Multiscale Modelling of Fluid and Drug Transport in Vascular Tumours

A model for fluid and drug transport through the leaky neovasculature and porous interstitium of a solid tumour is developed. The transport problems are posed on a micro-scale characterized by the inter-capillary distance, and the method of multiple scales is used to derive the continuum equations describing fluid and drug transport on the length scale of the tumour (under the assumption of a spatially periodic microstructure). The fluid equations comprise a double porous medium, with coupled Darcy flow through the interstitium and vasculature, whereas the drug equations comprise advection–reaction equations; in each case the dependence of the transport coefficients on the vascular geometry is determined by solving micro-scale cell problems.     

Numerical simulation of blood and interstitial flow through a solid tumor

A theoretical framework is presented for describing blood flow through the irregular vasculature of a solid tumor. The tumor capillary bed is modeled as a capillary tree of bifurcating segments whose geometrical construction involves deterministic and random parameters. Blood flow along the individual capillaries accounts for plasma leakage through the capillary walls due to the transmural pressure according to Sterling’s law. The extravasation flow into the interstitium is described by Darcy’s law for a biological porous medium. The pressure field developing in the interstitium is computed by solving Laplace’s equation subject to derived boundary conditions at the capillary vessel walls. Given the arterial, venous, and tumor surface pressures, the problem is formulated as a coupled system of integral and differential equations arising from the interstitium and capillary flow transport equations. Numerical discretization yields a system of linear algebraic equations for the interstitial and capillary segment pressures whose solution is found by iterative methods. Results of numerical computations document the effect of the interstitial hydraulic and vascular permeability on the fractional plasma leakage. Given the material properties, the fractional leakage reaches a maximum at a particular grade of the bifurcating vascular tree.     

Pressure-flow behavior of pulmonary interstitium

A method to measure the pressure-flow behavior of the interstitium around large pulmonary vessels is presented. Isolated rabbit lungs were degassed, and the air spaces and vasculature were inflated with a silicon rubber compound. After the rubber had hardened the caudal lobes were sliced into 1-cm-thick slabs. Two chambers were bonded to opposite sides of a slab enclosing a large blood vessel and were filled with saline containing 3 g/dl albumin. The flow through the interstitium surrounding the vessel was measured at a constant driving pressure of 5 cmH2O and at various mean interstitial pressures. Flow decreased with a reduction of mean interstitial pressure and reached a limiting minimum value at approximately -9 cmH2O. The pressure-flow behavior was analyzed under the assumptions that the interstitium is a porous material described by a single permeability constant that increases with hydration and that the expansion of the interstitium with interstitial pressure was due to the elastic response of the surrounding rubber compound. This resulted in an interstitial resistance (reciprocal of permeability constant) of 1.31 +/- 1.03 (SD) cmH2O.h.cm-2 and a ratio of interstitial cuff thickness to vessel radius of 0.022 +/- 0.007 (SD), n = 11. The phenomenon of flow limitation was demonstrated by holding the upstream pressure constant at 15 cmH2O and measuring the flow while the downstream pressure was reduced. The flow was limited at downstream pressures below -10 cmH2O.     

A one tube flow problem arising in physiology

A mathematical analysis, including existence and uniqueness, is given for some boundary value problems which model the flow of a fluid-solute mixture in a tube which is placed in an interstitium. The model permits an interchange of fluid and solute across the tube walls.     

A Model for Fluid Drainage by the Lymphatic System

This study investigates the fluid flow through tissues where lymphatic drainage occurs. Lymphatic drainage requires the use of two valve systems, primary and secondary. Primary valves are located in the initial lymphatics. Overlapping endothelial cells around the circumferential lining of lymphatic capillaries are presumed to act as a unidirectional valve system. Secondary valves are located in the lumen of the collecting lymphatics and act as another unidirectional valve system; these are well studied in contrast to primary valves. We propose a model for the drainage of fluid by the lymphatic system that includes the primary valve system. The analysis in this work incorporates the mechanics of the primary lymphatic valves as well as the fluid flow through the interstitium and that through the walls of the blood capillaries. The model predicts a piecewise linear relation between the drainage flux and the pressure difference between the blood and lymphatic capillaries. The model describes a permeable membrane around a blood capillary, an elastic primary lymphatic valve and the interstitium lying between the two.     

人体纤维结缔组织网络中的主动界面液体流动新现象及其机制研究进展

认识和理解组织液在凝胶样组织间隙中的流动规律和力学机制是近百年来的重大医学科学问题之一。本文回顾了生理学中关于组织液流动的若干代表性研究、颅内血管周围间隙中的液体流动研究及经络示踪成像研究等,并介绍了已有的系列研究结果,即通过追踪人体和动物体内不同部位组织间隙中的示踪剂流动,发现组织液沿动脉和静脉血管外膜、神经和皮肤等至少四种解剖结构进行长程而有序的流动;其中的皮肤组织液长程流动通路与人体四肢远端穴位区相连接;组织液长程流动通路的内部结构是由有序纤维集合而成的纵向纤维束,组织液以纤维等固相结构为轨道而流动,形成贯穿全身的"组织液界面流动网络",具有重要的医学科学和中西医结合研究价值。     

Perialveolar interstitial resistance and compliance in isolated rat lung

We have developed a method to characterize fluid transport through the perialveolar interstitium using micropuncture techniques. In 10 experiments we established isolated perfused rat lung preparations. The lungs were initially isogravimetric at 10 cmH2O arterial pressure, 2 cmH2O venous pressure, and 5 cmH2O alveolar pressure. Perialveolar interstitial pressure was determined by micropuncture at alveolar junctions by use of the servo-null technique. Simultaneously a second micropipette was placed in an alveolar junction 20-40 microns away, and a bolus of albumin solution (3.5 g/100 ml) was injected. The resulting pressure transient was recorded for injection durations of 1 and 4 s in nonedematous lungs. The measurements were repeated after gross edema formation induced by elevated perfusion pressure. We model the interstitium as a homogeneous linearly poroelastic material and assume the initial pressure distribution due to the injection to be Gaussian. The pressure decay is inversely proportional to time, with time constant T, where T is a measure of the ratio of interstitial tissue stiffness to interstitial resistance to fluid flow. A linear regression was performed on the reciprocal of the pressure for the decaying portion of the transients to determine T. Comparing pressure transients in nonedematous and edematous lungs, we found that T was 4.0 +/- 1.4 and 1.4 +/- 0.6 s, respectively. We have shown that fluid transport through the pulmonary interstitium on a local level is sensitive to changes in interstitial stiffness and resistance. These results are consistent with the decreased stiffness and resistance in the perialveolar interstitium that accompany increased hydration.     

Two-phase pulsatile flows through porous conical tubes of small diameters. Modelisation of the blood microcirculation

A theoretical study concerning two-component fluid pulsating flow through porous conical ducts is presented. The model corresponds to blood flows through small diameter porous conical vessels. This approach is based on a finite difference method. The physical hypothesis used were based on findings from simultaneous visualization methods. The influence of geometrical, hydrodynamical and structural parameters is systematically examined and related to velocity profiles, hydrostatic pressure.     

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.     

Xanthan degragation by biofilm in porous media

Biological activity in oil reservoirs can cause significant problems such as souring and plugging. This study focuses on the problem of polymer degradation and permeability reduction due to biofilm formation during polymer injection for improved oil recovery. Polymers are included in injection fluids to increase their viscosity. Results relating biological processes and polymer degradation to fluid‐dynamic conditions in a laboratory model porous medium are presented.

A transparent flow cell with an etched two‐dimensional network of pores served as a model porous medium. A sterile xanthan polymer and natural sea water solution were continuously injected into the porous medium. A bacterial culture capable of xanthan degradation was introduced into the cell by a single injection. Some of the cells from this culture attached to the pore walls forming an immobile bacterial culture, termed biofilm. The development of this biofilm, its xanthan degradation and its effect on permeability were measured.

The effects of injection rate and rate transitions were analyzed. Injection fluid viscosity was reduced by 30% after 5 min flow through the porous medium at the maximum steady state degradation rate observed. Permeability was significantly reduced by the xanthan degrading biofilm, causing an increase in pressure drop through the porous medium of up to 80%. Polymer injection in oil reservoirs may, therefore, have negative effects on oil recovery, unless efficient biofouling control is applied. The methodology presented may serve as a tool in the development of biofouling control measures in porous media.     

A fully coupled porous media and channels flow approach for simulation of blood and bile flow through the liver lobules

Two dimensional, steady state, and incompressible blood and bile flows through the liver lobules are numerically simulated. Two different geometric models A and B are proposed to study the effects of lobule structure on the fluid flow behaviour. In Model A, the lobule tissue is represented as a hexagonal shape porous medium with a set of flow channels at its vertices accounting for the hepatic artery, portal and central veins along with bile ductules. Model B is a channelized porous medium constructed by adding a set of flow channels, representing the bile canaliculies and lobule sinusoids, to Model A. The bile and blood flow through the lobule is simulated by the finite element approach, based on the Darcy/Brinkman equations in the lobule tissue and the Navier-Stokes (or Stokes) equations in the flow channels. In Model B, a transmission factor on the boundaries of the bile canaliculies is introduced to connect the bile and blood flows. First, a single regular lobule is utilized to exhibit the fluid flow pattern through the liver lobule represented by proposed geometric models. Then, the model is extended to a group of liver lobules to demonstrate the flow through a liver slice represented by irregular lobules. Numerical results indicate that the Darcy and Brinkman equations provide nearly the same solutions for Model A and similar solutions with a little difference for Model B. It is shown that the existence of sinusoids and bile canaliculies inside the liver lobules has noticeable effects on its fluid flow pattern, in terms of pressure and velocity fields.     

Instrumentation for advanced microseparations in pharmaceutical analysis and proteomics

Small-volume chromatographic columns are only able to generate narrow peaks when flow rates, injection volume and instrument components, such as detector, connecting tubing and fittings, are matched to the peak dispersion from the column. Criteria for the proper design of chromatographic instrumentation are therefore derived from a general model on total dispersion. The performance of such a system is then experimentally evaluated from applications run on narrow-bore, small-volume columns. In order to achieve flow rates that match the dimensions of such columns, a new concept for electronic flow control (EFC) is introduced. A theoretical optimization of column efficiency and throughput is discussed and the results verified with practical examples on short, narrow-bore columns packed with small, porous and superficially porous particles. For complex sample mixtures, the concept of peak capacity is introduced and applied to orthogonal separation principles in multiple chromatographic dimensions through column switching techniques.     

Elastohydrodynamic Lubrication of Porous Substrates. Application to Synovial Joint Analysis

This work presents computational results for an elastohydrodynamic model with a porous wall. The model - inspired by the cartilage exudation phenomenon arising in living joints - allows the lubricating fluid to be transferred from the porous walls to the flow channel and viceversa. The system is represented through a strongly coupled set of nonlinear equations which involves hydrodynamic lubrication, plane elasticity and continuity in porous media. The solution technique employed is based on the finite element method, Newton iteration and parametric continuation processes. The predictions obtained uncover mechanisms that might be responsible for the extremely low friction coefficients found in synovial joints.     

A new approach to model the spatiotemporal development of biofilm phase in porous media

Bacteria can exist within biofilms that are attached to the solid matrix of a porous medium. Under certain conditions, the biomass can fully occupy the pore space leading to reduced hydraulic conductivity and mass transport. Here, by treating biofilm as a growing, high-viscosity phase, a novel macroscopic approach to model biofilm spatial expansion and its corresponding effects on porous medium hydraulic properties is presented. The separate yet coupled flow of the water and biofilm phases is handled by using relative permeability curves that allow for biofilm movement within the porous medium and bioclogging effects. Fluid flow is governed by Darcy's law and component transport is set by the convection-diffusion equation reaction terms for each component. Here, the system of governing equations is solved by using a commercial multiphase flow reservoir simulator, which is used to validate the model against published laboratory experiments. A comparison of the model and experimental observations reveal that the model provides a reasonable means to predict biomass development in the porous medium. The results reveal that coupled flow of water and movement of biofilm, as described by relative permeability curves, is complex and has a large impact on the development of biomass and consequent bioclogging in the porous medium.     

A critical parameter for transcapillary exchange of small solutes in countercurrent systems

Small solute transport by a countercurrent capillary loop was studied using a theoretical model. In the model, the afferent and the efferent limbs of the loop share a common interstitial space, with which exchange of solute occurs. Sources of solute, epithelial cells, exist near capillaries and secret solute into the interstitial fluid. Parameters based on experimental measurements on young Sprague-Dawley rats were used in the model, and asymptotic solutions were derived. Comparison of the solute distribution in the interstitium between a capillary loop and a single capillary reveals that the ratio of the product of permeability (P(1)) and surface area (A(1)) to flow (F(1)) of the afferent limb, gamma(1)=P(1)A(1)/F(1) is a critical parameter for the countercurrent exchange system. It alone determines whether the countercurrent arrangement of capillaries facilitates clearance of solute from the interstitial fluid, a greater axial gradient of solute in the interstitium from the base to the tip of the capillary loop and a greater effect of flow, F, upon this gradient. The properties of the efferent limb affect the results, but it is gamma(1) that determines the characteristic difference between a capillary loop and a single capillary.     

A model of blood flow distribution during filtration-reabsorption processes in capillaries

A model of blood flow in a capillary was constructed, which takes into account the movement of plasma through its porous wall. The functions of changes in pressure and the rate of blood flow along the capillary were calculated. It was shown that, in the general case, the distribution of hemodynamic parameters as a result of filtration-reabsorption processes is nonlinear. Possible mechanism of tissue edema resulting from the disturbance of the filtration-reabsorption equilibrium were analyzed.     

Modeling tree water flow as an unsaturated flow through a porous medium

The electric circuit analogy has had a profound influence on how tree physiologists measure, model and think about tree water flow. For example, previous models that attempt to account for changes in saturation use the electric circuit analogy to define capacitance as the change in saturation per change in pressure. Given that capacitance is constant, this relationship implies that subjecting a block of wood to a pressure of -2.5 MPa for 2 min results in the same change in saturation as subjecting the same block to the same pressure for 2 days. Given the definition of capacitance, it is unclear how the electric circuit analogy could be used to predict changes in saturation separately from changes in pressure. The inadequacies in the electric circuit analogy discussed in this paper necessitate a new theory of tree water flow that recognizes the sapwood as being a porous medium and explicitly deals with the full implications of the unsaturated flow occurring in the sapwood. The theory proposed in this paper combines the Cohesion theory with a mathematical theory of multiphase flow through porous media. Based on this theory, both saturated and unsaturated tree water flow models are presented. Previous partial differential equation models of tree water flow based on the electric circuit analogy are shown to be mathematically equivalent to the model of saturated porous flow. The unsaturated model of tree water flow explicitly models the pressure profile and the rates of change in saturation and specific interfacial area (a measure of how the water in the unsaturated sapwood is partitioned between mobile and immobile components). The unsaturated model highlights the differences between saturated and unsaturated flow and the need to measure the variables governing tree water flow at higher spatial and temporal resolutions.     

Concentrating engines and the kidney. III. Canonical mass balance equation for multinephron models of the renal medulla.

The canonical mass balance relation derived for the central core model of the renal medulla is extended to medullary models in which an arbitrary assemblage of renal tubules and vascular capillaries exchange with each other both directly and via the medullary interstitium and in which not all of the vascular loops or loops of Henle extend to the papilla. It is shown that if descending limbs of Henle and descending vasa recta enter the medulla at approximately plasma osmolality, the concentration ratio is given by: r = 1/[1 - ft(1 - fu)(1 - fw)], where ft is fractional solute transport out of ascending Henle's limb, fu is fractional urine flow, and fw is fractional dissipation; fw is a measure of the solute returned to the systemic circulation without its isotonic complement of water. A modified equation that applies to the diluting as well as the concentrating kidney is also derived. By allowing concentrations in interstitium and vascular capillaries to become identical at a given medullary level, conservation relations are derived for a multinephron central core model of the renal medulla.     

Mathematical model of blood and interstitial flow and lymph production in the liver

We present a mathematical model of blood and interstitial flow in the liver. The liver is treated as a lattice of hexagonal ‘classic’ lobules, which are assumed to be long enough that end effects may be neglected and a two-dimensional problem considered. Since sinusoids and lymphatic vessels are numerous and small compared to the lobule, we use a homogenized approach, describing the sinusoidal and interstitial spaces as porous media. We model plasma filtration from sinusoids to the interstitium, lymph uptake by lymphatic ducts, and lymph outflow from the liver surface. Our results show that the effect of the liver surface only penetrates a depth of a few lobules’ thickness into the tissue. Thus, we separately consider a single lobule lying sufficiently far from all external boundaries that we may regard it as being in an infinite lattice, and also a model of the region near the liver surface. The model predicts that slightly more lymph is produced by interstitial fluid flowing through the liver surface than that taken up by the lymphatic vessels in the liver and that the non-peritonealized region of the surface of the liver results in the total lymph production (uptake by lymphatics plus fluid crossing surface) being about 5 % more than if the entire surface were covered by the Glisson–peritoneal membrane. Estimates of lymph outflow through the surface of the liver are in good agreement with experimental data. We also study the effect of non-physiological values of the controlling parameters, particularly focusing on the conditions of portal hypertension and ascites. To our knowledge, this is the first attempt to model lymph production in the liver. The model provides clinically relevant information about lymph outflow pathways and predicts the systemic response to pathological variations.     

Mathematical analysis of multisolute renal flow in a single nephron model of the kidney

A single nephron model, which includes the Bowman's space, Cortical interstitium, and Pelvis as well-stirred baths, is investigated. A boundary value problem, which allows for pelvic reflux, is established for the fluid-multisolute flow in the nephron. The implicit function theorem is used to establish the existence and uniqueness of a solution of the boundary value problem for the case of small permeability coefficients and transport rates.Dedicated to Professor Emertius L. P. Burton, Auburn University, for the research guidance he gave this author thirty years agoSupported in part by NIH Grant 7-ROl-DK 38817