The numerical foundations of the brain's waterscape

The brain’s numerical waterscape
Marie E. Rognes
Simula Research Laboratory
Oslo, Norway
Séminaire du Laboratoire
Laboratoire Jacques-Louis Lions
April 9 2021
1 / 39

Brainphatics: understanding the brain’s waterways [Louveau et al, 2017 (Fig 2)]
[Paolo Mascagni, Vasorum
Lymphaticorum Corporis Humani
Historia et Ichnographia (1787)]
3 / 39

[Beta-amyloid plaques and tau in the brain, National Institute of Health]
4 / 39

Outline
How can brain physiology benefit from
mathematical modelling?
I Introduction to brainphatics
II Computational brainphatics
How can applied mathematics benefit from
brain physiology?
III The poroelastic brain (macroscale)
IV Bridging electrochemistry and
mechanics (microscale)
Core message
Mathematical models can give new insight
into physiology – and the human brain gives
an extraordinary rich setting for mathematics
and numerics!
5 / 39

I: Brainphatics: the brain’s waterscape
6 / 39

Solutes spread through the CSF and into the brain parenchyma
[Pizzichelli et al, Numerical study of intrathecal drug delivery to a permeable spinal cord: effect of catheter position and angle, CMBBE, 2018; Ringstad et al, 2018]
7 / 39

Solutes spread along perivascular spaces
[Helen Cserr (credit: R. Cserr), Ichimura et al, 1991]
[Iliff et al, 2012, Xie et al, 2013]
[Maiken Nedergaard]
8 / 39

Faster spread with sleep, exercise,
...
[Maiken Nedergaard]
8 / 39

Faster spread with sleep, exercise,
one glass of wine (but not two!).
[Maiken Nedergaard]
8 / 39

Controversy and key open questions
1. Are there forces and spaces sufficient to
create fluid pathways in relevant brain
compartments?
2. What are the mechanisms underlying
influx and clearance in the brain and
brain environment?
3. How do brain clearance affect
neurological and neurodegenerative
diseases?
9 / 39

II: Computational brainphatics: surfing and diving into the brain
11 / 39

ICP (gradients) pulsate in sync with cardiac and respiratory cycles
[Vinje et al, Respiratory influence on cerebrospinal fluid flow..., Scientific Reports, 2019]
[Eide and Sæhle, 2010]
12 / 39

ICP (gradients) pulsate in sync with cardiac and respiratory cycles
[Eide and Sæhle, 2010]
dICP(t) ≈ ac sin(2πfct)+ar sin(2πfrt)
12 / 39

Pulsating ICP gradients induce pulsating CSF flow
Incompressible Navier-Stokes
Velocity v and pressure p such that
ρ (v̇ + v · ∇v) − µ∆v + grad p = 0
div v = 0
Pressure given between inlet and outlet:
dp(t) = ac sin(2πfct) + ar sin(2πfrt)
13 / 39

ρ (v̇ + v · ∇v) − µ∆v + grad p = 0
div v = 0
Analytic solution(s) in axisymmetric pipe
Peak flux Ar, Ac and stroke volume Vr, Vc:
A = |πr2 ia
ρω

1 −
2
Λ
J1(Λ)
J0(Λ)

|
V = A(πf)−1
where ω = 2πf, Λ = αi3/2
, α Womersley...
13 / 39

ρ (v̇ + v · ∇v) − µ∆v + grad p = 0
div v = 0
In patients (cardiac vs respiratory)
I Average peak flow rates: 0.29 vs 0.32 mL/s
I Average stroke volumes: 70 mL vs 308 mL
I Good agreement with cardiac-gated PC-MRI
I Resolves clinical pressure vs flow mystery!
13 / 39

Is perivascular flow driven by arterial pulsations?
[Mestre et al, 2018 (Figs 1, 2)]
14 / 39

Is perivascular flow driven by arterial pulsations?
[Daversin-Catty et al, The mechanisms behind perivascular fluid flow, PLOS ONE, 2020]
[Mestre et al, 2018 (Figs 1, 2)]
Scale bar: 50 µm.
Stokes in moving PVS
ρvt − µ∇2
v + ∇p = 0 in Ωt
∇ · v = 0 in Ωt
[San Martin, 2009]
∂Ω0 7→ ∂Ωt by travelling wall wave
with wave speed c = λ/f = 1 m/s.
14 / 39

Arterial pulsations drive pulsatile flow but not net flow
[Daversin-Catty et al, The mechanisms behind perivascular fluid flow, PLOS ONE, 2020]
Wall pulsations dICP dICP(t) Rigid motions
E)
A) B)
F)
C) D)
G) H)
Velocity
magnitude
(μm/s)
Pressure
(Pa)
Velocity magnitude (μm/s)
15 / 39

CSF tracer distributes brain-wide and centripetally in humans
... unlikely that diffusion alone explains brain-wide
distribution.
[Ringstad et al (2017), Ringstad et al (2018)]
16 / 39

What are likely tracer distributions given uncertainty in diffusion and
convection hypotheses? [Croci et al, Uncertainty quantification of parenchymal tracer distribution..., FBCNS, 2019]
A B
C D
1.8M vertices,
9.7M cells,
3200 samples
17 / 39

Likely tracer evolution and distribution via diffusion alone
Find the concentration c = c(t, x, ω) s.t.:
∂tc + div(vc) − div(D∇c) = 0,
D̄ = 1.2 × 10−10
m/s2
, v = 0.
1h 8h 24h
18 / 39

Likely tracer evolution and distribution via diffusion alone
Find the concentration c = c(t, x, ω) s.t.:
∂tc + div(vc) − div(D∇c) = 0,
D̄ = 1.2 × 10−10
m/s2
, v = 0.
1h 8h 24h
[Croci et al, Uncertainty quantification of parenchymal tracer distribution..., FBCNS, 2019]
a) b)
c) d)
Key observations
I Expected amount in gray matter peaks
around 15h
I Expected amount in white matter still
increasing at 24h
I Substantial variation in all outputs for
uncertain diffusion (homogeneous,
heterogeneous)
18 / 39

Glymphatic-type velocities can enhance tracer transport (or not)
[Kaur et al (2020), Kiviniemi et al (2016)]
19 / 39

Local directionality
vV1(x, ω) = vavg(ω)∇×(vx, vy, vz)(x, ω)
Global directionality
vV2(x, ω) = vV1(x, ω) + vdir(x)
vx, vy, vz i.i.d. Matérn fields: ν = 2.5
and correlation length λ = 1020µm.
19 / 39

Local directionality
vV1(x, ω) = vavg(ω)∇×(vx, vy, vz)(x, ω)
Global directionality
vV2(x, ω) = vV1(x, ω) + vdir(x)
vx, vy, vz i.i.d. Matérn fields: ν = 2.5
and correlation length λ = 1020µm.
Key observations
I The glymphatic velocity
model did not enhance
transport into any region
I – unless augmented by
a flow field with a
large-scale directionality.
19 / 39

III: The poroelastic brain
20 / 39

Brain tissue is soft, heterogeneous and rheologically complex
Brain tissue is
soft (shear modulus ≈ 0.5–2.5 kPa)
stiffer with increasing strain/strain rates (nonlinear)
stiffer during loading than unloading (viscoelastic)
stiffer in compression than in tension (poroelastic)
stiffer in some regions than in others (heterogeneous)
Stiffness/Shear modulus (kPa)
[Budday et al (2015) (Fig 6), Budday et al (2019) (Fig1)]
21 / 39

Biot’s equations describe displacement and fluid pressure in a
poroelastic medium
Find the displacement u = u(x, t) and the pressure
p = p(x, t) over Ω × [0, T] such that:
− div (2µε(u) + λ div uI − pI) = f,
c0ṗ + div u̇ − div K grad p = g
[Biot (1941), Murad, Thomée and Loula (1992-1996), Phillips and Wheeler (2007-2008), and many others]
22 / 39

poroelastic medium
Low-storage, incompressible regime: c0 = 0, λ → ∞:
div u → 0, system decouples
− div K grad p = g (Darcy)
− div (2µε(u) + λ div uI) = f − grad p (Elasticity)
22 / 39

poroelastic medium
Low-storage, incompressible regime: c0 = 0, λ → ∞:
div u → 0, system decouples
− div K grad p = g (Darcy)
− div (2µε(u) + λ div uI) = f − grad p (Elasticity)
Low-storage, impermeable regime: c0 = 0, K → 0:
div u = 0
(Stokes)
[Biot (1941), Murad, Thomée and Loula (1992-1996), Phillips and Wheeler (2007-2008), and many others]
22 / 39

At the macroscale, the brain can be viewed as an elastic medium
permeated by multiple fluid-filled networks
The brain parenchyma includes multiple fluid networks (extracellular spaces (ECSs),
arteries, capillaries, veins, paravascular spaces (PVSs))
[Zlokovic (2011)]
Rat cerebral cortex with ECS in black
(Scale bar: ≈ 1µm)
[Nicholson (2001) (Fig. 2)]
The brain is (≈):
5-10% blood
20% ECS
70-75% brain cells
80% water [Budday et al (2019)]
23 / 39

Multiple-network poroelastic theory (MPET) is a macroscopic model
for poroelastic media with multiple fluid networks
[Bai, Elsworth, Roegiers (1993); Tully and Ventikos (2011)]
24 / 39

The multiple-network poroelasticity (MPET) equations describe
displacement and fluid pressures in generalized poroelastic media
Find the displacement u = u(x, t) and J (network) pressures
pj = pj(x, t) for j = 1, . . . , J such that
− div(2µε(u) + λ div u I −
P
j αjpj I) = f,
cjṗj + αj div u̇ − div Kj grad pj + Sj = gj j = 1, . . . , J.
Fluid exchange between networks:
Sj =
P
i si←j =
P
i ξj←i(pj − pi).
J = 1 corresponds to Biot’s equations, J = 2 Barenblatt-Biot.
λ → ∞, cj → 0, ξj←i 1, ξj←i 1 and K → 0 interesting regimes.
[Biot, 1941; Bai, Elsworth and Roegiers, Water Resources Research, 1993]
[Tully and Ventikos, Jour Fluid Mech., 2011; Lee, Piersanti, Mardal, R., SISC, 2019]
25 / 39

More robust discretization of (nearly) incompressible MPET
[Lee et al, Mixed finite elements for MPET, SISC, 2019]
MPET structure with α = (α1, . . . , αJ ), p = (p1, . . . , pJ )T
:

− div(2µ + λ tr)ε − grad α
αT
div ∂
∂t
E

u
p

=

f
g

(2)
where E = − div K grad +S + C ∂
∂t
.
26 / 39

(BE, Taylor-Hood.)
:

αT
div ∂
∂t
E

u
p

=

f
g

(2)
∂t
.
26 / 39

(BE, Taylor-Hood.)
:

αT
div ∂
∂t
E

u
p

=

f
g

(2)
∂t
.
Key idea: introduce the total pressure (inspired by [Lee, Mardal, Winther, 2017])
p0 = λ div u − α · p,
26 / 39

(BE, Taylor-Hood.)
:

αT
div ∂
∂t
E

u
p

=

f
g

(2)
∂t
.
p0 = λ div u − α · p,
transforming (2) into


− div 2µε − grad 0
div −λ−1
−λ−1
α
0 λ−1
αT ∂
∂t
Ẽ + λ−1
ααT ∂
∂t




u
p0
p

 =


f
0
g


26 / 39

(BE, Taylor-Hood.)
:

αT
div ∂
∂t
E

u
p

=

f
g

(2)
∂t
.
p0 = λ div u − α · p,
transforming (2) into the more robust as λ → ∞


div −λ−1
−λ−1
α
0 λ−1
αT ∂
∂t
Ẽ + λ−1
ααT ∂
∂t




u
p0
p

 =


f
0
g


26 / 39

(BE, Taylor-Hood.)
:

αT
div ∂
∂t
E

u
p

=

f
g

(2)
∂t
.
p0 = λ div u − α · p,
transforming (2) into the more robust as λ → ∞


div −λ−1
−λ−1
α
0 λ−1
αT ∂
∂t
Ẽ + λ−1
ααT ∂
∂t




u
p0
p

 =


f
0
g


See also [Piersanti et al, Parameter robust preconditioning for MPET, 2021].
26 / 39

Convergence estimates for total-pressure MPET formulation
Continuous variational formulation
Given (compatible, and sufficiently regular) u0
and p0
j , f and gj for j = 1, . . . , J, find
u ∈ H1
(0, T; H1
(Ω)d
), p0 ∈ H1
(0, T; L2
(Ω)), and pj ∈ H1
(0, T; H1
(Ω)) such that
h2µε(u), ε(v)i + hp0, div vi = hf, vi
hdiv u, q0i − hλ−1
p0, q0i − hλ−1
α · p, q0i = 0
hcjṗj + αjλ−1
˙
p0 + αjλ−1
α · ṗ + Sj, qji + hKj grad pj, grad qji = hgj, qji, j = 1, . . . , J,
for all v ∈ H1
(Ωd
), q0 ∈ L2
(Ω), qj ∈ H1
(Ω).
27 / 39

Continuous Semi-discrete variational formulation
and p0
uh ∈ H1
(0, T; Vh), p0,h ∈ H1
(0, T; Q0,h), and pj,h ∈ H1
(0, T; Qh) such that
h2µε(uh), ε(v)i + hp0,h, div vi = hf, vi
hdiv uh, q0i − hλ−1
p0,h, q0i − hλ−1
α · ph, q0i = 0
hcjṗj,h + αjλ−1
ṗ0,h + αjλ−1
α · ˙
ph + Sj,h, qji + hKj grad pj,h, grad qji = hgj, qji, j = 1, . . . , J,
for all v ∈ Vh, q0 ∈ Q0,h, qj ∈ Qh.
27 / 39

and p0
uh ∈ H1
(0, T; Vh), p0,h ∈ H1
(0, T; Q0,h), and pj,h ∈ H1
p0,h, q0i − hλ−1
α · ph, q0i = 0
ṗ0,h + αjλ−1
α · ˙
Assumptions
A1 Vh × Q0,h is Stokes-stable (in the
Brezzi sense).
A2 Qj,h is H1
-conforming for
j = 1, . . . , J.
27 / 39

and p0
uh ∈ H1
(0, T; Vh), p0,h ∈ H1
(0, T; Q0,h), and pj,h ∈ H1
p0,h, q0i − hλ−1
α · ph, q0i = 0
ṗ0,h + αjλ−1
α · ˙
Theorem, Taylor-Hood type Vh × Qh × QJ
ku − uh(t)kH1 + kp0 − p0,h(t)kL2
. Eh
0 + hl+1
(kuk, kp0k) +
P
j hlj +1
kpjk,
P
j kpj − pj,hkL2H1 . Eh
0 + hl+1
(kuk, kp0k) +
P
j hlj
kpjk + . . . ,
independent of h, λ, cj, ξji.
Assumptions
A1 Vh × Q0,h is Stokes-stable (in the
Brezzi sense).
A2 Qj,h is H1
-conforming for
j = 1, . . . , J.
27 / 39

The total-pressure MPET formulation yields optimal convergence,
including near incompressibility [Lee et al, Mixed finite elements for MPET, SISC, 2019]
ku − uhkL∞(0,T,L2) Rate ku − uhkL∞(0,T,H1) Rate
h 6.27 × 10−2
1.46 × 100
h/2 7.28 × 10−3
3.11 3.95 × 10−1
1.88
h/4 8.70 × 10−4
3.06 1.01 × 10−1
1.97
h/8 1.07 × 10−4
3.02 2.55 × 10−2
1.99
h/16 1.33 × 10−5
3.01 6.38 × 10−3
2.00
Optimal 3 2
kp1 − p1,hkL∞(0,T,L2) Rate kp1 − p1,hkL∞(0,T,H1) Rate
h 1.58 × 10−1
1.68 × 100
h/2 4.22 × 10−2
1.90 8.65 × 10−1
0.96
h/4 1.08 × 10−2
1.97 4.35 × 10−1
0.99
h/8 2.70 × 10−3
1.99 2.18 × 10−1
1.00
h/16 6.76 × 10−4
2.00 1.09 × 10−1
1.00
Optimal 2 1
Smooth test case inspired by [Yi,
2017] with moderately high λ and
cj = 0 with total-pressure
augmented Taylor-Hood
Pd
2 × PJ+1
1 discretization.
28 / 39

IV: Bridging brain electrochemistry and mechanics
29 / 39

Ion concentrations is the common denominator for brain signalling
and brain fluid mechanics
Ion concentrations underpin brain signalling via
generating electrical potentials and induce fluid
movement via osmosis.
[Khan Academy - the neuron and nervous system]
The extracellular ion composition
changes with local neuronal activity
and across brain states
[Rasmussen et al, 2021]
30 / 39

Cortical Spreading Depression (CSD) is a slowly propagating wave
of depolarization of brain cells
CSD is a fundamental pattern of brain
signaling that provides an opportunity for
greater understanding of nervous system
physiology...
Substantial controversy regarding
mechanisms.
[Pietrobon and Moskowitz, 2014]
31 / 39

A mathematical (Mori) framework for brain ion and fluid movement
[Mori, 2015; Zhu et al, 2020; Nicholson (2001) (Fig. 2).]
In a (homogenized tissue) domain Ω ⊂ Rd
with
compartments r ∈ R (e.g. extracellular, neuronal,
glial spaces), and (ion) species k ∈ K (e.g. sodium
(Na+
), potassium (K+
), chloride (Cl−
) and glutamate
(Glu)).
Rat cerebral cortex with ECS in black (Scale bar:
≈ 1µm).
32 / 39

A mathematical (Mori) framework for brain ion and fluid movement
[Mori, 2015; Zhu et al, 2020; Nicholson (2001) (Fig. 2).]
In a (homogenized tissue) domain Ω ⊂ Rd
with
compartments r ∈ R (e.g. extracellular, neuronal,
glial spaces), and (ion) species k ∈ K (e.g. sodium
(Na+
), potassium (K+
), chloride (Cl−
) and glutamate
(Glu)).
For each compartment r and species k, x ∈ Ω, t 0,
find the
I concentrations [k]r(x, t),
I electrical potentials φr(x, t),
I volume fractions αr(x, t),
I hydrostatic pressures pr(x, t),
I fluid velocities ur(x, t).
Communication via the extracellular space (ECS, e).
Rat cerebral cortex with ECS in black (Scale bar:
≈ 1µm).
32 / 39

The Mori framework (without intra-compartment fluid flow) [Mori, 2015]
Change in cell volume is proportional to water movement across cell membrane
(3)
Ions move by diffusion and electrical forces, and across cell membranes
(4)
Electroneutrality
(5)
33 / 39

∂αr
∂t
= −γrewre r 6= e,
∂αe
∂t
=
P
r6=e γrewre, (3)
(4)
Electroneutrality
(5)
33 / 39

∂αr
∂t
∂αe
∂t
=
P
r6=e γrewre, (3)
where the transmembrane water flux wre is driven by osmotic pressure differences
wre ∝
P
k[k]e − [k]r + ae
αe
− ar
αr
(4)
Electroneutrality
(5)
33 / 39

∂αr
∂t
∂αe
∂t
=
P
r6=e γrewre, (3)
wre ∝
P
k[k]e − [k]r + ae
αe
− ar
αr
∂(αr[k]r)
∂t
= − div Jk
r − γreJk
re(·) r 6= e,
∂(αe[k]e)
∂t
= − div Jk
e +
P
r6=e γreJk
re(·). (4)
Electroneutrality
(5)
33 / 39

∂αr
∂t
∂αe
∂t
=
P
r6=e γrewre, (3)
wre ∝
P
k[k]e − [k]r + ae
αe
− ar
αr
∂(αr[k]r)
∂t
= − div Jk
r − γreJk
re(·) r 6= e,
∂(αe[k]e)
∂t
= − div Jk
e +
P
r6=e γreJk
re(·). (4)
with ion flux Jk
r driven by diffusion and drift in the electrical field:
Jk
r ∝ ∇[k]r + βr[k]r∇φr.
and transmembrane ion fluxes Jk
re subject to modelling (ODEs).
Electroneutrality
(5)
33 / 39

∂αr
∂t
∂αe
∂t
=
P
r6=e γrewre, (3)
wre ∝
P
k[k]e − [k]r + ae
αe
− ar
αr
∂(αr[k]r)
∂t
= − div Jk
r − γreJk
re(·) r 6= e,
∂(αe[k]e)
∂t
= − div Jk
e +
P
r6=e γreJk
re(·). (4)
with ion flux Jk
r driven by diffusion and drift in the electrical field:
Jk
r ∝ ∇[k]r + βr[k]r∇φr.
and transmembrane ion fluxes Jk
re subject to modelling (ODEs).
Electroneutrality
P
k zk div Jk
r = −
P
k zkγreJk
re r 6= e,
P
k zk div Jk
e =
P
k zk
P
r6=e γreJk
re. (5)
33 / 39

Imitating experimental methods successfully induces model CSD
Enger et al (2015). Dynamics of ionic shifts
in cortical spreading depression
Ik = g(φne − Ek), k ∈ {Na, K, Cl}
Leao et al (1944). Spreading depression of activity in the cerebral
cortex
Ipump =
Imax

1 +
mK
[K]e
2
1 +
mNa
[Na]n
3
Dreier (2011). The role of spreading depression, spreading
depolarization and spreading ischemia in neurological disease
34 / 39

Neuron depolarization and breakdown of the ionic homeostasis
spreads through the tissue [Ellingsrud et al, Validating a computational framework for ionic electrodiffusion, in prep., 2021]
35 / 39

What is the mean speed of the CSD wave?
[Ellingsrud et al, Accurate numerical simulation of ionic electrodiffusion, arXiv, 2021]
Discretization: Strang splitting, BDF2, ESDIRK4, N elements in space, ∆t time step.
36 / 39

The Mori framework is numerically challenging: hard to harvest
benefits from higher-order methods [Ellingsrud et al, Accurate numerical simulation of ionic electrodiffusion, preprint, 2021]
The Mori framework with physiologically relevant compartments and membrane
mechanisms (Jk
re) define a coupled system of time-dependent, nonlinear PDE and ODEs.
I Godunov or Strang splitting scheme
(PDEs vs ODEs)
I Finite element spatial discretization:
αr,h(t) ∈ Sh ⊂ L2
,
[k]r,h(t) ∈ Vh ⊂ H1
,
φr,h(t) ∈ Th ⊂ H1
,
I Finite difference PDE time
discretization: BE, CN, BDF2...
I Runge-Kutta type ODE discretizations... Better methods??
37 / 39

Collaborators
Nicolas Boullé (Oxford)
Matteo Croci (Oxford)
Cécile Daversin-Catty (Simula)
Per Kristian Eide (Oslo University Hospital)
Rune Enger (Oslo)
Ada J. Ellingsrud (Simula)
Patrick E. Farrell (Oxford)
Michael B. Giles (Oxford)
Jeonghun J. Lee (Baylor)
Erika K. Lindstrøm (Oslo)
Kent-André Mardal (Oslo)
Klas Pettersen (Oslo)
Eleonora Piersanti (Simula)
Geir Ringstad (Oslo University Hospital)
Travis B. Thompson (Oxford)
Lars Magnus Valnes (Oslo University Hospital)
Vegard Vinje (Simula)
... and others
Core message
Mathematical models can give new insight
into medicine, – and the human brain gives
an extraordinary rich setting for mathematics
and numerics!
This research is supported by the European Research Council (ERC) under the European
Union’s Horizon 2020 research and innovation programme under grant agreement 714892
(Waterscales), by the Research Council of Norway under grant #250731 (Waterscape),
and by the EPSRC Centre For Doctoral Training in Industrially Focused 706 Mathematical
Modelling (EP/L015803/1).
39 / 39

The numerical foundations of the brain's waterscape

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