elements of mathematical oncology -...

Elements of Mathematical Oncology

Franco Flandoli, University of Pisa

Padova 2015, Lecture 1

Franco Flandoli, University of Pisa () Elements of Mathematical Oncology Padova 2015, Lecture 1 1 / 30

Structure of the course

1 Part I: Macroscopic models (lectures 1, 2, 3)

2 Part II: Microscopic models (lectures 4, 5, 6)


Structure of the course: Part I

1 Introduction: an example of reaction diffusion system

2 FKPP model

3 Back to more complex models


Structure of the course: Part II

4 Introduction to interacting systems of cells; comments on in situ case;how to model proliferation

5 Short, long and intermediate range interactions and their macroscopiclimits (heuristics)

6 Rigorous details on mean field (long range) and intermediate rangeproblems.


Introduction to a few elements of biology of cancer

Cancer starts from a single normal cell. It requires a sequence ofgenetic modifications.

Main new feature of a cancer cell: uncontrolled duplication

Tumor mass starts to be detectable when there are around 109 cancercells.



A tumor can be in different phases.

The initial phase is called in situ.

Growth without free diffusion.

Then there is the invasive phase, with diffusion.

We shall see also the angiogenic cascade (or phase).


Macroscopic and microscopic models

Cancer is a phenomenon at cell level, a few micron

which become visible and dangerous at the scale of a few millimetersor centimeters, the tissue level.

These two levels, cell and tissue ones, will be called microscopic andmacroscopic levels.

Stochastic particle systems to describe cell motion and interaction(microscopic)

Deterministic partial differential equations to describe a tumor fromthe macroscopic viewpoint.

Link between the two levels: macroscopic limit.


An advanced model of invasive tumor with angiogenesis

Lectures 1 and 3 are devoted to a reaction-diffusion model for invasive +angiogenetic phases.It is taken from:

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.

It is just one of the very many models, devoted to different phases,different phenomena etc.It is a system of 7 equations (PDEs and ODEs), for 7 quantities.


Normoxic, hypoxic and apoptotic cells

This model splits the category of cancer cells in three classes:

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

2 hypoxic cells: quiescent tumor cells, with poor oxygen supply3 apoptotic cells: death or programmed to death tumor cells


PDE for normoxic cells

The model prescribes the following 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

Until now: N = normoxic cell density, V = N +H+A+ E +m =total density (m, o,H: later).


ODE for hypoxic cells

Hypoxic cells do not move and do not proliferate:

dHdt

= αN→H1o≤oHN︸︷︷︸normoxic → hypoxic

− αH→N 1o≥oHH︸︷︷︸hypoxic → normoxic

− αH→A1o≤oAH︸︷︷︸hypoxic → apoptotic

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

Until now: N , V ; H = hypoxic cell density (m, o: later).


ODE for apoptotic cells and for ECM

These cells are programmed to death. It means that they dissolve in aregulated way, not by necrosis and causing infections. The number ofapoptotic cells can only increase, due to the hypoxic cells that deteriorate:

dAdt

= αH→A1o≤oAH︸︷︷︸hypoxic → apoptotic

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

Extracellular Matrix can only deteriorate, due to the invasion of normoxiccells:

dmdt= − βmN︸︷︷︸

degradation by normoxic cells

(m (t, x) = ExtraCellular Matrix)

Until now: N , V , H; A = apoptotic cell density, m = ExtraCellularMatrix (o: later).


Crowding-driven diffusion

In the equation for normoxic cells:

div (σ (N )∇N )

called crowding-driven diffusion.

The prescription for σ (N ) in the paper:

σ (N ) = max (N −N0, 0) .

When the density of normoxic cells passes the threshold N0, anadditional diffusion starts.

Similar to σ (N ) = N which corresponds to

12

∆N 2

also called porous media diffusion.


The endothelial cascade

Angiogenesis: new vascularization to supply the tumor. Messenger fromhypoxic cells to endothelial cells: VEGF (Vascular Endothelial GrowthFactor).


The endothelial cascade

See also the movies at the web sites.


PDEs for the VEGF concentration

VEGF is a density of object at molecular level. These proteins have alwaysa diffusion (opposite to cells, which diffuse only under specialcircumstances):

∂g∂t= k4∆g︸︷︷︸

diffusion

(g (t, x) = VEGF concentration)

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

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

and the constant k4 is much bigger than k1.

Until now: N , V , H, A; g = VEGF concentration (o, E : later).


PDEs for endothelial ramification

Endothelial cells do not diffuse as isolated individuals but they propagateas microvessels.However, keeping track of their topological structure is diffi cult andperhaps not so important.Endothelial cell density, or density of endothelial ramification

∂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

Until now: N , V , H, A, g ; E = density of endothelial ramification(o: later).


PDEs for oxygen concentration

Oxygen is also molecular-level hence it diffuses, with k3 is much biggerthan k1

∂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

The system for N , V , H, A, g , E , o is complete.


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)


Numerical study, from the paper:

Blue: normoxic; green: ECM; dotted black: hypoxic; red: endothelial.Franco Flandoli, University of Pisa () Elements of Mathematical Oncology Padova 2015, Lecture 1 21 / 30

Stochastic dynamics associated to some element of theprevious model

The next last slides of Lecture 1 have the purpose to show the easiestmathematical elements of a microscopic, cell-level, description of theprevious PDE system.

Some elements of the previous model have a straightforwardtranslation into a microscopic, cell level, random dynamics.


SDE and Fokker-Planck equation

(Wt )t≥0 = Brownian motion in Rk , defined on a probability space(Ω,F ,P).b : [0,T ]×Rd → Rd , σ : [0,T ]×Rd → Rd×k continuous functionssuch that ∃ Lb , Lσ > 0,

|b (t, x)− b (t, y)| ≤ Lb |x − y ||σ (t, x)− σ (t, y)| ≤ Lσ |x − y |

∀x , y ∈ Rd , ∀t ∈ [0,T ]SDE in Rd

dXt = b (t,Xt ) dt + σ (t,Xt ) dWt

Xt = X0 +∫ t

0b (s,Xs ) ds +

∫ t

0σ (s,Xs ) dWs

Theorem: Existence and uniqueness of strong solutions.



Parabolic PDE on [0,T ]×Rd

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

called Fokker-Planck equation. Here

a = σσT .

Notations: 〈µt , φ〉 =∫

Rd φ (x) µt (dx); C∞c

(Rd), C 0c

([0,T ]×Rd

).

Measure-valued solution: family of Borel probability measures(µt )t∈[0,T ] on Rd such that t 7→ 〈µt , ϕ (t, .)〉 is measurable for allϕ ∈ C 0c

([0,T ]×Rd

)and

〈µt , φ〉− 〈µ0, φ〉 =12

d

∑i ,j=1

∫ t

0〈µs , aij (s, .) ∂i∂jφ〉 ds+

∫ t

0〈µs , b (s, .) · ∇φ〉 ds

for every φ ∈ C∞c

(Rd).



TheoremThe law µt of Xt is a measure-valued solution of the the Fokker-Planckequation.

Under suitable assumptions of non-degeneracy of aij , if µ0 has a density p0then also µt has a density p (t, ·), often with some regularity gained by theparabolic structures.


Non-interacting particles and Fokker-Planck asmacroscopic limit

Consider the sequence of (independent) SDEs in Rd

dX nt = b (t,Xnt ) dt + σ (t,X nt ) dW

nt

where W nt , n ∈N, is a sequence of independent Brownian motions in

Rk , X n0 given independent Rd -r.v.’s, F0-measurable.The processes X nt have the same law µt , solution of FP. Moreover,they are independent.Consider, for each N ∈N, the random probability measure, calledempirical measure,

SNt :=1N

N

∑n=1

δX nt

namely, for φ ∈ C 0c(Rd),⟨SNt , φ

⟩= 1

N ∑Nn=1 φ (X nt ).

The following simple theorem is our first example of macroscopic limitof a system of microscopic particles.


Non-interacting particles and Fokker-Planck asmacroscopic limit

Theorem

For every t ∈ [0,T ] and φ ∈ C 0c(Rd), a.s.

limN→∞

⟨SNt , φ

⟩= 〈µt , φ〉 .

In other words, SNt converges weakly, a.s., to a measure-valued solution ofthe Fokker-Planck equation.

Proof.Since φ (X nt ) are bounded i.i.d., by the strong LLN we have, a.s.,

limN→∞

⟨SNt , φ

⟩= E

[φ(X 1t)]= 〈µt , φ〉 .

A.s. convergence is uniform in φ, by a countable density argument.


Easy fragments of microscopic model associated to thePDE system: diffusion and transport

∂N∂t

= k1∆N︸︷︷︸background diffusion

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

+ ...

XNi (t), i = 1, 2, ... positions of independent normoxic cells

dXNi (t) = χ1∇m(t,XNi (t)

)dt +

√2k1dWNi (t) .

N should be the weak limit of

SNn (t) :=1n

n

∑i=1

δXNi (t).

Similarly for the diffusion and transport of endothelial cells.


Example of problem

The equation for ECM

dmdt= − βmN︸︷︷︸

degradation by normoxic cells

requires a function N (t, x), not the distribution 1n ∑n

i=1 δXNi (t).

One has to define a function Nn (t, x) based on 1n ∑n

i=1 δXNi (t), for

instance the convolution with a smooth mollifierθn (x) = ε−dn θ

(ε−1n x

)Nn (t, x) =

(θn ∗ SNn (t)

)(x) =

1n

n

∑i=1

θn(x − XNi (t)

).


Major other problems

nonlinear diffusionsdiv (σ (N )∇N )︸︷︷︸

crowding-driven diffusion

change of typeαH→N 1o≥oHH︸︷︷︸hypoxic → normoxic

proliferationc1N (Vmax − V)︸︷︷︸

proliferation


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