Analysis of coupled intra- and extraluminal flows for single and multiple capillaries

Matched asymptotic expansions are used to study a model of the coupled fluid flow in the capillaries and tissue of the microcirculation. These capillaries are long, narrow cylindrical tubes embedded in a uniform tissue space. The capillary, or intraluminal, flow is assumed to be that of an incompressible Navier-Stokes fluid wherein colloids are represented as dilute solute; the extraluminal flow in the tissue is according to Darcy's law. Central to this fluid exchange is the boundary condition on the fluid radial velocity at the semipermeable wall of the capillary. This boundary condition, involving the local hydrostatic and colloidal osmotic pressures in both the capillary and the tissue, together with the radial gradient of the tissue hydrostatic pressure, couples the intra- and extraluminal flow fields. With this model we investigate the relationship between transport properties, hydrostatic pressures, and flow exchange for a single capillary, and describe the fluid transport in the tissue space produced by an array of such capillaries.     

A microstructurally based continuum model of cartilage viscoelasticity and permeability incorporating measured statistical fiber orientations

The remarkable mechanical properties of cartilage derive from an interplay of isotropically distributed, densely packed and negatively charged proteoglycans; a highly anisotropic and inhomogeneously oriented fiber network of collagens; and an interstitial electrolytic fluid. We propose a new 3D finite strain constitutive model capable of simultaneously addressing both solid (reinforcement) and fluid (permeability) dependence of the tissue’s mechanical response on the patient-specific collagen fiber network. To represent fiber reinforcement, we integrate the strain energies of single collagen fibers—weighted by an orientation distribution function (ODF) defined over a unit sphere—over the distributed fiber orientations in 3D. We define the anisotropic intrinsic permeability of the tissue with a structure tensor based again on the integration of the local ODF over all spatial fiber orientations. By design, our modeling formulation accepts structural data on patient-specific collagen fiber networks as determined via diffusion tensor MRI. We implement our new model in 3D large strain finite elements and study the distributions of interstitial fluid pressure, fluid pressure load support and shear stress within a cartilage sample under indentation. Results show that the fiber network dramatically increases interstitial fluid pressure and focuses it near the surface. Inhomogeneity in the tissue’s composition also increases fluid pressure and reduces shear stress in the solid. Finally, a biphasic neo-Hookean material model, as is available in commercial finite element codes, does not capture important features of the intra-tissue response, e.g., distributions of interstitial fluid pressure and principal shear stress.     

Effects of vasomotion and venous pressure elevation on capillary-tissue fluid exchange across hetero-porous membrane

Flow autoregulation in the arteriolar network serves to maintain the capillary-tissue fluid balance by regulation of capillary pressure. In the present study, we have examined theoretically the effects of arteriolar vasomotion and venous pressure elevation on the capillary fluid exchange, the interstitial fluid pressure, and the interstitial osmotic pressure during capillary pressure regulation. We used Starling's hypothesis and extended it to include a consideration of a parallel hetero-porous pathway and to determine the effects of plasma protein filtration on interstitial fluid pressure and osmotic pressure. We have found that arteriolar vasomotion plays a primary role in protecting the capillary-tissue fluid balance during the elevation of capillary flow and that it is a secondary mechanism for the regulation of capillary arterial pressure.     

Implications of Transvascular Fluid Exchange in Nonlinear,Biphasic Analyses of Flow-Controlled Infusion in Brain

A nonlinear, coupled biphasic-mass transport model that includes transvascular fluid exchange is proposed for flow-controlled infusions in brain tissue. The model accounts for geometric and material nonlinearities, a hydraulic conductivity dependent on deformation, and transvascular fluid exchange according to Starling’s law. The governing equations were implemented in a custom-written code assuming spherical symmetry and using an updated Lagrangian finite-element algorithm. Results of the model indicate that, using normal physiological values of vascular permeability, transvascular fluid exchange has negligible effects on tissue deformation, fluid pressure, and transport of the infused agent. As vascular permeability may be increased artificially through methods such as administering nitric oxide, a parametric study was conducted to determine how increased vascular permeability affects flow-controlled infusion. Increased vascular permeability reduced both tissue deformation and fluid pressure, possibly reducing damage to tissue adjacent to the infusion catheter. Furthermore, the loss of fluid to the vasculature resulted in a significantly increased interstitial fluid concentration but a modestly increased tissue concentration. From a clinical point of view, this increase in concentration could be beneficial if limited to levels below which toxicity would not occur. However, the modestly increased tissue concentration may make the increase in interstitial fluid concentration difficult to assess in vivo using co-infused radiolabeled agents.     

Model study of the effects of interactions between systemic and peripheral circulation on interstitial fluid balance

Interstitial fluid balance is severely altered in microgravity, but the mechanisms underlying the fluid shift from lower to upper body are still partially unclear. A lumped parameter model of the arterial tree with active and non linear modulation of peripheral resistances and capillary fluid exchange was adopted to simulate the response of microcirculation to pulsatility and edema. Results suggest that myogenic regulation not only impinges on arteriolar radius, but it also indirectly affects interstitial fluid balance. Non linear dynamics of blood pressure (BP) and flow in capillary beds are influenced by systemic pulsatility, hinting that local activity is involved in the response to peripheral edema as well.     

A study of nutritional transport in capillary-tissue exchange system

A two region flow model has been developed for a capillary-tissue exchange unit. Nutrient dissolved in plasma enters into the tissue from capillary through diffusion, filtration and osmosis. The governing non-linear coupled partial differential equations in the two regions (capillary and tissue) have been solved separately with suitable boundary and matching conditions. The results for the variation of Taylor's Diffusivity Coefficient and concentration of a nutrient in the tissue region have been brought out for various values of the parameters involved in the analysis and discussed. It has been particularly noted that the penetration depth in the tissue for the nutritional transport can be considered as an important diagnostic parameter for many cardiovascular diseases.     

Pulmonary edema: ischemia reperfusion endothelial injury and its reversal by c-AMP.

A review of the factors that oppose pulmonary edema formation (alveolar flooding) when capillary pressure is elevated are presented for a normal capillary endothelial barrier and for damaged endothelium associated with ischemia/reperfusion in rabbit, rat, and dog lungs. Normally, tissue pressure, the plasma protein osmotic pressure gradient acting across the capillary wall and lymph flow (Edema Safety Factors) increase to prevent the build-up of fluid in the lung's interstitium when capillary pressure increases. No measureable alveolar edema fluid accumulates until capillary pressure exceeds 30 mmHg. When the capillary wall has been damaged, interstitial edema develops at lower capillary pressures because the plasma protein osmotic pressure will not change greatly to oppose capillary filtration, but lymph flow increases to very high levels to remove the increased filtrate and the result is that capillary pressures can increase to 20-25 mmHg before alveolar flooding results. In addition, the mechanisms responsible for producing pulmonary endothelial damage with ischemia/reperfusion are reviewed and the effects of O2 radical scavengers, neutrophil depletion or altering their adherence to the endothelium, and increasing cAMP on reversing the damage to the pulmonary endothelium is presented.     

Starling forces that oppose filtration after tissue oncotic pressure is increased

We tested the hypothesis that the effective oncotic force that opposes fluid filtration across the microvessel wall is the local oncotic pressure difference across the endothelial surface glycocalyx and not the global difference between the plasma and tissue. In single frog mesenteric microvessels perfused and superfused with solutions containing 50 mg/ml albumin, the effective oncotic pressure exerted across the microvessel wall was not significantly different from that measured when the perfusate alone contained albumin at 50 mg/ml. Measurements were made during transient and steady-state filtration at capillary pressures between 10 and 35 cmH(2)O. A cellular-level model of coupled water and solute flows in the interendothelial cleft showed water flux through small breaks in the junctional strand limited back diffusion of albumin into the protected space on the tissue side of the glycocalyx. Thus oncotic forces opposing filtration are larger than those estimated from blood-to-tissue protein concentration differences, and transcapillary fluid flux is smaller than estimated from global differences in oncotic and hydrostatic pressures.     

Evaluation of Starling forces in the equine digit

A pump-perfused extracorporeal digital preparation was used to evaluate blood flow, arterial pressure, venous pressure, isogravimetric capillary filtration coefficient, capillary pressure, and vascular compliance in six normal horses. From these data, pre- and postcapillary resistances and pre- and postcapillary resistance ratios were determined. Vascular and tissue oncotic pressures were estimated from plasma and lymph protein concentrations, respectively. By use of the collected and calculated data, tissue pressure in the digit was calculated using the Starling equation. In the isolated equine digit, isogravimetric capillary pressure averaged 36.7 mmHg, plasma and lymph oncotic pressures averaged aged 19.12 and 6.6 mmHg, respectively, interstitial fluid pressure averaged 25.6 mmHg, and the capillary filtration coefficient averaged 0.0013 ml.min-1.mm-1.100 g-1. Our results indicate that digital capillary pressure in the laterally recumbent horse is much higher than in analogous tissues in other species such as dog and human. However, the potential edemagenic effects of this high digital capillary pressure are opposed by at least two mechanisms: 1) a high tissue pressure and 2) a low microvascular surface area for fluid exchange and/or a low microvascular permeability to filtered fluid.     

Microvascular fluid filtration capacity (Kf) assessed with cumulative small venous pressure steps and with various degrees of tilt

Tilt procedures are frequently used to test central and peripheral cardio-vascular reflexes. We have previously used venous congestion strain gauge plethysmography for measurement of fluid filtration capacity (Kf) in human legs and have shown that, providing small cumulative venous congestion pressure steps are applied, venous congestion pressure can be increased to arterial diastolic pressure without activating peripheral vasoconstrictor mechanisms. We have also studied the effect of passive tilting on Kf and have shown that the procedure does not influence the measured value Kf indicating that passive tilting does not after the total surface area available for fluid filtration, but rather the blood flow in the microvessels of the tissue under study. In the present protocol we compared the fluid filtration (Jv) resulting from small (7-10 mmHg) cumulative pressure steps with those obtained by altering hydrostatic load with progressive increases and decreases of head down tilt of -8 degrees -15 degrees and -30 degrees, followed by a similar pattern of 15 degrees, 30 degrees and 70 degrees of head up tilt. The values of Jv obtained in response to these procedures were compared with those deduced from the relationship between fluid filtration and venous congestion pressure (Pcuff) obtained during the small cumulative pressure step protocol. It was reasoned that reflex activation, by the tilt induced pressure load, would cause a reduction in local blood flow and enhanced microvascular fluid extraction. The resulting local increase in colloid osmotic pressure would give rise to lower values of Jv than those predicted on the basis of the Kf slope.     

Wear particle diffusion and tissue differentiation in TKA implant fibrous interfaces

In the context of mechanical loosening, we studied the hypothesis that wear-particle migration in the fibrous membrane under tibial plateaus after total knee arthroplasty can be explained by the pumping effects of the interstitial fluid in the tissue. Further, as a secondary objective we investigated the possibility that interface-tissue differentiation is influenced by interstitial fluid flow and strain, as mechanical effects of interface motions. For comparative reasons, we analyzed a previously published simplified two-dimensional finite-element model, this time assuming biphasic tissue properties. We wanted to determine hydrostatic pressure and flow velocities in the fluid phase, in addition to stresses and strains, for time-dependent loading of the plateau. We found that fluid flow in the interface was extremely slow, except in the periphery. Hence, loosening due to particle-induced bone resorption appears improbable. The results, however, do support the idea that particles migrate with fluid flow, when such flow occurs. Where fibrous tissue tends to be prominent in reality, the fluid is repeatedly extruded and reabsorbed in the model. Where these values are low, fibrocartilage is commonly found. When material properties were varied to subsequently represent fibrocartilage and two stages of mineralization, the strains and fluid velocities is reduced. Fluid pressure, however, did not change. Our results refute the hypothesis that wear particles are pumped through the interface tissue below a TKA but support the hypothesis that interface tissue type and loosening processes are influenced by mechanical tissue variables such as tissue strain and interstitial fluid velocity.     

Control and Modulation of Fluid Flow in the Rat Kidney

We have developed a mathematical model of the rat’s renal hemodynamics in the nephron level, and used that model to study flow control and signal transduction in the rat kidney. The model represents an afferent arteriole, glomerular filtration, and a segment of a short-loop nephron. The model afferent arteriole is myogenically active and represents smooth muscle membrane potential and electrical coupling. The myogenic mechanism is based on the assumption that the activity of nonselective cation channels is shifted by changes in transmural pressure, such that elevation in pressure induces vasoconstriction, which increases resistance to blood flow. From the afferent arteriole’s fluid delivery output, glomerular filtration rate is computed, based on conservation of plasma and plasma protein. Chloride concentration is then computed along the renal tubule based on solute conservation that represents water reabsorption along the proximal tubule and the water-permeable segment of the descending limb, and chloride fluxes driven by passive diffusion and active transport. The model’s autoregulatory response is predicted to maintain stable renal blood flow within a physiologic range of blood pressure values. Power spectra associated with time series predicted by the model reveal a prominent fundamental peak at ～165 mHz arising from the afferent arteriole’s spontaneous vasomotion. Periodic external forcings interact with vasomotion to introduce heterodynes into the power spectra, significantly increasing their complexity.     

Numerical Modeling of Interstitial Fluid Flow Coupled with Blood Flow through a Remodeled Solid Tumor Microvascular Network

Modeling of interstitial fluid flow involves processes such as fluid diffusion, convective transport in extracellular matrix, and extravasation from blood vessels. To date, majority of microvascular flow modeling has been done at different levels and scales mostly on simple tumor shapes with their capillaries. However, with our proposed numerical model, more complex and realistic tumor shapes and capillary networks can be studied. Both blood flow through a capillary network, which is induced by a solid tumor, and fluid flow in tumor’s surrounding tissue are formulated. First, governing equations of angiogenesis are implemented to specify the different domains for the network and interstitium. Then, governing equations for flow modeling are introduced for different domains. The conservation laws for mass and momentum (including continuity equation, Darcy’s law for tissue, and simplified Navier–Stokes equation for blood flow through capillaries) are used for simulating interstitial and intravascular flows and Starling’s law is used for closing this system of equations and coupling the intravascular and extravascular flows. This is the first study of flow modeling in solid tumors to naturalistically couple intravascular and extravascular flow through a network. This network is generated by sprouting angiogenesis and consisting of one parent vessel connected to the network while taking into account the non-continuous behavior of blood, adaptability of capillary diameter to hemodynamics and metabolic stimuli, non-Newtonian blood flow, and phase separation of blood flow in capillary bifurcation. The incorporation of the outlined components beyond the previous models provides a more realistic prediction of interstitial fluid flow pattern in solid tumors and surrounding tissues. Results predict higher interstitial pressure, almost two times, for realistic model compared to the simplified model.     

Steady propagation of a liquid plug in a two-dimensional channel

In this study, we investigate the steady propagation of a liquid plug within a two-dimensional channel lined by a uniform, thin liquid film. The Navier-Stokes equations with free-surface boundary conditions are solved using the finite volume numerical scheme. We examine the effect of varying plug propagation speed and plug length in both the Stokes flow limit and for finite Reynolds number (Re). For a fixed plug length, the trailing film thickness increases with plug propagation speed. If the plug length is greater than the channel width, the trailing film thickness agrees with previous theories for semi-infinite bubble propagation. As the plug length decreases below the channel width, the trailing film thickness decreases, and for finite Re there is significant interaction between the leading and trailing menisci and their local flow effects. A recirculation flow forms inside the plug core and is skewed towards the rear meniscus as Re increases. The recirculation velocity between both tips decreases with the plug length. The macroscopic pressure gradient, which is the pressure drop between the leading and trailing gas phases divided by the plug length, is a function of U and U2, where U is the plug propagation speed, when the fluid property and the channel geometry are fixed. The U2 term becomes dominant at small values of the plug length. A capillary wave develops at the front meniscus, with an amplitude that increases with Re, and this causes large local changes in wall shear stresses and pressures.     

Peripheral-layer viscosity and microstructural effects on the capillary-tissue fluid exchange

A theoretical investigation of capillary-tissue fluid exchange has been studied including the characteristics and influence of the boundaries and media through which the fluid flows. Filtration from a cylindrical capillary into the concentrically surrounding tissue space and flow from a capillary into the tissue across a thin membrane are analyzed in detail. In has been observed that the filtration efficiency of the functional unit decreases as the viscosity of the peripheral layer increases. Contrary to the results of Apelblat [17], the slip velocity at the porous boundary plays a dominant role in filtration efficiency. It has also been noticed that the filtration efficiency decreases as the slip velocity at the porous boundary increases.     

Positron emission tomography in patients with clinically diagnosed Alzheimer's disease.

Fourteen patients who had clinically diagnosed Alzheimer''s disease with mild to severe dementia (mean age 69.1 years) were evaluated by calculation of local cerebral metabolic rate for glucose (LCMR-gl) based on uptake of 18F-2-fluoro-2-deoxyglucose (FDG) detected with positron emission tomography (PET). PET scanning showed that the patients had significantly lower LCMR-gl values than 11 age-matched neurologically normal volunteers (mean age 66.3 years). The differences were most marked in the temporal cortex, followed by the frontal, parietal and occipital cortex. In each case the LCMR-gl value was below the lowest control value in at least one cortical area and usually in several; the reduction in LCMR-gl and the number of regions involved in the patients increased with the severity of the dementia. Deficits noted in neuropsychologic testing generally correlated with those predicted from loss of regional cortical metabolism. The patients with Alzheimer''s disease were also examined with magnetic resonance imaging, computed tomography or both; the degree of atrophy found showed only a poor correlation with the neuropsychologic deficit. Significant atrophy was also noted in some of the controls. A detailed analysis of LCMR-gl values in selected cerebral regions of various sizes refuted the hypothesis that the reduction in cortical glucose metabolism in Alzheimer''s disease is due to the filling by metabolically inert cerebrospinal fluid of space created by tissue atrophy.     

Multiscale modeling of lymphatic drainage from tissues using homogenization theory

Lymphatic capillary drainage of interstitial fluid under both steady-state and inflammatory conditions is important for tissue fluid balance, cancer metastasis, and immunity. Lymphatic drainage function is critically coupled to the fluid mechanical properties of the interstitium, yet this coupling is poorly understood. Here we sought to effectively model the lymphatic-interstitial fluid coupling and ask why the lymphatic capillary network often appears with roughly a hexagonal architecture. We use homogenization method, which allows tissue-scale lymph flow to be integrated with the microstructural details of the lymphatic capillaries, thus gaining insight into the functionality of lymphatic anatomy. We first describe flow in lymphatic capillaries using the Navier-Stokes equations and flow through the interstitium using Darcy's law. We then use multiscale homogenization to derive macroscale equations describing lymphatic drainage, with the mouse tail skin as a basis. We find that the limiting resistance for fluid drainage is that from the interstitium into the capillaries rather than within the capillaries. We also find that between hexagonal, square, and parallel tube configurations of lymphatic capillary networks, the hexagonal structure is the most efficient architecture for coupled interstitial and capillary fluid transport; that is, it clears the most interstitial fluid for a given network density and baseline interstitial fluid pressure. Thus, using homogenization theory, one can assess how vessel microstructure influences the macroscale fluid drainage by the lymphatics and demonstrate why the hexagonal network of dermal lymphatic capillaries is optimal for interstitial tissue fluid clearance.     

The response of subpleural pulmonary capillary endothelium to hydrothorax in rats

The principal focus of this study was to evaluate the hypothesis that increased interstitial fluid pressures served to stimulate de novo vesicle formation in pulmonary capillary endothelium. Direct measurements of interstitial fluid pressures within the alveolar septa pose great technical difficulty. The pleural space and subpleural capillaries are easily accessible, and thus, provide a more feasible model to test this hypothesis. After hydrostatic pressure of pleural space fluid was increased by periodic saline infusions into the pleural cavity, vesicle numerical densities were significantly increased in portions of the subpleural capillary endothelium. Those segments of the endothelium that directly apposed the interstitium of the visceral pleura displayed de novo vesicle formation. The endothelial segments located immediately adjacent to the alveolar epithelium were not affected by the elevated interstitial fluid pressures. In addition to the increased vesiculation, those same segments of the endothelium were characterized by increased attenuation of their cytoplasmic compartments. These conformational changes in the plasmalemma of portions of the subpleural capillary endothelium provide support to the tentative hypothesis, however, whether the increased numbers of vesicles contribute to a potential transendothelial transport system or expand a possible static network of membrane invaginations remains uncertain.