The brain's waterscape/Hjernens vannveier

Hjernens vannveier
Marie E. Rognes
Simula Research Laboratory
NTVA, Bergen, Norway
Jan 30 2018
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Act 1: Introduction to the brain’s numerical waterscape
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Neurodegenerative diseases such as dementia represent an immense societal
challenge for Europe and the World
Alzheimer's Horizon20200
50
100
150
200
250
300
350
400
450
BillionEUR/year
Europe
Rest of world
The estimated yearly cost of Alzheimer’s disease worldwide is over 400 billion EUR.
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Several neurodegenerative diseases are associated with abnormal
aggregations of proteins in the brain nerve cells
Alzheimer’s disease: extracellular
aggregations of β-amyloid in the gray
matter of the brain
[Glenner and Wong, Biochem Biophys ..., 1984;
Selkoe, Neuron, 1991]
Parkinson’s disease/Dementia with Lewy
bodies: aggregations of α-synuclein in
neurons
[Dawson and Dawson, Science, 2003;
McKeith et al, Lancet, 2004]
Lewy-body (brown) in Parkinson’s disease
[Wikimedia Commons]5 / 55

The lymphatic system clears the body of metabolic waste, but what plays
the role of the lymphatics in the brain?
[Wikimedia Commons, Morris et al [2014]]
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Tracer experiments in mice indicate an increase in extracellular space and
more rapid tracer transport during sleep
[Xie et al., Science, 2013]
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Structural properties of brain tissue (such as e.g. the ratio of intracellular to
extracellular space, arterial stiﬀness) change with age
[Sykova et al., Proc Natl Aca Sci, 2005]
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Cerebrospinal ﬂuid (CSF) circulates in the spaces surrounding the brain
(subarachnoid, ventricles, aqueduct)
[Wikimedia Commons]
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Perivascular spaces and glial cells are key structures at the meso/microscale
[Zlokovic, 2011]
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Is there a convective ﬂow of ﬂuid in spaces surrounding the vasculature?
Distribution of solutes in rat brain. Left: solutes distribute perivascularly, scale bar 125 µ m. Right: Volumes of
solute distributions. [Hadaczek et al, 2006]
[Rennels, 1985; Stoodley, 1997; Hadaczek et al, 2006; ...]
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Iliff/Nedergaard groups argue that CSF flows rapidly through paravascular
spaces and drives a convective ISF flow through the extracellular space
[Iliff et al., Sci Trans Med, 2012, ...]
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Carare/Weller group argues instead that there is perivascular flow in the
opposite direction
Fact: in cerebral amyloid angiopathy
patients, protein fragments (amyloid-beta)
accumulate in the arterial walls.
Implications for fluid flow directionality?
[Carare et al, 2008; Bakker et al, 2016, ...]
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Developing the next generation simulation technology to solve problems
aﬀecting human health and disease
[Massing et al., Numer. Math., 2014] [Farrell et al., SISC, 2013] [Valen-Sendstad et al, 2015]
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Can computational modelling give sorely needed insight on the removal and
aggregation of waste in the brain?
Ambition: to establish mathematical, numerical and computational
foundations for predictively modeling fluid flow and solute transport
through the brain across scales - from the cellular to the organ level.
Topics:
New mathematical models: multiple-network porous media flow,
mixed dimensional models coupling tissue and (peri-)vasculature,
electrochemical-mechanical models;
New numerical methods with better properties (stability,
convergence, robustness, efficiency);
New simulation software with tailored features for both forward
and inverse computational studies and uncertainty quantification
Funding:
ERC Starting Grant (Waterscales), 2017-2022
RCN Young Research Talent (Waterscape), 2016-2019
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Waterscales and Waterscape combined form an interdisciplinary and diverse
research team and environment
Core research team
Collaborators and research environments
Simula Research Laboratory (Scientiﬁc Computing, Applied Mathematics)
G. Einevoll, Norwegian University of Life Sciences (Neuroscience)
L. Formaggia/P. Zunino, Politechnico di Milano (Mathematics)
Y. Ventikos, University College London (Bioengineering)
E. Nagelhus, University of Oslo (Physiology)
P. K. Eide/K. E. Emblem, Oslo University Hospital (Neurosurgery, Neuroradiology)
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Societal impact: bridging brain ﬂuid dynamics and electrophysiology to
create a new avenue for investigating neurological disease (and more!)
vt + v · v − ν∆v + p = f
· v = 0
vt − · M v = −Iion
st = F(v, s)
utt − · σ(u) + p = f
· ut − · κ p = g23 / 55

Act 2: A zoo of on-going studies
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Iliff/Nedergaard groups argue that CSF flows rapidly through paravascular
spaces and drives a convective ISF flow through the extracellular space
[Iliff et al., Sci Trans Med, 2012, ...]
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Convective ﬂow through extracellular space seem unlikely as driver of
interstitial waste clearance
[Holter et al, PNAS, 2017]
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In the context of cerebral fluid flow and transport, the brain can be viewed
as a multi-porous and elastic medium
A small region of rat cerebral cortex with
black areas representing extracellular space
(Scale bar: ≈ 1µm)
[Nicolson, 2001]
.
The brain parenchyma includes multiple fluid networks
(extracellular spaces, arteries, capillaries, veins, paravascular
spaces)
[Zlokovic, 2011]
. 29 / 55

Multiple-network poroelastic theory (MPET) is a macroscopic model for
poroelastic media with multiple ﬂuid networks
[Bai, Elsworth, Roegiers (1993); Tully and Ventikos (2011)]
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The MPET equations describe displacement and A-network pressures under
balance of momentum and mass
Given A ∈ N, f and ga, ﬁnd the displacement u(x, t) and the pressures pa = pa(x, t)
for a = 1, . . . , A such that
− div(σ∗
− a αapa I) = f,
ca ˙pa + αa div ˙u − div Ka grad pa + Sa = ga, a = 1, . . . , A,
Transfer between networks:
Sa = b sa←b = b ξa←b(pa − pb),
Linearly elastic (isotropic) tissue matrix/pseudo-stress:
σ∗
(u) = 2µε(u) + λ div u I,
A = 1 corresponds to Biot’s equations, A = 2 Barenblatt-Biot. λ → ∞, ca → 0 and
K → 0 interesting regimes.
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Finite elements is a numerical method for solving (partial) diﬀerential
equations dating back to the 1940s
Original problem
Find u ∈ V , a function of space and/or
time, that solves
( u, v) = (f, v)
for all v ∈ V .
Finite element approximation
Find uh ∈ Vh that solves
( uh, v) = (f, v)
for all vh ∈ Vh.
[Courant, Feng, Zienkiewicz, Ciarlet, many others...]
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Finite element methods still active research topic after 60-80 years
Consider a stove top, heated by some heat source and with the temperature ﬁxed on its
boundary, how does the temperature distribute on the top?
( u, v) = (1, v) u, v ∈ H1
0 (Ω)
Wrong Right
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Standard finite element approximations of poroelasticity equations typically
suffer from different types of numerical instabilities
[Haga et al, 2012]
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In 1991, the Sleipner A platform unexpectedly sank during towing, resulting
in catastrophic losses
[Engineers Australia Pty Limited]
The reasons for the weaknesses [...] were recognized to be: Unfavorable
geometrical shaping of some ﬁnite elements in the global analysis. [...] this
lead to an underestimation of the shear forces [...] by some 45%.
[Jakobsen and Rosendahl, The Sleipner Platform Accident, 1994]
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Standard ﬁnite element formulation suﬀers from Poisson locking (loss of
convergence) in the incompressible limit λ → ∞
u − uh L∞(0,T,L2) Rate u − uh L∞(0,T,H1) Rate
h 0.169 2.066
h/2 0.040 2.09 0.980 1.08
h/4 0.010 2.04 0.480 1.03
h/8 0.002 2.03 0.235 1.03
h/16 0.001 2.09 0.110 1.10
Optimal 3 2
Test case A: Smooth manufactured solution test case inspired by [Yi, 2017 [Section 7.1]] with
div u → 0 as λ → ∞, but linear in time, A = 2, T = 0.5, µ = 0.33, λ = 1.67 × 104
, S = 0, all
other parameters 1, Crank-Nicolson discretization in time. Standard Taylor-Hood
(uh, ph) ∈ Pd
2 × PA
1 discretization in space:
(σ∗
(u), grad v) − a(αapa, div v) = (f, v) ∀ v ∈ Vh
(ca ˙pa + αa div ˙u + Sa, qa) + (Ka grad pa, grad qa) = (ga, qa) ∀ qa ∈ Qa,h
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Introducing a total-pressure-augmented variational formulation of the
MPET equations
Key idea: inspired by [Lee, Mardal, Winther, 2017]: introduce the total pressure
p0 = λ div u − A
a=1 αapa ↔ div u = λ−1
α · p
with α = (1, α1, . . . , αA), p = (p0, . . . , pA).
Variational formulation of total-pressure augmented quasi-static MPET
Given ga for a = 1, . . . , A, ﬁnd u ∈ H1
((0, T]; V ) and pa ∈ H1
((0, T]; Qa) for
a = 0, . . . , A such that
(2µε(u), ε(v)) + (p0, div v) = 0 ∀ v ∈ V
(div u, q0) − (λ−1
α · p, q0) = 0 ∀ q0 ∈ Q0
(ca ˙pa + αaλ−1
α · ˙p + Sa, qa) + (Ka grad pa, grad qa) = (ga, qa) ∀ qa ∈ Qa, a = 1, . . . , A
[Lee, Piersanti, Mardal, R., in prep., 2018]
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The total-pressure augmented formulation restores optimal convergence,
including in the incompressible limit and zero saturation coeﬃcient
u − uh L∞(0,T,L2) Rate u − uh L∞(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
p1 − p1,h L∞(0,T,L2) Rate p1 − p1,h L∞(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
Test case inspired
by [Yi, 2017]
with moderately
high λ and
ca = 0 with
total-pressure
augmented
Taylor-Hood
Pd
2 × PA+1
1
discretization.
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Investigating paravascular fluid flow driven by subarachnoid space arterial
pressure variations in the murine brain
Pressure pulsation on outer boundary: σ · n = psas · n with psas = 0.008 × 133 sin(2πt) (Pa)
Semi-permeable outer and inner membranes: −K grad p · n = β(p − pin).
Displacement Paravascular fluid pressure Paravascular fluid velocity
[Simulations courtesy of M. Croci; Mesh courtesy of Allen Brain Atlas, Alexandra Diem and Janis Grobovs]
[Vinje, Croci, Mardal, R., in prep., 2018]
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How will a highly uncertain permeability field affect peak paravascular flow?
(u, p) quasi-static MPET solutions, v = −Kφ−1
grad p.
Assume that K is a stochastic field:
K(x, (α, β)) = K0eu(x,α)
10v(β)
, x ∈ Ω
u(x, ·) ∼ N(µ, σ2
) | E[eu(x,·)
] = 1, V[eu(x,·)
]1/2
= 0.2
v(·) ∼ U(−2, 2),
Output functional: F(α, β) = C max
t∈(0,T] ROI
|v(t)| dx (α, β)
Preliminary estimates: P(F ∈ [0, 8 × 10−5
] mm/s) ≈ 0.95
E[F] ≈ 2.5 × 10−5
mm/s
V[F] ≈ (3.2 × 10−5
mm/s)2
Techniques: MLMC, white noise on non-nested mesh hierarchies. [Croci, R., Farrell, Giles, in prep., 2018]
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Interaction between paravascular and tissue fluid flow is a key mechanism
Multiple hypotheses involve convective tissue fluid flow and transport in cerebral
paravascular spaces (sheaths surrounding blood vessels) interacting with water
movement in the intra/extracellular spaces.
[Iliff et al, 2012; Morris et al, 2014]
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Next steps
Derive and study 3D-1D type models describing
potential interactions between the vasculature,
paravasculature and interstitium aiming to answer
physiological questions relating to ﬂow directionality,
rate and pathologies.
[Iliﬀ et al., 2012]
[D’Angelo and Quarteroni, 2008]
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The paradigm in computational cardiac modelling has been classical
homogenized models (bidomain, monodomain)
vt − div(Mi grad v + Mi grad ue) = −Iion(v, s)
div(Mi grad v + (Mi + Me) grad ue) = 0
st = F(v, s)
Heart cell: h ≈ 100µm
FEM cell: h ≤ 100µm (!)
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[The Guardian, Feb 2 2014]
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The emerging EMI framework use a geometrically explicit representation of
the cellular domains
Find the intracellular potential ui in Ωi = ∪kΩk
i ⊂ Rd
, the extracellular potential ue in
Ωe ⊂ Rd
and the membrane potential v (and other membrane variables) on
Γ = Γ1 ∪ Γ2 ∪ Γ1,2.
Ω1
i Ω2
i
Ωe
Γ1 Γ2
Γ1,2
[Stinstra et al, 2010; Agudelo-Toro and Neef, 2013]
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The EMI equations describe the dynamics of extracellular, membrane and
intracellular electrical potentials
Find the electric potential u(t) : Ω → R, and the transmembrane potential
v(t) : Γ → R s.t:
div σ grad u = f in Ωi, Ωe
Cmvt + σ grad u · n|Ωi + Iion(v) = 0 on Γ,
with interface conditions on Γ
u = v, (v is the jump in u)
σ grad u · n = 0 (continuity of ﬂux/current)
Boundary condition u = 0 on ∂Ω and initial conditions for v.
For now, Iion(v) = v. (Generally Iion = I(v, s) highly non-linear.)
[Stinstra et al, 2010; Agudelo-Toro and Neef, 2013; Tveito et al, 2017]
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Next steps
Derive and study EMI-type models to
answer physiological questions related to
the role of astrocytes and AQP4 (water
channels in cell membranes) in brain water
homeostasis and ﬂow mechanisms.
[Zlokovic, 2011; Nagelhus and Ottersen, 2013]
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Collaborators
Matteo Croci (University of Oxford/Simula)
Cécile Daversin-Catty (Simula)
Ada Ellingsrud (Simula)
Per-Kristian Eide (Oslo University Hospital)
Patrick E. Farrell (University of Oxford)
Mike Giles (University of Oxford)
Janis Grobovs (University of Oslo)
Karoline Horgmo Jæger (Simula)
Miroslav Kuchta (University of Oslo)
Jeonghun Lee (ICES/UT Austin)
Kent-Andre Mardal (University of Oslo)
Eleonora Piersanti (Simula)
Geir Ringstad (Oslo University Hospital)
Travis Thompson (Simula)
Aslak Tveito (Simula)
Vegard Vinje (Simula)
Core message
New models and methods required for
insight into the brain’s waterscape – which
is a beautiful setting for mathematics and
numerics!
Thank you!
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The brain's waterscape/Hjernens vannveier

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