Franco Flandoli, University of Pisa Stochastic Analysis ... · Franco Flandoli, University of Pisa...

Introduction to some topics in Mathematical Oncology

Franco Flandoli, University of Pisa

Stochastic Analysis and applications in Biology, Finance and Physics,Berlin 2014

Franco Flandoli, University of Pisa () Mathematical OncologyStochastic Analysis and applications in Biology, Finance and Physics, Berlin 2014 1

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The field received considerable attentions in the past 10 years

One of the plenary talks at ICM 2014 was on this subject (B.Perthame)

Some people believe it could be a revolution in the next 10 years ifproperly developed (see Forbes)

For people interested in stochastic systems it is an opportunity totouch an applied field of basic importance for society wherestochasticity plays a role

The following discussion is the outcome of an initial investigation withMichele Coghi, Mauro Maurelli, Manuela Benedetti and other youngcollaborators.


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Most common models

1 Macroscopic models (tissue level)2 microscopic models (cell level)3 mixed or multi-scale models.

Remarks.

Usually, in Physics, microscopic means molecular. Cell scale is in asense a meso-scale, but I will call it microscopic.

Cell motion is stochastic to a large extent.


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Objects studied by the models

1 Macroscopic models: density of cells, oxygen concentration etc.2 microscopic models: single cells3 multi-scale models: single cells and oxygen concentration, etc.


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Equations

1 Macroscopic models: Fokker-Planck equations with nonlinear reactionterms

2 microscopic models: interacting particle systems (based on SDE ordiscrete models)

3 mixed or multi-scale models: both


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Example of Macroscopic model

P. Hinow, P. Gerlee, L. J. McCawley, V. Quaranta, M. Ciobanu, S.Wang, J. M. Graham, B. P. Ayati, J. Claridge, K. R. Swanson, M.Loveless, A. R. A. Anderson, A spatial model of tumor-hostinteraction: application of chemoterapy, Math. Biosc. Engin. 2009.

Among several other models I have chosen this one due to three facts:

it is a very good paper

the team is a leading one in quantitative oncology (e.g. KristineSwanson)

it does not require deep biological training.

This model is made of 7 coupled PDE-ODE. I will spend some time on itin order to explain the level of complexity and realism that is usuallyreached in such papers.


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Normoxic, hypoxic and apoptotic cells

normoxic cells: healthy, proliferating tumor cells, with normal oxygensupply

hypoxic cells: quiescent tumor cells, with poor oxygen supply

apoptotic cells: death or programmed to death tumor cells


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The PDE for normoxic cells

∂N∂t

= k1∆N︸︷︷︸background diffusion

(N (t, x) = normoxic cell density)

div (σ (N )∇N )︸︷︷︸crowding-driven diffusion

+ c1N (Vmax − V)︸︷︷︸proliferation

− χ1 div (N∇m)︸︷︷︸transport along ECM gradient

− αN→H1o≤oHN︸︷︷︸normoxic → hypoxic

+ αH→N 1o≥oHH︸︷︷︸hypoxic → normoxic

I will come back later to the diffusion and crowding-driven diffusion.

V = N +H+A+ E +m = total volume occupied by cells and ECM


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ODEs for hypoxic and apoptotic cells and for ECM

dHdt

= αN→H1o≤oHN︸︷︷︸normoxic → hypoxic

− αH→N 1o≥oHH︸︷︷︸hypoxic → normoxic

− αH→A1o≤oAH︸︷︷︸hypoxic → apoptotic

(H (t, x) = hypoxic cell density)

dAdt

= αH→A1o≤oAH︸︷︷︸hypoxic → apoptotic

(A (t, x) = apoptotic cell density)

dmdt= − βmN︸︷︷︸

degradation by normoxic cells

(m (t, x) = ExtraCellular Matrix)


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The endothelial cascade

hypoxic cells: need more oxygen to survive. They initiate a cascade ofcellular interactions. The result is angiogenesis: new vascularizationto supply the tumor (microvessels branching from main vessels in thedirection of the tumor).

Messanger from hypoxic cells to endothelial cells: VEGF (VascularEndothelial Growth Factor)


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The endothelial cascade

hypoxic cells: need more oxygen to survive. They initiate a cascade ofcellular interactions. The result is angiogenesis: new vascularizationto supply the tumor (microvessels branching from main vessels in thedirection of the tumor).Messanger from hypoxic cells to endothelial cells: VEGF (VascularEndothelial Growth Factor)


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PDEs for the endothelial cascade

∂g∂t= k4∆g︸︷︷︸

diffusion

(g (t, x) = VEGF concentration)

+ αH→gH︸︷︷︸production by hypoxic cells

− αg→MEg︸︷︷︸uptake by endothelial cells

∂E∂t= k2∆E︸︷︷︸

diffusion

(E (t, x) = density of endothelial ramification)

− χ2 div (E∇g)︸︷︷︸transport along VEGF gradient

+ c2Eg (Vmax − V)︸︷︷︸proliferation under VEGF presence


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PDEs for oxygen concentration

∂o∂t= k3∆o︸︷︷︸

diffusion

(o (t, x) = oxygen concentration)

+ c3E (omax − o)︸︷︷︸production by endothelial cells

− αo→N ,H,E (N +H+ E) o︸︷︷︸uptake by all living cells

− γo︸︷︷︸oxygen decay


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Summary of variables

N (t, x) = density of normoxic cellsH (t, x) = density of hypoxic cellsA (t, x) = density of apoptotic cellsE (t, x) = density of endothelial cells (or density of vasculature)o (t, x) = oxygen concentrationg (t, x) = angiogenic growth factor (VEGF) concentrationm (t, x) = ECM (ExtraCellular Matrix)


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Summary of constants (1)

k1 = background random motility coeffi cient of normoxic cellsk2 = random motility coeffi cient of endothelial cellsk3 = diffusion coeffi cient of oxygenk4 = diffusion coeffi cient of angiogenic factorχ1 = transport coeffi cient of normoxic cells along ECM gradientχ2 = transport coeffi cient of endothelial cells along VEGF gradientVcr = threshold for crowding-driven diffusionVmax = limit to total volume of cells and ECMc1 = proliferation rate of normoxic cellsc2 = proliferation rate of endothelial cellsc3 = production rate of oxygen


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Summary of constants (2)

αN→H = decay rate from normoxic to hypoxic cellsαH→N = restoration rate from hypoxic to normoxic cellsαH→A = decay rate from hypoxic to apoptotic cellsαH→g = production rate of VEGF from hypoxic cellsαo→N ,H,E = uptake rate of oxygen from all living cellsαg→E = uptake rate of VEGF from endothelial cellsomax = maximum oxygen concentrationoH = oxygen threshold for transition normoxic ↔ hypoxicoA = oxygen threshold for transition hypoxic ↔ apoptoticβ = rate of ECM degradationγ = oxygen decay rate


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Blue: density of normoxic cells; light blue (green): extracellular matrixDotted black: density of hypoxic cells; red: density of endothelialramification.


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"Although this does not give any new insight into the dynamics of tumorinvasion it nevertheless gives us a reference with which we can comparethe different treatment strategies we apply."

Cytostatic drugs inhibit cell division or some other function of tumoror host cells (e.g. angiogenesis).

Cytotoxic drugs actively kill proliferating tumor (and healthy) cells.

Drugs that specifically target proliferation of the endothelial cells couldbe more effi cient than agents which reduce chemotaxis.

Under very weak supply of cytotoxic therapy the tumor could evenincrease (only the boundary of normoxic cells is destroyed so morehypoxic cells became again normoxic).


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"Although this does not give any new insight into the dynamics of tumorinvasion it nevertheless gives us a reference with which we can comparethe different treatment strategies we apply."

Cytostatic drugs inhibit cell division or some other function of tumoror host cells (e.g. angiogenesis).

Cytotoxic drugs actively kill proliferating tumor (and healthy) cells.

Drugs that specifically target proliferation of the endothelial cells couldbe more effi cient than agents which reduce chemotaxis.Under very weak supply of cytotoxic therapy the tumor could evenincrease (only the boundary of normoxic cells is destroyed so morehypoxic cells became again normoxic).


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Other macroscopic models

There are a lot of models, for specific tumors, for specific phenomena.Among the differences, let us only emphasize the variety of diffusion terms:

div (A (x)∇C) diffusion of cells C in a realistic medium (Fick)

∑ ∂i∂j (Aij (x) C) Fokker-Planck instead of Fick

div (max ((C − Cmin) ∧ 0)∇C) crowding-driven diffusion

div ((Vmax − V)∇C) diffusion limited by volume constraint

A problem around which we concentrate at present our attention is acareful discussion of the diffusion terms and their microscopic(stochastic) justification.


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Example of realistic medium, complex geometry


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Microscopic models

By microscopic models we mean models at the cell level.They are always stochastic.They can be discrete or continuous:

exclusion models, Potts models, etc.

stochastic differential equations (SDEs) for a single typical cell or forinteracting cells

Let me discuss the second possibility, just because of my personalbackground.


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Microscopic models

A first natural strategy (it was our first idea) to build a microscopic modelbased on SDEs, is to apply the well known relation between Fokker-Planckequations and SDEs: the solution p (t, ·) of the PDE

∂p∂t=12 ∑ ∂i∂j (aijp)− χ div (p∇g) , p|t=0 = p0

is the law of the solution Xt of the SDE

dXt = ∇g (t,Xt ) dt + σ (t,Xt ) dWt , a = σσT

if p0 is the density of the law of X0. Reaction terms (proliferation, death,change of type) can be described by birth-death-like processes added tothis SDE.


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The SDE associated to the PDE

This viewpoint, of building (mathematically) the SDE associated to thePDE may be useful for numerical purposes and may throw some initiallight into the microscopic phenomena. However, it is basically an artificialexercise.It nevertheless arises interesting questions. Let me mention one. Mostpapers assume Fick structure of the diffusion term:

div (a (t, x)∇u) (Fick type).

This is not the term we have in a Fokker-Planck equation:

∑ ∂i∂j (aij (t, x) u) (Fokker-Planck type).

Which one is correct?


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Direct microscopic description

A better approach is to investigate directly the microscopic system andbuild a model (discrete or continuous). Let us discuss only normoxic cells.Normoxic cells are free to move (opposite to normal cells which arebounded by adhesion constraints; hypoxic cells do not move)

1 the motion of a normoxic cell has a random component (it isattracted at random in all possible directions by chemical impulses,like an animal who look at random for food)

2 it has also a systematic component in the direction of chemical ornutrient gradients (like ECM)

3 (this point is specific of the model above) it has an additionaldiffusivity due to crowding: it tends to escape more crowded regions(crowding is also caused by proliferation).


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Non-interacting particles

If we take into account only 1 and 2 (random component and nutrientgradients) we model the position of cell i by the SDE

dX it =√2k1dW i

t︸︷︷︸background diffusion

+ χ1∇m(t,X it

)dt︸︷︷︸

transport along ECM gradient

which corresponds to the PDE (N (t, x)= density of normoxic cells)

∂N∂t

= k1∆N︸︷︷︸background diffusion

− χ1 div (N∇m)︸︷︷︸transport along ECM gradient


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Fisher-Kolmogorov model (with transport)

If we add proliferation, in the sense that we have N (t) particles

X 1t , ...,XN (t)t , where N (t) is a non-homogeneous Poisson process with

rate c1 (Nmax −N (t)), subject to the equationsdX it = χ1∇m

(t,X it

)dt +

√2k1dW i

t

and with a suitable rule for the initial position of the new particles, in asuitable limit as Nmax → ∞ the empirical density

1Nmax

N (t)

∑i=1

δX it

converges (in the weak topology of measures) to the solution of theequation

∂N∂t

= k1∆N + c1N (Nmax −N )− χ1 div (N∇m) .This is already an interesting model with a traveling wave structure ofsolutions (similar to the simulations above).


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Interactions: short range, long range, contact ones

Let us try to model also the interactions between particles. A cell mayreact to messages arriving from other cells, of two classes:

short range (namely from cells composing the same tissue or veryclose tissues; e.g. VEGF)

long range (namely from cells belonging to other parts of the body;e.g. hormones).

Moreover, it reacts to contact inputs:

adhesion (but we exclude it for normoxic cells)

exclusion (two cells cannot occupy the same space).

The crowding-driven diffusion is an enhanced form of exclusion, atendency to go apart other cells in order not to compete with them foroxygen and other nutrients.


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Modelling the crowding-driven diffusion

The purpose of this last part of the talk is more specialized: how to model,at the level of interacting particles, the crowding-driven diffusion?Let us repeat its features: it includes the exclusion constraint but also withsome tendency to go apart from very nearest neighbor cells.Let us introduce two convolution kernels:

kε (x) = (2π)−d/2 ε−d/2 exp(− 12ε|x |2

)(heat kernel)

Kε (x) = −∇kε (x) =xεkε (x) , Kε : Rd → Rd .


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The first kernel,

kε (x) = (2π)−d/2 ε−d/2 exp(− 12ε|x |2

)when applied to the position of particles,

∑j 6=ikε

(X it − X jt

)measures the number of particles X jt very close to X

it .


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RecallKε (x) = −∇kε (x) =

xεkε (x) , Kε : Rd → Rd .

The second kernel, when applied to the position of particles,

∑j 6=iKε

(X it − X jt

)= ∑

j 6=ikε

(X it − X jt

) X it − X jtε

gives us a force acting on X it in the direction opposite to Xjt , but only

when X jt is very close to Xit . It is a form of crowding-driven force.


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The limit kernels

We have

limε→0

∫kε (x − y)N (t, y) dy = N (t, x)

limε→0

∫Kε (x − y)N (t, y) dy = −∇N (t, x)

if N (t, x) is of class C 1. This means: if we replace the empirical density1

Nmax ∑N (t)i=1 δX it by a continuum density, the first kernel acts (in the limit as

ε→ 0) as pointwise evaluation, the second one as a derivative.Again we see the interpretation of the second kernel:

1Nmax

∑j 6=iKε

(X it − X jt

)∼ −∇N

(t,X it

)(large Nmax, small ε). It gives us a force in the direction of decreasingconcentration of cells.


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Interacting cells with crowding-driven motility. Mean fieldlimit

dX it =√2k1dW i

t︸︷︷︸background diffusion

+ χ1∇m(t,X it

)dt︸︷︷︸

transport along ECM gradient

+1

Nmax∑j 6=iKε

(X it − X jt

)dt︸︷︷︸

motion in direction opposite to nearby particles

The natural conjecture (we still have to prove it) is that, in a suitable limitwhen ε→ 0 and Nmax → ∞, each particle is subject to the nonlinear SDE(of McKean type)

dXt =√2k1dWt + χ1∇m (t,Xt ) dt −∇N (t,Xt ) dt

where N (t, x) is the law of Xt itself. Hence∂N∂t

= k1∆N − χ1 div (N∇m) + div (N∇N )

(plus reaction terms if imposed by birth-death rules).Franco Flandoli, University of Pisa () Mathematical Oncology

Stochastic Analysis and applications in Biology, Finance and Physics, Berlin 2014 32/ 36

Different nonlinear diffusion terms

We have found the nonlinear diffusion term

div (N∇N ) .

The model at the beginning prescribed

div (max ((N −Nmin) ∧ 0)∇N )

which is similar. Even if not equal, it is of Fick type!Which one is more correct is not clear but I tend to vote formax ((N −Nmin) ∧ 0). If so, we have found a microscopic interactingmodel which is very similar to the expected one but not exactly.We are working to find microscopic models which are more flexible, toproduce

div (g (N )∇N )for suitable functions g .Remark. It is known that hydrodynamic limits of discrete exclusion-typemodels also give rise to such Fick terms but perhaps not so flexible again.


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Concluding remarks and open questions

This is not like fluid mechanics, or electromagnetism etc. where thegoverning equations are know.

The models illustrated above are just examples, among several othersbased on different elements.

What makes a mathematical model potentially better than classicalapproach to medicine?

the possibility to change the parameters and study the differences inoutcomes of computer simulations

in particular, to compare the effi cacy of different anti-cancer treatmentprotocols (control theory)

At the qualitative level this is already achieved by some models.

Gompertz law is a more classical example of qualitative law describedin all books of clinical oncology.


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This is not like fluid mechanics, or electromagnetism etc. where thegoverning equations are know.

The models illustrated above are just examples, among several othersbased on different elements.

What makes a mathematical model potentially better than classicalapproach to medicine?

the possibility to change the parameters and study the differences inoutcomes of computer simulationsin particular, to compare the effi cacy of different anti-cancer treatmentprotocols (control theory)

At the qualitative level this is already achieved by some models.

Gompertz law is a more classical example of qualitative law describedin all books of clinical oncology.


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At the quantitative level, the predictions of these models are notapplicable in practice yet, to patients.

Several elements are missing:

each model discards several parallel phenomena

think for instance to the side effects of chemioterapy, or to thereactions by the immune systemeach model contains several constants which are poorly knownthe "geometry" where the equations take place is very complexgenetic mutations play a fundamental role and are unpredictable

Nevertheless the importance of the topic justify to persist inimproving the existing models.


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each model discards several parallel phenomenathink for instance to the side effects of chemioterapy, or to thereactions by the immune system

each model contains several constants which are poorly knownthe "geometry" where the equations take place is very complexgenetic mutations play a fundamental role and are unpredictable



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each model discards several parallel phenomenathink for instance to the side effects of chemioterapy, or to thereactions by the immune systemeach model contains several constants which are poorly known

the "geometry" where the equations take place is very complexgenetic mutations play a fundamental role and are unpredictable



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each model discards several parallel phenomenathink for instance to the side effects of chemioterapy, or to thereactions by the immune systemeach model contains several constants which are poorly knownthe "geometry" where the equations take place is very complex

genetic mutations play a fundamental role and are unpredictable



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each model discards several parallel phenomenathink for instance to the side effects of chemioterapy, or to thereactions by the immune systemeach model contains several constants which are poorly knownthe "geometry" where the equations take place is very complexgenetic mutations play a fundamental role and are unpredictable



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Thank you for your attention


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