Kinetic modelling of cell migration in the ECM.

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1 Kinetic modelling of cell migration in the ECM. Nadia Loy, Andrea Tosin, Luigi Preziosi Politecnico di Torino-Università degli studi di Torino July 6, 2018

2 Jed Johnson, M. Oskar Nowicki, Carol H. Lee, E. Antonio Chiocca, Mariano S. Viapiano, Sean E. Lawler, and John J. Lannutti Quantitative Analysis of Complex Glioma Cell Migration on Electrospun Polycaprolactone Using Time-Lapse Microscopy,TISSUE ENGINEERING: Part C Volume 15, Number 4, 2009

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4 Cells population p = p(t, x, v), t 0, x Ω R n, v = ˆvv, (ˆv, v) V = S n 1 + [ U, U]. Macroscopic quantities (per unit volume) * Density * Momentum * Mean velocity * Pressure tensor ρ(x, t) = p(x, v, ˆv, t) dvdˆv. V ρ(x, t)u(x, t) = vˆvp(x, v, ˆv, t) dvdˆv. V vˆvp(x, v, ˆv, t) dvdˆv V U(x, t) =. p(x, v, ˆv, t) dvdˆv V D = (vˆv U) (vˆv U)pdvdˆv. V

5 Matrix m = m(x, n), x Ω, n S n 1 + * Matrix density * Matrix mean direction * Matrix diffusion tensor D m = M(x) = m(x, n)dn. S + n 1 m(x, n) Û M = S + M(x) ndn n 1 n n nm(x, n)dn. M(x) S + n 1

6 Anisotropy index n i=1 (λ i λ) F A = n 2 n i=1 λ i Figure: Surulescu et Al. Figure: Hagmann et Al.

7 The model p (t, x, v) + v xp(t, x, v) = Jm[p](t, x, v)) + Jc[p](t, x, v)) t J m[p](t, x, v)) = µ m(t, x)(ρ(t, x)ψ m(v) m(x, ˆv) p(t, x, v)), M(x) µm(t, x) = (1 d(t, x))f(m) J c[p](t, x, v) = µ c(t, x)(ρ(t, x) Hillen 05, Chauviere, Hillen, Preziosi 10 H Theorem (Lods, Toscani, Bisi) p(0, x, ) in L 1 (Ω) L 2 (V ) with finite mass and energy, as t for a.e. x in V. =ψ m(v) m(x,ˆv) M(x) {}}{ ψ c(v) p(t, x, v)), µ c(t, x) = η c(t, x)ρ(t, x) p (µ cψ c + µ mψ m m M ) L 2 (V ) 0

8 Index of directional persistence ψ d = sgn(ūm v )ÛM ˆv = { ±1 if v ÛM 0 if v ÛM Mean squared displacement (Alt, Othmer 88) [ ] 2D 1 t D(t) = µ(1 ψ d ) µ(1 ψ d ) (1 e µ(1 ψ d)t Dt 2 if ψ d = 1 if ψ d 1

9 Momentum of the cells population Momentum of the matrix/mean velocity of cells Mean post-collision velocity U M = U M Û M U c = ψ cdndv V U(x, t) U 0 exp( (µ m+µ c)t)+(u M +U c)[1 exp( (µ m+µ c)t)] (U M +U c) a.e. Diffusion tensor D c = (v U) (v U)ψ c(v)dvdˆv V D M = (v U) (v U)ψ m m(v) V M dvdˆv

10 Properties of ψ m ψ m(v ˆv) : [ U, U] R + * U U ψm(v)dv = 1 * U U ψm(v)vdv = U M * U U ψm(v)v2 dv = D M ψ m even: diffusion case; ψ m not even: drift case.

11 Macroscopic models parabolic scaling: τ = ɛ 2 t, ˆx = ɛx diffusion case ; hyperbolic scaling: τ = ɛt, ˆx = ɛx drift case; moments closure.

12 Nondimensionalization * V m mean velocity, X macroscopic length * l 0 c, µ 0 c mean free path/ frequency for cells binary collisions * lm 0 = d0 X 1 d 0, µ 0 m mean free path/frequency for cell-fiber interaction * ɛ = l0 c X : Knudsen number. * µ 0 c = Vm l 0 * µ 0 m = Vm l 0 m µ 0 m l0 m = µ0 c l0 c V m = d 0 Xf(M 0 ). ˆx = x X, ˆv = v, ˆt = t V m τ, ˆρ = ρ, ˆp = p ρ ρ 0, ˆψ = ψu n, ˆm = m M, ˆM = 0 Vm n M 0 M 0 Macroscopic time scales T drift = X V m = 1 µ 0 c ɛ. D = V 2 m µ0 m 1 d 0 T diff = X2 D = µ0 m (1 d 0) ɛ 2 µ 02 c

13 Parabolic scaling - Diffusive limit α = µ0 m µ 0 c ɛ 2 p t (t, x, v) + ɛαv xp(t, x, v) = α2 J m[p] + αj c[p] Diffusion dominated equation ( ) ρ 0 t α x 1 µ cd c + αµ md M [ x ρ 0 ] = 0 αµ m + µ c αµ m + µ c Covergence: Hillen 00.

14 Hyperbolic scaling - Chapman Enskog ɛ p t (t, x, v) + ɛα(1 d 0)v xp(t, x, v) = J m[p] + J c[p] Drift dominated equation ( ) ρ 0 t + x 1 ρ 0 (αµ mu M + µ cu c) = 0, αµ m + µ c

15 Loss of details!! Spatially homogeneous case: Microscopic approach ρ t = 0 Parallelism between microscopic approach and velocity jump approach.

16 Microscopic collision approach-2d, spatially homogeneous f = f(τ, θ, v), m = m(θ n), θ, θ n [0, π), v [ U, U] θ θ = ɛ(θ n θ) + ɛθ (D θ θm 2 ) v v = ɛ(u M v) + ɛθ (D v UM 2 ) π U d ϕ(θ, v)f(τ, θ, v)dθdv = dτ 0 U π π U ( ϕ(θ, v ) ϕ(θ, v) ) f(τ, θ, v)m(θ n)dθ ndθdv 0 0 U

17 Scaling: t = tτ, p(t, θ, v) = f(τ, θ, v) Quasi-invariant limit f(t,θ,v) t = θ [f(θ M θ)] + v[f(u M v)] + 2 θθ [f(d θ θ 2 M )] θv [f D θ θ 2 M D v U 2 M ] + 2 vv [f(dv U 2 M )] Stationary state U M constant ( ) f = ρc ( ( (UM v) 2 + Dv) 2 (θm θ) 2 + Dθ) 2

18 L.Pareschi, M.Zanella Structure preserving schemes for nonlinear Fokker-Planck equations and applications, Journal of Scientific Computing Figure: Random initial cells, aligned fibers θ = 3π/8,, U M = 0 Figure: Random initial cells, aligned fibers θ = 3π/8,, U M = 0.5

19 Figure: Random initial cells, aligned fibers θ = 3π/8,, U M = 0 Figure: Random initial cells, aligned fibers θ = 3π/8,, U M = 0.5

20 Figure: Uniform fibers, U M = 0 Figure: Aligned fibers θ = 3π/8,, U M = 0.5

21 Thank you for your attention!