Hybrid Modeling of Tumor Induced Angiogenesis

Transcription

1 University of Iceland Reykjavik, Iceland July 9, 25 Hybrid Modeling of Tumor Induced Angiogenesis L. L. Bonilla, M. Alvaro, V. Capasso, M. Carretero, F. Terragni Gregorio Millán Institute for Fluid Dynamics, Nanoscience, and Industrial Mathematics Universidad Carlos III de Madrid, 289 Leganés, Spain Università degli Studi di Milano, ADAMSS, 233 Milan, Italy L. L. Bonilla et al. Modeling Angiogenesis / 29

2 Outline 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

3 Outline 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

4 The physiological process of angiogenesis The formation of blood vessels is essential for organ growth & repair Above: postnatal retinal vascularization Gariano & Gardner, Nature (25) L. L. Bonilla et al. Modeling Angiogenesis 4 / 29

5 Angiogenesis mechanisms Above: molecular basis of vessel branching Carmeliet & Jain, Nature (2) L. L. Bonilla et al. Modeling Angiogenesis 5 / 29

6 Angiogenesis treatment Experimental dose-effect analysis is routine in biomedical laboratories, but these still lack methods of optimal control to assess effective therapies Above: angiogenesis on a rat cornea E. Dejana et al. (25) L. L. Bonilla et al. Modeling Angiogenesis 6 / 29

7 Modeling angiogenesis There are different models, from deterministic (e.g., reaction-diffusion equations) to stochastic (e.g., stochastic differential equations, cellular Potts models) Many are multiscale models, combining randomness at the natural microscale/mesoscale with numerical solutions of PDEs at the macroscale 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 (29) 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

8 Outline 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

9 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 existing vessel (vi) blood circulation & other processes: ignored L. L. Bonilla et al. Modeling Angiogenesis 9 / 29

10 Some notation Let N(t) denote the number of tips at time t (N() = N ) N is the (fixed) number of tips expected during an experiment Let X i (t) R 2 be the location of the i-th tip at time t Let v i (t) R 2 be the velocity of the i-th tip at time t The i-th tip branches at a random time T i and disappears (after reaching the tumor or by anastomosis) at a later random time Θ i The network of endothelial cells (existing vessels) is X(t) = N(t) i= { } X i (s), T i s min{t,θ i } L. L. Bonilla et al. Modeling Angiogenesis / 29

11 Tip branching Vessels branch out of capillary tips (not from mature vessels) The probability that a tip branches from an existing one (birth) in the interval (t,t+dt] is measured by N(t) ( ) α C(t,X i C (t)) dt, with α(c) = α C R +C, i= where C R is a reference value for the TAF concentration C(t,x) emitted by the tumor (α is a coefficient). Then, a tip located in x actually branches whenever its velocity is close to a fixed non-random velocity v, generating a new tip with initial state (X N(t)+,v N(t)+ ) = (x,v ) L. L. Bonilla et al. Modeling Angiogenesis / 29

12 Vessel extension Vessel extension is modeled by tracking the trajectories of capillary tips Description is done in terms of the Langevin equations dx i (t) = v i (t)dt dv i (t) = kv i (t) }{{} friction ( dt + F ) C(t,X i (t)) } {{ } force due to TAF dt + σdw i (t) }{{} random noise for t > T i, where W i 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 (k, σ, d, γ are parameters) F(C) = d +γ C xc L. L. Bonilla et al. Modeling Angiogenesis 2 / 29

13 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 v i of the i-th tip in a infinitesimal region around it The evolution equation is N(t) C(t,x) = d2 xc(t,x) ηc(t,x) ( (t)) t v i (t)δ x X i N, where d 2 and η are parameters. i= an initial Gaussian-like concentration C(, x) is considered the production of C(t, x) due to the tumor is incorporated through a TAF flux boundary condition at the tumoral source L. L. Bonilla et al. Modeling Angiogenesis 3 / 29

14 Anastomosis If a tip meets an existing vessel, they join at that point and time the tip stops the evolution y/l The empirical measure T N counts tips with any velocity that are at time s in a spatial region A as T N(s)(A) = N N(s) i= ǫ X i (s)(a). The death rate (of capillary tips) at location x and time t should be proportional to t N(s) ds δ(x X i (s)). N i= L. L. Bonilla et al. Modeling Angiogenesis 4 / 29

15 Numerical simulation: setting The 2D spatial domain is given by x = (x,y) [,L] [, ] The primary vessel is at x =, the tumor is at x = L The boundary conditions for the TAF field are C(t,x,± ) =, C x (t,,y) =, C (t,l,y) = f(y), x where f(y) is the TAF flux emitted by the tumor y/l y/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 5 / 29

16 Vessel network, TAF field, x - component of TAF force Time = 8 h Number of tips = 28.5 Time = 6 h Number of tips = 47.5 Time = 24 h Number of tips = y/l.. y/l.. y/l y/l y/l y/l L. L. Bonilla et al. Modeling Angiogenesis 6 / 29

17 Outline 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 7 / 29

18 Few ideas Derivation of a mean field equation for the vessel tip density, as N Itō 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 given by the empirical measure Q N Q N(t)(B) = N N(t) i= ǫ (X i (t),v i (t)) (B) = N N(t) i= B ( ) ( ) δ x X i (t) δ v v i (t) dxdv If N is sufficiently large, Q N may admit a density by laws of large numbers Q N(t)(d(x,v)) p(t,x,v)dxdv Thus, as N N N(t) i= g(x i (t),v i (t)) g(x,v)p(t,x,v)dxdv Tip branching & anastomosis are added as source & sink terms to the obtained equation for the tip density p(t,x,v) L. L. Bonilla et al. Modeling Angiogenesis 8 / 29

19 Limiting equations Bonilla, Capasso, Alvaro, Carretero, Phys. Rev. E 9, 24 As N, the tip density p(t,x,v) satisfies the Fokker-Planck equation t p(t,x,v) = α(c(t,x))p(t,x,v)δ(v v ) }{{} birth term (tip branching) t γ p(t,x,v) ds p(s,x,v )dv } {{ } death term (anastomosis) [ + k v v xp(t,x,v) }{{} transport [ v F(C(t,x))p(t,x,v) } {{ } force due to TAF ] vp(t,x,v) } {{ } friction ] + σ2 2 vp(t,x,v) }{{} diffusion (γ is the coefficient of anastomosis) L. L. Bonilla et al. Modeling Angiogenesis 9 / 29

20 Limiting equations Bonilla, Capasso, Alvaro, Carretero, Phys. Rev. E 9, 24 As N, the TAF reaction-diffusion equation becomes where t C(t,x) = d 2 xc(t,x) ηc(t,x) j(t,x) 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 2 / 29

21 Limiting equations Bonilla, Capasso, Alvaro, Carretero, Phys. Rev. E 9, 24 As N, the TAF reaction-diffusion equation becomes where t C(t,x) = d 2 xc(t,x) ηc(t,x) j(t,x) 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 2 / 29

22 Boundary conditions for the tip density Since p has 2nd-order derivatives in v, we impose p(t,x,v) as v Which (spatial) BCs for p? (p has st-order derivatives only in x) L. L. Bonilla et al. Modeling Angiogenesis 2 / 29

23 Boundary conditions for the tip density Since p has 2nd-order derivatives in v, we impose p(t,x,v) as v Which (spatial) BCs for p? (p has st-order derivatives only in x) At any time, we expect to know the marginal tip density at the tumor at x = L p(t,l,y) = p L (t,y) where p(t,x) = p(t,x,v )dv the normal tip density flux injected at the primary vessel at x = n j(t,,y) = j (t,y) Using these values & assuming that p is closed to a local equilibrium distribution at the boundaries, compatible BCs for p + at x = and p at x = L are imposed (see charge transport in semiconductors; Bonilla & Grahn, Rep. Prog. Phys., 25) L. L. Bonilla et al. Modeling Angiogenesis 2 / 29

24 Boundary conditions for p p + (t,,y,v,w) = e k v v 2 σ 2 v e k v v 2 σ 2 dv dw [ j (t,y) ] v p (t,,y,v,w )dv dw p (t,l,y,v,w) = e k v v 2 σ 2 e k v v 2 σ 2 dvdw [ p L (t,y) ] p + (t,l,y,v,w )dv dw where p + = p for v > and p = p for v < v = (v,w); v is the average velocity of the tips at x =, L σ 2 /k is the boundary temperature of the equilibrium distribution L. L. Bonilla et al. Modeling Angiogenesis 22 / 29

25 Outline 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

26 Stochastic vs. mean field approximation For comparison, results of 5 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 the stochastic p(t, x) (right) condition p as v may be roughly satisfied in the numerical code L. L. Bonilla et al. Modeling Angiogenesis 24 / 29

27 Stochastic vs. mean field approximation the profiles p(t,x,y = ) 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 Time = h Number of tips = 2 5 Time = 8 h Number of tips = 43 2 Time = h Number of tips = 2 25 Time = 8 h Number of tips = Time = 6 h Number of tips = 45 5 Time = 24 h Number of tips = 5 25 Time = 6 h Number of tips = 5 2 Time = 24 h Number of tips = L. L. Bonilla et al. Modeling Angiogenesis 25 / 29

28 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: involved coefficient is being estimated time (hours) time (hours) L. L. Bonilla et al. Modeling Angiogenesis 26 / 29

29 Pulse solution for the tip density Approximate p k πσ 2 p(t,x)e k v v 2 /σ 2 Use perturbation technique to obtain p t = k α t πσ 2 p γ p p(s,x)ds σ2 x (F p)+ k 2k 2 x p Ignore x p, assume slow variation of C(t,x) and p = p(x ct). We find (B. Birnir): ( kc k2 α 2 ) [ p(x ct) = 2(F x kc) γπ 2 σ 4 +2K sech 2 k 2 α ] 2 γπ 2 σ 4 +2K k(x ct+ξ ) 2(F x kc) L. L. Bonilla et al. Modeling Angiogenesis 27 / 29

30 Pulse solution for the tip density Approximate p k πσ 2 p(t,x)e k v v 2 /σ 2 Use perturbation technique to obtain p t = k α t πσ 2 p γ p p(s,x)ds σ2 x (F p)+ k 2k 2 x p Ignore x p, assume slow variation of C(t,x) and p = p(x ct). We find (B. Birnir): ( kc k2 α 2 ) [ p(x ct) = 2(F x kc) γπ 2 σ 4 +2K sech 2 k 2 α ] 2 γπ 2 σ 4 +2K k(x ct+ξ ) 2(F x kc) This is the KdV soliton solution with scaling factor speed c. K is a constant with units of frequency. k2 α 2 γπ 2 σ 4 +2K k 2(F x kc) and L. L. Bonilla et al. Modeling Angiogenesis 27 / 29

31 Outline 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

32 Summary Stochastic description of angiogenesis involving tip branching, vessel extension, 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 stochastic model upon averaging over many random experiments L. L. Bonilla et al. Modeling Angiogenesis 29 / 29

33 Summary Stochastic description of angiogenesis involving tip branching, vessel extension, 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 stochastic model upon averaging over many random experiments open: control: inhibit or enhance vessel network open: developing a hybrid model: TAF field equation with averaged quantities with stochastic tip branching & growth open: including new ingredients, as equations for fibronectin and matrix-degrading enzyme, blood circulation,... L. L. Bonilla et al. Modeling Angiogenesis 29 / 29