Objective Recent methods for imaging microvascular structures provide geometrical data on networks containing thousands of segments. Prediction of functional properties, such as solute transport, requires information on blood flow rates also, but experimental measurement of many individual flows is difficult. Here, a method is presented for estimating flow rates in a microvascular network based on incomplete information on the flows in the boundary segments that feed and drain the network. Methods With incomplete boundary data, the equations governing blood flow form an underdetermined linear system. An algorithm was developed that uses independent information about the distribution of wall shear stresses and pressures in microvessels to resolve this indeterminacy, by minimizing the deviation of pressures and wall shear stresses from target values. Results The algorithm was tested using previously obtained experimental flow data from four microvascular networks in the rat mesentery. With two or three prescribed boundary conditions, predicted flows showed relatively small errors in most segments and fewer than 10% incorrect flow directions on average. Conclusions The proposed method can be used to estimate flow rates in microvascular networks, based on incomplete boundary data and provides a basis for deducing functional properties of microvessel networks.

We have developed a highly detailed mathematical model of solute transport in the renal medulla of the rat kidney to study the impact of the structured organization of nephrons and vessels revealed in anatomic studies. The model represents the arrangement of tubules around a vascular bundle in the outer medulla and around a collecting duct cluster in the upper inner medulla. Model simulations yield marked gradients in intrabundle and interbundle interstitial fluid oxygen tension (PO2), NaCl concentration, and osmolality in the outer medulla, owing to the vigorous active reabsorption of NaCl by the thick ascending limbs. In the inner medulla, where the thin ascending limbs do not mediate significant active NaCl transport, interstitial fluid composition becomes much more homogeneous with respect to NaCl, urea, and osmolality. Nonetheless, a substantial PO2 gradient remains, owing to the relatively high oxygen demand of the inner medullary collecting ducts. Perhaps more importantly, the model predicts that in the absence of the three-dimensional medullary architecture, oxygen delivery to the inner medulla would drastically decrease, with the terminal inner medulla nearly completely deprived of oxygen. Thus model results suggest that the functional role of the three-dimensional medullary architecture may be to preserve oxygen delivery to the papilla. Additionally, a simulation that represents low medullary blood flow suggests that the separation of thick limbs from the vascular bundles substantially increases the risk of the segments to hypoxic injury. When nephrons and vessels are more homogeneously distributed, luminal PO2 in the thick ascending limb of superficial nephrons increases by 66% in the inner stripe. Furthermore, simulations predict that owing to the Bohr effect, the presumed greater acidity of blood in the interbundle regions, where thick ascending limbs are located, relative to that in the vascular bundles, facilitates the delivery of O2 to support the high metabolic requirements of the thick limbs and raises NaCl reabsorption.

Acute kidney injury, a prevalent complication of cardiac surgery performed on cardiopulmonary bypass (CPB), is thought to be driven partly by hypoxic damage in the renal medulla. To determine the causes of medullary hypoxia during CPB, we modeled its impact on renal hemodynamics and function, and thus oxygen delivery and consumption in the renal medulla. The model incorporates autoregulation of renal blood flow and glomerular filtration rate and the utilization of oxygen for tubular transport. The model predicts that renal medullary oxygen delivery and consumption are reduced by a similar magnitude during the hypothermic (down to 28°C) phase of CPB. Thus, the fractional extraction of oxygen in the medulla, an index of hypoxia, is increased only by 58% from baseline. However, during the rewarming phase (up to 37°C), oxygen consumption by the medullary thick ascending limb increases 2.3‐fold but medullary oxygen delivery increases only by 33%. Consequently, the fractional extraction of oxygen in the medulla is increased 2.7‐fold from baseline. Thus, the renal medulla is particularly susceptible to hypoxia during the rewarming phase of CPB. Furthermore, autoregulation of both renal blood flow and glomerular filtration rate is blunted during CPB by the combined effects of hemodilution and nonpulsatile blood flow. Thus, renal hypoxia can be markedly exacerbated if arterial pressure falls below its target level of 50 mmHg. Our findings suggest that tight control of arterial pressure, and thus renal oxygen delivery, may be critical in the prevention of acute kidney injury associated with cardiac surgery performed on CPB.

In striated muscle, the number of capillaries containing moving red blood cells increases with increasing metabolic demand. This phenomenon, termed capillary recruitment, has long been recognized, but its mechanism has been unclear. Here, a theoretical model for metabolic blood flow regulation in a heterogeneous network is used to test the hypothesis that capillary recruitment occurs as a result of active control of arteriolar diameters, combined with unequal partition of hematocrit at diverging microvascular bifurcations. The network structure is derived from published observations of hamster cremaster muscle in control and dilated states. The model for modulation of arteriolar diameters includes length-tension characteristics of vascular smooth muscle and responses of smooth muscle tone to myogenic, shear-dependent, and metabolic stimuli. Blood flow is simulated including nonuniform hematocrit distribution. Convective and diffusive oxygen transport in the network is simulated. Oxygen-dependent metabolic signals are assumed to be conducted upstream from distal vessels to arterioles. With increasing oxygen demand, arterioles dilate, blood flow increases, and the numbers of flowing arterioles and capillaries, as defined by red blood cell flux above a small threshold value, increase. Unequal hematocrit partition at diverging bifurcations contributes to recruitment and enhances tissue oxygenation. The results imply that capillary recruitment, as observed in the hamster cremaster preparations, can occur as a consequence of local control of arteriolar tone and the resulting nonuniform changes in red blood cell fluxes, and provide an explanation for observations of sequential recruitment of individual capillaries in response to modulation of terminal arteriolar diameter.