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
Introduction Stochastic Model Mean Field Equations Simulations Conclusions
Outline
1 Introduction
2 Stochastic Model
3 Mean Field Equation for the Tip Density
4 Numerical Results
5 Concluding Remarks
L. L. Bonilla et al. | Modeling Angiogenesis | 2 / 29
Introduction Stochastic Model Mean Field Equations Simulations Conclusions
Outline
1 Introduction
2 Stochastic Model
3 Mean Field Equation for the Tip Density
4 Numerical Results
5 Concluding Remarks
L. L. Bonilla et al. | Modeling Angiogenesis | 3 / 29
Introduction Stochastic Model Mean Field Equations Simulations Conclusions
The physiological process of angiogenesis
The formation of blood vessels is essential for organ growth & repair
Above: postnatal retinal vascularization – Gariano & Gardner, Nature (2005)
L. L. Bonilla et al. | Modeling Angiogenesis | 4 / 29
Introduction Stochastic Model Mean Field Equations Simulations Conclusions
Angiogenesis mechanisms
Above: molecular basis of vessel branching – Carmeliet & Jain, Nature (2011)
L. L. Bonilla et al. | Modeling Angiogenesis | 5 / 29
Introduction Stochastic Model Mean Field Equations Simulations Conclusions
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)
L. L. Bonilla et al. | Modeling Angiogenesis | 6 / 29
Introduction Stochastic Model Mean Field Equations Simulations Conclusions
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 ←
L. L. Bonilla et al. | Modeling Angiogenesis | 7 / 29
Introduction Stochastic Model Mean Field Equations Simulations Conclusions
Outline
1 Introduction
2 Stochastic Model
3 Mean Field Equation for the Tip Density
4 Numerical Results
5 Concluding Remarks
L. L. Bonilla et al. | Modeling Angiogenesis | 8 / 29
Introduction Stochastic Model Mean Field Equations Simulations Conclusions
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
L. L. Bonilla et al. | Modeling Angiogenesis | 9 / 29
Introduction Stochastic Model Mean Field Equations Simulations Conclusions
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}
}
L. L. Bonilla et al. | Modeling Angiogenesis | 10 / 29
Introduction Stochastic Model Mean Field Equations Simulations Conclusions
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)
L. L. Bonilla et al. | Modeling Angiogenesis | 11 / 29
Introduction Stochastic Model Mean Field Equations Simulations Conclusions
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)
L. L. Bonilla et al. | Modeling Angiogenesis | 12 / 29
Introduction Stochastic Model Mean Field Equations Simulations Conclusions
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
L. L. Bonilla et al. | Modeling Angiogenesis | 13 / 29
Introduction Stochastic Model Mean Field Equations Simulations Conclusions
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)) .
L. L. Bonilla et al. | Modeling Angiogenesis | 14 / 29
Introduction Stochastic Model Mean Field Equations Simulations Conclusions
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)
L. L. Bonilla et al. | Modeling Angiogenesis | 15 / 29
Introduction Stochastic Model Mean Field Equations Simulations Conclusions
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
L. L. Bonilla et al. | Modeling Angiogenesis | 16 / 29
Introduction Stochastic Model Mean Field Equations Simulations Conclusions
Outline
1 Introduction
2 Stochastic Model
3 Mean Field Equation for the Tip Density
4 Numerical Results
5 Concluding Remarks
L. L. Bonilla et al. | Modeling Angiogenesis | 17 / 29
Introduction Stochastic Model Mean Field Equations Simulations Conclusions
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)
L. L. Bonilla et al. | Modeling Angiogenesis | 18 / 29
Introduction Stochastic Model Mean Field Equations Simulations Conclusions
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)
L. L. Bonilla et al. | Modeling Angiogenesis | 19 / 29
Introduction Stochastic Model Mean Field Equations Simulations Conclusions
Limiting equations
Bonilla, Capasso, Alvaro, Carretero, Phys. Rev. E 90 , 2014
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 .
L. L. Bonilla et al. | Modeling Angiogenesis | 20 / 29
Introduction Stochastic Model Mean Field Equations Simulations Conclusions
Limiting equations
Bonilla, Capasso, Alvaro, Carretero, Phys. Rev. E 90 , 2014
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
L. L. Bonilla et al. | Modeling Angiogenesis | 20 / 29
Introduction Stochastic Model Mean Field Equations Simulations Conclusions
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)
L. L. Bonilla et al. | Modeling Angiogenesis | 21 / 29
Introduction Stochastic Model Mean Field Equations Simulations Conclusions
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)
L. L. Bonilla et al. | Modeling Angiogenesis | 21 / 29
Introduction Stochastic Model Mean Field Equations Simulations Conclusions
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
L. L. Bonilla et al. | Modeling Angiogenesis | 22 / 29
Introduction Stochastic Model Mean Field Equations Simulations Conclusions
Outline
1 Introduction
2 Stochastic Model
3 Mean Field Equation for the Tip Density
4 Numerical Results
5 Concluding Remarks
L. L. Bonilla et al. | Modeling Angiogenesis | 23 / 29
Introduction Stochastic Model Mean Field Equations Simulations Conclusions
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
L. L. Bonilla et al. | Modeling Angiogenesis | 24 / 29
Introduction Stochastic Model Mean Field Equations Simulations Conclusions
Stochastic vs. mean field approximation
⋆ 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
Time = 0 h – Number of tips = 20
0 0.2 0.4 0.6 0.8 10
100
200
300
400
500
x/L
Time = 8 h – Number of tips = 43
0 0.2 0.4 0.6 0.8 10
100
200
300
400
x/L
Time = 16 h – Number of tips = 45
0 0.2 0.4 0.6 0.8 10
100
200
300
400
500
x/L
Time = 24 h – Number of tips = 50
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
250Time = 8 h – Number of tips = 47
x/L
0 0.2 0.4 0.6 0.8 10
50
100
150
200
250Time = 16 h – Number of tips = 50
x/L0 0.2 0.4 0.6 0.8 1
0
50
100
150
200Time = 24 h – Number of tips = 56
x/L
L. L. Bonilla et al. | Modeling Angiogenesis | 25 / 29
Introduction Stochastic Model Mean Field Equations Simulations Conclusions
Stochastic vs. mean field approximation
⋆ 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)
L. L. Bonilla et al. | Modeling Angiogenesis | 26 / 29
Introduction Stochastic Model Mean Field Equations Simulations Conclusions
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)
]
L. L. Bonilla et al. | Modeling Angiogenesis | 27 / 29
Introduction Stochastic Model Mean Field Equations Simulations Conclusions
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.
L. L. Bonilla et al. | Modeling Angiogenesis | 27 / 29
Introduction Stochastic Model Mean Field Equations Simulations Conclusions
Outline
1 Introduction
2 Stochastic Model
3 Mean Field Equation for the Tip Density
4 Numerical Results
5 Concluding Remarks
L. L. Bonilla et al. | Modeling Angiogenesis | 28 / 29
Introduction Stochastic Model Mean Field Equations Simulations Conclusions
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
L. L. Bonilla et al. | Modeling Angiogenesis | 29 / 29
Introduction Stochastic Model Mean Field Equations Simulations Conclusions
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, ...
L. L. Bonilla et al. | Modeling Angiogenesis | 29 / 29