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Delivery of drugs to the posterior segment of the eye is a significant challenge in the field of opthalmic pharmaceuticals. Several restrictive barriers hinder drug delivery to this district. Static barriers include tissues and limiting membranes, while dynamic barriers include drug clearance mechanism from blood and lymphatics. Strategies for delivering drugs to the posterior segment most often consist in topical ocular medications or systemic administrations, but dose/response profiles are generally very poor. Intravitreal injections and transscleral delivery are new emerging techniques with promising results. Purpose of this study is to develop a mathematical model to assess drug levels subsequent to a transscleral drug implant. Both computational and analytical techniques are adopted. The model comprises sclera, choroid, retina and vitreous along with the retina pigment epithelium at the choroid-retina boundary and the inner blood retinal barrier of the retinal vessels. Darcy equations are used to compute the filtration velocity of the interstitial fluid and a fictitious velocity field is added to model active pumping from the retinal pigmented epithelium. Convective-diffusive-reactive equations for drug concentration are then solved. Permeability parameters and partition coefficients simulate the presence of internal membranes and barriers, with possible different values in outward and inward directions. An important result of the model is the evaluation of the roles of the different physical parameters, which offers key points to improve drug delivery techniques. Namely, the sensitivity study suggests that diffusion in tissue, clearance rates, membrane permeabilities and active pumping play important roles in determining drug peak concentration and time-to-peak. However, their relative influence can be dramatically different depending on the rate-limiting parameter.

Microvessels -blood vessels with diameter less than 200 µm-form large, intricate networks organized into arterioles, capillaries and venules. In these networks, the distribution of flow and pressure drop is a highly interlaced function of single vessel resistances and mutual vessel interactions. Since, it is often impossible to quantify all these aspects when collecting experimental measures, in this paper we propose a mathematical and computational model to study the behavior of microcirculatory networks subjected to different conditions. The network geometry, which can be derived from digitized images of experimental measures or constructed in silico on a computer by mathematical laws, is simplified for computational purposes into a graph of connected straight cylinders, each one representing a vessel. The blood flow and pressure drop across the single vessel, further split into smaller elements, are related through a generalized Ohm's law featuring a conductivity parameter, function of the vessel cross section area and geometry, which undergo deformations under pressure loads. The membrane theory is used for the description of the deformation of vessel lumina, tailored to the structure of thick-walled arterioles and thin-walled venules. In addition, since venules can possibly experience negative values of transmural pressure (difference between luminal and interstitial pressure), a buckling model is also included to represent vessel collapse. The complete model including arterioles, capillaries and venules represents a nonlinear coupled system of PDEs, which is approached numerically by finite element discretization and linearization techniques. As an example of application, we use the model to simulate flow in the microcirculation of the human eye retina, a terminal system with a single inlet and outlet. After a phase of validation against experimental measurements of the correctness of the blood flow and pressure fields in the network, we compute the network response to different interstitial pressure values. Such a study is carried out both for global and localized variations of the interstitial pressure. In both cases, significant redistributions of the blood flow in the network arise, highlighting the importance of considering the single vessel behavior along with its position and connectivity in the network.

This paper examines the behavior of microvascular networks subjected to the coupled action of external (interstitial) and luminal pressure loads and under a prescribed inlet/outlet pressure drop. A network of 1D distensible tubes is used to represent microcirculatory districts. The thin shell theory is used to compute the geometry of the deformed tube section under the pressure loads. Blood flow and pressure drop in each tube are then related through a generalized Ohm's law featuring a conductivity parameter, function of the area and shape of the tube cross section. The model is used to study the response of representative arterial and venous bifurcations for different external and outlet pressures. Several features of the system response due to the interaction of the vessels of the network emerge from the simulations. Flow plateau, choking and flow diversion from one branch of the system to the other arise when buckling occurs. These results highlight the importance of considering the network connectivity along with the geometrical and mechanical characteristics of the vessels

This paper presents a mathematical model and a computational framework for the resolution of large networks of distensible blood microvessels characterized by a hierarchy of different diameters. We analyze the contribution of different distensibility relations, representing both active and passive responses in the case of the microvascular network in the eye retina. Model predictions indicate that, depending on the geometrical (hierarchical) location in the network, distensibility effects may be significantly different and lead, over long times, to possible geometrically-driven remodeling processes.

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