a pde system modeling biological network formationjcarrill/cuba/lectures/haskovec.pdf ·...
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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