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Dr. Andrea Dziubek, Dr. Edmond Rusjan, Dr. William Thistleton

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Dziubek, A., Guidoboni, G., Harris, A., Hirani, A. N., Rusjan, E., & Thistleton, W. (2016). Effect of ocular shape and vascular geometry on retinal : a computational model. Biomechanics And Modeling In Mechanobiology, (4), 893. doi:10.1007/s10237-015-0731-8

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Effect of ocular shape and vascular geometry on retinal hemodynamics: a computational model

Andrea Dziubek · Giovanna Guidoboni · Alon Harris · Anil N. Hirani · Edmond Rusjan · William Thistleton

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Abstract A computational model for retinal hemody- retinal hemodynamics is influenced by the ocular shape namics accounting for ocular curvature is presented. (sphere, oblate spheroid, prolate spheroid and barrel The model combines: (i) a hierarchical Darcy model for are compared) and vascular architecture (four vascular the flow through small arterioles, capillaries and small arcs and a branching vascular tree are compared). The venules in the retinal tissue, where blood vessels of dif- model predictions show that changes in ocular shape ferent size are comprised in different hierarchical levels induce non-uniform alterations of blood pressure and of a porous medium; and (ii) a one-dimensional net- velocity in the . In particular, we found that: (i) work model for the blood flow through retinal arteri- the temporal region is affected the least by changes in oles and venules of larger size. The non-planar ocular ocular shape, and (ii) the barrel shape departs the most shape is included by: (i) defining the hierarchical Darcy from the hemispherical reference geometry in terms of flow model on a two-dimensional curved surface embed- associated pressure and velocity distributions in the ded in the three-dimensional space; and (ii) mapping retinal microvasculature. These results support the clin- the simplified one-dimensional network model onto the ical hypothesis that alterations in ocular shape, such curved surface. The model is solved numerically using as those occurring in myopic , might be associated a finite element method in which spatial domain and with pathological alterations in retinal hemodynamics. hierarchical levels are discretized separately. For the fi- Keywords mathematical modeling · retinal hemo- nite element method we use an exterior calculus based dynamics · hierarchical porous medium · ocular implementation which permits an easier treatment of curvature · vascular network · finite element exterior non-planar domains. Numerical solutions are verified calculus against suitably constructed analytical solutions. Nu- merical experiments are performed to investigate how Mathematics Subject Classification (2000) MSC 76S05 · MSC 93A30 · MSC 65N30 · MSC 76M10 · A. Dziubek, E. Rusjan, W. Thistleton Mathematics, State University of New York Polytechnic In- stitute, 100 Seymour Rd, Utica, NY 13501, USA 1 Introduction E-mail: [email protected] G. Guidoboni The rapid advance of imaging technologies in ophthal- Mathematical Sciences, Indiana University Purdue University mology is making available a continually increasing num- Indianapolis, Indianapolis, IN 46202, USA LABEX IRMIA, Universit´ede Strasbourg, Strasbourg, F- ber of data regarding retinal morphology, hemodynam- 67000, France ics and metabolism. An accurate and efficient interpre- A. Harris tation of such data is the key to advancing the under- Eugene and Marilyn Glick Institute, Indiana University standing of ocular diseases and their treatment. How- School of , Indianapolis, IN 46202 ever, the clinical interpretation of retinal measurements A. N. Hirani is still very challenging. Many systemic and ocular fac- Mathematics, University of Illinois at Urbana Champaign, tors combine to give rise to the observed data and it is Urbana, IL 61801 extremely difficult to single out their individual contri- 2 Andrea Dziubek et al. butions during clinical and animal studies. Mathemati- dimensional curved surface embedded in the three-di- cal modeling is gaining more and more attention in the mensional space, and the one-dimensional network mod- ophthalmic science, as it may help providing a quanti- el is mapped onto the curved surface. In particular, tative representation of the biophysical processes in the here we compare four different geometries of clinical eye and their interwoven . Here we use math- interest: sphere, oblate spheroid, prolate spheroid and ematical modeling to theoretically investigate how and barrel [36]. to what extent different ocular shapes and vascular ar- We use finite elements to discretize the hierarchical chitectures will influence the model predictions for the variable and reduce the hierarchical Darcy flow equa- retinal perfusion, i.e. hemodynamics in the microvascu- tion to a coupled system of Darcy flow equations. We lature nourishing the retinal tissue. then use finite element exterior calculus to discretize The choice to study changes in ocular shape and vas- the spatial variable and couple the system of hierarchi- cular architecture is motivated by clinical observations. cal Darcy flow equations to the one-dimensional net- Alterations in ocular shape, as it occurs for example in work model via delta functions. Finite element exterior , are often associated with vascular abnormali- calculus provides a convenient framework for problems ties [5,34,42] and with higher risks of severe ocular con- on curved surfaces. In particular, we use the PyDEC [4] ditions, such as [18,30], retinal detatchment library for our numerical experiments. [32,50] and maculopathy [17,27]. Architectural changes The mathematical model is described in Section 2 in the retinal vasculature are also often associated with and the numerical method adopted to solve the model ocular diseases, including glaucoma [22,29,46,48], dia- is presented in Section 3. Numerical solutions are veri- betic [9,8,24] and myopia [16,31]. fied against suitably constructed analytical solutions in Several theoretical approaches have been used to Section 4. The numerical results are presented and dis- capture the details of the retinal vasculature, includ- cussed in Section 5. Conclusions and future directions ing network models, fractal models (based on Murray’s are outlined in Section 6. law [37,41]), and recently multi-fractal analysis [45]. Interestingly, various modeling approaches have been proposed to model retinal blood flow [2,23,39,44], and leverage information from fundus images to improve 2 Mathematical Model physically based models [19,20,33]. To the best of our knowledge, however, none of the 2.1 Hierarchical Darcy flow model currently available models describing retinal hemody- namics accounts for the curvature of the retinal surface, which is indeed the main novelty of our study from both We describe the retinal tissue as a thin two-dimensional the modeling and clinical viewpoints. curved surface Ω embedded in a three-dimensional space. Therefore, x ∈ Ω denotes a vector of spatial coordinates In this paper, we describe retinal perfusion via a in 3. multiscale approach. The blood flow in retinal arte- R rioles and venules of larger size is described using a The retinal microvasculature through the tissue is one-dimensional network model. The blood flow in the composed of a network of vessels of different diameter microcirculation, comprising small arterioles, capillar- and is modeled as a hierarchical porous medium [47]. ies and small venules, is described using a hierarchical In this framework, blood vessels are considered as pores Darcy flow model where blood vessels of different size in the tissue and a hierarchical variable θ, defined on correspond to hierarchical levels of different permabil- the interval [0, 1], is introduced to represent the vari- ity [10,47]. We remark that indeed a poro-elastic model ations in diameters of the pores. Small arterioles and would be more suitable for the description of retinal small venules, corresponding to the levels θ = 1 and perfusion, since the retina is directly exposed to the ac- 0, respectively, are coupled with the one-dimensional tion of the which might alter vas- network described in Section 2.3. cular diameters and, consequently, retinal blood flow Blood flow takes place both in the spatial direction [23]. In this perspective, the Darcy model for the flow (through vessels of comparable radius) and the hier- through a porous (rather than poro-elastic) medium archical direction (from arterioles to venules), so that utilized here should be seen as an informative and nec- model parameters and unknowns are functions of both essary first step towards the development of more com- spatial and hierarchical variables x and θ. plex models. Here we consider the stationary problem, which con- In order to account for the non-planar ocular shape, sists of finding spatial velocity v = v(x, θ), hierarchical the hierarchical Darcy flow model is defined on a two- velocity ω = ω(x, θ), and pressure p = p(x, θ), by solv- Effect of ocular shape and vascular geometry on retinal hemodynamics: a computational model 3 ing the following system: ∂ ∇ · (n v) + (n ω) = 0 x ∈ Ω, θ ∈ (0, 1) (1a) b ∂θ b nbv = −K∇p x ∈ Ω, θ ∈ (0, 1) (1b) ∂p n ω = −α x ∈ Ω, θ ∈ (0, 1) (1c) b ∂θ subject to the boundary conditions

(nbv)(x, θ) · n = 0, x ∈ ∂Ω, θ ∈ [0, 1] (2a)   (n w)(x, 0) = −G p(x, 0) − p (x) , x ∈ Ω (2b) b v v Fig. 1 Six regions of particular clinical interest in the retina:   head (ONH), inferior quadrant (I), superior quad- (nbw)(x, 1) = −Ga pa(x) − p(x, 1) , x ∈ Ω. (2c) rant (S), nasal quadrant (N), temporal quadrant (T) and fovea. Here nb is the tissue porosity or fluid volume frac- tion (ratio between blood volume and total volume of blood and tissue in the reference volume of the porous 2.2 Domain geometry medium), −nbω is the tissue perfusion (volume of blood flowing down in the hierarchy per unit tissue), K is The reference geometry for the domain Ω is the hemi- the spatial permeability tensor, and α is the hierarchical spherical surface with radius R = 12 mm, representing permeability. In general, porosity and permeability may a normal eye. In addition, we consider the following depend both on x and θ. In the boundary conditions, shapes (also depicted in Figure 2): n represents the vector normal to the outer boundary of the spatial tissue domain Ω, and Gv and Ga are the 1. an oblate spheroid with semi-axes R = 12 mm and hydraulic conductances between lowest hierarchy and H = 10 mm, representing an eye with shorter axial draining veins, and between the highest hierarchy and length (condition often associated with hyperopia); feeding arteries, respectively. Following [10], we will ex- 2. a prolate spheroid with semi-axes R = 12 mm, H = press the hydraulic conductances as 14 mm, representing an eye with longer axial length (condition often associated with myopia); G = α δ (x) and G = α δ (x) (3) v v v a a a 3. a barrel-like surface described by a polynomial of 2 3 where αv and αa are venous and arterial conductances degree three p3(Z) = a1 +a2Z +a3Z +a4Z where p 2 2 0 and δv(x) and δa(x) are delta functions which differ Z = x + y satisfying the conditions p3(0) = 0, R from zero only where the feeding arteries and draining p3(0) = H, p3( 2 ) = H − 1.3 mm, p3(R) = 0; repre- veins are located, respectively. The intravascular pres- senting an eye with longer axial length and flattened sure in the feeding arteries and draining veins are de- posterior (another condition often associated noted by pv(x) and pa(x), respectively, and αv and αa with myopia). are scalar functions possibly varying with x.

The well-posedness of equations (1a)-(1c) with bound- ary conditions (2a)-(2c) with the delta function data of equation (3) is discussed in [10]. In this paper we solve system (1) with the boundary conditions (2) on domains of different shape and for vascular tree of dif- ferent geometries, as described in the following sections. In order to facilitate the comparison between vas- cular trees of different geometry, it is useful to identify six regions of clinical interest in the retina, as schema- tized in Figure 1. The major retinal vessels depart from the optic nerve head (ONH) and virtually divide the Fig. 2 (Left) Representation of three hierarchical levels, retina into four quadrants, referred to as the inferior where four main arterioles are included as sources in the ar- terial level and four main venules are included as sinks in the (I), superior (S), nasal (N) and temporal (T) quadrants. venous level; (Right) spherical, oblate ellipse, prolate ellipse, The fovea, located in the temporal region, provides the and barrel shapes. sharpest vision and is avascular. The retinal image is taken from the DRIVE data base [43]. 4 Andrea Dziubek et al.

2.3 Arterial and venous vascular tree geometry At each branching point of the dichotomous tree, the relationship between the radius R1 of the mother Larger arterioles and venules are described using sim- branch and the radii R2,1 and R2,2 of the two daughter plified one-dimensional models. Let us denote by Λa branches is assumed to be: and Λv the collection of centerlines of the arterial and m m m venous networks in the flat plane, respectively. The cen- R1 = R2,1 + R2,2, (4) terlines are then mapped onto the spherical surface as with m = 2.85 [37]. The length Lj of each individual shown in Figure 3. More precisely, every point (x0, y0) branch characterized by the radius Rj is calculated as in Λa or Λv is mapped into a point on the sphere p 2 2 (x0, y0, z0 = x0 + y0). The resulting Γa = γa(Λa) and η Lj(Rj) = β Rj , (5) Γv = γv(Λv) lay on the curved surface, meaning that 3 3 Γa ⊂ Ω ⊂ R and Γv ⊂ Ω ⊂ R . We rotate the geom- where η = 1.15 and β = 7.4 [44]. etry of the remapped vascular networks by π/4 about In each vessel, we assume that: (i) the blood be- the z axis followed by a rotation of π/6 about the x axis, haves as a Newtonian viscous fluid of viscosity µ; (ii) the so that the arcs originate√ from a point located at x = 0, flow is laminar, stationary and axially-symmetric; (iii) y = −R/2 and z = R 3/2, representing the location no-slip boundary conditions are imposed at the lateral of the optic nerve head (ONH). The same rotation is vessel boundary. These assumptions allow us to write applied to the arterial and venous networks. the blood pressure p inside each vessel as a function of the curvilinear coordinate s along the vessel center- line and to use Hagen-Poiseuille’s law to calculate the pressure p as ! 8µjLj p(s) = p0 − 4 Qj s for s ∈ [0,Lj] (6) πRj

where s = 0 and s = Lj indicate the upstream and downstream ends of the vessel, p0 is the upstream pres- sure and Qj is the volumetric flow rate. We recall that Fig. 3 (Left) Collection of binary trees joined at their roots mass conservation at each branch implies that Q = on a flat domain; (Right) binary trees mapped onto the spher- g ical surface and meshed with Gmsh. 2Qg+1, where g is the generation of a branch. Viscosity values per branch segment are linearly in- terpolated from viscosity values reported in [2], where In this paper, we consider two vascular architectures µ = 2.28 cPa in the larger arterioles and venules, and embodying different levels of detail. The simplest ar- µ = 2.08 cPa in the smaller arterioles and venules. chitecture includes only four major feeding arteries and Pressure values at the four main branches of the cen- draining veins modeled as four arcs, see Figure 2, along tral retinal artery (CRA) and the central retinal vein which a constant pressure is imposed. Here we consider (CRV) are assumed to be p = p = 33.75 mmHg p = 33.75 mmHg for arteries and p = 16.4 mmHg for 0a a a v and p = p = 16.4 mmHg, respectively, which is in veins. This simple vascular architecture is used to com- 0v v agreement with [2]. The average blood flow rate in the pare the numerical simulations obtained when varying retina is assumed to be Q¯ = 36.11 µl/min, as re- the ocular shape, see Section 5.2. tot ported in the Table 5.1. For the blood flow rate after A more realistic vascular model is obtained by ap- the first division into the four major branches we as- plying the dichotomous tree approach proposed by Taka- sume Q = Q¯ /4. hashi et al in [44] starting from the four major arcs 0 tot Pressure values at the branching nodes of the bi- mentioned above. Figure 3 (Left) shows the centerlines nary tree are computed via (6) using the length of the of a collection of binary trees joint at the root. The first branches in the flat plane and are summarized in Ta- four branches are 3.6 mm and 2.4 mm long. Branches ble 1. Note that 760.0 mmHg = 101.325 kPa. of the following generations are scaled by 0.8, spatially arranged with angles of π/6, π/8, π/10, π/12 between parent/child and child/child branches. The angles are Branching node 0 1 2 3 4 chosen to ensure that (i) the domain is well covered by Arterial pressure 33.5 31.51 29.56 27.64 25.8 Venous pressure 16.4 18.39 20.34 22.26 24.1 the vascular tree, (ii) the foveal area is free of vessels, (iii) the ends of the binary trees are not too close to Table 1 Pressures in mmHg at nodes of the vascular tree. each other, see also Figure 1. Effect of ocular shape and vascular geometry on retinal hemodynamics: a computational model 5

The computational mesh for the domain Ω is gen- where erated with the Delaunay algorithm using Gmsh [21], n with additional functionality to ensure that the center- X Z 1 K (x) = K(x, θ ) ϕ (θ) ϕ (θ) ϕ (θ) dθ (11a) lines Γa and Γv are along the edges of the triangulation ij k k i j 0 while avoiding a distorted mesh. k=0 n Z 1 X 0 0 αij(x) = α(x, θk) ϕk(θ) ϕi(θ) ϕj(θ) dθ (11b) k=0 0 3 Numerical Method + δi0δj0Gv + δinδjnGa 3.1 Discretization of the hierarchical variable θ fi(x) = δi0Gv pv + δinGa pa (11c) To discretize the system (1) and (2) in the hierarchical variable we follow the approach described in [47]. We and where pj(x) = p(x, θj) and δij is the Kronecker first write (1) in direct form with only pressure as the delta. For each i and j, Kij(x) is still a 2 × 2 matrix to unknown: potentially account for anisotropies in the tissue per- meability. In this paper though, as in [10], we neglect ∂  ∂p −∇ · (K∇p) − α = 0 (x, θ) ∈ Ω × (0, 1). such anisotropy and we assume that permability and ∂θ ∂θ conductivity depend only on θ. Then K(θ) = K(θ)I, (7) which implies Kij = KijI. The hierarchical discretization leads to a tridiagonal Let ϕ = ϕ(θ) be a smooth function defined on [0, 1]. To system for the pressures p (x) at each hierarchical node. obtain the weak formulation, we multiply (7) by a test j In the case of three nodes, namely j = 0, 1, 2, we obtain function ϕ(θ), integrate by parts over [0, 1] and utilize the boundary conditions in θ to obtain:      −K00∆ + α00 −K01∆ + α01 0 p0 f0 Z 1 −K10∆ + α10 −K11∆ + α11 −K12∆ + α12p1=f1 ∂p(x, θ) 0 −∇·(K(x, θ)∇p(x, θ)) ϕ(θ)+α(x, θ) ϕ (θ) dθ 0 −K21∆ + α21 −K22∆ + α22 p2 f2 0 ∂θ (12) + ϕ(1)Ga p(x, 1) + ϕ(0)Gv p(x, 0)

= ϕ(0)Gvpv(x) + ϕ(1)Gapa(x). (8) where the scalar Kij is the ij entry of the matrix

Now, to obtain a semi-discrete model in the hier-   3K0 + K1 K0 + K1 0 archical variable θ, we introduce piecewise linear basis 1  K0 + K1 K0 + 6K1 + K2 K1 + K2  , functions {ϕk}, with k = 0, .., n on [0, 1] corresponding 24 0 K1 + K2 K1 + 3K2 to nodes {θk}. We write permeability, conductance and pressure as piecewise linear interpolations at the nodes the scalar α is the ij entry of the matrix of the discrete hierarchical variable, namely ij

n   X α0 + α1 + Gv −(α0 + α1) 0 K(x, θ) = K(x, θk)ϕk(θ), (9a) −(α0 + α1) α0 + 2α1 + α2 −(α1 + α2) , k=0   n 0 −(α1 + α2) α1 + α2 + Ga X α(x, θ) = α(x, θk)ϕk(θ), (9b) k=0 and the scalar fi is the i component of the vector n X p(x, θ) = p(x, θ )ϕ (θ). (9c) k k  T k=0 Gvpv, 0,Gapa .

Substituting expressions (9a)-(9c) into (8) and consid- Here α , α , α and K ,K ,K are the conductivities ering a specific basis function ϕ = ϕ , we obtain the 0 1 2 0 1 2 i and permeabilities at the three hierarchical levels and semi-discrete weak form are given in equation (24) and (26). n n X X The homogeneous Neumann boundary conditions −∇ · (Kij(x)∇pj(x)) + αij(x)pj(x) = fi(x) (2a) complete the system. Note that the Robin pressure j=0 j=0 conditions (2b) and (2c) contribute to the two terms αij (10) and fi. 6 Andrea Dziubek et al.

3.2 Spatial discretization using Exterior Calulus δinGapa on the right hand side are the prescribed ve- nous and arterial pressures at the vertices correspond- Equation (12) can be written as a system for each θ ing to the edges of the triangulation of veins and ar- level i (using Einstein notation) teries. In the discretization we consider the values at vertices to be piecewise constant, with each such con- −K ∆ p + α p = f (13) ij j ij j i stant value extending from the vertex to the midpoint and the systems obtained for each i have to be solved of all edges containing that vertex. The terms δi0δj0Gv simultaneously as indicated in equation (12). and δinδjnGa are treated the same way. This is achieved To carry out the discretization of the spatial opera- by defining the matrix E to be a diagonal matrix of size tor ∆ in (13), there are several alternatives even within N0 × N0. The nonzero diagonal entry corresponding to the framework of finite element methods. The domain Ω a vertex is 0.5 times the sum of lengths of edges con- is a curved surface embedded in R3 and so the Laplacian taining that vertex. Then the terms αijpj on the left operator ∆ in equation (13) is the Laplace-Beltrami op- hand side of (13) are discretized as erator. One approach toward discretizing the Laplace-   Beltrami operator is via surface finite element methods ∗0αijpj + E δi0δj0αv + δinδjnαa pj (15) [11,12,14]. Another systematic framework is the use of and f on right hand side of (13) is discretized as exterior calculus discretization and this is the one we i follow in this paper. Exterior calculus is a generaliza-   δ α p + δ α p (16) tion of calculus to manifolds and allows the definition of E i0 v v in a a differential operators in a coordinate invariant manner, where pa and pv contains the values of prescribed pres- which can then be discretized by intrinsic computation sures sampled at the vertices. Here ∗0 is the usual mass of quantities in the triangles approximating the surface. matrix of standard piecewise linear finite elements. Thus For computational purposes exterior calculus has the complete discretization of (13) is been discretized as finite element exterior calculus [3] and Discrete Exterior Calculus [13,25]. These discretiza- n X T tions are useful either when a mixed method (involving Kijd0 ∗1d0pj both velocities and pressures) is to be implemented or j=0 when the domain is not flat, as is the case in this paper. + ∗0αij pj + E(δi0δ0jαv + δinδnjαa)pj They have been implemented in the software package   = δ α p + δ α p . (17) PyDEC [4], which we used for our computations. E i0 v v in a a We discretize the first term on the left hand side One of the authors recently showed that a formulation of equation (13) using finite element exterior calculus. based on exterior calculus is very effective in simulating Since equation (13) involves a pressure-only formula- Darcy flow on curved surfaces [26]. However, the com- tion, the resulting matrix is identical to the one ob- putational approach in the present article differs from tained by a standard finite element formulation. The that discussed in [26] since here we consider: (i) an ex- matrix corresponding to −Kij∆pj in the left hand side tension of the Darcy equations to the flow through a of equation (13) is hierarchical porous medium, thereby requiring suitable T discretization of the corresponding hierarchical variable Kij d0 ∗1d0 pj. (14) θ; (ii) source terms distributed along the centerlines of T Here d0 ∗1d0 is the usual stiffness matrix of standard the vascular trees, thereby requiring suitable discretiza- piecewise linear finite element method. The matrix d0 tion of the corresponding delta functions. In addition, has N1 rows and N0 columns, where N1 is the number we remark that the method in [26] relies on a mixed of edges in the triangulation approximation of Ω and formulation of the problem, whereas here we solve for N0 is the number of vertices in the triangulation. Each the pressure as the sole primal variable. column corresponds to a vertex and the entries are +1, −1 or 0, depending on if that vertex belongs to the head of an edge, its tail, or not at all. The matrix ∗1 is of size 3.3 Pullback of the velocity field to the sphere N1 × N1 and is obtained by computing inner products of Whitney forms, also known as edge elements. See [4] To compare the velocity fields defined on different do- for details. main geometries, we pull back all velocity fields to the  T We now describe the discretization of the δi0δj0Gv sphere. The velocities u = u1, u2, u3 from the oblate and δinδjnGa terms in equation (11b) and the discretiza- and the prolate ellipsoidal surface are pulled back to the  H T tion of the right hand side fi. The terms δi0Gvpv and spherical surface by a simple scaling v = u1, u2, u3 R . Effect of ocular shape and vascular geometry on retinal hemodynamics: a computational model 7

2 Sphere S Barrel B2 When performing the semi-discretization of the prob- lem, these boundary conditions become part of the right V ∈ T S2 p ∈ B2 P Φ hand side for a differential problem in space, as shown P ∈ S2 2 v ∈ TpB in (13). Thus, in order to test the accuracy of the nu- merical method, it is enough to consider the following γ ψ Poisson problem with a Dirac source term: S = ψ ◦ Φ ◦ γ−1 2 2 Q ∈ R q ∈ R ∆p = δΓc in Ωc (18) U u p = 0 on ∂Ωc

Fig. 4 Map from sphere to barrel shaped surface where, for simplicity, we assume that Ωc is the planar circle of radius Rc centered at the origin, and that Γc is its horizontal diameter. Thus we can write To pull the velocities from the barrel shaped surface back to the sphere we extend the map from the sphere 2 2 2 Ωc = {x = (x1, x2) | x1 + x2 < Rc }, to the barrel shaped surface φ : S2 → B2 to a map in 2 three dimensions Φ : P(Q) → p(q), where Q, q ∈ R Γc = {x ∈ Ωc | x2 = 0}. and P, p ∈ 3. Tangent vectors are pushed forward by R An analytical solution to problem (18) can be con- the differential of the map v = Φ (V) = DΦ (V) = ∗ structed using standard techniques in partial differen- FV. The map and its differential are given by tial equations [15]. The solution p is written as the sum X  x = X  1  of two parts, namely p = v + u, where v accounts for Φ : Y  →  y = Y  DΦ =  1  the Dirac source in Ωc and u accounts for the boundary ∂z Z z = p3(Z) ∂Z conditions on ∂Ωc. More precisely, v is the convolution of the fundamental solution k(x) = 1 log |x| with the where z = p3(Z) is the polynomial of degree three de- 2π source f(x) = δ , namely scribing the barrel shape, see Figure 4. Note, that the Γc ∂z Z map is not invertible for some values ∂Z = 0, so we cannot compute the inverse of F to pull back the barrel v(x) = k(x − y)f(y) dy Ωc velocity to the sphere but instead need to go ’back- Z 1 wards’: = log |x − y|δΓ (y) dy, 2π c   Ωc v1 V = Φ∗ (v) = Dγ−1 ◦ DS−1 ◦ Dψ (v) =  v2  . √−v1x−v2y  R2−x2−y2 and u satisfies the Laplace’s equation ∆u = 0 in Ωc, with u = −v on ∂Ωc, which can be solved with the Poisson formula 4 Verification of the Numerical Strategy 2 2 Z Rc − |x| g(ξ) u(x) = 2 d(∂Ωa) We construct here two ad-hoc analytical solutions to 2π |ξ|=Rc |ξ − x| the coupled problem (1) and (2) in order to test our numerical strategy. In particular we want to verify the correctness of our numerical implementation with re- using polar coordinates ξ = hRc cos ψ, Rc sin ψi. In sum- spect to: mary, the analytical solution p = v + u is given by 1. the boundary conditions in (2) along the vessels Γ v 1 Z   and Γ involving delta functions δ and δ ; p(x) = log (x − y )2 + x2 a Γv Γa 4π 1 1 2 2. combining the hierarchical and spatial discretization Γc 2 2 Z using finite elements and discrete exterior calculus Rc − |x| g(ξ) + 2 dξ. (19) matrices. 2π ∂Ωc |ξ − x| To test our numerical solver, we compute a numer- 4.1 Testcase to verify the discretization of the ical solution of (18) by solving (17) on a planar cir- 2 boundary conditions along the vessels Γv and Γa cle with Rc = 12 mm, K = 1 mm /(s kPa), αij = 0 (s kPa)−1, and with homogeneous boundary conditions. involving delta functions δΓv and δΓa A colorplot of the analytical solution (19) is reported in The boundary conditions involving delta functions are Figure 5 (Left), whereas the difference between analyt- those pertaining to the θ-direction, see (2b) and (2c). ical and numerical solutions (obtained for an average 8 Andrea Dziubek et al.

defined in Qc, where f is a given scalar function. We construct an analytical solution to (20) subject to the following boundary conditions: ∂p = 0 on Σ ,Σ \ Γ and Σ \ Γ , ∂r l b v t a p = pv on Γv, (21)

p = pa on Γa. Fig. 5 (Left) Colorplot of the analytical solution (19) for the pressure satisfying problem (18); (Center) Colorplot of If the function f in (20) is given by the difference between analytical and numerical solution for b(θ)−a(θ) the pressure satisfying problem (18); (Right) logarithm of the f(r, ψ, θ) = K(θ) r2 sin(2ψ) (22) Frobenius norm of the difference between analytical and nu-  ∂2a 2 ∂2b 2  merical solutions for the pressure satisfying problem (18). − α(θ) ∂θ2 cos ψ + ∂θ2 sin ψ

where α(θ) = αaθ + (1−θ)αv, K(θ) = Kaθ + (1−θ)Kv, 2 3 edge length of 3 mm) is portayed in Figure 5 (Right). a(θ)=p ¯+ 3(pa−p¯)θ − 2(pa−p¯)θ ,p ¯ = (pa + pv)/2, and 2 3 We notice that the largest errors are located around b(θ)=pv + 3(¯p−pv)θ − 2(¯p−pv)θ , the problem admits the points (−R, 0) and (R, 0). This is due to the fact the exact solution that the analytical solution embodies the delta source p(ψ, θ) = a(θ) cos2 ψ + b(θ) sin2 ψ. (23) term on the whole interval [−R,R], whereas the numer- ical solution accounts for the delta source term only on We set pa = 20 mmHg, pv = 10 mmHg, Rc = 12 2 the interval [−(R − cl),R − cl], where cl is the aver- mm, Ka = Kv = 1 mm /(s kPa), and αa = αv = 1 −1 age edge length (i.e. excluding the two boundary points (s kPa) . where homogeneous dirichlet boundary conditions are imposed). Figure 5 (Center) shows that the logarithm of the Frobenius norm of the difference between numerical and analytical solutions is decreasing with the average mesh size. Here we have considered average edge length of 12, 9, 6 and 3 mm. The slope is roughly −1, which is consistent with a first order method.

4.2 Testcase to verify the hierarchical and spatial discretization using finite elements and discrete exterior calculus matrices Fig. 6 (Top Left) Colorplot of the numerical solution com- puted for problem (20) with the boundary conditions (21) and Let us consider again Ω = Ωc, where Ωc is the planar right hand side(22), in the case of 3 θ-levels. Difference be- circle of radius Rc. Thus, in polar coordinates we can tween numerical and analytical solutions computed for prob- define the cylinder Qc = Ωc × (0, 1) as lem (20) with the boundary conditions (21) and right hand side (22), in the case of 3 θ-levels (Top Right), 5 θ-levels (Bottom Left) and 9 θ-levels (Bottom Right). Qc = {(r, ψ, θ) | 0 ≤ r < Rc, 0 ≤ ψ < 2π, 0 < θ < 1}.

Let us denote by Σl the lateral surface of the cylinder, To obtain a numerical solution we solve (17) with and by Σb and Σt the bottom and top circular surfaces, the right hand side vector (22). We assume Kij = 1 respectively. Let us define the lines Γv ⊂ Σb and Γa ⊂ 2 −1 mm /(s kPa) and αij = αv = αa = 1 (s kPa) .A Σt as colorplot of the numerical solution is reported in Fig- ure 6 (Top Left) in the case of 3 θ-levels. The remain- Γv = {(r, ψ, θ) | 0 ≤ r ≤ Rc, ψ = π/2 or ψ = 3π/2, θ = 0}, ing panels in Figure 6 show that the difference be- Γa = {(r, ψ, θ) | 0 ≤ r ≤ Rc, ψ = 0 or ψ = π, θ = 1}. tween numerical and analytical solutions is less than 2% when 3, 5, and 9 θ-levels are considered. Conver- Then, we consider the following simplified problem gence was decided to be obtained when further mesh refinements do no change the solution. Average mesh 1 ∂  ∂p 1 ∂2p ∂2p −K(θ) r + − α(θ) = f (20) element length is 0.5 mm. For visualization of the re- r ∂r ∂r r2 ∂ψ ∂θ2 sults we used Mayavi [38]. Effect of ocular shape and vascular geometry on retinal hemodynamics: a computational model 9

5 Results of the Numerical Experiments again in (s kPa)−1. The permeability values at each hierarchical level In this Section, we utilize our computational model K(θi) = Ki are chosen as to obtain physiologically rea- to theoretically investigate some interesting features of sonable velocity distribution at the capillary levels of retinal hemodynamics. We consider three hierarchical the hierarchy when the spherical geometry is consid- levels corresponding to arterioles (θ = 1), capillaries ered, see Figure 2. Here we used (θ = 1/2), and venules (θ = 0), see also Figure 2. Val- ues for material and physical properties pertaining to K2 = 2,K1 = 0.1,K0 = 6, (26) each level are provided in Section 5.1. Numerical ex- 2 periments investigating how retinal hemodynamics is measured in mm /(s kPa). influenced by the domain geometry and vascular archi- tecture are presented in Sections 5.2 and Sections 5.3. 5.2 Influence of domain geometry

5.1 Porosity, Conductance and Permeability We report in this Section solutions to (1) and (2) ob- tained with the parameters described in Section 5.1 in We utilize the theory of capillary models for porous me- the case of the four different domain geometries de- dia [6,28] and we write the porosity at each hierarchical scribed in Section 2.2. Here, the arterial and venous level as vascular networks are modeled as four major arcs, as 2 np(θ)πD(θ) L(θ) described in Section 2.3 and depicted in Figure 2. nb(θ) = 4Vtot Simulated pressure fields and velocity fields in the where np is the number of pores in the volume, D is reference case of hemispherical geometry are reported the pore diameter, L is the length of pores, and Vtot = in Figure 7. For all the three hierarchical levels corre- 4πR2h/2 is the total tissue volume, with R denoting sponding to arterioles, capillaries and venules, pressures the eye radius and h denoting the tissue thickness. We and velocities are within physiological ranges and, in have assumed here that np and D are constant in space particular, are of the same order of magnitude as those and depend only on the hierarchical level θ. Using the reported in [2]. values from [2] reported also in Table 5.1, we obtain the Differences in pressure fields and velocity fields be- following estimates for nb(θi) = nbi: tween the non-spherical geometries and the hemispher- ical reference model are reported in Figure 8. We focus nb2 = 0.0040, nb1 = 0.0393, nb0 = 0.0085. here on the capillary level, since it is the most important level for tissue perfusion and metabolism. We remark Conductances between adjacent hierarchical levels that the velocity fields are obtained by subtracting ve- can be estimated knowing average pressure drops ∆p¯ i locity field pullbacks to the sphere, as described in Sec- and mean tissue perfusion ω between levels as t tion 3.3. Even though the distributions of velocity and ∆θi pressure for the four cases look similar, interesting sim- αi ' ωt ∆p¯i ilarities and differences can be noticed when they are compared to the sphere, see Figure 8. with ω = Q¯ /V defined as the average blood vol- t tot tot Overall, the blood velocity in the temporal region is ume flow rate (Q¯ ) per unit tissue volume (V ). Us- tot tot less impacted by changes in domain geometry, i.e. oc- ing again the values reported in Table 5.1, we obtain ular shape, since negligible differences are predicted in ω = 0.007 s−1 and, as a consequence, observing that t the velocity fields between non-spherical and spherical ∆θ = 1/2, the following estimates for α in (s kPa)−1 i i geometry for all cases, see Figure 8. The largest differ- can be deduced: ence from the velocity field calculated for the sphere ωt ωt is reported for the barrel geometry, see Figure 8 (bot- α2 = = 0.003, α1 = = 0.005, 2(¯p2 − p¯1) p¯2 − p¯0 tom), where nasal, inferior and superior regions exhibit ωt differences up to 0.045 mm/s, corresponding to a 30% α0 = = 0.007. (24) 2(¯p1 − p¯0) increase with respect to the sphere. Pressure differences are largest near the ONH, where Venous and arterial conductances αv and αa are es- timated analogously, to obtain the maximal pressure difference reaches 0.015 mmHg for barrel geometry. The precision of the computed pres- ωt ωt αa = = 0.006, αv = = 0.023, sure values in the capillary level is 1e-6 mmHg, and 2(¯pa − p¯2) 2(¯p0 − p¯v) therefore the predicted differences are significant from (25) the numerical viewpoint. 10 Andrea Dziubek et al.

Parameters Arteries Capillaries Veins Parameters Value (i = 2) (i = 1) (i = 0) Hierarchy (θ) [1] 1 1/2 0 Eye radius (R) [mm] 12 Number of pores (np) [1] 40 187890 40 Tissue thickness (h)[µm] 100 3 Pore diameter (D)[µm] 47.2 6.0 68.5 Total tissue volume (Vtot) [mm ] 90.48 Pore length (L) [cm] 0.52 0.067 0.52 Arterial pressure (pa) [mmHg ] 33.75 Porosity (nb) [1] 0.0040 0.0393 0.0085 Venous pressure (pv) [mmHg] 16.4 Average velocity (¯v) [cm/s] 0.86 0.011 0.41 Average blood flow rate (Q¯tot)[µl/min] 36.11 Average pressure (¯p) [mmHg] 28.75 21.23 17.56 Average tissue perfusion (ωt) [1/s] 0.007 Conductance (α) [1/(s kPa)] 0.003 0.005 0.007 Arterial conductance (αa) [1/(s kPa)] 0.006 2 Permeability (K) [mm /(s kPa)] 2 0.1 6 Venous conductance (αv) [1/(s kPa)] 0.023

Table 2 Values of material and physical parameters in the hierarchical porous media model for the retinal circulation.

Fig. 7 Pressure fields in mmHg (left) and velocity fields in Fig. 8 Pressure differences in mmHg (left) and differences in mm/s (right) for arterial (top), capillary (middle) and venous the magnitude of velocities in mm/s (right) at the capillary (bottom) level: hemispherical surface with major feeding ar- level between (top to bottom) oblate spheroid and sphere, teries as four source terms in the arterial level with pa = 33.75 prolate spheroid and sphere, barrel and sphere. mmHg and the major draining veins as four source terms in the venous level with pv = 16.4 mmHg. (resp. vein) as depicted in Figure 3. For the above de- scribed domain geometry and vascular architecture, we 5.3 Influence of vascular architecture consider two different modeling choices for the pressure distribution along the network: The four arcs model for the arterial and venous network utilized in the previous Section is indeed an extreme 1. the pressure is constant in arterioles and venules, simplification of real vascular architectures. It is rea- pa = 33.75 mmHg and pv = 16.4 mmHg, respec- sonable to hypothesize that different modeling choices tively. Results are reported in Figure 9; for the vascular architecture would influence retinal per- 2. the pressure decreases linearly along the network fusion. In this Section, we compute pressure and veloc- according to the Poiseuille law (6), see also table 1. ity fields obtained in the case of reference hemispherical Results are reported in Figure 10. geometry with the arterial (resp. venous) network mod- Figures 9 and 10 show similar pressure and velocity eled as binary trees stemming from (resp. merging into) ranges when compared with the reference model with the four major branches of the central retinal artery the four arcs, see Figure 7. Velocities are higher when Effect of ocular shape and vascular geometry on retinal hemodynamics: a computational model 11 the pressure is assumed constant along the vascular net- rently being extended to distinguish between arterioles work, see Figure 9, when compared to the case with and venules as required for simulations [40]. Once ac- linear pressure distribution, see Figure 10. quired, the extracted vascular geometry must then be mapped to the curved retinal surface.

Fig. 9 Pressure fields in mmHg (left) and velocity fields in mm/s (right) for arterial, capillary and venous level. Binary Fig. 10 Pressure fields in mmHg (left) and velocity fields in tree model with same pressure on all arc segments (pa = 33.75 mm/s (right) for arterial, capillary and venous level. Binary mmHg for arterioles and pv = 16.4 mmHg for venules). tree with linear pressure distribution in each generation.

It is very encouraging to find that reasonable pres- sure and velocity fields are obtained not only in the simple case of the four arcs, but also when a more complex vascular network is considered. The next step would be considering image-based vascular network to test the viability of patient-specific simulations. Pho- tographically acquired fundus images are widely used in the clinical setting to help determine the health of the retina, and to track retinal changes over time. Sup- port for the automatic processing of fundus images is available in a variety of computational environments, including MATLAB [35], ImageJ [1] and ITK [49]. Fig. 11 Vascular Extraction: Original and Extracted Image An example of image-based vascular network recon- structed by our group using matched filters and local entropy thresholding, as in [7], is reported in Figure 11. Image extraction was performed on retinal images ob- 6 Conclusions and Future Perspectives tained from those archived in the DRIVE data base and represents images from diabetic subjects [43]. In We have developed a computational framework to ana- order to integrate patient-specific vascular networks ex- lyze the effect of ocular shape and vascular architecture tracted from images into the retinal perfusion model on the retinal blood perfusion. The model predictions two steps must still be completed. The algorithm is cur- show that changes in ocular shape induce non-uniform 12 Andrea Dziubek et al. alterations of pressure and velocity in the retina. In Acknowledgements We thank Alexandra Benavente-Perez particular, we found that (i) the temporal region is af- and Muhammed Ali Alan for helpful discussions. We thank fected the less by changes in ocular shape, and that (ii) the students who participated in the SUNY STEM summer program 2014. This work has been partially supported by the the barrel shape departs the most from the hemispher- NSF DMS-1224195, NSF CCF-1064429, NIH 1R21EY022101- ical reference geometry in terms of associated pressure 01A1, a grant from Research to Prevent Blindness (RPB, NY, and velocity distributions in the retinal microvascula- USA), an Indiana University Collaborative Research Grant of ture. These results support the clinical hypothesis that the Office of the Vice President for Research and the Chair Gutenberg funds. alterations in ocular shape, such as those occurring in myopic eyes, might be associated with pathological al- terations in retinal hemodynamics. Physiologically rea- References sonable results are also obtained in the case of more complex vascular geometries when the pressure is as- 1. Abramoff, M., Niemeijer, M., Suttorp-Schulten, M., sumed to decrease linearly along the network. 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