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June 2005 Lisette de Pillis HMC Mathematics
Introduction to Mathematical Modeling in Biology with ODEs
Lisette de Pillis
Department of Mathematics
Harvey Mudd College
June 2005 Lisette de Pillis HMC Mathematics
Mathematical Modeling and Mathematical Biology
• What is Mathematical Modeling …and how do you spell it?
– “Mathematics consists of the study and development of methods for prediction”
– The aim of Biology is “to find useful and verifiable descriptions and explanations of phenomena in the natural world”
– Modeling = The use of mathematics as a tool to explain and make predictions of natural phenomena
– Mathematical Biology involves mathematically modeling biological phenomena
Thanks: Cliff Taubes, 2001
June 2005 Lisette de Pillis HMC Mathematics
Mathematical Modeling Philosophy
• Why are models useful:– Formulating precise ideas implicit
assumpltions less likely to “slip by”– Mathematics = concise language that
encourages clarity of communication– Mathematical theorems and computational
resources can be accessed
June 2005 Lisette de Pillis HMC Mathematics
Mathematical Modeling Philosophy
• Why are models useful (cont):– Can safely test hypotheses (eg, drug
treatment), and confirm or reject– Can predict system performance under
untested or untestable conditions
• How models can be limited (trade-offs):– Easy math Unrealistic model– Realistic model Too many parameters– Caution: unrealistic conclusions possible
June 2005 Lisette de Pillis HMC Mathematics
The Modeling Process
Model World
Mathematical Model(Equations)
Real World
Occam’s Razor*
Interpret and Test(Validate)
Formulate ModelWorld Problem
Model Results
Mathematical Analysis
Solutions,Numerics
*Occams’s Razor: “Entia non sunt multiplicanda praeter necessitatem”“Things should not be multiplied without good reason”
June 2005 Lisette de Pillis HMC Mathematics
Model World
Components of the Model World
Things whose effects are neglected
Things that affect the model but whose behavior the model is not designed to study (exogenous or independent variables)
Things the model is designed to study (endogenous or dependent variables)
June 2005 Lisette de Pillis HMC Mathematics
The Five Stages of Modeling
1. Ask the question.
2. Select the modeling approach.
3. Formulate the model.
4. Solve the model. Validate if possible.
5. Answer the question.
June 2005 Lisette de Pillis HMC Mathematics
Introduction to Continuous Models
• One of simplest experiments in biology: Tracking cell divisions (eg, bacteria) over time.
• Analogous dynamics for tumor cell divisions (what they learn in med school):
Thanks: Leah Keshet, Ami Radunskaya
A tumor starts as one cell The cell divides and becomes two cells
June 2005 Lisette de Pillis HMC Mathematics
Introduction to Continuous Modeling
24 cells
Cell divisions continue…
22 cells
23 cells
June 2005 Lisette de Pillis HMC Mathematics
Ordinary Differential Equations (ODEs)
• Mathematical equations used to study time dependent phenomena
• A “differential equation” of a function = an algebraic equation involving the function and its derivatives
• A “derivative” is a function representing the change of a dependent variable with respect to an independent variable. (Often thought of as representing a slope.)
June 2005 Lisette de Pillis HMC Mathematics
Ordinary Differential Equations (ODEs)
• Ex: If N (representing, eg, bacterial density, or number of tumor cells) is a continuous function of t (time), then the derivative of N with respect to t is another function, called dN/dt, whose value is defined by the limit process
• This represents the change is N with respect to time.
t
tNttN
dt
dNt
)()(lim
0
June 2005 Lisette de Pillis HMC Mathematics
• Let N(t) = bacterial density over time• Let K = the reproduction rate of the bacteria per
unit time (K > 0) • Observe bacterial cell density at times t and
(t+Dt). Then
N(t+Dt) ≈ N(t) + K N(t) Dt
• Rewrite: (N(t+Dt) – N(t))/Dt ≈ KN(t)
Our Cell Division Model: Getting the ODE
Total density at time t+Dt
Total density at time t + increase in density due to reproduction during time interval Dt
≈
June 2005 Lisette de Pillis HMC Mathematics
Our Cell Division Model: Getting the ODE
• Take the limit as Dt → 0
“Exponential growth” (Malthus:1798) • Analytic solution possible here.
• Implication: Can calculate doubling time
)N(N
eNtN Kt
0
)(
0
0
KNdTdN
June 2005 Lisette de Pillis HMC Mathematics
• Find “population doubling time” t:
• Point: doubling time inversely proportional to reproductive constant K
Analysis of Cell Division Model: Exponential
KteNtN 0)( Ke2
2)( 0 NN
K)2ln( K/)2ln(
and
imply
Taking logs and solving for t gives
June 2005 Lisette de Pillis HMC Mathematics
Exponential Growth Implications: 1 Day Doubling
• Doubling time t=ln(2)/K• Suppose K=ln(2), so t=1, ie, cell popn doubles in 1 day. • : In 30 days, 1 cell →→ detectable population• is about a sphere (bag)• is about a 100 grams (1/10 kilo) of tumor (bag)• Tumor will reach 100 grams between days 36 and 37. • One week later, tumor weighs a kilo (at around cells)
and is lethal.• 90% tumor removal of cells leaves 10 billion cells. • 99% removals leaves 1 billion cells. • Every cancer cell must be killed to eliminate the tumor
910
930 102
1210
1110
31cm
1110
June 2005 Lisette de Pillis HMC Mathematics
Exponential Growth: Realistic?
June 2005 Lisette de Pillis HMC Mathematics
Extending the Growth Model: Additional Assumptions + New System
• Reproductive rate K is proportional to the nutrient concentration, C(t): so K(C)=kC
• a units of nutrient are consumed in producing 1 unit of pop’n increment → system of equations:
• Simplify the system of ODEs (collapse):
• Logistic Growth Law!
• Note: equiv to assuming K=K(N)=C0- aN, ie K is density dependent.
CNdtdNdtdC
CNdtdN
NNCdtdN 0
June 2005 Lisette de Pillis HMC Mathematics
• Solution:
• N0 = initial population
• kC0 = intrinsic growth rate
• C0/a = carrying capacity
• For small popn levels N, N grows about “exponentially”, with growth rate r ≈ kC0
• As time t → ∞, N → N(∞)=C0/a
• This “self limiting” behavior may be more realistic for longer times
tCeN
CN
CNtN 0
00
00
0)(
Analysis of Logistic Model for Cell Growth
June 2005 Lisette de Pillis HMC Mathematics
Exponential versus Logistic Growth
June 2005 Lisette de Pillis HMC Mathematics
Logistic Growth: Initial Conditions, Stability
June 2005 Lisette de Pillis HMC Mathematics
Other Growth Models
• Power Law:
• Gompertz:
• Von Bertlanffy:
baNdtdN
)0(,1
)))(1(()(
:Solution
010
)1(1
NNbNC
CatbtNb
b
bNaNdtdN
agdtdg
gNdtdN
1ln:Alt
bbNtN at exp0)(:Solution
1 bNaNdtdN
)0(
)exp()exp(1
1)(:Solution
0
1
00
NNb
avttaNN
btN
June 2005 Lisette de Pillis HMC Mathematics
Intrinsic Cell Growth Models: Comparisons
Von
Ber
tala
nffy
Logi
stic
Gom
pert
zP
ower
Law
June 2005 Lisette de Pillis HMC Mathematics
Used to represent Inter- and Intra-Species Competition
Dynamic Population Model Formulation: General Approach
• Balance (Conservation):
• Law of Mass Action: Encounters between populations occur randomly, and the number of encounters is proportional to the product of the populations, eg,
PopulationChange in Time
= Stuff Going In – Stuff Going Out
dNMkMcMdtdM
bNMkNaNdtdN
M
N
1:Predator
1 :Prey
June 2005 Lisette de Pillis HMC Mathematics
Formulating a 2-Population Model: Tumor-Immune Interactions
• Step 1 - Ask the Question:
How does the immune system affect tumor cell growth? Could it be responsible for “dormancy” followed by aggressive recurrence?
• Step 2 - Select the Modeling Approach:
Track tumor and immune populations over time → Employ ODEs
June 2005 Lisette de Pillis HMC Mathematics
Formulating a 2-Population Model: Tumor-Immune Interactions
• Step 3 - Forumlate the Model:• Identify important quantities to track:
– Dependent Variables:• E(t)=Immune Cells that kill tumor cells (Effectors) (#cells or density)• T(t)=Tumor cells (#cells or density)
– Independent Variable: t (time)
• Specify Basic Assumptions:– Effectors have a constant source– Effectors are recuited by tumor cells– Tumor cells can deactivate effectors (assume mass action law)– Effectors have a natural death rate– Tumor cell population grows logistically (includes death already)– Effector cells kill tumor cells (assume mass action law)
June 2005 Lisette de Pillis HMC Mathematics
• Rate parameters (units)• s=constant immune cells source rate (#cells/day)• s=steepness coefficient (#cells)• r=Tumor recruitment rate of effectors (1/day)• c1=Tumor deactivation rate of effectors (1/(cell*day))• d=Effector death rate (1/day)• a=intrinsic tumor growth rate (1/day)• 1/b=tumor population carrying capacity (#cells)• c2=Effector kill rate of tumor cells (1/(cell*day))
A Two Population System
dEETcsdt
dET
rET 1)(
ETcbTaTdt
dT2)1(
June 2005 Lisette de Pillis HMC Mathematics
Model Elements
dEETcsdt
dET
rET 1)(
ETcbTaTdt
dT2)1(
Stuff going in Stuff going outPopulation change in time
June 2005 Lisette de Pillis HMC Mathematics
Model Elements
dEETcsdt
dET
rET 1)(
ETcbTaTdt
dT2)1(
Logistic GrowthMassAction
Michaelis-Menten
June 2005 Lisette de Pillis HMC Mathematics
Step 4: Solve the System
• Must treat system as a whole
• In general, a closed-form solution does not exist
• Solution approaches: • Dynamical systems analysis (find general system
features) • Numerical (find example system solutions)
• Next up: Finding general system features
June 2005 Lisette de Pillis HMC Mathematics
Dynamical Systems Analysis: When we cannot solve analytically
• Find equilibrium points (set ODEs to 0): plot nullclines and find intersections
• Find stability properties of equilbrium points (if nonlinear: must linearize)
• Trace possible trajectories in phase diagram
June 2005 Lisette de Pillis HMC Mathematics
Dynamical Systems Analysis: When we cannot solve analytically
• Set ODEs to 0:
• Therefore:
• Solve for E and T curves (nullclines). Find points of overlap (intersections).
0
0
dtdT
dtdE
01
0
2
1
ETcbTaT
dEETcTrETs
Find equilibrium points
June 2005 Lisette de Pillis HMC Mathematics
Analysis: the equilibria are determined by setting both differential equations to zero.
E-equation = 0T-equation = 0
June 2005 Lisette de Pillis HMC Mathematics
Each stable equilibrium point has a basin of attraction
June 2005 Lisette de Pillis HMC Mathematics
Step 5: Answer the Question
• Question: Do we see dormancy?
• Question: Do we see aggressive regrowth in this model?
• Not yet: How about with different parameters? Let’s see…
June 2005 Lisette de Pillis HMC Mathematics
Alternate Parameters: Tumor Dormancy with Immune System Evident
• Four equilibria - two stable• Dormancy: stable spiral
I m m u n e
Tumor
June 2005 Lisette de Pillis HMC Mathematics
Alternate Parameters – Dangerous Regrowth with Immune System
• Creeping through to dangerous equilibrium:
Tumor
I m m u n e
June 2005 Lisette de Pillis HMC Mathematics
Step 5: Answer the Question
• Question: Now do we see dormancy?
• Question: Now do we see aggressive regrowth in this model?
Yes!
Yes!
June 2005 Lisette de Pillis HMC Mathematics
Step 1: Ask a New Question
• New Question: In the clinic, what causes asynchronous response to chemotherapy?
• Note: The current 2 population model does not answer this question…We need to extend the model.
Continue the modeling cycle…
June 2005 Lisette de Pillis HMC Mathematics
Extend the Model Further - More Realism: Adding Normal Cells (Competition)
• Turn the two population model into a three population model (dePillis and Radunskaya, 2001, 2003)
• Why: Gives more realistic response to chemotherapy treatments, eg, allows for delayed response to chemotherapy
June 2005 Lisette de Pillis HMC Mathematics
Three Population Mathematical Model
• Combine Effector (Immune), Tumor,
Normal Cells
TNcNbNrdtdN
TNcETcTbTrdtdT
EdETcTAETsdtdE
422
3211
11
)1(
)1(
)(
Note: There is always a tumor-free equilibrium at (s/d,0,1)
Stuff going in Stuff going outPopulation change in time
June 2005 Lisette de Pillis HMC Mathematics
Analysis: Finding Null Surfaces
Trb
c
bNNdtdN
Nrb
cE
rb
c
bTTdtdT
rTTAdTATc
TAsEdtdE
22
4
2
11
3
11
2
1
11
1or 00
1or 00
)()(
)(0
• Curved Surface:
• Planes
June 2005 Lisette de Pillis HMC Mathematics
Null surfaces: Immune, Tumor, Normal cells
June 2005 Lisette de Pillis HMC Mathematics
Analysis: Determining Stability of Equilibrium Points
• Linearize ODE’s about (eg, tumor-free) equilibrium point
• Solve for system eigenvalues:
Negativeor Positive
Negative Always 0
Negative Always 0
221313
2222
11
bcdscr
bcr
d
June 2005 Lisette de Pillis HMC Mathematics
CoExisting Equilibria Map: Paremeter Space s
Region 4: Stable @ (E=0.4, T=0.6, N=0.4) Unstable @ (E=0.8, T=0.2, N=0.8)
June 2005 Lisette de Pillis HMC Mathematics
Time Series Plots
• Creeping Through to Dangerous Equilibrium:
June 2005 Lisette de Pillis HMC Mathematics
Evolution in Time: Increasing Initial Immune Strength
Initial Immune Strength Range: 0.0 < E(0) < 0.3Basin Boundary Range: 0.12 < E(0) < 0.15
Stable Equilibrium - Co-Existing: E=0.4, T=0.6, N=0.4Stable Equilibrium - Tumor Free: E=1.65, T=0, N=1.0
Time vs Tumor Time vs Normal Time vs Immune
June 2005 Lisette de Pillis HMC Mathematics
Cell Response to Chemotherapy
• Idea: Add drug response term to each DE, create DE describing drug
Amount of cell kill for given amount of drug u: )1()( ku
i eauF
June 2005 Lisette de Pillis HMC Mathematics
• Four populations:
• Goal: control dose to minimize tumor• See: “A Mathematical Tumor Model with Immune Resistance and Drug
Therapy: an Optimal Control Approach”, Journal of Theoretical Medicine, 2001
Normal,Tumor & Effector cells with Chemotherapy
udtvdtdu
NeaTNcNbNrdtdN
TeaTNcETcTbTrdtdT
EeaEdETcTArETsdtdE
u
u
u
2
3422
23211
111
)(
)1()1(
)1()1(
)1()(
)(tv
June 2005 Lisette de Pillis HMC Mathematics
Continuing the Modeling Process
• Ask new questions: Example – Are there better treatments that can cure when traditional treatments do not?
• How to use our model: Experiment with timings, Apply optimal control.
June 2005 Lisette de Pillis HMC Mathematics
Tumor Growth - No Medication
E(0) = 0.1E(0) = 0.15
June 2005 Lisette de Pillis HMC Mathematics
Tumor Growth - Traditional Pulsed Chemotherapy
I(0) = 0.15 I(0) = 0.1
June 2005 Lisette de Pillis HMC Mathematics
Compare to chemotherapy based on Optimal Control Theory:
I(0) = 0.15 I(0) = 0.1
June 2005 Lisette de Pillis HMC Mathematics
Continuing the Modeling Process
• More questions:
• Can we validate the model?
• Are there experimental data against which we can compare model components?
• If we find data, can we modify our dynamics?
June 2005 Lisette de Pillis HMC Mathematics
Conventional effector-target interaction term:cNTcellsNK-by Lysis CellTarget of Rate
Tumor Cell Lysis by NK-Cells: Fit to Mouse Data
Mass Action Law: Does it Fit Data?
June 2005 Lisette de Pillis HMC Mathematics
Tumor Cell Lysis by CD8+T-Cells: Fit to Mouse DataConventional product (power) form not necessarily a good fit for CD8+T-Target interactions
TTLs
TLd
eL
eL
)/(
)/(cells-Tby Lysis CellTarget of Rate
NEW EFFECTOR to TARGET LYSIS LAW:
Rational Law a Better Fit for CD8+T Cells
June 2005 Lisette de Pillis HMC Mathematics
Ratio Dependence: A Predator-Prey Model
Refs:Akcakaya et al. Ecology Apr 1995Abrams and Ginzberg TREE Aug 2000
LLLTgfdT
dL
LLTgTTfdtdT
)),,((
),()(
2
1
g(T,L) = “Functional Response”
f2(T,L) = “Numerical Response”
Ratio Dependence
June 2005 Lisette de Pillis HMC Mathematics
Ratio Dependence: A Predator-Prey Model
Our “Functional Response:”
LTL
sT
LdLTg
),(
Our “Numerical Response:”
A Michaelis-Menten type function of g(T,L).
Ratio Dependence
June 2005 Lisette de Pillis HMC Mathematics
Tumor Cell Lysis by CD8+T-Cells: Fit to Mouse DataConventional power form not necessarily a good fit for CD8+T-Target interactions
Close Up: POWER vs RATIONAL LAWS:
Power vs Rational:Non-Ligand-Transduced Ligand-Transduced
Power vs Rational:
Ratio Dependence: Good Fit to Data
June 2005 Lisette de Pillis HMC Mathematics
CD8-Tumor Lysis Equations: Error Comparison
Goodness of Fit for CD8+T-cell Lytic Activity: Comparing the residuals (error) of the conventionalconventional product formproduct form with the new rational form.
June 2005 Lisette de Pillis HMC Mathematics
CD8-Tumor Cell Lysis Equations: Fit to Human Data
TTLs
TLd
eL
eL
)/(
)/(cells-Tby Lysis CellTarget of Rate
NEW EFFECTOR to TARGET LYSIS LAW applies to HUMAN DATA:
June 2005 Lisette de Pillis HMC Mathematics
Continuing the Modeling Process
• The new dyamics require a new model:
• Develop model with different populations to track:
• Specific Immune Cells (CD8+T, Rational Kill)• Nonspecific Immune Cells (NK, Mass-Action Kill)• Tumor Cells
• Test new treatments
June 2005 Lisette de Pillis HMC Mathematics
T
sD
qLTLDk
DjmL
dt
dL
pNTNTh
TgfNe
dt
dN
dDcNTbT)aT(1dt
dT
eLT
L
2
2
2
2
Where eLT
L
Logistic Growth
NK-Tumor Kill:Power Law
CD8-Tumor Kill:Rational Law
Immune Recruitment:Michaelis-Menten Kinetics
New Model Equations: Two Immune Populations, Ratio Dependent Kill Term
June 2005 Lisette de Pillis HMC Mathematics
Parameters a, b, c, d, s, and eL were fit from published experimental data. All other parameters were estimated or taken from the literature.
Circulating lymphocytes
Rate of drug administration and decay
No IL2
IL-2 boost
System of Model Equations: Additional Treatment
June 2005 Lisette de Pillis HMC Mathematics
Treatment: Chemotherapy Alone, Cancer Escapes
Healthy Immune System.
Twice Tumor Burden T0=2x107
Multiple Chemotherapy Doses.
Simulation parameters: human, with chemo, no vaccine, u small
Bolus chemotherapy every 10 days
June 2005 Lisette de Pillis HMC Mathematics
Treatment: Vaccine Therapy Alone, Cancer Escapes
Healthy Immune System.
Twice Tumor Burden T0=2x107
Single Vaccine Dose.
Simulation parameters: human, vaccine alone, u small
June 2005 Lisette de Pillis HMC Mathematics
Treatment: Vaccine and Chemo Combined
Cancer Is Controlled
Healthy Immune System.
Twice Tumor Burden T0=2x107
Single Vaccine Dose.
Simulation parameters: human, with chemo, with vaccine, u small
Three Chemotherapy Doses
June 2005 Lisette de Pillis HMC Mathematics
Equilibrium points of 4-population system (no treatment) are found at points where the values of LE1 from equation (20) intersect with the solutions L of equation (21). These points of intersection can be found numerically, yielding equilibrium
point(s) (TE,NE,LE,CE).
June 2005 Lisette de Pillis HMC Mathematics
Stability: Zero Tumor Equilibrium
June 2005 Lisette de Pillis HMC Mathematics
Stability: Specific Parameter Set
• With the specific parameter set:– The zero tumor equilibrium is unstable – There is only one non-zero tumor equilibrium,
and it is stable.
• Point: – The tumor-free equilibrium is unstable, while
the high-tumor equilibrium is stable: Only a change in system parameter values may permit permanent removal of the tumor → Immunotherapy/Vaccine is one way to do this
June 2005 Lisette de Pillis HMC Mathematics
Bifurcation diagram: the effect of varying the NK-kill rate, c.
June 2005 Lisette de Pillis HMC Mathematics
Sensitivity to Initial Conditions after Bifurcation Point. C*=0.9763
June 2005 Lisette de Pillis HMC Mathematics
Bifurcation Diagram: CD8+T Parameter j
June 2005 Lisette de Pillis HMC Mathematics
Basin of Attraftion of zero−tumor and high−tumor equilibria
June 2005 Lisette de Pillis HMC Mathematics
Bifurcation Analysis: Basins of Attraction
The barrier separates system-states which evolve towards the low-tumor-burden equilibrium from those which evolve towards the high tumor-burden state.
With Immunotherapy
With Chemotherapy
No therapy
This barrier moves with therapy
June 2005 Lisette de Pillis HMC Mathematics
•Add spatial heterogeneity: non-uniform tissue, morphology-dependent.
•Cellular automata: discrete, probabilistic, and/or hybrids.
Additional Model Extensions – Extending from ODEs to PDEs and Cellular Automata
June 2005 Lisette de Pillis HMC Mathematics
Deterministic & Probabilistic:2D and 3D
Image Courtesy http://www.ssainc.net/images/melanoma_pics.GIF
http://www.lbah.com/Rats/rat_mammary_tumor.htm
http://www.lbah.com/Rats/ovarian_tumor.htm
http://www.loni.ucla.edu/~thompson/HBM2000/sean_SNO2000abs.html
Spatial Tumor Growth Modeling
June 2005 Lisette de Pillis HMC Mathematics
Microenvironment Simulations: Entire System.
Modeling Tumor Growth and TreatmentL.G. de Pillis & A.E. RadunskayaL.G. de Pillis & A.E. Radunskaya
June 2005 Lisette de Pillis HMC Mathematics
Final Thoughts on Modeling
• “All models are wrong…some are useful”, Box and Draper, 1987
• “All decisions are based on models…and all models are wrong”, Sterman, 2002
• “Although knowledge is incomplete, nonetheless decisions have to be made. Modeling…takes place in the effort to plan clinical trials or understand their results. Formal modeling should improve that effort, but cautious consideration of the assumptions is demanded”, Day, Shackness and Peters, 2005
June 2005 Lisette de Pillis HMC Mathematics
Thanks for listening!
Lisette de Pillis
June 2005 Lisette de Pillis HMC Mathematics
BLANK
June 2005 Lisette de Pillis HMC Mathematics
CA Simulation: Movie - a snapshot every 20 days for 200 days showing tumor growth and necrosis.
QuickTime™ and aH.263 decompressor
are needed to see this picture.
June 2005 Lisette de Pillis HMC Mathematics
QuickTime™ and aDV/DVCPRO - NTSC decompressor
are needed to see this picture.
The tumor affects the acidity of the micro-environment:
June 2005 Lisette de Pillis HMC Mathematics
•Add nano-vaccines (based on mouse models).
•What are some of the questions under discussion? (Dose, treatment scheduling, where to administer the vaccine)
•What parameters might be good indicators of successful response to treatment?
Modeling Tumor Growth and TreatmentA.E. RadunskayaA.E. Radunskaya
What next ?
What do we have?
•A mathematical model which simulates some of the main features of tumor growth: hypoxia, high acidity, necrosis.
•A model which allows the addition of other cells (immune cells) and/or small molecules (drugs, vaccines).