∫H( e−λ0tn(t,x)
Φ(x)
)dµ(x) +H
( ∫ e−λ0tn(t,x)Φ(x) dµ(x)
)]≤ 0,
for all convex function H. The statements (i) and (ii) follow from this inequality.
8.8 General Relative Entropy for parabolic equation
Consider the parabolic equation, set in a domain Ω that can be the whole space or on a bounded
domain with Dirichlet, NEumann or mixed boundary conditions,
∂
∂tn−∆n+ b(x).∇n+R(x)n = 0.
Assume there is a triplet of eigenelements (λ0,Φ > 0,Ψ > 0)
−∆Φ + b(x).∇Φ +R(x)Φ = λ0Φ,
−∆Ψ− b(x).∇Ψ +R(x)Ψ = λ0Ψ.
Show thatd
dt
∫Ω
Ψ(x)Φ(x)H(e−λ0tn(t, x)
Φ(x)
)dx := −DH(t) ≤ 0,
DH(t) =
∫Ω
Ψ(x)Φ(x)H ′′(e−λ0tn(t, x)
Φ(x)
) ∣∣∣∣∇e−λ0tn(t, x)
Φ(x)
∣∣∣∣2 dx.Poincare inequalities are typically related to the square entropy.
Case 1. The standard Poincare inequality states that for a bounded domain∫Ωu2 ≤ ν
∫Ω|∇u|2, ∀u ∈ H1
0 (Ω)
and is used for the Laplace equation with Dirichlet boundary condition [30].
Case 2. The Poincare-Wirtinger inequality is used for the Neumann boundary condition and
states that ∫Ωu2 ≤ ν
∫Ω|∇u|2, ∀u ∈ H1(Ω),
∫Ωu = 0.
98
This case is closer to what we have used above. The Laplace equation with Neumann boundary
condition admits the first eigenelements triplet (λ0 = 0,Φ = Ψ = 1).
Case 3. In the full space, one cantreat similarly the Fokker-Planck equation
∂
∂tn−∆n− div(n∇V ) = 0.
The eigenelements are (λ0 = 0,Φ = e−V , ψ = 1).
Assuming that V has superquadratic growth at infinity, one has the Poincare inequality (see
[45] for instance) ∫Rd
Φu2 ≤ ν∫Rd
Φ|∇u|2, ∀u ∈ C1comp,
∫ΩuΦ = 0.
Here one has in mind that u = n/Φ.
99
100
Bibliography
[1] L. Ambrosio, Transport equation and Cauchy problem for BV vector fields. Invent. Math.,
158 (2004), 227–260.
[2] J. C. Arciero, T. L. Jackson and D. E. Kirschner, A mathematical model of tumor-immune
evasion and siRNA treatment. DCDS(B), 4(1), 2004, 39–58.
[3] T. Balois, C. Chatelain and M. Ben Amar, Patterns in melanocytic lesions: impact of the
geometry on growth and transport inside the epidermis. J. R. Soc. Interface 11: 20140339.
http://dx.doi.org/10.1098/rsif.2014.0339
[4] Claude Basdevant, Jean Clairambault and Francis Levi, Optimisation of time-scheduled
regimen for anti-cancer drug infusion. ESAIM: M2AN, Vol. 39, No 6, 2005, 1069–1086.
[5] M. Ben Amar, C. Chatelain, and P. Ciarletta, Contour Instabilities in Early Tumor Growth
Models. PRL 106, 148101 (2011).
[6] Eshel Ben-Jacob, Donald S. Coffey and Herbert Levine, Bacterial survival strategies suggest
rethinking cancer cooperativity, Trends in Microbiology, 2012, Vol. 20, No. 9, 403–410.
[7] Benzekry, S. Modeling and mathematical analysis of anti-cancer therapies for metastatic
cancers. PhD thesis University of Aix-Marseille (2011).
[8] M. Bertsch, R. Dal Passo and Masayasu Mimura. A free boundary problem arising in a
simplified tumour growth model of contact inhibition. Interfaces and Free Boundaries 12
(2010), 235250
[9] Michiel Bertsch, Danielle Hilhorst, Hirofumi Izuhara and Masayasu Mimura. Segregation
property in a tumour growth PDE model with contact inhibition
[10] Bertsch, M. ; Hilhorst, D. ; Izuhara, H. ; Mimura, M. ; Wakasa, T. Travelling wave solutions
of a parabolic-hyperbolic system for contact inhibition of cell-growth. European J. Appl.
Math. 26 (2015), no. 3, 297–323.
101
[11] N. Bessonov, F. Crauste, I. Demin, V. Volpert, Dynamics of erythroid progenitors and
erythroleukemia. Math. Model. Nat. Phenom. 4(3), 2009, 210–232.
[12] Didier Bresch, Thierry Colin, Emmanuel Grenier, Benjamin Ribba, and Olivier Saut. Com-
putational modeling of solid tumor growth: the avascular stage.
[13] Didier Bresch, Thierry Colin, Emmanuel Grenier, Benjamin Ribba, and Olivier Saut. A
viscoelastic model for avascular tumor growth.
[14] H. M. Byrne and M. A. J. Chaplain, Growth of nonnecrotic tumours in the presence and
absence of inhibitors, Mathematical Biosciences, 181 (1995), 130-151.
[15] H. Byrne and M. Chaplain, Growth of necrotic tumors in the presence and absence of
inhibitors, Math. Biosci. 135 (1996), 187-216.
[16] H. Byrne and D. Drasdo, Individual based and continuum models of growing cell popula-
tions: a comparison, J. Math. Biol. (2009).
[17] H. M. Byrne, J. R. King, D. L. S. Mc Elwain and L. Preziosi. A two-phase model of solid
tumour growth. Applied Mathematics Letters 16 (2003) 567–573.
[18] H. Byrne and L. Preziosi, Modelling solid tumour growth using the theory of mixtures.
Math. Med. Biol., 20(4):341–366, 2003.
[19] Chaplain, M., Matzavinos, A.. Mathematical modelling of spatio-temporal phenomena in
tumour immunology. In: Friedman, A. (Ed.), Tutorials in Mathematical Biosciences III:
Cell Cycle, Proliferation, and Cancer. Lecture Notes in Mathematics, vol. 1872. Springer,
Berlin, 131–183 (2006).
[20] C. Y. Chen, H. M. Byrne, J. R. King. The influence of growth-induced stress from the
surrounding medium on the development of multicell spheroids. Journal of Mathematical
Biology 43(3) 191–220 (2001).
[21] J. Clairambault, S. Gaubert, and T. Lepoutre. Circadian rhythm and cell population
growth. Mathematical and Computer Modelling, 53:1558–1567, 2011.
[22] Th. Colin, A. Iollo, D. Lombardi and O. Saut. System indentification in tumor growth mod-
eling using semi-empirical eigenfunctions. Mathematical Models and Methods in Applied
Sciences Vol. 22, No. 6 (2012) 1250003 (30 pages). DOI: 10.1142/S0218202512500030
[23] V. Cristini, J. Lowengrub and Q. Nie. Nonlinear simulation of tumor growth, J. Math.
biology 46 (2003) 191–224.
102
[24] J. Dambrine, N. Meunier, B. Maury, A. Roudneff-Chupin. A congestion model for cell
migration,
[25] De Angelis E., Delitala M., Marasco A., and Romano A., Bifurcation analysis for a mean
field modeling of tumor and immune system competition, Math. Comp. Model. 37, 1131–
1142, (2003).
[26] De Angelis E., and Jabin P.E.,Qualitative Analysis of a mean field model of tumor-immune
system competition, Mathematical Models and Methods in Applied Sciences, 13, 187–206
(2003).
[27] DiPerna, R. J. and Lions, P.-L. Ordinary differential equations, transport theory and
Sobolev spaces. Invent. Math. 98 (1989), no. 3, 511–547.
[28] Drasdo, D., Hohme, S. A single-cell-based model of tumor growth in vitro: monolayers and
spheroids. Phys Biol. Jul 12;2(3):133-47 (2005).
[29] Dyson, J., Sanchez, E., Villella-Bressan, R., and Webb, G. An age and spatially structured
model of tumor invasion with haptotaxis. DCDS(B) Vol. 8(1) 45–60 (2007).
[30] Evans, L. C. Partial Differential Equations, Graduate Studies in Mathematics, Vol. 19,
American Mathematical Society (1998).
[31] J. Folkman. Antiangiogenesis : new concept for therapy of solid tumors. Ann. Surg.,
175:409416, 1972.
[32] Francoise, J.-P., Oscillations en biologie. Series: Mathmatiques et Applications, Vol. 46.
Springer 2005.
[33] Friedman, A. A hierarchy of cancer models and their mathematical challenges. DCDS(B)
4(1) (2004), 147–159.
[34] Friedman, A. Mathematical analysis and challenges arising from models of tumor growth,
M3AS, Vol. 17, Supplement (2007), 1751–1772.
[35] Friedman A. and Reithich F., Analysis of a mathematical model for the growth of tumors.
J. Math. Biol. 38, 262–284 (1999).
[36] Gatenby R. A. and Gawlinski E. T. The glycolytic phenotype in carcinogenesis and tumor
invasion: insights through mathematical models Cancer Res. 63 3847–54 (2003).
[37] Gatenby R. A., Gawlinski E. T., Gmitro A. F., Kaylor B. and Gillies R. J. Acid-mediated
tumor invasion: a multidisciplinary study Cancer Res. 66 5216–5223 (2006 ).
103
[38] Greenspan, H. P. Models for the growth of a solid tumor by diffusion. Stud. Appl. Math. 51
(1972), no. 4, 317–340.
[39] P. Hahnfeldt, D. Panigraphy, J. Folkman, and L. Hlatky. Tumor development under angio-
genic signaling : a dynamical theory of tumor growth, treatment, response and postvascular
dormancy. Cancer Research, 59(19):4770–4775, 1999.
[40] Th. Hillen, H. Enderling and P. Hahnfeld. The Tumor Growth Paradox and Immune System-
Mediated Selection for Cancer Stem Cells. Bull. Math. Biol. To appear.
[41] Itik, M. and Banks, S.P., Chaos in a Three-dimensional Cancer Model, Int. Journal of
Bifurcation and Chaos, 20 (1), 71-79, 2010.
[42] D. Kirschner and J. C. Panetta, Modeling immunotherapy of the tumor-immune interaction.
J. Math. Biol. (1998) 37: 235–252.
[43] Isaac Klapper and Jack Dockery, Mathematical Description of Microbial Biofilms. SIAM
Review, Vol. 52, No. 2, 221–265 (2010).
[44] V. Kuznetsov, I. Makalkyn, M. Taylor, A. Perelson, Nonlinear dynamics of immunogenic
tumors: parameter estimation and global bifurcation analysis, Bulletin of Mathematical
Biology, 1994, Vol. 56, No. 2, pp 295321.
[45] Ledoux M. The concentration of measure phenomenon. AMS math. surveys and monographs
89. Providence 2001.
[46] Urszula Ledzewicz, Helmut Maurer and Heinz Schattler, Optimal and suboptimal protocols
for a mathematical model for tumor anti-angiogenesis in combination with chemotherapy.
Math. Biosc. Engin. To appear
[47] U. Ledzewicz and H. Schaettler, Singular controls and chattering arcs in optimal control
problems arising in biomedicine, Control and Cybernetics, 38, (2009), 1501–1523.
[48] Urszula Ledzewicz, Mohammad Naghnaeian and Heinz Schattler, Optimal response to
chemotherapy for a mathematical model of tumor-immune dynamics. J. Math. Biol. (2012)
64, 557–577.
[49] C. Letellier, F. Denis, L.A. Aguirrec, What can be learned from a chaotic cancer model?
Journal of Theoretical Biology, Vol. 322 (2013) 7–16.
[50] Liu, D., Ruan, S., Zhu, D. Stable periodic osccilations in a two-stage cancer model of tumor
and immune system interactions. Math. biosciences and Eng. 9(2) (2012) 347–368.
104
[51] Lowengrub, J. S.; Frieboes H. B.; Jin, F.; Chuang, Y.-L.; Li, X.; Macklin, P.; Wise, S.
M.; Cristini, V. Nonlinear modelling of cancer: bridging the gap between cells and tumours.
Nonlinearity 23 (2010) R1–R91.
[52] Nikos V Mantzaris, Steve Webb, and Hans G Othmer. Mathematical modeling of tumor-
induced angiogenesis. Journal of Mathematical Biology, 49(2), February 2004.
[53] B. Maury, A. Roudneff-Chupin, F. Santambrogio, A macroscopic crowd motion model of
gradient flow type, Math. Mod. Meth. Appl. Sci.,
[54] Perthame B. Transport equations in biology. Series ’Frontiers in Mathematics’, Birkhauser
(2007).
[55] B. Perthame, F. Quiros, and J.-L. Vazquez. The Hele-Shaw asymptotics for mechanical
models of tumor growth. Arch. Ration. Mech. Anal., Vol. 212, No 1 (2014) 93–127.
[56] B. Perthame, F. Quiros, M. Tang and N. Vauchelet. Derivation of a Hele-Shaw type system
from a cell model with active motion. Interfaces and Free Boundaries 16 (2015), 489–508.
DOI 10.4171/IFB/327. HAL-UPMC : hal-00906168
[57] B. Perthame, M. Tang, and N. Vauchelet. Traveling wave solution of the Hele-Shaw model
of tumor growth with nutrient. M3AS Vol. 24, No. 13, 2601–2626 (2014).
[58] Preziosi, L.; Tosin, A. Multiphase modelling of tumour growth and extracellular matrix
interaction: mathematical tools and applications. J. Math. Biol. 58 (2009), no. 4-5, 625–
656.
[59] Ranft, J.; Basana, M.; Elgeti, J.; Joanny, J.-F.; Prost, J.; Julicher, F. Fluidization of tissues
by cell division and apoptosis. Proc. Natl. Acad. Sci. USA (2010), no. 49, 20863–20868.
[60] Rohrschneider M., Scheuermann G., Hoehme, S. and Drasdo D. (2007) Shape Charac-
terization of Extracted and Simulated Tumor Samples using Topological and Geometric
Measures. IEEE Engineering in Medicine and Biology Society, 2007, 6271–6277.
[61] T. Roose, S. J. Chapman and P. K. Maini. Mathematical models of avascular tumor growth.
SIAM Review Vol. 49, No . 2, 2007, 179–208.
[62] R. K. Sachs, L. R. Hlatky, and P. Hahnfeldt. Simple ode models of tumor growth and
anti-angiogenic or radiation treatment. Mathematical and Computer Modelling, 33(12)
:12971305, 2001.
105
[63] Olivier Saut, Jean-Baptiste Lagaert, Thierry Colin, and Hassan M Fathallah-Shaykh. A
multilayer grow-or-go model for gbm : effects of invasive cells and anti-angiogenesis on
growth. Bulletin of mathematical biology, 76(9) :2306–2333, 2014.
[64] Heinz Schattler and Urszula Ledzewicz. Geometric Optimal Control, Theory, Methods, and
Examples. Springer, to appear.
[65] Agnese Seminara, Thomas E. Angelini, James N. Wilking, Hera Vlamakis, Senan Ebrahim,
Roberto Kolter, David A. Weitz, and Michael P. Brenner, Osmotic spreading of Bacillus
subtilis biofilms driven by an extracellular matrix. PNAS Vol. 109 (2012) 1116–1121.
[66] Serre D., Les matrices, theorie et pratique, Dunod (2001).
[67] Jonathan A. Sherratt and Mark A. J. Chaplain, A new mathematical model for avascular
tumour growth. J. Math. Biol. 43, 291–312 (2001).
[68] Tang, M.; Vauchelet, N.; Cheddadi, I.; Vignon-Clementel, I.; Drasdo, D.; Perthame, B.
Composite waves for a cell population system modelling tumor growth and invasion. Chin.
Ann. Math. Ser. B 34 (2013), no. 2, 295–318.
[69] van Kampen, N. G. Stochastic processes in physics and chemistry. Lecture Notes in Math-
ematics, 888. North-Holland Publishing Co., Amsterdam-New York, 1981.
[70] Alexei Tsygvintsev, Simeone Marino and Denise E. Kirschner A mathematical model of
Gene Therapy for the Treatment of Cancer, paper in edited book ”Mathematical Models
and Methods in Biomedicine”, Springer, 2012, to appear.
[71] Vazquez, J.-L. The Porous Medium Equation; Mathematical Theory. Clarendon Press
Oxford Mathematical Monographs (2006).
106
![A Tumor-Immune Interaction Model for Synergistic Combinations of Anti PD-L1 … · 2020. 8. 31. · shortcomings of monotherapy [10–13]. There are mathematical models of immune-tumor](https://img.dokumen.tips/doc/110x75/610aef398de2215ffc480f9c/a-tumor-immune-interaction-model-for-synergistic-combinations-of-anti-pd-l1-2020.jpg)
