Mathematical modeling of local perfusion in large distensible microvascular networks

Microvessels -blood vessels with diameter less than 200 microns- 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. In this paper we propose a mathematical and computational model to study the behavior of microcirculatory networks subjected to different conditions. The network geometry is composed of 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 to describe 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 transmural pressures, a buckling model is also included to represent vessel collapse. The complete model including arterioles, capillaries and venules represents a nonlinear system of PDEs, which is approached numerically by finite element discretization and linearization techniques. 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, we simulate 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

Mathematical methods for modeling the microcirculation

The microcirculation plays a major role in maintaining homeostasis in the body. Alterations or dysfunctions of the microcirculation can lead to several types of serious diseases. It is not surprising, then, that the microcirculation has been an object of intense theoretical and experimental study over the past few decades. Mathematical approaches offer a valuable method for quantifying the relationships between various mechanical, hemodynamic, and regulatory factors of the microcirculation and the pathophysiology of numerous diseases. This work provides an overview of several mathematical models that describe and investigate the many different aspects of the microcirculation, including geometry of the vascular bed, blood flow in the vascular networks, solute transport and delivery to the surrounding tissue, and vessel wall mechanics under passive and active stimuli. Representing relevant phenomena across multiple spatial scales remains a major challenge in modeling the microcirculation. Nevertheless, the depth and breadth of mathematical modeling with applications in the microcirculation is demonstrated in this work. A special emphasis is placed on models of the retinal circulation, including models that predict the influence of ocular hemodynamic alterations with the progression of ocular diseases such as glaucoma

Nanotopography and microconfinement impact on primary hippocampal astrocyte morphology, cytoskeleton and spontaneous calcium wave signalling

Astrocytes' organisation affects the functioning and the fine morphology of the brain, both in physiological and pathological contexts. Although many aspects of their role have been characterised, their complex functions remain, to a certain extent, unclear with respect to their contribution to brain cell communication. Here, we studied the effects of nanotopography and microconfinement on primary hippocampal rat astrocytes. For this purpose, we fabricated nanostructured zirconia surfaces as homogenous substrates and as micrometric patterns, the latter produced by a combination of an additive nanofabrication and micropatterning technique. These engineered substrates reproduce both nanotopographical features and microscale geometries that astrocytes encounter in their natural environment, such as basement membrane topography, as well as blood vessels and axonal fibre topology. The impact of restrictive adhesion manifests in the modulation of several cellular properties of single cells (morphological and actin cytoskeletal changes) and the network organisation and functioning. Calcium wave signalling was observed only in astrocytes grown in confined geometries, with an activity enhancement in cells forming elongated agglomerates with dimensions typical of blood vessels or axon fibres. Our results suggest that calcium oscillation and wave propagation are closely related to astrocytic morphology and actin cytoskeleton organisation

Micropatterning of substrates for the culture of cell networks by stencil-assisted additive nanofabrication

The fabrication of in vitro neuronal cell networks where cells are chemically or electrically connected to form functional circuits with useful properties is of great interest. Standard cell culture substrates provide ensembles of cells that scarcely reproduce physiological structures since their spatial organization and connectivity cannot be controlled. Supersonic Cluster Beam Deposition (SCBD) has been used as an effective additive method for the large-scale fabrication of interfaces with extracellular matrix-mimicking surface nanotopography and reproducible morphological properties for cell culture. Due to the high collimation of SCBD, it is possible to exploit stencil masks for the fabrication of patterned films and reproduce features as small as tens of micrometers. Here, we present a protocol to fabricate micropatterned cell culture substrates based on the deposition of nanostructured cluster-assembled zirconia films by stencil-assisted SCBD. The effectiveness of this approach is demonstrated by the fabrication of micrometric patterns able to confine primary astrocytes. Calcium waves propagating in the astrocyte networks are shown

Blood flow repartition in distensible microvascular networks: Implication of interstitial and outflow pressure conditions

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

Blood flow mechanics and oxygen transport and delivery in the retinal microcirculation: multiscale mathematical modeling and numerical simulation

The scientific community continues to accrue evidence that blood flow alterations and ischemic conditions in the retina play an important role in the pathogenesis of ocular diseases. Many factors influence retinal hemodynamics and tissue oxygenation, including blood pressure, blood rheology, oxygen arterial permeability and tissue metabolic demand. Since the influence of these factors on the retinal circulation is difficult to isolate in vivo, we propose here a novel mathematical and computational model describing the coupling between blood flow mechanics and oxygen ((Formula presented.)) transport in the retina. Albeit in a simplified manner, the model accounts for the three-dimensional anatomical structure of the retina, consisting in a layered tissue nourished by an arteriolar/venular network laying on the surface proximal to the vitreous. Capillary plexi, originating from terminal arterioles and converging into smaller venules, are embedded in two distinct tissue layers. Arteriolar and venular networks are represented by fractal trees, whereas capillary plexi are represented using a simplified lumped description. In the model, (Formula presented.) is transported along the vasculature and delivered to the tissue at a rate that depends on the metabolic demand of the various tissue layers. First, the model is validated against available experimental results to identify baseline conditions. Then, a sensitivity analysis is performed to quantify the influence of blood pressure, blood rheology, oxygen arterial permeability and tissue oxygen demand on the (Formula presented.) distribution within the blood vessels and in the tissue. This analysis shows that: (1) systemic arterial blood pressure has a strong influence on the (Formula presented.) profiles in both blood and tissue; (2) plasma viscosity and metabolic consumption rates have a strong influence on the (Formula presented.) tension at the level of the retinal ganglion cells; and (3) arterial (Formula presented.) permeability has a strong influence on the (Formula presented.) saturation in the retinal arterioles