A Model of the Blood Flow in the Retina of the Eye

Transcription

1 A Model of the Blood Flow in the Retina of the Eye Edmond Rusjan with Andrea Dziubek, Giovanna Guidoboni, Alon Harris, Anil Hirani, William Thistleton SUNY Poly, IUPUI, University of Illinois at Urbana-Champaign Vanderbilt, April 14, 2017

2 Abstract: Understanding the blood flow in the retina of the eye may provide insights into several eye pathologies and ultimately lead to better treatments. We model the flow as a hierarchical Darcy flow on a curved surface and solve the model numerically using discrete exterior calculus and finite element exterior calculus. Results support the hypothesis that changes in the shape of the retina cause significant changes in the ocular blood flow, which may play a role in the dynamics of open angle glaucoma.

3 Outline Modeling the Eye: Goals and Motivation A Multiscale Model of Blood Flow in the Retina Hierarchical Darcy Flow Vascular Tree Architecture Computational Method Hierarchical Discretization Spatial Discretization Curved Surfaces Discrete Exterior Calculus (DEC) and Finite Element Exterior Calculus (FEEC) Results Ocular Curvature May Play a Role in Glaucoma Summary and Future Plans Modeling the Eye: Goals and Motivation 3

4 The Eye The eye is the organ of sight: the sensory part is the retina. Modeling the Eye: Goals and Motivation 4

5 Goals Advance the theory: most accurate models become theory Help reason: simplified models help us think more clearly Simulate experiments: experiments may be hard, or even impossible Competing goals: tension between simplicity and accuracy Solution: a collection of models at various levels of refinement Modeling the Eye: Goals and Motivation 5

6 Motivation Preserve the vision Window into the body Part of the brain Opportunity to study flow on curved surfaces Modeling the Eye: Goals and Motivation 6

7 Outline Modeling the Eye: Goals and Motivation A Multiscale Model of Blood Flow in the Retina Hierarchical Darcy Flow Vascular Tree Architecture Computational Method Hierarchical Discretization Spatial Discretization Curved Surfaces Discrete Exterior Calculus (DEC) and Finite Element Exterior Calculus (FEEC) Results Ocular Curvature May Play a Role in Glaucoma Summary and Future Plans A Multiscale Model of Blood Flow in the Retina 7

8 Fundus Images A healthy right eye (left) and a healthy left eye (right) A Multiscale Model of Blood Flow in the Retina 8

9 Regions of Particular Clinical Interest in the Retina optic nerve head (ONH) inferior quadrant (I) superior quadrant (S) nasal quadrant (N) temporal quadrant (T) fovea i At the level of capillaries and small arterioles and venules, we don t see the details of the vascular tree architecture The tissue appears to be a porous medium Think of the vascular tree as a hierarchical structure Parameterize these hierarchical levels by a continuous parameter ϑ A Multiscale Model of Blood Flow in the Retina 9

10 Hierarchical Darcy Flow Consider averaged (smeared) quantities at position x in some spatial domain Ω for each hierarchy level ϑ Velocity v(x, ϑ) Hierarchical velocity ω(x, ϑ) Pressure p(x, ϑ) Model each ϑ level by Darcy flow equations and couple vertically with hierarchical flow (n b v) + ϑ (n bω) = 0 n b v = K p n b ω = α p ϑ x Ω, ϑ (0,1) x Ω, ϑ (0,1) x Ω, ϑ (0,1) A Multiscale Model of Blood Flow in the Retina 10

11 Parameters and Boundary Conditions Parameters Tissue porosity n b (x, ϑ) Tissue perfusion n b ω Tissue permeability tensor K(x, ϑ) Hierarchical permeability α(x, ϑ) Hydraulic conductivities G v, G a Venous and arterial pressures p v(x) and p a(x) Boundary conditions n b ω(x, 0) = G v (p(x, 0) p v (x)) n b ω(x, 1) = G a (p a (x) p(x, 1)) n b v(x, ϑ) n = 0 x Ω x Ω x Ω, ϑ [0,1] Couple to feeding arteries and draining veins G v = α v δ v (x) and G a = α a δ a (x) α v and α a are venous and arterial conductances A Multiscale Model of Blood Flow in the Retina 11

12 Vascular Tree Architecture δ v (x) and δ a (x) are delta functions which differ from zero only where the feeding arteries and draining veins are located, respectively Need to extract feeding arteries and draining veins from fundus images Alternatively, model the vascular tree Young (1808): Compare parent and daughters vessels Assume two identical daughters r Radius ratio: p r d = = 1.26 A Area ratio: p A d = Murray (1926): Generalize to asymmetric trees Similar to Pythagora, except cubes: r 3 p = r 3 d Derives from optimality (variational principle) Forgotten for many years, then rediscovered A Multiscale Model of Blood Flow in the Retina 12

13 Optimal Transport = Murray s Law P f power needed to maintain the blood flow (viscous losses) P m power needed to metabolically maintain the blood and the vessel P t = P f + P m total power Minimize P t! Assume cylindrical vessels: f = cp f is volumetric flow rate p is pressure difference c is conductance coefficient Assume Poiseuille flow: c = πr4 8µl µ is blood viscosity l is vessel length Min P f = pf = af 2 r 4 = r? a = 8µl π Min P m = mv = mπr 2 l = br 2 = r 0? m is a metabolic coefficient b = mπl Min P t = dpt dr = 0 = 4af 2 r 5 + 2br = 0 = f = kr 3! k = ( b 2a )1/2 A Multiscale Model of Blood Flow in the Retina 13

14 Recent Experimental Data and Models Takahashi (2009, 2013): f = kr m Theory (Murray): m = 3 Data (Takahashi 2009): m 2.85 Data (Takahashi 2013): significant deviations in m between various organs In addition to radius, interested in length and angle Theory: open problem! Data: rather scarce - open problem! Model (Takahashi 2009): L = 7.4r 1.15 Open question: what optimality? A Multiscale Model of Blood Flow in the Retina 14

15 Outline Modeling the Eye: Goals and Motivation A Multiscale Model of Blood Flow in the Retina Hierarchical Darcy Flow Vascular Tree Architecture Computational Method Hierarchical Discretization Spatial Discretization Curved Surfaces Discrete Exterior Calculus (DEC) and Finite Element Exterior Calculus (FEEC) Results Ocular Curvature May Play a Role in Glaucoma Summary and Future Plans Computational Method 15

16 Hierarchical Discretization Standard FE procedure Rewrite in weak form Introduce piecewise linear basis functions {ϕ i } with i = 0,.., n on [0, 1] corresponding to nodes {θ i } and we let ϕ = ϕ i. We assume permeabilities and pressure are piecewise linear interpolations of the hierarchical variable n n K(x, θ) = K(x, θ k )ϕ k (θ), α(x, θ) = α(x, θ k )ϕ k (θ), n j=0 Ω k=0 p(x, θ) = k=0 n p(x, θ k )ϕ k (θ) k=0 ( ) K ij (x) p j (x), q(x) dx n j=0 Ω α ij (x) p j (x) q(x) dx Computational Method = f i (x) q(x) dx 16 Ω

17 Hierarchical Discretization Cont. where K ij (x) = α ij (x) = n K(x, θ k ) ϕ k (θ) ϕ i (θ) ϕ j (θ) dθ [0,1] n α(x, θ k ) ϕ k (θ) ϕ i(θ) ϕ j (θ) dθ (1) [0,1] θ θ k=0 k=0 f i (x) = α v p v δ i0 + α a p a δ in The hierarchical discretization leads to a tridiagonal system for the pressures p i (x, θ i ). Computational Method 17

18 Example: 3 layer model For three levels i = 0, 1, 2 and with isotropic and space independent permeability K ij = K ij I per level we have K ij = K ij This yields K 00 + α 00 K 01 + α 01 K 10 + α 10 K 11 + α 11 K 12 + α 12 p 0 p 1 = K 21 + α 21 K 22 + α 22 p 2 Permeability matrix K ij, conductivity matrix α ij, and right hand side vector f i as given in equation (1). f 0 f 1 f 2. (2) Computational Method 18

19 The K and α matrices in the 3 layer model The scalar K ij is the ij entry of the matrix 3K K 1 K 0 + K 1 0 K 0 + K 1 K 0 + 6K 1 + K 2 K 1 + K 2, 24 0 K 1 + K 2 K 1 + 3K 2 The scalar α ij is the ij entry of the matrix α 0 + α 1 + G v (α 0 + α 1 ) 0 (α 0 + α 1 ) α 0 + 2α 1 + α 2 (α 1 + α 2 ), 0 (α 1 + α 2 ) α 1 + α 2 + G a The scalar f i is the i component of the vector [ Gv p v, 0, G a p a ] T. Here α 0, α 1, α 2 and K 0, K 1, K 2 are the conductivities and permeabilities at the three hierarchical levels and are given in equation (1). Computational Method 19

20 Exterior Algebra Motivation: generalize the cross product from R 3 to R n V an n dimensional real vector space with elements u, v,... u v exterior (wedge) product of u and v anti-symmetric: u v = v u linear in each factor: (au + bv) w = au w + bv w 0 V = R 1 V = V 2 V = V V, 2-vectors continue recursively, demanding associativity p V p-vectors, dim( p V ) = ( ) n p dim( n V ) = 1 V inner product space: natural isomorphism between V and its dual V flat: : V V, u (v) = (u, v) sharp: : V V, = 1 inner product on V induces inner product on p V (the notion of length induces notions of area, volume, p-volume) Computational Method 20

21 Hodge Motivation: u v is a 2-vector, so not quite u v yet Solution: so map u v to u v - call this map Hodge Then: u v = (u v) V oriented inner product space: : p V (n p) V natural isomorphism Let {e 1, e 2,..., e n } be a basis for V Let σ = e 1 e 2 e n be a chosen orientation Hodge : u v = ( u, v)σ, v (n p) V Intuition: complementary n p vector to the given p vector orthogonal consistent with the orientation Side result: u v = (u, v) = (u v) = (v u) Computational Method 21

22 Hodge Example V = R 2 : V = R 3 : 1 = e 1 e 2 e 1 = e 2 e 2 = e 1 (e 1 e 2 ) = 1 1 = e 1 e 2 e 3 e 1 = e 2 e 3... =... (e 1 e 2 ) = e 3... =... (e 1 e 2 e 3 ) = 1 Computational Method 22

23 Exterior Calculus Differential forms - generalizations of differentials Let M be a manifold 0-forms are functions 1-forms are differentials k-forms at a point P are k vectors (elements in k TP M) Exterior Derivative d df = f dx i x i d(fdx 1 dx k ) = df dx 1 dx k Adjoint derivative: δ = d, with (δα, β) = (α, dβ) Laplacian: = (d + δ) 2 Interior Product: (i X α)(v 2,..., v k ) = α(x, v 2,..., v k ) Lie Derivative (using Cartan s Magic Formula): L X α = di X α + i X dα Computational Method 23

24 Curved Surfaces Numerically, a major issue is: retina is a curved surface Flat: ( x i ) = g ij dx j, (v ) i = g ij v j Sharp: (dx i ) = g ij x j Hodge : ( α) i1,i 2,...,i n k = 1 k! αj1,...,j k det g ɛ j1,...,j k,i 1,...,i n k Computational Method 24

25 Algebraic Topology Approach to Space Discretization Simplices Simplicial Complexes Chains Cochains Boundary operator Discrete exterior derivative D = T Discrete version of Stokes theorem Computational Method 25

26 M DEC : DEC Hodge * 1 Marsden, Hirani, Desbrun, et al.: σ σ α = 1 σ σ α [M DEC ] ij = σk i σ k i δ ij Advantage: M DEC diagonal Disadvantage: M DEC not positive definite, because circumcenter may be outside the simplex Example: standard 2-simplex; vertices: (0, 0), (1, 0), (0, 1) 1 M1 DEC = Computational Method 26

27 M W hit : FEEC Hodge * Arnold, Falk, Winther. Whitney 0-forms are barycentric coordinates: Whitney 1-forms: η k = λ i dλ j λ j dλ i [M W hit ] ij = η i η j da Advantage: M W hit is positive definite, because barycenter is always inside the simplex Disadvantage: M W hit not diagonal, (M W hit ) 1 not sparse Example: standard 2-simplex; vertices: (0, 0), (1, 0), (0, 1) M W hit 1 = Computational Method 27

28 PyDEC - A Python Implementation of DEC Authors: Hirani and Bell Efficient operator implementation in terms of sparse matrices Data structures Simplicial Complex Regular Cube Complex Operators Discrete Exterior Derivative Diagonal Sparse Matrix Discrete Hodge First Order Whitney Hodge Provides a nice playground for numerical experiments Computational Method 28

29 Data Figure: Retina Fundus Image webvision.med.utah.edu/book/part-i-foundations/simple-anatomy-of-the-retina/ Computational Method 29

30 Spatial Discretization Figure: Triangularization by gmsh. Computational Method 30

31 Example: 3 layer model geometry Figure: Three layer discretization The θ = 1 layer represents arterioles, the θ = 1 2 capillaries and θ = 0 represents venules. layer represents Computational Method 31

32 Outline Modeling the Eye: Goals and Motivation A Multiscale Model of Blood Flow in the Retina Hierarchical Darcy Flow Vascular Tree Architecture Computational Method Hierarchical Discretization Spatial Discretization Curved Surfaces Discrete Exterior Calculus (DEC) and Finite Element Exterior Calculus (FEEC) Results Ocular Curvature May Play a Role in Glaucoma Summary and Future Plans Results 32

33 Capillary Velocity Field Figure: Capillary Blood Velocity Results 33

34 Application: Myopia Figure: Capillary Blood Velocity in Myopia The myopic eye is extended along the optical axis. Results 34

35 Eye Deformations Figure: Eye Shapes Results 35

36 Velocity Field Deformations Figure: Velocity Difference Caution: to compare the velocity field difference in a meaningful way, all velocity fields are first pulled back to the sphere. Results 36

37 Outline Modeling the Eye: Goals and Motivation A Multiscale Model of Blood Flow in the Retina Hierarchical Darcy Flow Vascular Tree Architecture Computational Method Hierarchical Discretization Spatial Discretization Curved Surfaces Discrete Exterior Calculus (DEC) and Finite Element Exterior Calculus (FEEC) Results Ocular Curvature May Play a Role in Glaucoma Summary and Future Plans Summary and Future Plans 37

38 Summary The hierarchical Darcy equation models blood flow in the retina as a hierarchical porous medium. Exterior Calculus generalizes vector calculus to manifold and is convenient for formulating and solving models on surfaces. The discrete exterior derivative d is unique and determined by the discrete Stokes theorem. The discrete Hodge * operator is an open question and a topic of current research. DEC and FEEC are related; they differ in the Hodge * discretization. DEC and FEEC can be mixed with traditional FEs. Curvature may play an important role in the development of glaucoma. Summary and Future Plans 38

39 Future Plans Generalize the model to space dependent parameters. Couple the model with elasticity. Use the model to make predictions for other eye pathologies. The eye is the window into the body. What can we learn about conditions in the rest of the body? Personalized medicine: couple the model with image analysis front end. Summary and Future Plans 39

40 Acknowledgments NSF DMS and NSF CCF NIH 1R21EY A1 Slocum Dickson Foundation Research to Prevent Blindness (RPB, NY, USA) Indiana University Collaborative Research Grant of the Office of the Vice President for Research Chair Gutenberg funds SUNY STEM Summer Research SUNY Poly School of Arts and Sciences 40

41 References Abraham, Marsden, Ratiu, Manifolds, Tensor Analysis and Applications. Flanders, Differential Forms with Applications to the Physical Sciences. Frankel, The Geometry of Physics. T. Sherman, On Connecting Large Vessels to Small, The Meaning of Murray s Law, J. Gen. Phisiol, 78, , (1981). T. Takahashi et. al., A mathematical model for the distribution of hemodynamic parameters in the human retinal microvascular network, J. Biorrheol, 23, 77-86, (2009). Desbrun, A. N. Hirani, J. E. Marsden: Discrete Exterior Calculus for Variational Problems in Computer Graphics and Vision, Proceedings of the 42nd IEEE Conference on Decision and Control, Honolulu 5 (2003), Arnold D., Falk R., Winther R. : Finite Element Exterior Calculus: from Hodge Theory to Numerical Stability, Bull. Amer. Math. Soc. 47,2 (2010),

42 References Cont. Anil N. Hirani and Kaushik Kalyanaraman: Numerical Experiments for Darcy Flow on a Surface Using Mixed Exterior Calculus Methods, E-print arxiv: v1 [cs.na] on arxiv.org (2011). A. N. Hirani, K. B. Nakshatrala, J. H. Chaudhry: Numerical method for Darcy flow derived using Discrete Exterior Calculus, E-print arxiv: [math.na] on arxiv.org (2008) Nathan Bell and Anil N. Hirani: PyDEC: Software and Algorithms for Discretization of Exterior Calculus, ACM Transactions on Mathematical Software, Vol 39 1 (2012), A. Dziubek, E. Rusjan, W. Thistleton: Challenges in Blood Flow Simulation: Numerical Methods and Image Processing Tools, Proceedings of the ASME/FDA, 1st Annual Frontiers in Medical Devices, Washington DC (2013). A. Dziubek, G. Guidoboni, A. Harris, A.H. Hirani, E. Rusjan, W. Thistleton, Effect of ocular shape and vascular geometry on retinal hemodynamics: a computational model, Biomechanics and Modeling in Mechanobiology (2015),

43 Thank You! 43