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The brain's waterscape/Hjernens vannveier

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Hjernens vannveier. Introduction for a general, technical audience to key aspects of the brain's waterscape and the Waterscales and Waterscape research projects. Talk by Marie E. Rognes at Norges Tekniske Vitenskapsakademi (Norwegian Technical Academy of Science) in Bergen, Norway on January 30 2018

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The brain's waterscape/Hjernens vannveier

  1. 1. Hjernens vannveier Marie E. Rognes Simula Research Laboratory NTVA, Bergen, Norway Jan 30 2018 1 / 55
  2. 2. Act 1: Introduction to the brain’s numerical waterscape 2 / 55
  3. 3. 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. 3 / 55
  4. 4. 4 / 55
  5. 5. 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
  6. 6. 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]] 6 / 55
  7. 7. 7 / 55
  8. 8. 8 / 55
  9. 9. Tracer experiments in mice indicate an increase in extracellular space and more rapid tracer transport during sleep [Xie et al., Science, 2013] 9 / 55
  10. 10. 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] 10 / 55
  11. 11. 11 / 55
  12. 12. Cerebrospinal fluid (CSF) circulates in the spaces surrounding the brain (subarachnoid, ventricles, aqueduct) [Wikimedia Commons] 12 / 55
  13. 13. Perivascular spaces and glial cells are key structures at the meso/microscale [Zlokovic, 2011] 13 / 55
  14. 14. 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; ...] 14 / 55
  15. 15. 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, ...] 15 / 55
  16. 16. 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, ...] 16 / 55
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  19. 19. 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] 19 / 55
  20. 20. 20 / 55
  21. 21. 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 21 / 55
  22. 22. 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 21 / 55
  23. 23. 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 21 / 55
  24. 24. 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 21 / 55
  25. 25. 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) 22 / 55
  26. 26. 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
  27. 27. 24 / 55
  28. 28. Act 2: A zoo of on-going studies 25 / 55
  29. 29. 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, ...] 26 / 55
  30. 30. Convective flow through extracellular space seem unlikely as driver of interstitial waste clearance [Holter et al, PNAS, 2017] 27 / 55
  31. 31. 28 / 55
  32. 32. 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
  33. 33. Multiple-network poroelastic theory (MPET) is a macroscopic model for poroelastic media with multiple fluid networks [Bai, Elsworth, Roegiers (1993); Tully and Ventikos (2011)] 30 / 55
  34. 34. 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. 31 / 55
  35. 35. 32 / 55
  36. 36. 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...] 33 / 55
  37. 37. 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 34 / 55
  38. 38. Standard finite element approximations of poroelasticity equations typically suffer from different types of numerical instabilities [Haga et al, 2012] 35 / 55
  39. 39. 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] 36 / 55
  40. 40. 37 / 55
  41. 41. 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 38 / 55
  42. 42. 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] 39 / 55
  43. 43. 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. 40 / 55
  44. 44. 41 / 55
  45. 45. 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] 42 / 55
  46. 46. 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] 43 / 55
  47. 47. 44 / 55
  48. 48. 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] 45 / 55
  49. 49. 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] 46 / 55
  50. 50. 47 / 55
  51. 51. 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 (!) 48 / 55
  52. 52. [The Guardian, Feb 2 2014] 49 / 55
  53. 53. 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] 50 / 55
  54. 54. 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] 51 / 55
  55. 55. 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] 52 / 55
  56. 56. 53 / 55
  57. 57. 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! 54 / 55

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