Invited presentation by Marie E. Rognes at the IEEE Brain Discovery Workshop on Brain, Mind and Body at University of California San Diego, Nov 2022.

1. Such stuff as dreams are made on: a computational tale
of optimal transport and brain clearance
Marie E. Rognes
Chief Research Scientist
Simula Research Laboratory
Oslo, Norway
Fulbright Visiting Scholar
Institute of Engineering in Medicine
University of California San Diego
IEEE Brain Discovery Workshop on Brain, Mind, and Body: Cognitive Neuroengineering for Health and Wellness
Nov 10 2022
1 / 24

2. We are such stuff
As dreams are made on
[Act 4, The Tempest, W. Shakespeare]
2 / 24

3. MRI reveals human brain-wide tracer enhancement and clearance
[Ringstad et al (2017), Ringstad et al (2018)]
3 / 24

4. Subject-specific finite element models of tracer transport in the brain
Find c = c(x, t) for x ∈ Ω, t > 0:
ct − div D grad c = 0 in Ω
c = C(I) on ∂Ω
Continuous linear finite elements
1 − 5M cells: h ≈ 0.2 − 5.6mm
Implicit in time, τ ≈ 0.5h
[Vinje, Zapf, Ringstad, Eide, R., Mardal (FBCNS, 2023)]
C(I)
Ωgray ∪ Ωwhite
tr(D)
N = 11 + 7
4 / 24

5. Transport by diffusion alone underestimates brain influx and efflux
Z
Ωgray
c(x, t) dx
Z
Ωwhite
c(x, t) dx
0 10 20 30 40 50
time (hours)
0.00
0.02
0.04
0.06
0.08
0.10
tracer
(mmol)
night night
data & simulation (white)
data & simulation (gray)
ct − div D grad c = 0
[Vinje et al (2023), ]
5 / 24

6. Transport by diffusion alone underestimates brain influx and efflux
Z
Ωgray
c(x, t) dx
Z
Ωwhite
c(x, t) dx
0 10 20 30 40 50
time (hours)
0.00
0.02
0.04
0.06
0.08
0.10
tracer
(mmol)
night night
data & simulation (white)
data & simulation (gray)
ct − div D grad c = 0
0 10 20 30 40 50
time (hours)
0.00
0.02
0.04
0.06
0.08
0.10
tracer
(mmol)
night night
1 2 3 4 5
ct − div αD grad c = 0
[Vinje et al (2023), Iliff et al (2012), Hladky and Barrand (2014), Smith et al (2017), Croci et al (2019), Troyetsky et al (2021), Ray et al (2021)]
5 / 24

7. Is there a quantifiable (bulk) flow of interstitial fluid in brain tissue?
[Helen Cserr (credit: R. Cserr)]
[Kwong et al (Fig 1, 2020)]
[Cserr et al (1986), Hladky and Barrand (2014), Smith et al (2017)]
[Abbott et al (2018), Hablitz and Nedergaard (2021), ...]
6 / 24

8. Is there a quantifiable (bulk) flow of interstitial fluid in brain tissue?
[Helen Cserr (credit: R. Cserr)]
[Kwong et al (Fig 1, 2020)]
[Cserr et al (1986), Hladky and Barrand (2014), Smith et al (2017)]
[Abbott et al (2018), Hablitz and Nedergaard (2021), ...]
6 / 24

9. Is there a quantifiable (bulk) flow of interstitial fluid in brain tissue?
[Helen Cserr (credit: R. Cserr)]
[Kwong et al (Fig 1, 2020)]
[Cserr et al (1986), Hladky and Barrand (2014), Smith et al (2017)]
[Abbott et al (2018), Hablitz and Nedergaard (2021), ...]
With a velocity field φ : Ω → Rd
, find
c : Ω × [0, T] → R:
ct + div(cφ) − div D grad c = 0 in Ω,
c = C(I) on ∂Ω
A few observations C(I) on Ω available
24h 48 h
6 / 24

10. Determining a fluid flow field from images via optical flow methods
c1 at t1
c2 at t2
τ = t2 − t1
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11. Determining a fluid flow field from images via optical flow methods
c1 at t1
c2 at t2
τ = t2 − t1
Given c1, c2 : Ω → R at t1, t2, α > 0, and
f = ct + φ · grad c,
R = k grad φkL2(Ω),
find an optimal φ ∈ H1
(Ω; Rd
):
min
φ
Z t2
t1
Z
Ω
f2
dx dt + α2
R2
,
e.g. via the Euler-Lagrange equations.
[Horn and Schunck (1981), Wildes et al (2000)]
7 / 24

12. Determining a fluid flow field from images via optical flow methods
c1 at t1
c2 at t2
τ = t2 − t1
Given c1, c2 : Ω → R at t1, t2, α > 0, and
f = ct + φ · grad c,
R = k grad φkL2(Ω),
find an optimal φ ∈ H1
(Ω; Rd
):
min
φ
Z t2
t1
Z
Ω
f2
dx dt + α2
R2
,
e.g. via the Euler-Lagrange equations.
[Horn and Schunck (1981), Wildes et al (2000)]
Monge-Kantorovich mass transfer:
R = kcφ · φkL1([t1,t2])×L1(Ω)
[..., Benamou and Bernier (2000), Mueller et al (2013), ...]
7 / 24

13. Determining a fluid flow field from images via optical flow methods
c1 at t1
c2 at t2
τ = t2 − t1
Given c1, c2 : Ω → R at t1, t2, α > 0, and
f = ct + φ · grad c,
R = k grad φkL2(Ω),
find an optimal φ ∈ H1
(Ω; Rd
):
min
φ
Z t2
t1
Z
Ω
f2
dx dt + α2
R2
,
e.g. via the Euler-Lagrange equations.
[Horn and Schunck (1981), Wildes et al (2000)]
Monge-Kantorovich mass transfer:
R = kcφ · φkL1([t1,t2])×L1(Ω)
[..., Benamou and Bernier (2000), Mueller et al (2013), ...]
7 / 24

14. Determining a fluid flow field from images via optical flow methods
c1 at t1
c2 at t2
τ = t2 − t1
Given c1, c2 : Ω → R at t1, t2, α > 0, and
f = ct + φ · grad c,
R = k grad φkL2(Ω),
find an optimal φ ∈ H1
(Ω; Rd
):
min
φ
Z t2
t1
Z
Ω
f2
dx dt + α2
R2
,
e.g. via the Euler-Lagrange equations.
[Horn and Schunck (1981), Wildes et al (2000)]
Monge-Kantorovich mass transfer:
R = kcφ · φkL1([t1,t2])×L1(Ω)
[..., Benamou and Bernier (2000), Mueller et al (2013), ...]
But not a viable approach here!
Sensitive to regularization α
Discrete derivatives of non-smooth
data (∇hc1, ∆hc1)
Optical flow ignores diffusion
7 / 24

15. Recovering approximations of fluid flow fields accounting for diffusion
c1 at t1
c2 at t2
τ = t2 − t1
Optimal Convection-Diffusion (OCD)
Given c1, c2 : Ω → R at t1, t2, α 0,
optimize
min
c,φ
kc(t2) − c2k2
L2(Ω) + α2
R2
,
subject to c(t1) = c1 and for t ∈ (t1, t2]:
ct + div(cφ) − div D grad c = 0. (2)
[∼ Andreev et al (2015), Glowinski et al (2022)]
Let ct ≈ δτ c ≡ 1
τ
(c − c1)
[Alnæs et al (2015), Farrell et al (2013), Zapf et al, unpublished (2022)]
8 / 24

16. Recovering approximations of fluid flow fields accounting for diffusion
c1 at t1
c2 at t2
τ = t2 − t1
Optimal Convection-Diffusion (OCD)
Given c1, c2 : Ω → R at t1, t2, α 0,
optimize
min
c,φ
kc(t2) − c2k2
L2(Ω) + α2
R2
,
subject to c(t1) = c1 and for t ∈ (t1, t2]:
ct + div(cφ) − div D grad c = 0. (2)
[∼ Andreev et al (2015), Glowinski et al (2022)]
Let ct ≈ δτ c ≡ 1
τ
(c − c1)
Iterative optimization approach
Define the reduced functional
J : H(div, Ω) → R
J(φ) = kc(φ)(t2) − c2k2
L2(Ω) + α2
R2
where φ 7→ c(φ) by c solving (2).
Solve
min
φ
J(φ)
via e.g. scipy (L-BFGS), FEniCS,
Dolfin-adjoint.
[Alnæs et al (2015), Farrell et al (2013), Zapf et al, unpublished (2022)]
8 / 24

17. OCD approach yields robust approximation of average velocities
Average speed φ̄ = 1
|Ω|
Z
Ω
|φ| dx (µm/min):
n vs. α2
10−3
10−4
10−5
16 2.83 3.67 4.00
32 2.33 2.83 2.83
[Zapf et al, unpublished (2022)]
9 / 24

18. [Eide, Vinje et al (2021)]
10 / 24

19. [Eide, Vinje et al (2021)]
10 / 24

20. Estimate φ between 24h and 48h images for each patient.
Average flow speed φ̄ for α2
= 10−3
(dots), 10−4
(stars), 10−5
(circles) for low-res (light blue) and standard meshes (blue).
11 / 24

21. Convective velocities are reduced (by x2) after sleep-deprivation
Estimate φ between 24h and 48h images for each patient.
Average flow speed φ̄ for α2
= 10−3
(dots), 10−4
(stars), 10−5
(circles) for low-res (light blue) and standard meshes (blue).
Brain-wide Gray White Stem
0
1
2
3
4
5
6
Flow
speed
(
m/min)
N = 11 + 7
4.23 vs. 2.11 µm/min (p=0.0057)
11 / 24

22. This cold night will turn us all
to fools and madmen
[Act 3, King Lear, W. Shakespeare]
12 / 24

23. [Beta-amyloid plaques and tau in the brain, National Institute of Health]
13 / 24

24. Protein propagation by diffusion and clearance in the connectome
[Tarasoff-Conway et al (2015), Pereira et al (2018), Fornari et al (2019), Thompson et al (2020), Brennan et al (2022), ...]
14 / 24

25. Protein propagation by diffusion and clearance in the connectome
Given a connectome (graph) (V, E), find the toxic protein
concentration pi = pi(t) for i ∈ V , t 0 such that
∂
∂t pi + ρ
P
j∈V Lijpj = 0
pi(0) = p0
i
Anisotropic diffusion via the diffusion coefficient ρ, graph Laplacian
graph Laplacian Lij = Dij − Aij, weighted adjacency matrix A and
degree matrix D.
[Tarasoff-Conway et al (2015), Pereira et al (2018), Fornari et al (2019), Thompson et al (2020), Brennan et al (2022), ...]
14 / 24

26. Protein propagation by diffusion and clearance in the connectome
Given a connectome (graph) (V, E), find the toxic (tau) protein
concentration pi = pi(t) and clearance λi = λi(t) for i ∈ V ,
t 0 such that
∂
∂t pi + ρ
P
j∈V Lijpj = (λcrit − λi)pi − αp2
i
∂
∂t λi = −βipi(λi − λ∞
i )
pi(0) = p0
i , λi(0) = λ0
i
Anisotropic diffusion via the diffusion coefficient ρ, graph Laplacian
graph Laplacian Lij = Dij − Aij, weighted adjacency matrix A and
degree matrix D.
Clearance via kinetic constant βi 0, minimal clearance λ∞
i and
critical clearance λcrit.
[Tarasoff-Conway et al (2015), Pereira et al (2018), Fornari et al (2019), Thompson et al (2020), Brennan et al (2022), ...]
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27. Clearance level below critical clearance sets off the toxic cascade
Single node dynamical system
∂
∂t p = (λcrit − λ)p − αp2
∂
∂t λ = −βp(λ − λ∞
)
Fixed points
Healthy state(s) (stable iff λ λcrit)
(p, λ) = (0, λ), λ∞
≤ λ ≤ λ0
Unhealthy state (unconditionally stable)
(p, λ) = (α−1
(λcrit − λ∞
), λ∞
)
[Brennan et al (2022)]
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28. Clearance level below critical clearance sets off the toxic cascade
Single node dynamical system
∂
∂t p = (λcrit − λ)p − αp2
∂
∂t λ = −βp(λ − λ∞
)
Fixed points
Healthy state(s) (stable iff λ λcrit)
(p, λ) = (0, λ), λ∞
≤ λ ≤ λ0
Unhealthy state (unconditionally stable)
(p, λ) = (α−1
(λcrit − λ∞
), λ∞
)
[Brennan et al (2022)]
Critical clearance defines critical toxic seed pS
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29. Clearance level below critical clearance sets off the toxic cascade
Single node dynamical system
∂
∂t p = (λcrit − λ)p − αp2
∂
∂t λ = −βp(λ − λ∞
)
Fixed points
Healthy state(s) (stable iff λ λcrit)
(p, λ) = (0, λ), λ∞
≤ λ ≤ λ0
Unhealthy state (unconditionally stable)
(p, λ) = (α−1
(λcrit − λ∞
), λ∞
)
[Brennan et al (2022)]
Critical clearance defines critical toxic seed pS
Network connectivity increases toxic resilience
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30. Dynamic clearance landscapes alter disease progression
Higher clearance (λ0
EC = λcrit, λ0
i = λ0
)
delays spread
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31. Dynamic clearance landscapes alter disease progression
Higher clearance (λ0
EC = λcrit, λ0
i = λ0
)
delays spread
16 / 24

32. Dynamic clearance landscapes alter disease progression
Higher clearance (λ0
EC = λcrit, λ0
i = λ0
)
delays spread
Heterogeneous clearance alters spreading
patterns
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33. ¥÷*
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We are such stuff
As dreams are made on,
and our little life
Is rounded with a sleep
[Act 4, The Tempest, W. Shakespeare]
17 / 24

34. Solute transport Brain mechanics CSF flow Neurodegeneration
Ions and osmosis Model reduction Optimal control Software
18 / 24

35. 19 / 24

36. Collaborators
Georgia Brennan (Oxford)
Matteo Croci (UT Austin)
Per Kristian Eide (Oslo University Hospital)
Patrick Farrell (Oxford)
Alain Goriely (Oxford)
Martin Hornkjøl (Oslo/Simula)
Hadrien Oliveri (Oxford)
Kent-Andre Mardal (Oslo/Simula)
Geir Ringstad (Oslo University Hospital)
Travis B. Thompson (Oxford)
Lars Magnus Valnes (Oslo University Hospital)
Vegard Vinje (Simula)
Bastian Zapf (Oslo)
... a special thanks to Johannes Ring (Simula)
... and many 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 grants #250731 (Waterscape)
and #324239 (EMIx), and by the EPSRC Centre For Doctoral Training in Industrially
Focused 706 Mathematical Modelling (EP/L015803/1).
20 / 24

37. Sleep affects CSF outflux and tracer availability at brain surface
...the spread of tracer to the PVS of the brain surface was directly related to the amount
of tracer remaining at the basal cisterns and indirectly proportional to the amount that had
reached the systemic blood.
[Ma et al, Acta Neuropathologica, 2019 (p. 159, Fig 1)]
21 / 24

38. Increased CSF production (as when awake) limits brain tracer influx
[Hornkjøl et la, CSF circulation and dispersion yield rapid clearance from intracranial compartments, Front. Bioeng. Biotechnol, 2022]
Given CSF production g, find the CSF
velocity u and pressure p in ΩF
µ∇2
u − ∇p = 0,
∇ · u = g,
with restricted outflux at ∂Ωout
µ∇u · n − pn = −R0u · n,
and the concentration c in ΩF ∪ ΩB
φ
∂c
∂t
+ φu · ∇c − ∇ · (φαD∇c) = 0.
6 h 24 h
22 / 24

39. Increased CSF production (as when awake) limits brain tracer influx
[Hornkjøl et la, CSF circulation and dispersion yield rapid clearance from intracranial compartments, Front. Bioeng. Biotechnol, 2022]
Given CSF production g, find the CSF
velocity u and pressure p in ΩF
µ∇2
u − ∇p = 0,
∇ · u = g,
with restricted outflux at ∂Ωout
µ∇u · n − pn = −R0u · n,
and the concentration c in ΩF ∪ ΩB
φ
∂c
∂t
+ φu · ∇c − ∇ · (φαD∇c) = 0.
6 h 24 h
22 / 24