Hybrid Modelingof Tumor Induced Angiogenesisscala.uc3m.es/bonilla/pdf/LLB_Iceland_2015.pdf · Reykjavik, Iceland ∼ July 9, 2015 Hybrid Modelingof Tumor Induced Angiogenesis L. L

Introduction Stochastic Model Mean Field Equations Simulations Conclusions

University of Iceland

Reykjavik , Iceland ∼ July 9, 2015

Hybrid Modeling of Tumor InducedAngiogenesis

L. L. Bonilla †, M. Alvaro †, V. Capasso ‡, M. Carretero †, F. Terragni †

† Gregorio Millan Institute for Fluid Dynamics, Nanoscience, and Industrial MathematicsUniversidad Carlos III de Madrid, 28911 Leganes, Spain ([email protected])

‡ Universita degli Studi di Milano, ADAMSS, 20133 Milan, Italy

L. L. Bonilla et al. | Modeling Angiogenesis | 1 / 29


Outline

1 Introduction

2 Stochastic Model

3 Mean Field Equation for the Tip Density

4 Numerical Results

5 Concluding Remarks



Outline

1 Introduction

2 Stochastic Model


4 Numerical Results




The physiological process of angiogenesis

The formation of blood vessels is essential for organ growth & repair

Above: postnatal retinal vascularization – Gariano & Gardner, Nature (2005)



Angiogenesis mechanisms

Above: molecular basis of vessel branching – Carmeliet & Jain, Nature (2011)



Angiogenesis treatment

Experimental dose-effect analysis is routine in biomedical laboratories, butthese still lack methods of optimal control to assess effective therapies

Above: angiogenesis on a rat cornea – E. Dejana et al. (2005)



Modeling angiogenesis

⋆ There are different models, from deterministic (e.g., reaction-diffusionequations) to stochastic (e.g., stochastic differential equations, cellularPotts models)

⋆ Many are multiscale models, combining randomness at the naturalmicroscale/mesoscale with numerical solutions of PDEs at themacroscale

⋆ Some mathematical models: Chaplain, Bellomo, Preziosi, Byrne,Folkman, Sleeman, Anderson, Stokes, Lauffenburger, Wheeler

⋆ Some experiments: Jain, Carmeliet, Dejana, Fruttiger

⋆ Our approach: V. Capasso & D. Morale, J. Math. Biol. 58 (2009)

→ the endothelial cells proliferation is led by cells at the vessel tips;we track the trajectories & analyze the behavior of vessel tips ←



Outline

1 Introduction

2 Stochastic Model


4 Numerical Results




Features of our mathematical model

Formation of a tumor-driven vessel network involves various processes.

(i) tip branching: birth process of capillary tips

(ii) vessel extension: Langevin equations

(iii) chemotaxis in response to a generic tumor angiogenic factor (TAF),released by tumor cells: reaction-diffusion equation

(iv) haptotactic migration in response to fibronectin gradient (present within

the extracellular matrix & degraded/produced by endothelial cells): ignored

(v) anastomosis: death process of capillary tips that encounter an existingvessel

(vi) blood circulation & other processes: ignored



Some notation

X Let N(t) denote the number of tips at time t (N(0) = N0)

→ N is the (fixed) number of tips ‘expected’ during an experiment

X Let Xi(t) ∈ R2 be the location of the i-th tip at time t

X Let vi(t) ∈ R2 be the velocity of the i-th tip at time t

X The i-th tip branches at a random time T i and disappears (afterreaching the tumor or by anastomosis) at a later random time Θi

X The network of endothelial cells (existing vessels) is

X(t) =

N(t)⋃

i=1

{

Xi(s), T i ≤ s ≤ min{t,Θi}

}



Tip branching

Vessels branch out of capillary tips (not from mature vessels)

The probability that a tip branches from an existing one (birth) in theinterval (t, t+ dt] is measured by

N(t)∑

i=1

α(

C(t,Xi(t)))

dt , with α(C) = α1C

CR + C,

where CR is a reference value for the TAF concentration C(t,x) emitted bythe tumor (α1 is a coefficient).

Then, a tip located in x actually branches whenever its velocity is ‘close’ toa fixed non-random velocity v0, generating a new tip with initial state

(XN(t)+1,vN(t)+1) = (x,v0)



Vessel extension

Vessel extension is modeled by tracking the trajectories of capillary tips

Description is done in terms of the Langevin equations

dXi(t) = vi(t) dt

dvi(t) = −k vi(t)︸︷︷︸

friction

dt + F(

C(t,Xi(t)))

︸︷︷︸

force due to TAF

dt + σ dWi(t)︸︷︷︸

random noise

for t > T i , where Wi is a Brownian motion associated with the i-th tip.

The chemotactic force due to the underlying TAF field C(t,x) is given by

F(C) =d1

1 + γ1C∇xC

(k, σ, d1, γ1 are parameters)



The TAF field

The TAF diffuses & decreases where endothelial cells are present

Assuming that degradation is due only to tip cells, the TAF consumption is

proportional to velocity vi of the i-th tip in a infinitesimal region around it

The evolution equation is

∂

∂tC(t,x) = d2△xC(t,x)− η C(t,x)

∣∣∣∣∣∣

1

N

N(t)∑

i=1

vi(t) δ

(

x−Xi(t)

)

∣∣∣∣∣∣

,

where d2 and η are parameters.

X an initial Gaussian-like concentration C(0,x) is considered

X the production of C(t,x) due to the tumor is incorporated througha TAF flux boundary condition at the tumoral source



Anastomosis

If a tip meets an existing vessel,they join at that point and time→ the tip stops the evolution

x/L

y/L

The empirical measure TN counts tips with any velocity that are at time sin a spatial region A as

TN(s)(A) =1

N

N(s)∑

i=1

ǫXi(s)(A) .

The death rate (of capillary tips) at location x and time t should beproportional to

1

N

∫ t

0

ds

N(s)∑

i=1

δ(x−Xi(s)) .



Numerical simulation: setting

⋆ The 2D spatial domain is given by x = (x, y) ∈ [0, L]× [−∞,∞]

⋆ The primary vessel is at x = 0, the tumor is at x = L

⋆ The boundary conditions for the TAF field are

C(t, x,±∞) = 0 ,∂C

∂x(t, 0, y) = 0 ,

∂C

∂x(t, L, y) = f(y) ,

where f(y) is the TAF flux emitted by the tumor

00.2

0.40.6

0.81

−1

−0.5

0

0.5

10

0.2

0.4

0.6

0.8

1

1.2

1.4

x/L

y/L

00.2

0.40.6

0.81

−1

−0.5

0

0.5

1−0.2

0

0.2

0.4

0.6

0.8

x/L

y/L

00.2

0.40.6

0.81

−1

−0.5

0

0.5

1−5

0

5

x/L

y/L

Initial TAF (left) and x/y - components of the force due to TAF (middle / right)



Vessel network, TAF field, x - component of TAF force

Time = 8 h – Number of tips = 28

x/L

y/L

0 0.2 0.4 0.6 0.8 1−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5Time = 16 h – Number of tips = 47

x/L

y/L

0 0.2 0.4 0.6 0.8 1−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5Time = 24 h – Number of tips = 53

x/L

y/L

0 0.2 0.4 0.6 0.8 1−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

00.2

0.40.6

0.81

−1

−0.5

0

0.5

1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

x/Ly/L

00.2

0.40.6

0.81

−1

−0.5

0

0.5

1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

x/Ly/L

00.2

0.40.6

0.81

−1

−0.5

0

0.5

1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

x/Ly/L



Outline

1 Introduction

2 Stochastic Model


4 Numerical Results




Few ideas

Derivation of a mean field equation for the vessel tip density, as N →∞

⋆ Ito’s formula is applied for a smooth g(x,v) & the process in Langevin eqns

⋆ The scaled number of tips with positions & velocities at time t in B is givenby the empirical measure QN

QN (t)(B) =1

N

N(t)∑

i=1

ǫ(Xi(t),vi(t))(B) =1

N

N(t)∑

i=1

∫

B

δ(

x − Xi(t)

)

δ(

v − vi(t)

)

dx dv

⋆ If N is sufficiently large, QN may admit a density by laws of large numbers

QN (t) (d(x,v)) ∼ p(t,x,v) dx dv

⋆ Thus, as N → ∞

1

N

N(t)∑

i=1

g(Xi(t),vi(t)) ∼

∫

g(x,v) p(t,x,v) dx dv

⋆ Tip branching & anastomosis are added as source & sink terms to theobtained equation for the tip density p(t,x,v)



Limiting equations

Bonilla, Capasso, Alvaro, Carretero, Phys. Rev. E 90 , 2014

As N →∞, the tip density p(t,x,v) satisfies the Fokker-Planck equation

∂

∂tp(t,x,v) = α(C(t, x)) p(t,x,v) δ(v − v0)

︸︷︷︸

birth term (tip branching)

− γ p(t,x,v)

∫ t

0ds

∫

p(s,x,v′) dv′

︸︷︷︸

death term (anastomosis)

−v · ∇x p(t,x,v)︸︷︷︸

transport

+ k ∇v ·[

v p(t,x,v)]

︸︷︷︸

friction

−∇v ·[

F(C(t,x)) p(t,x,v)]

︸︷︷︸

force due to TAF

+σ2

2∆v p(t,x,v)

︸︷︷︸

diffusion

(γ is the coefficient of anastomosis)



Limiting equations


As N →∞, the TAF reaction-diffusion equation becomes

∂

∂tC(t,x) = d2 △xC(t,x)− η C(t,x) |j(t,x)|

where

j(t, x) =

∫

v′ p(t,x,v′) dv′

is the flux of tip density at time t and position x .



Limiting equations


As N →∞, the TAF reaction-diffusion equation becomes

∂

∂tC(t,x) = d2 △xC(t,x)− η C(t,x) |j(t,x)|

where

j(t, x) =

∫

v′ p(t,x,v′) dv′

is the flux of tip density at time t and position x .

→ the BCs for the TAF field have been discussed before



Boundary conditions for the tip density

⋆ Since p has 2nd -order derivatives in v, we impose

p(t,x,v) → 0 as |v| → ∞

⋆ Which (spatial) BCs for p ? (p has 1st-order derivatives only in x)



Boundary conditions for the tip density

⋆ Since p has 2nd -order derivatives in v, we impose

p(t,x,v) → 0 as |v| → ∞

⋆ Which (spatial) BCs for p ? (p has 1st-order derivatives only in x)

At any time, we expect to know

X the marginal tip density at the tumor at x = L

p(t, L, y) = pL(t, y) where p(t,x) =

∫

p(t,x,v′) dv′

X the normal tip density flux injected at the primary vessel at x = 0

−n · j(t, 0, y) = j0(t, y)

Using these values & assuming that p is closed to a local equilibrium distribution

at the boundaries, compatible BCs for p+ at x = 0 and p− at x = L are imposed

(see charge transport in semiconductors; Bonilla & Grahn, Rep. Prog. Phys., 2005)



Boundary conditions for p

p+(t, 0, y, v, w) =e−

k|v−v0|2

σ2

∫∞0

∫∞−∞ v′e

−k|v′−v0|2

σ2 dv′ dw′

[

j0(t, y)−

∫ 0

−∞

∫ ∞

−∞v′p−(t, 0, y, v′, w′)dv′dw′

]

p−(t, L, y, v, w) =e−

k|v−v0|2

σ2

∫ 0−∞

∫∞−∞

e−

k|v′−v0|2

σ2 dv dw

[

pL(t, y)−

∫ ∞

0

∫ ∞

−∞p+(t, L, y, v′, w′)dv′dw′

]

where

⋆ p+ = p for v > 0 and p− = p for v < 0

⋆ v = (v, w) ; v0 is the average velocity of the tips at x = 0, L

⋆ σ2/k is the boundary temperature of the equilibrium distribution



Outline

1 Introduction

2 Stochastic Model


4 Numerical Results




Stochastic vs. mean field approximation

For comparison, results of 50 stochastic experiments were averaged

⋆ the qualitative behavior of p(t,x) is similar in the two models

⋆ time evolution for the mean field p(t,x) (left) is slightly faster than for thestochastic p(t,x) (right)

⋆ condition p → 0 as |v| → ∞ may be roughly satisfied in the numerical code




⋆ the profiles p(t, x, y = 0) similarly evolve in time in the mean field (left)

and stochastic (right) models (discrepancies for p observed before)

⋆ tip generation & motion proceed as a pulse advancing towards the tumor at x = L

0 0.2 0.4 0.6 0.8 10

200

400

600

800

1000

x/L


0 0.2 0.4 0.6 0.8 10

100

200

300

400

500

x/L


0 0.2 0.4 0.6 0.8 10

100

200

300

400

x/L


0 0.2 0.4 0.6 0.8 10

100

200

300

400

500

x/L


0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

120Time = 0 h – Number of tips = 20

x/L0 0.2 0.4 0.6 0.8 1

0

50

100

150

200


x/L

0 0.2 0.4 0.6 0.8 10

50

100

150

200


x/L0 0.2 0.4 0.6 0.8 1

0

50

100

150


x/L




⋆ the number of tips over time is fairly comparable in the mean field (left)and stochastic (right) models

⋆ discrepancies are seemingly due to a different effect of anastomosis: involvedcoefficient is being estimated

0 4 8 12 16 20 2420

30

40

50

time (hours)

0 4 8 12 16 20 2420

30

40

50

60

time (hours)



Pulse solution for the tip density

⋆ Approximate

p ∼k

πσ2p(t,x)e−k |v−v0|

2/σ2

⋆ Use perturbation technique to obtain

∂p

∂t=

kα

πσ2p− γp

∫ t

0p(s,x)ds−

1

k∇x · (Fp) +

σ2

2k2△xp

⋆ Ignore △xp, assume slow variation of C(t,x) and p = p(x− ct). Wefind (B. Birnir):

p(x− ct) =kc

2(Fx − kc)

(k2α2

γπ2σ4+ 2K

)

sech2

[√

k2α2

γπ2σ4+ 2K

k(x− ct+ ξ0)

2(Fx − kc)

]



Pulse solution for the tip density

⋆ Approximate

p ∼k

πσ2p(t,x)e−k |v−v0|

2/σ2

⋆ Use perturbation technique to obtain

∂p

∂t=

kα

πσ2p− γp

∫ t

0p(s,x)ds−

1

k∇x · (Fp) +

σ2

2k2△xp

⋆ Ignore △xp, assume slow variation of C(t,x) and p = p(x− ct). Wefind (B. Birnir):

p(x− ct) =kc

2(Fx − kc)

(k2α2

γπ2σ4+ 2K

)

sech2

[√

k2α2

γπ2σ4+ 2K

k(x− ct+ ξ0)

2(Fx − kc)

]

This is the KdV soliton solution with scaling factor√

k2α2

γπ2σ4 + 2K k2(Fx−kc)

and

speed c. K is a constant with units of frequency.



Outline

1 Introduction

2 Stochastic Model


4 Numerical Results




Summary

♣ Stochastic description of angiogenesis involving tip branching, vesselextension, chemotaxis, and anastomosis

♣ Mean field equations as N →∞ for tip density & TAF concentration→ with novel boundary conditions for p(t,x,v) ←

♣ Agreement between the mean field approximation & the stochasticmodel upon averaging over many random experiments



Summary

♣ Stochastic description of angiogenesis involving tip branching, vesselextension, chemotaxis, and anastomosis

♣ Mean field equations as N →∞ for tip density & TAF concentration→ with novel boundary conditions for p(t,x,v) ←

♣ Agreement between the mean field approximation & the stochasticmodel upon averaging over many random experiments

X open: control : inhibit or enhance vessel network

X open: developing a hybrid model : TAF field equation with averagedquantities with stochastic tip branching & growth

X open: including new ingredients, as equations for fibronectin andmatrix-degrading enzyme, blood circulation, ...


Documents

Hybrid Modelingof Tumor Induced Angiogenesisscala.uc3m.es/bonilla/pdf/LLB_Iceland_2015.pdf · Reykjavik, Iceland ∼ July 9, 2015 Hybrid Modelingof Tumor Induced Angiogenesis L. L