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Such stuff as dreams are made on: a computational tale of optimal transport and brain clearance

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.

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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
We are such stuff
As dreams are made on
[Act 4, The Tempest, W. Shakespeare]
2 / 24
MRI reveals human brain-wide tracer enhancement and clearance
[Ringstad et al (2017), Ringstad et al (2018)]
3 / 24
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
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
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

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Such stuff as dreams are made on: a computational tale of optimal transport and brain clearance

  • 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 7 / 24
  • 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), ...] 14 / 24
  • 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)] 15 / 24
  • 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 15 / 24
  • 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 15 / 24
  • 30. Dynamic clearance landscapes alter disease progression Higher clearance (λ0 EC = λcrit, λ0 i = λ0 ) delays spread 16 / 24
  • 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 16 / 24
  • 33. ¥÷* ¥ ¥¥. ÷ ¥¥÷÷÷÷ ☒ ÷ ÷ : ÷ É ÷ ¥ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ 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
  • 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