A PDE System Modeling BiologicalNetwork Formation

Jan Haskovec

King Abdullah University of Science and Technology, KSA

in collaboration with

Peter Markowich (KAUST)Benoit Perthame (Paris VI)

Matthias Schlottbom (Münster)

(J. A. Carrillo, M. Fornasier, G. Albi, M. Artina, M. Burger, H. Jönsson, . . . )

Jan Haskovec PDE System Modelling Biological Network Formation

Biological transportation networks

Leaf venation Neural network Blood capillaries

Jan Haskovec PDE System Modelling Biological Network Formation

Discrete network models

Jan Haskovec PDE System Modelling Biological Network Formation

Discrete network models

Static/dynamic graph-based models, deterministic/random.Topological and geometric properties: loops/trees, connectivity, . . .[Barabasi&Albert’1999, Newman’2003, Watts&Strogatz’1998, . . . ]

The framework:Flow of a material through the network-graph (V, E)

Pressures Pj on vertices j ∈ VConductivities Ci on edges i ∈ E

Assume low Reynolds number (Poiseuille flow), then fluxes

Qi = −Ci(∆P)iLi

Conservation of mass - Kirchhoff law with sources Sj ,∑i∈e(j)

Qi =∑i∈e(j)

Ci

Pj − Pk(i ,j)

Li= Sj for each j ∈ V

Jan Haskovec PDE System Modelling Biological Network Formation

Discrete network model of [Hu&Cai’2014]

Energy cost functional

E :=∑i∈E

(Q2

i

Ci+ νCγi

)Li

consisting ofpumping power (Joule’s law: power = potential × current)

(∆P)iQi =Q2

i

CiLi

metabolic cost νCγiγ = 1/2 for blood flow1/2 ≤ γ ≤ 1 for leaf venation

Gradient flow of the energy E , constrained by the Kirchhoff law:

dCi

dt=

(Q2

i

C 2i

− νγCγ−1i

)Li

Jan Haskovec PDE System Modelling Biological Network Formation

Discrete network model of [Hu&Cai’2014]

Optimal networks exhibit a phase transition at γ = 1 withlooples tree for γ < 1 (a)"uniform sheet" for γ > 1 (b)

Jan Haskovec PDE System Modelling Biological Network Formation

Transition to continuumdescription

Jan Haskovec PDE System Modelling Biological Network Formation

Transition to continuum description

Consider the vector field m such that

P[m] = m ⊗m

is the permeability tensor in a porous medium.Identify the local conductivity Ci with the principal eigenvalueof P[m]

Ci∼= |m|2

Flux given by the Darcy’s law

Qi∼= q = −P[m]∇p = −(m · ∇p)∇p

Kirchhoff law Poisson equation

−∇ · [(m ⊗m)∇p] = S

Jan Haskovec PDE System Modelling Biological Network Formation

Transition to continuum description

Continuum analogue of the discrete energy

E =12

∫c2(m · ∇p)2 +

|m|2γ

γdx

and its formal L2 gradient flow constrained by the Poissonequation

∂tm = c2(m · ∇p)∇p − |m|2(γ−1)m

−∇ · [(m ⊗m)∇p] = S

The discrete model of [Hu&Cai’2014] is a finite differencediscretization of the PDE system for (m, p)

on an equidistant rectangular mesh

(... if you do it "right")

Jan Haskovec PDE System Modelling Biological Network Formation

Corrections of the PDE model

P1. The Poisson equation −∇ · [(m ⊗m)∇p] = S isdegenerate in directions m⊥ and, in general, unsolvable.Fix: Introduce isotropic background permeability of themedium r(x) ≥ r0 > 0,

P[m] := r(x)I + m ⊗m, −∇ · (P[m]∇p) = S

P2. The m-equation is local in x (no network growth).Fix: Introduce a linear diffusive term D2∆m,

∂tm = D2∆m + c2(m · ∇p)∇p − |m|2(γ−1)m

The energy functional has to be updated as

E =12

∫D2|∇m|2 + c2r(x)|∇p|2 + c2(m · ∇p)2 +

|m|2γ

γdx

Jan Haskovec PDE System Modelling Biological Network Formation

The PDE system

Jan Haskovec PDE System Modelling Biological Network Formation

The PDE system

Ω ⊆ Rd , d ≤ 3, bounded network domain (porous medium)S = S(x) - scalar valued source term (given)p = p(t, x) - scalar valued pressure

−∇ · [(r(x)I + m ⊗m)∇p] = S

m = m(t, x) - vector valued conductance

∂m

∂t= D2∆m︸ ︷︷ ︸

random effectsin the porous medium

+ c2(m · ∇p)∇p︸ ︷︷ ︸activation term

− |m|2(γ−1)m︸ ︷︷ ︸relaxation term

c2 > 0 - activation parameter, D2 ≥ 0 - diffusivity,γ ≥ 1/2 - relaxation exponentappropriate BC for m and p, and IC for m

Jan Haskovec PDE System Modelling Biological Network Formation

PDE simulation results (FEM)

Courtesy of M. Schlottbom (U. Münster)

Jan Haskovec PDE System Modelling Biological Network Formation

Mathematical problems

−∇ · [(I + m ⊗m)∇p] = S

∂m

∂t= D2∆m + c2(∇p ⊗∇p)m − |m|2(γ−1)m

Better regularity results (beyond Lax-Milgram) for

−∇ · (A(x)∇p) = S in Ω

p = 0 on ∂Ω

require at least A ∈ L∞(Ω).

Jan Haskovec PDE System Modelling Biological Network Formation

Existence of globalweak solutions

(γ > 1/2)

Jan Haskovec PDE System Modelling Biological Network Formation

Weak solutions for a regularized system

The regularized system

−∇ · [∇p + m((m · ∇p)∗ηε)] = S

∂m

∂t= D2∆m + c2[(m · ∇p)∗ηε]∇p − |m|2(γ−1)m

with the heat kernel ηε(x) := (4πε)−d/2 exp(−|x |2/4ε),preserves the energy dissipation structureadmits a weak solution (Leray-Schauder)

Why heat kernel as mollifier? Because, setting u := m · ∇p,∫Rd

(u ∗ η)u dx =

∫Rd

|u ∗ %|2 dx ≥ 0,

where % = F−1[(η)1/2] (... Parseval).

Jan Haskovec PDE System Modelling Biological Network Formation

The case γ = 1/2

relaxation term |m|2(γ−1)m := m|m| . . . singularity at m = 0

but relaxation energy R(m) :=∫

Ω |m| dx convex!

We prove the existence of a weak solution of

∂tm ∈ D2∆m + c2(m · ∇p[m])∇p[m]− ∂R(m)

with

∂R(m) = r ∈ L∞(Ω); r(x) = m(x)/|m(x)| if m(x) 6= 0,|r(x)| ≤ 1 if m(x) = 0

Conjecture: m is a slow solution, i.e.

m

|m|=

m|m| for m 6= 0

0 for m = 0

Compact support property of solutions (e.g., [Brezis’1974])Jan Haskovec PDE System Modelling Biological Network Formation

Local existence of mild solutions(γ > 1/2)

and their uniqueness(γ ≥ 1)

Jan Haskovec PDE System Modelling Biological Network Formation

Local existence and uniqueness of mild solutions

Global theory for nonlinear eigenvalue problems of compact operatorsby Krasnoselski and Rabinowitz for

T (c2,m) = m,

with T : R+ ×XT → XT given by

T (c2,m) = eLtm0 +

∫ t

0eL(t−s)

(c2F [m](s)− G [m](s)

)ds,

where L := D2∆ is the Dirichlet Laplacian, and

F [m] = (m · ∇p[m])∇p[m], G [m] = |m|2γ−1m,

and the Banach space

XT := L∞(0,T ;X) with X := L∞(Ω) ∩VMO(Ω).

Jan Haskovec PDE System Modelling Biological Network Formation

Elliptic regularity

[Marino’02]:Let A ∈ X, f ∈ Lq(Ω), S ∈ Lr (Ω), r := max1, dq/(d + q).Then

−∇ ·(A(x)∇p + f

)= S in Ω,

p = 0 on ∂Ω

has a unique weak solution p ∈W 1,q(Ω) such that

‖∇p‖Lq(Ω) ≤ C (‖A‖X)(‖f ‖Lq(Ω) + ‖S‖Lr (Ω)

).

[Meyers’63]: For some q > 2 the above estimate holds forA ∈ L∞ with C = C (‖A‖L∞(Ω)) . . . only useful in 2D.

Jan Haskovec PDE System Modelling Biological Network Formation

Stationary statesand network formation

Jan Haskovec PDE System Modelling Biological Network Formation

Stationary states with D > 0, γ ≥ 1

Trivial stationary state: m0 ≡ 0, −∆p0 = S

For large D2 > 0, only the trivial stationary state.

For D2 > 0 small enough, nontrivial stationary states areconstructed as solutions of the fixed point problem

m = βLm + F (m, β)

using the global bifurcation theorem by [Rabinowitz’71],with β := c2/D2 the bifurcation parameter.

Pattern (network) formation can be seen as a consequence of(linear) instability of the trivial solution.

Jan Haskovec PDE System Modelling Biological Network Formation

Construction of stationary solutions with D = 0, γ ≥ 1

For γ > 1:

c2(∇p ⊗∇p)m = |m|2(γ−1)m

so that m = λ∇p and p solves

−∇ ·[(

1 + c2

γ−1 |∇p(x)|2

γ−1χA(x))∇p]

= S ,

which is the Euler-Lagrange equation of a strictly convexenergy functional iff γ > 1.

For γ = 1:

c2(∇p ⊗∇p)m = m

leads to the (even more) nonlinear Poisson equation

−∇ ·[(

1 +λ2(x)

c2χc|∇p|=1(x)

)∇p]

= S

Jan Haskovec PDE System Modelling Biological Network Formation

Numerical results

Jan Haskovec PDE System Modelling Biological Network Formation

Numerical results: varying D

S ≡ −1, c = 50, γ = 1/2

D = 10−2 D = 5× 10−3 D = 10−3

Jan Haskovec PDE System Modelling Biological Network Formation

Numerical results: varying γ

S ≡ −1, c = 50, D = 0.025

γ = 1/2 γ = 3/4 γ = 1

Jan Haskovec PDE System Modelling Biological Network Formation

One source, two sinks

Jan Haskovec PDE System Modelling Biological Network Formation

Open problems in network formation

Understanding of the branching mechanism

Is the PDE model capable of producing networks with loops?

Does the PDE system admit fractal solutions (i.e., supportedon sets of Hausdorff dimension < d), in scaling limits asD → 0, r → 0 and/or c →∞?

More reliable numerical methods

Extensions of the model:Nonlinear (porous-medium-type) diffusion in the m-equationBrinkmann instead of Darcy for p (viscous resistance in thematerial flow - angiogenesis in cancer)Coupling with the Poisson-Nernst-Planck system as a modelfor neural networks

Jan Haskovec PDE System Modelling Biological Network Formation

Conclusions

Thank you for your attention!

Jan Haskovec PDE System Modelling Biological Network Formation