A Population-level Hybrid Model of Tumour-Immune System

A Population-level Hybrid Model ofTumour-Immune System Interplay: model

construction and analysis.

Giulio Caravagna

Dipartimento di Informatica, Sistemistica e Comunicazione,Universita degli Studi Milano-Bicocca.

giulio.caravagna@disco.unimib.it

Joint works on T-IS interplay with:

I A. d’Onofrio (European Institute of Oncology);

I R. Barbuti and P. Milazzo (University of Pisa).

References:

I G.Caravagna, A.d’Onofrio, P.Milazzo, R.Barbuti (2010)Antitumour Immune Surveillance Through Stochastic Oscillations.Journal of Theoretical Biology, 265 (3), 336-345, 2010.

I G.Caravagna, R.Barbuti, A.d’Onofrio, (2011)Fine-tuning anti-tumor immunotherapies via stochastic simulations.Submitted to BMC Bioinformatics.

Step 1: the bio-bakground, as understood by a computer scientist.

I IS-response triggered by specific (neo)antigens;

I antigens appear as a consequence of (epi)genetic events;

I tumours evolve to eventually evade the IS-control;

(“[..] naive T cells express neither BLIMP1(1) nor interleukin-2 (IL-2).

Following encounter with antigen, activated T cells secrete high quantities of

IL-2 and upregulate the IL-2 receptor (IL-2R). Engagement of IL-2R by IL-2

results in BLIMP1 transcription in early effector cells. BLIMP1 expression

increases in late effector cells and represses Il2 expression through an unknown

mechanism. [..] ”)(1) The B-lymphocyte-induced maturation protein 1 (BLIMP1) guides the fate of effector B and T cells.

Immune surveillance: a phenomenon concerning the control, andsometimes the eradication, of a tumour by the IS.

(“... is a theory that the IS patrols the body not only to recognize and destroy

invading pathogens but also host cells that become cancerous. Perhaps

potential cancer cells arise frequently throughout life, but the IS usually

destroys them as fast as they appear. There is some evidence for this attractive

notion. There is also evidence that the IS mounts an attack against established

cancers although it often fails.”)

Old hypothesis (≈ 1900) recently supported by:

I new molecular techniques;

I many epidemiological studies in cancer immunobiology.

Take-home lesson: the tumour may be suppressed by the IS.

Goal 1: population-level model of T-IS interplay + surveillance.

Step 2: a purely deterministic model.

Some basic background: the logistic growth function is the ODE

dy

dx= ry

(1− y

C

)where:

I y is the population size, x is time;

I r defines the growth rate;

I C is the carrying capacity of the organism.

Similarly, the Michaelis-menten kinetics is the ODE modeling the enzymekinetics of

E + Sk1→ ES

k2→ E + P

with reversible de-comeplexation

ESk−1

1→ E + S .

Building a model: the choice of the abstraction level is crucial.

Possible agents:

I microscopic: molecules, nucleotides, chromosome, proteins, ...

I macroscopic: cells, proteins, tissues, ...

Consequently involved events:

I microscopic: transcriptions, complexations, dimerization...

I macroscopic: apoptosis, coagulation, ...

This choice is crucial and affects

I the type of data necessary to analyze the model;

I the validity of the predictions model.

We define a macroscopic population-level model.

Agents as populations:

I people;

I animals;

I cells;

I ..

Consequently involved events:

I people move, get sick, become virus carriers, vaccinate, recover, ...

I cells proliferate, duplicate, cross membranes, die, ...

Example

I SIR models;

I ...

The deterministic population-level model that we consider appears in:Kirschner and Panetta, Journal of Mathematical Biology 37(3), 1998.

Why? It has been highly influential (i.e. ≈ 140 citations).

Cellular populations involved: Tumour cells (T) and Effector cells (E).

... plus a protein-level population: Interleukins IL-2 (I).

Mathematically, each population is modeled by a density, and itsinfinitesimal changes are ruled by an ODE. This results in a system of 3linked ODEs.

Tumour cells (T): (i) saturated logistic growth and (ii) cell death inducedby the effectors via Michaelis-menten kinetics.

T ′ = rT (1− bT )− aTE

g2 + T

Effector cells (E): (i) linear growth (cT , c ≡ antigenicity) models theresponse, (ii) Michaelis-menten proliferation induced by interleukins and(iii) death as linear apoptosis.

E ′ =p1IE

g1 + I− µ2E + cT

Interleukins IL-2 (I): (i) Michaelis-menten proliferation induced by theT-IS interplay and (ii) linear degradation.

I ′ =p2TE

g3 + T− µ3I

To analyze a model you need data....

Param. Val. Unit Description

r 0.18 days−1 baseline growth rate of the tumor

b 10−9 ml−1 carrying capacity of the tumor

a 1 ml · days−1 baseline strength of the killing rate by effectors

c 10−4 days−1 tumor antigenicity

gT 105 ml−1 50% reduction factor of the killing rate by effectors

gE 2 · 107 pg · l−1 50% reduction factor of IL-stimulated growth rate of effectors

gI 103 ml−1 50% reduction factor of production rate of ILs

pE 0.1245 days−1 baseline strength of the IL-stimulated growth rate of effectors

pI 5 pg · days−1 baseline strength of production rate of ILs

µE 0.03 days−1 inverse of average lifespan of effectors

µI 10 days−1 loss/degradation rate of IL2

... these are fitted on data pertaining to mice.

Analysis varying c , i.e. the average number of antigens expressed by atumour cells.

Fact: the equilibrium (0, 0, 0) is unstable.

(A) bounded growth (i.e. death);

(B,C) late-recurrent oscillations (i.e. undetectable miSS is different fromeradication since tumours eventually evade);

(D) dumped oscillations.

KP precludes eradication, but small numbers are predicted (T � 1).

Vexata quæstio: what if we consider (intrinsic) stochastic effects?

Step 2: towards hybrid models...

Wee need to move from densities to explicit numbers of cells ormolecules. We can do that by using the volume V .

We estimated volume V ∈ [1.16, 3.2]ml for a chimeric mouse:

- body weight ≈ 20gr (female) up to 40gr grams (male);

- blood volume [5.8, 8]ml per 100gr .

With V and the Avogadro number we can convert the KP modelto a stochastic Gillespie-based model.

Is a Gillespie-based model feasible?

I molecular weight of interleukins is 15 · 103Da;

I the average number of proteins is huge (#I ↑);

You should know that if A ↑ then 1/A ↓. What happens withGillespie’s equation

τ =1∑aj(x)

log(r−1)

when #I ↑?

A purely stochastic model is computationally intractable.Moreover, when numbers are big stochastic fluctuations areneglectable, and the ODE-approach is not that bad....

Deterministic models:

I densities/concentrations;

I changes yield differential equations (ODE, DDE, PDE);

I fluid variations.

What is in the middle?

Stochastic models:

I exact numbers;

I probability distributions;

I jump processes (piece-wise constant).

Deterministic models:

I densities/concentrations;

I changes yield differential equations (ODE, DDE, PDE);

I fluid variations.

Hybrid models:

I exact numbers + densities/concentrations;

I probability distributions + differential equations;

I jump processes (piece-wise constant) + fluid variations.

Stochastic models:

I exact numbers;

I probability distributions;

I jump processes (piece-wise constant).

([..] we may assume that the dynamics of I (t) is well approximated by a linear

ODE with randomly varying coefficients which, however, in the intervals

between two consecutive stochastic events are evidently constant [..] )

Hybrid model : with discrete E and T , and continuous I :

(i) Reaction-based events (a4 is time-dependent):

T 7→ T + 1 a1 = r2TT 7→ T − 1 a2 = r2bV

−1T 2

T 7→ T − 1 a3 = (aTE )/(gTV + T )E 7→ E + 1 a4(t) = (pEEI (t))/(g1 + I (t))E 7→ E − 1 a5 = µEEE 7→ E + 1 a6 = cT

(ii) interlinked with the ODE for I (t).

Next steps to achieve Goal 1:

1. find a way to simulate this model;

2. reproduce all the KP results;

3. investigate whether this model exhibits eradication.

Gillespie’s SSA states that, if

p(Rj fires at t + dt | x, t) = aj(x)dt

then the putative time follows

p(τ | x, t) = a0(x) exp(−a0(x)τ)

where

a0(x)τ =∑j

∫ t+τ

t

aj(x)dt .

This because the state is constant in [t, t + τ), hence∫ t+τ

t

aj(x)dt = aj(x)

∫ t+τ

t

dt = aj(x)τ .

For time-dependent propensity functions it becomes∫ t+τ

t

aj(t)dt

a function of time yielding a unknown density.

I E and T constant in any stochastic interval [ti ; ti+1);

I Without immunotherapies I (t) has analytical form

tn ≤ t < tn+1 I (t) = Bn + (I (tn)− Bn) e−µ3(t−tn)

where Bn is a constant function of (Tn,En).

I tn+1 = tn + τ obtained by generalizing Gillespie: let χ ∼ Exp(1) in∫ tn+τ

tn

a4(t)dt + τ

7∑i=1i 6=4

aj = χ and P(Rj) =aj∑7i=1 ai

Transcendental equation ≡ no analytical form and costly simulation.

Why is it costly: let us assume that we want to find a 0 of a function f ,i.e.

∃x . f (x) = 0 ?

Many techniques, we used the bisection method:

I find a such that f (a) < 0;

I find b such that f (b) > 0;

I pick the mid-point

f

(b − a

2

)and if it is smaller than 0 then assign

b − a

2to a, otherwise to b;

I repeat until f

(b − a

2

)is arbitrary close to 0.

The hybrid model is qualitatively equivalent to the KP....

0

5e+07

1e+08

1.5e+08

2e+08

2.5e+08

3e+08

3.5e+08

4e+08

4.5e+08

0 20 40 60 80 100 120 140 160 180 200

T

time (days)

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

110 115 120 125 130 135 140 145 150 155 160

density

eradication time

T (0) = 1 E(0) = I (0) = 0 c = .02

..however, (left) low-level oscillations yield eradication: T (≈ 140) = 0.

Probabilistic analysis: what is the empirical PDF of eradication?

I time-dependent property (i.e. model-checking a path);

I For PDFs (right) we averaged 103 runs;

I PDFs can not be obtained by KP.

PDF of eradication with variable E (0)/I (0) and fixed antigenicity.

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0 20 40 60 80 100 120 140 160 180

density

eradication time

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 20 40 60 80 100 120 140 160 180 200

density

eradication time

T (0) = 1 I (0) = 0 E(0) = 10i (left) i = 1, 2, 3 (right) i = 4, 5

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

110 120 130 140 150 160 170 180

density

eradication time

T (0) = 1 E(0) = 0 I (0) = 10i i = 1, 2, . . . , 6

Part I: (basic) model construction and analysis.

Part II: further model modifications and analysis.

Immunotherapy: therapeutical-driven modication of the T-IS interplay.

Therapy-related patient-dependent issues:

1. How should the IS be stimulated?

2. Which is a good scheduling for a therapy?

3. How much should be the duration of a single session of therapy?

Known facts, results of experimental/theoretical studies:

I fixing such parameters is non-trivial;

I scheduling plays a major role in the effectiveness of a therapy.

Two somewhat contrasting needs arise:

I avoiding that the therapy-related side-effects kill the patients;

I no tumour cells at the end of the therapy (i.e eradication).

Goal 2: fine-tuning immunotherapies by means of stochastic simulations.

We study (i) passive and (ii) active immunotherapies in the hybrid model.

(i) Adoptive cellular immunotherapy (ACI) → boost directly E

i.e. withdraw white blood cells, culture them, re-inject.

E ′ =pE IE

gE + I− µEE + cT+VσE (t)

(ii) IL-2-based immunotherapy: → boost directly I , indirectly E

i.e. inject interleukins in the patient.

I ′ =pIV

TE

gIV + T− µI I+σI (t)

We consider piecewise-constant/impulsive and daily/weekly deliveries.

θ0 +At0 ts teθ0 θ1 θk

d0

d1

dk

θ1 +A θk +A

In both cases the total injected drug is the same.

Mathematical modeling of therapies is pretty intuitive: for ts ≤ t ≤ te

I piecewise-constant: σ(t) =∑k

i=0 di[H(t − θi )− H(A− t + θi )

].

I impulsive: σ(t) =∑k

i=0 uiδ(t − θi ).

I this yields impulsive ODEs (i.e. x(θ+i ) = x(θ−i ) + ui ).

The simulation algorithm is slightly modified (2 time-dependent react.).

Simulation time spans from mins to days. (v.s. few mins for KP).

Asymptotic analysis: ODEs predicts eradication only if therapy →∞.

i.e. when the therapy stops, tumour restarts growing!

IL-2 therapy: {1,. . . ,30 }. T (0) = 105, E (0) = I (0) = 0.

0

500000

1e+06

1.5e+06

2e+06

0 3 6 9 12 15 18 21 24 27 30

T

E

0

5

10

15

20

15 15.25 15.5 15.75 16

0

500000

1e+06

1.5e+06

2e+06

0 3 6 9 12 15 18 21 24 27 30

T

E

0

5

10

15

20

22 22.25 22.5 22.75 23

Single-run of piecewise-constant (left) and impulsive (right).

(left) ter ≈ 16 days, Tmax ≈ 5 · T (0)

(right) ter ≈ 23 days, Tmax ≈ 20 · T (0)

Is the piecewise-constant more efficient?

ACI: {1,. . . ,30 }. T (0) = 105, E (0) = I (0) = 0.

0

20000

40000

60000

80000

100000

120000

140000

0 3 6 9 12 15 18 21 24 27 30

T

E

0

5

10

15

20

13 14 15 16

0

20000

40000

60000

80000

100000

120000

140000

0 3 6 9 12 15 18 21 24 27 30

T

E

0

5

10

15

20

13 14 15 16

Single-run of piecewise-constant (left) and impulsive (right).

(left) ter ≈ 15 days, Tmax ≈ 1.5 · T (0)

(right) ter ≈ 15 days, Tmax ≈ 1.5 · T (0)

Are the therapies equivalent?

Combined ACI & IL-2. T (0) = 105, E (0) = I (0) = 0.

0

50000

100000

150000

200000

250000

300000

350000

400000

450000

0 3 6 9 12 15 18 21 24 27 30 33 36 39

T

E

0

5

10

15

20

28 29 30 31

0

50000

100000

150000

200000

250000

300000

350000

400000

450000

0 3 6 9 12 15 18 21 24 27 30 33 36 39

T

E

0

5

10

15

20

38 39 40 41

Single-run of impulsive therapies: (left) synch., (right) asynch. deliveries.

ΘILi = i + 0.5 ΘE

i = i i = 1, . . . , 30

ΘILi = ΘE

i = i i = 1, . . . , 30

(left) ter ≈ 31 days, Emax ≈ 5 · 105

(right) ter ≈ 41 days, Emax ≈ 3 · 105

Is the IS response more efficient in the synchronous case?

ACI. {1,. . . ,30 }. E (0) = I (0) = 0 and bigger initial tumour.

0

500000

1e+06

1.5e+06

2e+06

2.5e+06

3e+06

0 3 6 9 12 15 18 21 24 27 30

T

E

0

5

10

15

20

23 23.5 24 24.5 25

0

2e+07

4e+07

6e+07

8e+07

1e+08

1.2e+08

1.4e+08

0 3 6 9 12 15 18 21 24 27 30

T

E

0

5

10

15

20

25 25.25 25.5 25.75 26

Single-run of impulsive therapies: (left) T (0) = 106, (left) T (0) = 107.

In (right) with the same injection rate of (left) T (t) ≈ 109 and ter ↑.

(left) ter ≈ 25 days, Tmax ≈ 2.5 · 106

(right) ter ≈ 25 days, Tmax ≈ 1.2 · 108

Does T (0)/rates affect the final outcome of the T-IS interplay?

ACI. {1,. . . ,30 }. T (0) = 105, E (0) = I (0) = 0.

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50 60 70

End of the therapy

52.5

21.5

1

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50 60 70

End of the therapy

52.5

21.5

1

Empirical PDF of piecewise-constant (left) and impulsive (right) ACIs.

Variable rates of injection: multiplicative terms w∗ ∈ {5, 2.5, 2, 1.5, 1}.

(left), (right) ter < 30 for w∗ ≥ 2, ter > 30 for w∗ ≤ 1.5

Is [1.5; 2] is a range for w∗ to have ter < 30 (as often desired)?

Conclusions

A hybrid model enforcing the following hypothesis:

(i) daily IL-based: piecewise-constant > impulsive;

(ii) daily ACI: piecewise-constant = impulsive;

(iii) impulsive ACI: weekly > daily 1

(iv) weekly ACI: impulsive > piecewise-constant;

(v) combined impulsive ACI+IL-2: synchronous > asynchronous.

What is next?

I realistic extrinsic noise (e.g. stochastic nonlinear equations);

I systematic exploration of the parameters space;

I more complex effects of therapies (e.g. delayed);

I more efficient simulation strategies;

I individual-based models.1metronomic delivery, recognized to be more effective for other anti-tumor

therapies such as anti-angiogenesis therapies and chemotherapies.