Elements of Mathematical Oncologyusers.dma.unipi.it/~flandoli/Padova_2015_lezione3.pdf · 3 (N+A)(t 0,x 0) = V max for some x 0 2Rd. Following Smoller book, consider the function

Elements of Mathematical Oncology

Franco Flandoli, University of Pisa

Padova 2015, Lecture 3

Franco Flandoli, University of Pisa () Elements of Mathematical Oncology Padova 2015, Lecture 3 1 / 31

Invariant regions for the full system?

The full system of Lecture 1 seems apparently based on the sameingredients of the FKPP equation.

It contains the proliferation term N (Vmax − V) similar to u (1− u):proliferation decreases when the total volume V approaches amaximum value Vmax.The constraint V ≤ Vmax is necessary, otherwise the termN (Vmax − V) becomes negative (no biological meaning).One can prove that the constraints N ≥ 0, H ≥ 0, A ≥ 0 etc. arefulfilled (e.g. by linearization).

But what about V ≤ Vmax?


A reduced system

To understand, consider the reduced system

∂N∂t

= ∆N +N (Vmax − V)− αN→ANdAdt

= αN→AN

with V = N +A.It deal only with normoxic cells which diffuse, proliferate and transforminto apoptotic after some time.The natural invariant region is now

Σ = {(N ,A) : N ≥ 0,A ≥ 0,N +A ≤ Vmax} .

One can show that N ≥ 0,A ≥ 0 is fulfilled.Let us investigate N +A ≤ Vmax.


The proof of invariant regions fails for the reduced system

Assume we are dealing with the equation modified by h (u).Assume there is t0 ∈ [0,T ] with the following properties:

1 t0 > 02 (N ,A) (t, x) ∈ Σ for every x ∈ Rd and every t ∈ [0, t0]3 (N +A) (t0, x0) = Vmax for some x0 ∈ Rd .

Following Smoller book, consider the function V = N +A and compute∂V∂t :

∂V∂t= ∆N +N (Vmax − V) + h

(the terms −αN→AN+αN→AN compensate).


The proof of invariant regions fails for the reduced system

We deduce∂V∂t(t0, x0) ≥ 0

∇V (t0, x0) = 0, ∆V (t0, x0) ≤ 0and from the equation (recall h)

∂V∂t(t0, x0) < ∆N (t0, x0) .

But we have no reason to claim that ∆N (t0, x0) ≤ 0 and deduce acontradiction.


Idea of a counter-example

This argument is not only logical: it indicates what goes wrong. When andwhere V approaches the threshold, ∆N may remain > 0. Some portion ofN diffuses there instead of diffusing away. N may continue to increasewhen and where V = Vmax.Take at time t = 0

N (0, x) = Vmax · 1[5,10] (x) + Vmax · 1[−10,−5] (x)A (0, x) = Vmax · 1[−5,5] (x) .

The constraint N +A ≤ Vmax is fulfilled.But a second later part of the density N will be diffused in the region[−5, 5], because of the term k1∆N , while A cannot go down in thatregion. Thus we shall have points in the interval [−5, 5] whereN +A > Vmax.


Simulations

Normoxic in black, apoptotic in red, total volume in green:


Invariant regions for different diffusion operators

How could we modify the system to have Σ invariant?Consider, as a model problem, the equation

∂u∂t= Du + u (1− u)

where

D1u = ∆uD2u = (1− u)∆uD3u = div ((1− u)∇u)D4u = ∆ ((1− u) u) .

One can show that u ≥ 0 is always preserved. The problem is u ≤ 1.



D1u = ∆uD2u = (1− u)∆uD3u = div ((1− u)∇u)D4u = ∆ ((1− u) u) .

TheoremThe region [0, 1] is invariant in cases 1,2,3. In case 4 solutions remainpositive but the constraint u ≤ 1 is not necessarily preserved.

Preliminary relations:

D3u = ∇ (1− u) · ∇u +D2u = − |∇u|2 +D2u

D4u = D2u − u∆u − 2 |∇u|2 .Franco Flandoli, University of Pisa () Elements of Mathematical Oncology Padova 2015, Lecture 3 9 / 31



1 t0 > 02 u (t, x) ∈ [0, 1] for every x ∈ Rd and every t ∈ [0, t0]3 u (t0, x0) ∈ {0, 1} for some x0 ∈ Rd .

Let us analyze the case u (t0, x0) = 1.



We deduce:

1 ∇u (t0, x0) = 0 and ∆u (t0, x0) ≤ 02 ∂u

∂t (t0, x0) ≥ 0 (because u (t, x0) ≤ 1 for t ∈ [0, t0] andu (t0, x0) = 1)

3 From the relations above:

D2u (t0, x0) = 0D3u (t0, x0) = 0

D4u = −u (t0, x0)∆u (t0, x0) ≥ 0

Hence, from the equation, ∂u∂t (t0, x0) < 0 in cases 1, 2, 3, but not in case

4.



Consider now the following operators, all (except the first one) havingsomething to do with the idea that the diffusion increases when thedensity is larger (like for pressure-driven diffusion):

D5u = u∆u

D6u = div (u∇u) =12

∆u2

TheoremThe region [0, 1] is invariant in both cases 5,6.

Preliminary relation:

D6u = div (u∇u) = |∇u|2 +D5u.




1 t0 > 02 u (t, x) ∈ [0, 1] for every x ∈ Rd and every t ∈ [0, t0]3 u (t0, x0) ∈ {0, 1} for some x0 ∈ Rd .

Let us analyze the case u (t0, x0) = 1.



We deduce:

1 ∇u (t0, x0) = 0 and ∆u (t0, x0) ≤ 02 ∂u

∂t (t0, x0) ≥ 0 (because u (t, x0) ≤ 1 for t ∈ [0, t0] andu (t0, x0) = 1).

3

D5u (t0, x0) ≤ 0D6u (t0, x0) ≤ 0

by the relation D6u = |∇u|2 +D5u. Hence ∂u∂t (t0, x0) < 0 in both

cases 5, 6.

Remark. The intuition about the constraint u ≤ 1 in the group 2-3 isopposite to the case of group 5-6 but the result is the same. 2-3: thefactor (1− u) damps the diffusion when we approach u = 1. 5-6: thefactor u increases the diffusion when u is larger.


Back to the system normoxic+apoptotic

Let us, for instance, modify ∆N into (Vmax − V)∆N :

∂N∂t

= (Vmax − V)∆N +N (Vmax − V)− αN→ANdAdt

= αN→AN

TheoremThe region

Σ = {(N ,A) : N ≥ 0,A ≥ 0,N +A ≤ Vmax}

is invariant.

(Recall that the problem was only N +A ≤ Vmax)



Proof. Assume we are dealing with the equation modified by h (u).Assume there is t0 ∈ [0,T ] with the following properties:

1 t0 > 02 (N ,A) (t, x) ∈ Σ for every x ∈ Rd and every t ∈ [0, t0]3 (N +A) (t0, x0) = Vmax for some x0 ∈ Rd .

Consider the function V = N +A. We have∂V∂t = (Vmax − V)∆N +N (Vmax − V) + hFrom 1-3 we deduce ∂V

∂t (t0, x0) ≥ 0, ∇V (t0, x0) = 0, ∆V (t0, x0) ≤ 0and from the equation (recall h) ∂V

∂t (t0, x0) < 0.



Normoxic in black, apoptotic in red, total volume in green:




Brainstorming on the problem of bounds on cell density

1 The bound on the total cell density V ≤ Vmax is something new, notcommon when we deal with fluids or gases of molecules. For cells, ithas a meaning.

2 Is necessary to impose it strictly, or a mild form is suffi cient? Livingtissues may deform, stretch, may accommodate higher density a littlebit. How to describe mathematically this mild accommodationpossibility?

3 For a single equation, preservation of bounds like

0 ≤ u ≤ 1

hold also for certain diffusion operators different from ∆u. See above.4 But for systems, if we want to preserve V (t, x) ≤ Vmax, diffusionterms have to be modified, ∆u is not suffi cient, as we have showedabove.


Role of transport terms on bounds on cell density

For a single equation in transport form

∂u∂t= ∆u + b · ∇u + cu

a constraint of the formu ≤ umax

holds when c = 0 or more generally c ≤ 0. Otherwise, we only have

u (t, x) ≤ sup u0 · esup[c ]+t .



But an equation of Fokker-Planck type (divergence form)

∂u∂t= ∆u − div (bu)

is of the form∂u∂t= ∆u − b · ∇u − u div b

(namely c = − div b) hence u ≤ umax is sure only when

[div b]− = 0

becauseu (t, x) ≤ sup u0 · esup[− div b]

+t .

If the vector field b has negative divergence, concentration may happens.



1 Even worse for systems, where transport terms may couple differentvariables:

div (N∇m) , div (E∇g)2 If, instead of having a smooth distribution of endothelial cells, wemore realistically assume they are concentrated in vessels, this couldbe more dangerous.


Example of concentration due to transport

∂u∂t= ∆u − div (bu) , b (x) = −5 (x − x0) e−0.08|x−x0 |

2

flux lines move in the direction of x0 from both sides:


Remark on Keller-Siegel model

Chemotaxis equations, in particular the Keller-Siegel model (see alsomodels under the name "aggregation models") may even lead to blow-up.An example is the system, in dimension ≥ 2

∂u∂t= ∆u − χ div (u∇v)

∆v = 1− u.

It is known that there is a value χ∗ > 0 such that, for all χ > χ∗, radiallysymmetric positive solutions can be constructed which blow-up in finitetime.However, blow-up does not happen for small values of χ and in dimension1.


Simulations about normoxic+hypoxic+apoptotic

Normoxic (blue), hypoxic (red), apoptotic (black), total volume (green)(no angiogenic cascade):


Simulations about the full system

Above the profiles of normoxic (blue), hypoxic (grey), apoptotic (black),endothelial (red) with the initial profile of endothelial (orange) forcomparison.Below, the profile of oxygen (red) ECM (yellow), VEGF (grey).


Simulations about the full system


Fick or Fokker-Planck?

Another unclear issue: which one is more correct?

div (a (t, x)∇u (t, x)) (1)

∆ (a (t, x) u (t, x)) . (2)

Form (1) is the celebrated in-homogeneous Fick law, so often used in thebiological literature.Form (2) is the Fokker-Planck type of diffusion, immediately related to amicroscopic model of SDEs.[Notice that Fick law, when a (t, x) = 1− u (t, x) preserves u ≤ 1, notFokker-Planck.]



A natural question for probabilists is: which SDE corresponds to Fick law?Consider the model problem, in Fick form

∂p∂t=12

∂

∂x

(σ2

∂

∂xp).

Obviously∂p∂t=12

∂2

∂x2(σ2p

)− ∂

∂x

(σ′σp

).

Hence, recalling the theorem on Fokker-Planck equation in Chapter 1, wemay associate the SDE

dXt = σ′ (t,Xt ) σ (t,Xt ) dt + σ (t,Xt ) dBt

to the Fick diffusion.Is this a natural SDE?



Is there any physical justification of the drift term σ′ (t,Xt ) σ (t,Xt ) dt?Let me only remark that this is not Stratonovich equation

dXt =√2σ (t,Xt ) ◦ dBt .

If it were, since Stratonovich equations are for good reasons more physicalthan Itô equations, we would have a wonderful reason to prefer Fickdiffusion. But it is not so.The rewriting of our Stratonovich equation in Itô form is

dXt =12

σ′ (t,Xt ) σ (t,Xt ) dt + σ (t,Xt ) dBt .

Just by the factor 12 , we miss this very interesting interpretation.


List of (open?) problems

Existence, uniqueness, regularity, for the system of 7 equations.

Invariant regions for the system of 7 equations.

Microscopic modelling of the term div (σ (N )∇N ).Choose between Fick or Fokker-Planck. In general, which diffusionoperators are more appropriate.

[Microscopic modelling of proliferation and change of type terms.]

[Microscopic modelling of "controlled" proliferation, as in FKPP.]

Re-start from cell-level and deduce PDEs. Also to discover whichdiffusion operators are more appropriate.

Model different phases (in situ versus invasive).


Documents

Elements of Mathematical Oncologyusers.dma.unipi.it/~flandoli/Padova_2015_lezione3.pdf · 3 (N+A)(t 0,x 0) = V max for some x 0 2Rd. Following Smoller book, consider the function