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

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Dr. Andrea Dziubek, Dr. Edmond Rusjan, Dr. William Thistleton Faculty and Staff Publication: Dr. Andrea Dziubek, Dr. Edmond Rusjan, Dr. William Thistleton Pre-Print Citation for Final Published Work: Dziubek, A., Guidoboni, G., Harris, A., Hirani, A. N., Rusjan, E., & Thistleton, W. (2016). Effect of ocular shape and vascular geometry on retinal hemodynamics: a computational model. Biomechanics And Modeling In Mechanobiology, (4), 893. doi:10.1007/s10237-015-0731-8 https://search.proquest.com/docview/1811187248 Biomechanics and Modeling in Mechanobiology manuscript No. (will be inserted by the editor) 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 Received: date / Accepted: date 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 retina. 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 eyes, 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 Eye Institute, Indiana University standing of ocular diseases and their treatment. How- School of Medicine, 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 physiology. 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 myopia, 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 glaucoma [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 retinopathy [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 2 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 intraocular pressure 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 ! = !(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: @ r · (n v) + (n !) = 0 x 2 Ω; θ 2 (0; 1) (1a) b @θ b nbv = −Krp x 2 Ω; θ 2 (0; 1) (1b) @p n ! = −α x 2 Ω; θ 2 (0; 1) (1c) b @θ subject to the boundary conditions (nbv)(x; θ) · n = 0; x 2 @Ω; θ 2 [0; 1] (2a)
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