Non-Newtonian flow-induced deformation from pressurized cavities in absorbing porous tissues

We investigate the behavior of a spherical cavity in a soft biological tissue modeled as a deformable porous material during an injection of non-Newtonian fluid that follows a power law model. Fluid flows into the neighboring tissue due to high cavity pressure where it is absorbed by capillaries and lymphatics at a rate proportional to the local pressure. Power law fluid pressure and displacement of a solid in the tissue are computed as function of radial distance and time. Numerical solutions indicate that shear thickening fluids exhibit less fluid pressure and induce small solid deformation as compared to shear thinning fluids. Absorption in the biological tissue increases as a consequence of flow-induced deformation for power law fluids. In most cases non-Newtonian results are compared with the viscous fluid case to magnify the differences.     

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.     

Poro-elastic modeling of Syringomyelia – a systematic study of the effects of pia mater,central canal,median fissure,white and gray matter on pressure wave propagation and fluid movement within the cervical spinal cord

Syringomyelia, fluid-filled cavities within the spinal cord, occurs frequently in association with a Chiari I malformation and produces some of its most severe neurological symptoms. The exact mechanism causing syringomyelia remains unknown. Since syringomyelia occurs frequently in association with obstructed cerebrospinal fluid (CSF) flow, it has been hypothesized that syrinx formation is mechanically driven. In this study we model the spinal cord tissue either as a poro-elastic medium or as a solid linear elastic medium, and simulate the propagation of pressure waves through an anatomically plausible 3D geometry, with boundary conditions based on in vivo CSF pressure measurements. Then various anatomic and tissue properties are modified, resulting in a total of 11 variations of the model that are compared. The results show that an open segment of the central canal and a stiff pia (relative to the cord) both increase the radial pressure gradients and enhance interstitial fluid flow in the central canal. The anterior median fissure, anisotropic permeability of the white matter, and Poisson ratio play minor roles.     

A Mathematical Model of Spatial Self-Organization in a Mechanically Active Cellular Medium

A general continual model of a medium composed of mechanically active cells is proposed. The medium is considered to be formed by three phases: cells, extracellular fluid, and an additional phase that is responsible for active interaction forces between cells and, for instance, may correspond to a system of protrusions that provide the development of active contractile forces. The deformation of the medium, which is identified with the deformation of the cell phase, consists of two components: elastic deformation of individual cells and cell rearrangements. The elastic deformation is associated with stresses in the cell phase. The spherical component of the stress tensor describes the nonlinear resistance of the cellular medium, which leads to the impossibility of its excessive compression. The constitutive equation for pressure in the cell phase is taken in the form of a nonlinear dependence on the volume cell density. The rearrangement of cells is considered as a flow controlled by stresses in the cell phase, active stresses, and fluid pressure. The tensor of active stresses is assumed to be spherical and nonlocally dependent on the cell density. Assuming that the process of biological tissue deformation is slow, we obtained a reduced model that neglects the elastic deformation of cells, compared to the inelastic deformation. A linear stability analysis of a spatially uniform steady-state solution was performed. The hydrostatic pressure of fluid is present among the parameters that are responsible for the loss of stability of the steady-state solution: an increase in it has a destabilizing effect owing to the action of the component of the interphase interaction force that is determined by the fluid pressure. The model we obtained can be used to describe the process of cavity formation in an initially homogeneous cell spheroid. The role of local and nonlocal mechanisms of active stress generation in the formation of cavity is investigated.     

An elongation model of left ventricle deformation in diastole

A numerical method of the left ventricle (LV) deformation, an elongation model, was put forth for the study of LV fluid mechanics in diastole. The LV elongated only along the apical axis, and the motion was controlled by the intraventricular flow rate. Two other LV models, a fixed control volume model and a dilation model, were also used for model comparison and the study of LV fluid mechanics. For clinical sphere indices (SIs, between 1.0 and 2.0), the three models showed little difference in pressure and velocity distributions along the apical axis at E-peak. The energy dissipation was lower at a larger SI in that the jet and vortex development was less limited by the LV cavity in the apical direction. LV deformation of apical elongation may represent the primary feature of LV deformation in comparison with the secondary radial expansion. The elongation model of the LV deformation with an appropriate SI is a reasonable, simple method to study LV fluid mechanics in diastole.     

The pathogenesis of normal pressure hydrocephalus: A theoretical analysis

Hydrocephalus is an abnormal accumulation of cerebrospinal fluid (CSF) in the cerebral ventricles, usually caused by impaired absorption of the fluid into the bloodstream. Despite obstructed absorption and continued secretion of CSF into the ventricles at a near normal rate, the ventricular CSF pressure (VCSFP) is often normal. We attempt to understand how hydrocephalus can exist with normal VCSFP by exploring the role of the brain parenchyma in absorbing CSF in hydrocephalus. We test three theories: (1) the ventricular wall is impermeable to CSF; (2) ventricular CSF seeps into the parenchyma, from which it is efficiently absorbed; and (3) ventricular CSF seeps into the parenchyma but is absorbed inefficiently. We model the brain as a thick spherical shell consisting of a porous, elastic, solid matrix, containing interstitial fluid and blood. We modify the equations of poroelasticity, which describe flow of fluid through porous solids, to allow for parenchymal absorption. For each of the three theories we calculate the steady state changes in VCSFP and in parenchymal fluid pressure caused by an incremental defect in CSF absorption. We also calculate the steady state changes in fluid content, tissue volume, tissue displacement, and stresses caused by a small increment of VCSFP. We conclude that only the second theory—seepage of CSF with efficient parenchymal absorption—accounts for the clinical features of normal pressure hydrocephalus. These features include sustained ventricular dilatation despite normal VCSFP, increased periventricular fluid content, and localized periventricular white matter damage.     

A nonlinear biphasic model of flow-controlled infusion in brain: Fluid transport and tissue deformation analyses

A biphasic nonlinear mathematical model is proposed for the concomitant fluid transport and tissue deformation that occurs during constant flow rate infusions into brain tissue. The model takes into account material and geometrical nonlinearities, a hydraulic conductivity dependent on strain, and nonlinear boundary conditions at the infusion cavity. The biphasic equations were implemented in a custom written code assuming spherical symmetry and using an updated Lagrangian finite element algorithm. Results of the model showed that both, geometric and material nonlinearities play an important role in the physics of infusions, yielding important differences from infinitesimal analyses. Geometrical nonlinearities were mainly due to the significant enlargement of the infusion cavity, while variations of the parameters that describe the degree of nonlinearity of the stress–strain curve yielded significant differences in all distributions. For example, a parameter set showing stiffening under tension yielded maximum values of radial displacement and porosity not localized at the infusion cavity. On the other hand, a parameter set showing softening under tension yielded a slight decrease in the fluid velocity for a three-fold increase in the flow rate, which can be explained by the substantial increase of the infusion cavity, not considered in linear analyses. This study strongly suggests that significant enlargement of the infusion cavity is a real phenomenon during infusions that may produce collateral damage to brain tissue. Our results indicate that more experimental tests have to be undertaken in order to determine material nonlinearities of brain tissue over a range of strains. With better understanding of these nonlinear effects, clinicians may be able to develop protocols that can minimize the damage to surrounding tissue.     

On the infusion of a therapeutic agent into a solid tumor modeled as a poroelastic medium

The direct infusion of an agent into a solid tumor, modeled as a spherical poroelastic material with anisotropic dependence of the tumor hydraulic conductivity upon the tissue deformation, is treated both by solving the coupled fluid/elastic equations, and by expressing the solution as an asymptotic expansion in terms of a small parameter, ratio between the driving pressure force in the fluid system, and the elastic properties of the solid. Results at order one match almost perfectly the solutions of the full system over a large range of infusion pressures. Comparison with experimental results is acceptable after the hydraulic conductivity of the medium is properly calibrated. Given the uncertain estimates of some model constants, the order zero solution of the expansion, for which fluid and porous matrix are decoupled, yields acceptable values and trends for all the physical fields of interest, rendering the coupled analysis (in the limit of small displacements) of little use. When the deformation of the tissue becomes large nonlinear elasticity theory must be resorted to.     

The structure and adhesive mechanism of octopus suckers

Octopus suckers consist of a tightly packed three-dimensionalarray of muscle with three major muscle fiber orientations:1) radial muscles that traverse the wall; 2) circular musclesarranged circumferentially around the sucker; and 3) meridionalmuscles oriented perpendicular to the circular and radial muscles.The sucker also includes inner and outer fibrous connectivetissue layers and an array of crossed connective tissue fibersembedded in the musculature. Adhesion results from reducingthe pressure inside the sucker cavity. This can be achievedby the three-dimensional array of muscle functioning as a muscular-hydrostat.Contraction of the radial muscles thins the wall, thereby increasingthe enclosed volume of the sucker. If the sucker is sealed toa surface the cohesiveness of water resists this expansion.Thus, the pressure of the enclosed water decreases instead.The meridional and circular muscles antagonize the radial muscles.The crossed connective tissue fibers may store elastic energy,providing an economical mechanism for maintaining attachmentfor extended periods. Measurements using miniature flush-mountedpressure transducers show that suckers can generate hydrostaticpressures below 0 kPa on wettable surfaces but cannot do soon non-wettable surfaces. Thus, cavitation, the failure of waterin tension, may limit the attachment force of suckers. As depthincreases, however, cavitation will cease to be limiting becauseambient pressure increases with depth while the cavitation thresholdis unchanged. Structural differences between suckers will thendetermine the attachment force.     

Optimising Cell Aggregate Expansion in a Perfused Hollow Fibre Bioreactor via Mathematical Modelling

The need for efficient and controlled expansion of cell populations is paramount in tissue engineering. Hollow fibre bioreactors (HFBs) have the potential to meet this need, but only with improved understanding of how operating conditions and cell seeding strategy affect cell proliferation in the bioreactor. This study is designed to assess the effects of two key operating parameters (the flow rate of culture medium into the fibre lumen and the fluid pressure imposed at the lumen outlet), together with the cell seeding distribution, on cell population growth in a single-fibre HFB. This is achieved using mathematical modelling and numerical methods to simulate the growth of cell aggregates along the outer surface of the fibre in response to the local oxygen concentration and fluid shear stress. The oxygen delivery to the cell aggregates and the fluid shear stress increase as the flow rate and pressure imposed at the lumen outlet are increased. Although the increased oxygen delivery promotes growth, the higher fluid shear stress can lead to cell death. For a given cell type and initial aggregate distribution, the operating parameters that give the most rapid overall growth can be identified from simulations. For example, when aggregates of rat cardiomyocytes that can tolerate shear stresses of up to are evenly distributed along the fibre, the inlet flow rate and outlet pressure that maximise the overall growth rate are predicted to be in the ranges to (equivalent to to ) and to (or 15.6 psi to 15.7 psi) respectively. The combined effects of the seeding distribution and flow on the growth are also investigated and the optimal conditions for growth found to depend on the shear tolerance and oxygen demands of the cells.     

Evidence for increased de novo synthesis of NAD in immune-activated RAW264.7 macrophages: a self-protective mechanism?

Chondrogenesis in cartilage development and repair and cartilage degeneration in arthritis can be regulated by mechanical-load-induced physical factors such as tissue deformation, interstitial fluid flow and pressure, and electrical fields or streaming potentials. Previous animal and tissue explant studies have shown that time-varying dynamic tissue loading can increase the synthesis and deposition of matrix molecules in an amplitude-, frequency-, and spatially dependent manner. To provide information on the cell-level physical factors which may stimulate chondrocytes to increase production and export of aggrecan, the main proteoglycan component of the cartilage matrix, we characterized local changes in aggrecan synthesis within cyclically loaded tissue explant disks and compared those changes to values of predicted local physical factors. Aggrecan synthesis following a 23-h compression/radiolabel protocol was measured with a spatial resolution of approximately 0.1 mm across the 1.5-mm radius of explanted disks using a quantitative autoradiography method. A uniform stimulation of aggrecan synthesis was observed at an intermediate frequency of 0.01 Hz, while, at a higher frequency of 0.1 Hz, stimulation was only seen at peripheral radial positions. Profiles of radial solid matrix deformation and interstitial fluid pressure and velocity predicted to be occurring across the radius of the disk during sinusoidal loading were estimated using a composite poroelastic model. Tissue regions experiencing high interstitial fluid velocities corresponded to those displaying increased aggrecan synthesis. These results reinforce the role of load-induced flow of interstitial fluid in the stimulation of aggrecan production during dynamic loading of cartilage.     

The direct effect of intraorbital pressure on orbital growth in the anophthalmic piglet.

Tissue expanders placed within the orbit can have a positive effect on orbital and ipsilateral midfacial growth. To date, there is no precise method for controlling and monitoring expansion to induce normal growth in the developing facial skeleton. The present study was undertaken to determine the optimal physiologic pressure required to stimulate normal orbital growth and to determine whether above-normal growth could be achieved with higher intraorbital pressures. Using a neonatal swine model, an accurate method of monitoring intraorbital pressure, precisely controlling intraorbital expansion, and achieving normal orbital growth was explored. Sixteen male, 3-week-old Yorkshire piglets were randomly divided into three surgical groups. In each group, the left orbit was the experimental side, and the contralateral right orbit served as an untreated control. Group 1 (n = 6) underwent enucleation only. Group 2 (n = 5) underwent enucleation and orbital expansion at a near-normal physiologic pressure of 20 mmHg. Group 3 (n = 5) underwent enucleation and orbital expansion at a supernormal pressure of 60 mmHg. Spherical tissue expanders (10 cc) with a separate injection port were utilized as the orbital expanders. Pressure was monitored by an electronic manometer that was calibrated daily. Morphology of the orbits was documented by photography, the dimensions of the orbits were quantitated by three-dimensional mechanical digitization, and orbital volumes were calculated. In the unexpanded, anophthalmic control group, a significant reduction in radial growth after evisceration was seen. In group 2, the orbit stimulated with a consistent pressure of 20 mmHg, just above the physiologic normal pressure of 17 mmHg, showed an increase in radial dimension of 8 percent compared with the unoperated side. In the high-pressure group of 60 mmHg, an increase of 16 percent in the radius was observed over the 4-week period. This led to a corresponding increase in orbital volumes with increased pressure. Utilizing a paired t test, these differences in the radial and volumetric growth of the orbit were statistically significant (p < 0.005). The results obtained demonstrated a direct relationship between intraorbital pressure and the growth of the bony orbit in the radial dimension. On the basis of this study, we concluded that orbital expansion maintained at normal physiologic pressure can stimulate normal orbital growth in the neonatal facial skeleton. In addition, application of above-normal pressures for expansion can induce accelerated orbital growth.     

"Whither flows the fluid in bone?" An osteocyte's perspective

Bone represents a porous tissue containing a fluid phase, a solid matrix, and cells. Movement of the fluid phase within the pores or spaces of the solid matrix translates endogenous and exogenous mechanobiological, biochemical and electromechanical signals from the system that is exposed to the dynamic external environment to the cells that have the machinery to remodel the tissue from within. Hence, bone fluid serves as a coupling medium, providing an elegant feedback mechanism for functional adaptation. Until recently relatively little has been known about bone fluid per se or the influences governing the characteristics of its flow. This work is designed to review the current state of this emerging field. The structure of bone, as an environment for fluid flow, is discussed in terms of the properties of the spaces and channel walls through which the fluid flows and the influences on flow under physiological conditions. In particular, the development of the bone cell syncytium and lacunocanalicular system are presented, and pathways for fluid flow are described from the systemic to the organ, tissue, cellular and subcellular levels. Finally, exogenous and endogenous mechanisms for pressure-induced fluid movement through bone, including mechanical loading, vascular derived pressure gradients, and osmotic pressure gradients are discussed. The objective of this review is to survey the current understanding of the means by which fluid flow in bone is regulated, from the level of the skeletal system down to the level of osteocyte, and to provide impetus for future research in this area of signal transduction and coupling. An understanding of this important aspect of bone physiology has profound implications for restoration of function through innovative treatment modalities on Earth and in space, as well as for engineering of biomimetic replacement tissue.     

Perfusion characteristics of the human hepatic microcirculation based on three-dimensional reconstructions and computational fluid dynamic analysis

The perfusion of the liver microcirculation is often analyzed in terms of idealized functional units (hexagonal liver lobules) based on a porous medium approach. More elaborate research is essential to assess the validity of this approach and to provide a more adequate and quantitative characterization of the liver microcirculation. To this end, we modeled the perfusion of the liver microcirculation using an image-based three-dimensional (3D) reconstruction of human liver sinusoids and computational fluid dynamics techniques. After vascular corrosion casting, a microvascular sample (±0.134 mm(3)) representing three liver lobules, was dissected from a human liver vascular replica and scanned using a high resolution (2.6 μm) micro-CT scanner. Following image processing, a cube (0.15?×?0.15?×?0.15 mm(3)) representing a sample of intertwined and interconnected sinusoids, was isolated from the 3D reconstructed dataset to define the fluid domain. Three models were studied to simulate flow along three orthogonal directions (i.e., parallel to the central vein and in the radial and circumferential directions of the lobule). Inflow and outflow guidances were added to facilitate solution convergence, and good quality volume meshes were obtained using approximately 9?×?10(6) tetrahedral cells. Subsequently, three computational fluid dynamics models were generated and solved assuming Newtonian liquid properties (viscosity 3.5 mPa s). Post-processing allowed to visualize and quantify the microvascular flow characteristics, to calculate the permeability tensor and corresponding principal permeability axes, as well as the 3D porosity. The computational fluid dynamics simulations provided data on pressure differences, preferential flow pathways and wall shear stresses. Notably, the pressure difference resulting from the flow simulation parallel to the central vein (0-100 Pa) was clearly smaller than the difference from the radial (0-170 Pa) and circumferential (0-180 Pa) flow directions. This resulted in a higher permeability along the central vein direction (k(d,33)?=?3.64?×?10(-14) m(2)) in comparison with the radial (k(d,11)?=?1.56?×?10(-14) m(2)) and circumferential (k(d,22)?=?1.75?×?10(-14) m(2)) permeabilities which were approximately equal. The mean 3D porosity was 14.3. Our data indicate that the human hepatic microcirculation is characterized by a higher permeability along the central vein direction, and an about two times lower permeability along the radial and circumferential directions of a lobule. Since the permeability coefficients depend on the flow direction, (porous medium) liver microcirculation models should take into account sinusoidal anisotropy.     

Peripheral vascular decoupling in porcine endotoxic shock

Cardiac output measurement from arterial pressure waveforms presumes a defined relationship between the arterial pulse pressure (PP), vascular compliance (C), and resistance (R). Cardiac output estimates degrade if these assumptions are incorrect. We hypothesized that sepsis would differentially alter central and peripheral vasomotor tone, decoupling the usual pressure wave propagation from central to peripheral sites. We assessed arterial input impedance (Z), C, and R from central and peripheral arterial pressures, and aortic blood flow in an anesthetized porcine model (n = 19) of fluid resuscitated endotoxic shock induced by endotoxin infusion (7 μg·kg?1·h?1 increased to 14 and 20 μg·kg?1·h?1 every 10 min and stopped when mean arterial pressure <40 mmHg or Sv(O?) < 45%). Aortic, femoral, and radial artery pressures and aortic and radial artery flows were measured. Z was calculated by FFT of flow and pressure data. R and C were derived using a two-element Windkessel model. Arterial PP increased from aortic to femoral and radial sites. During stable endotoxemia with fluid resuscitation, aortic and radial blood flows returned to or exceeded baseline while mean arterial pressure remained similarly decreased at all three sites. However, aortic PP exceeded both femoral and radial arterial PP. Although Z, R, and C derived from aortic and radial pressure and aortic flow were similar during baseline, Z increases and C decreases when derived from aortic pressure whereas Z decreases and C increases when derived from radial pressure, while R decreased similarly with both pressure signals. This central-to-peripheral vascular tone decoupling, as quantified by the difference in calculated Z and C from aortic and radial artery pressure, may explain the decreasing precision of peripheral arterial pressure profile algorithms in assessing cardiac output in septic shock patients and suggests that different algorithms taking this vascular decoupling into account may be necessary to improve their precision in this patient population.     

Theoretical analysis of pressure pulse propagation in arterial vessels.

An original mathematical model of viscous fluid motion in a tapered and distensible tube is presented. The model equations are deduced by assuming a two-dimensional flow and taking into account the nonlinear terms in the fluid motion equations, as well as the nonlinear deformation of the tube wall. One distinctive feature of the model is the formal integration with respect to the radial coordinate of the Navier-Stokes equations by power series expansion. The consequent computational frame allows an easy, accurate evaluation of the effects produced by changing the values of all physical and geometrical tube parameters. The model is employed to study the propagation along an arterial vessel of a pressure pulse produced by a single flow pulse applied at the proximal vessel extremity. In particular, the effects of the natural taper angle of the arterial wall on pulse propagation are investigated. The simulation results show that tapering considerably influences wave attenuation but not wave velocity. The substantially different behavior of pulse propagation, depending upon whether it travels towards the distal extremity or in the opposite direction, is observed: natural tapering causes a continuous increase in the pulse amplitude as it moves towards the distal extremity; on the contrary, the reflected pulse, running in the opposite direction, is greatly damped. For a vessel with physical and geometrical properties similar to those of a canine femoral artery and 0.1 degree taper angle, the forward amplification is about 0.9 m-1 and the backward attenuation is 1.4 m-1, so that the overall tapering effect gives a remarkably damped pressure response. For a natural taper angle of 0.14 degrees the perturbation is almost extinct when the pulse wave returns to the proximal extremity.     

Computational simulation of hemodynamic-driven growth and remodeling of embryonic atrioventricular valves

Embryonic heart valves develop under continuous and demanding hemodynamic loading. The particular contributions of fluid pressure and shear tractions in valve morphogenesis are difficult to decouple experimentally. To better understand how fluid loads could direct valve formation, we developed a computational model of avian embryonic atrioventricular (AV) valve (cushion) growth and remodeling using experimentally derived parameters for the blood flow and the cushion stiffness. Through an iterative scheme, we first solved the fluid loads on the axisymmetric AV canal and cushion model geometry. We then applied the fluid loads to the cushion and integrated the evolution equations to determine the growth and remodeling. After a set time of growth, we updated the fluid domain to reflect the change in cushion geometry and resolved for the fluid forces. The rate of growth and remodeling was assumed to be a function of the difference between the current stress and an isotropic homeostatic stress state. The magnitude of the homeostatic stress modulated the rate of volume addition during the evolution. We found that the pressure distribution on the AV cushion was sufficient to generate leaflet-like elongation in the direction of flow, through inducing tissue resorption on the inflow side of cushion and expansion on the outflow side. Conversely, shear tractions minimally altered tissue volume, but regulated the remodeling of tissue near the cushion surface, particular at the leading edge. Significant shear and circumferential residual stresses developed as the cushion evolved. This model offers insight into how natural and perturbed mechanical environments may direct AV valvulogenesis and provides an initial framework on which to incorporate more mechano-biological details.     

Mathematical modeling of the circulation in the liver lobule

In this paper, we develop a mathematical model of blood circulation in the liver lobule. We aim to find the pressure and flux distributions within a liver lobule. We also investigate the effects of changes in pressure that occur following a resection of part of the liver, which often leads to high pressure in the portal vein. The liver can be divided into functional units called lobules. Each lobule has a hexagonal cross-section, and we assume that its longitudinal extent is large compared with its width. We consider an infinite lattice of identical lobules and study the two-dimensional flow in the hexagonal cross-sections. We model the sinusoidal space as a porous medium, with blood entering from the portal tracts (located at each of the vertices of the cross-section of the lobule) and exiting via the centrilobular vein (located in the center of the cross-section). We first develop and solve an idealized mathematical model, treating the porous medium as rigid and isotropic and blood as a Newtonian fluid. The pressure drop across the lobule and the flux of blood through the lobule are proportional to one another. In spite of its simplicity, the model gives insight into the real pressure and velocity distribution in the lobule. We then consider three modifications of the model that are designed to make it more realistic. In the first modification, we account for the fact that the sinusoids tend to be preferentially aligned in the direction of the centrilobular vein by considering an anisotropic porous medium. In the second, we account more accurately for the true behavior of the blood by using a shear-thinning model. We show that both these modifications have a small quantitative effect on the behavior but no qualitative effect. The motivation for the final modification is to understand what happens either after a partial resection of the liver or after an implantation of a liver of small size. In these cases, the pressure is observed to rise significantly, which could cause deformation of the tissue. We show that including the effects of tissue compliance in the model means that the total blood flow increases more than linearly as the pressure rises.