The choriocapillaris is a capillary bed located in a thin layer adjacent to the outer retina and is part of the oxygen delivery system to the photoreceptors of the eye. The blood flow is approximately planar and is serviced by microvessels, which join the choriocapillaris through inlets perpendicular to its plane. Capillaries are densely organised and separated by avascular septal posts, which direct the blood flow. The capillary bed is composed of a juxtaposition of tessellating vascular units called lobules, which are filled and drained independently from each other. A theoretical analysis of the blood flow in an idealised model of a lobule of the choriocapillaris is developed and studied. Lobules are modelled as tessellating polygonal prisms, where the upper and lower surfaces correspond to planar parallel membranes. The septae are modelled as cylinders randomly distributed between the two membranes. Feeding arterioles and draining venules are modelled as inlets and outlets connecting at the lower surface of the prism perpendicularly to the plane of the lobule. An inlet is placed inside the lobule, while an outlet is placed at each of the vertices. The polygonal prism can be formally subdivided into a set of triangular prisms with one inlet and two outlets, each of them located at one of the vertices. The triangular prisms are taken to be isosceles, and are therefore characterised by a vertex angle ${\it\omega}$ at the inlet and a span $L$. The flow is viscously dominated, and is investigated in the lubrication limit, in which the characteristic thickness of the prism is much smaller than the diameter of the cylinders. As a result of the geometry, a stagnation point is located midway between the outlets. A separation streamline joins the inlet and the stagnation point. The pressure drop ${\rm\Delta}\tilde{p}$ and the average fluid particle residence time $\langle \tilde{T}\rangle$ are analysed as a function of the angle at the inlet ${\it\omega}$ and the septae volume fraction ${\it\Phi}$. When no cylinders are present (${\it\Phi}=0$), an analytical expression for the pressure field is calculated by conformal mapping. Close to the triangle walls, the flow is quasi-parallel and characterised by a shorter fluid particle residence time. In the vicinity of the stagnation point, the velocity decreases and the residence time diverges logarithmically with the distance to the stagnation streamline. The minimum in pressure drop corresponds to a maximum in residence time, and is obtained for ${\it\omega}={\rm\pi}/2$. Asymptotic expressions for the pressure drop and average residence time are formulated in both the limits $\Vert {\it\omega}\Vert \ll 1$ and $\Vert {\rm\pi}-{\it\omega}\Vert \ll 1$. The impact of ${\it\Phi}$ on the flow is characterised by solving the equations for the flow numerically and using the Darwin drift framework. We show that the pressure drop is approximately proportional to $1+2{\it\Phi}$ for relatively small ${\it\Phi}$, and that $\langle \tilde{T}\rangle$ is proportional to $1-{\it\Phi}$ regardless of the void fraction or shape of the septae. In the case ${\it\Phi}=0$, the average residence time equals the volume of the domain divided by the volumetric flux. This analysis provides a new perspective on the blood flow dynamics within the choriocapillaris. Lobules form systems, where perfusion and corpuscle transport are a function of the angle that any two venular openings make with an arteriolar opening, the surface area perfused and the void volume fraction. The blood flow velocity and residence time are significantly heterogeneous, which may be responsible for the high degree of selective localisation observed in the pathogenesis of some inflammatory and degenerative diseases of the eye.
Topography of Capillary Density, Glucose Metabolism, and Microvascular Function within the Rat Inferior Colliculus
