Comparison of models for flow induced deformation of soft biological tissue

Poroelastic Characterization and Modeling of Subcutaneous Tissue Under Confined Compression

Subcutaneous tissue mechanics are important for drug delivery. Yet, even though this material is poroelastic, its mechanical characterization has focused on its hyperelastic response. Moreover, advancement in subcutaneous drug delivery requires effective tissue mimics such as hydrogels for which similar gaps of poroelastic data exist. Porcine subcutaneous samples and gelatin hydrogels were tested under confined compression at physiological conditions and strain rates of 0.01%/s in 5% strain steps with 2600 s of stress relaxation between loading steps. Force-time data were used in an inverse finite element approach to obtain material parameters. Tissues and gels were modeled as porous neo-Hookean materials with properties specified via shear modulus, effective solid volume fraction, initial hydraulic permeability, permeability exponent, and normalized viscous relaxation moduli. The constitutive model was implemented into an isogeometric analysis (IGA) framework to study subcutaneous injection. Subcutaneous tissue exhibited an initial spike in stress due to compression of the solid and fluid pressure buildup, with rapid relaxation explained by fluid drainage, and longer time-scale relaxation explained by viscous dissipation. The inferred parameters aligned with the ranges reported in the literature. Hydraulic permeability, the most important parameter for drug delivery, was in the rangek 0 ∈ [ 0.142 , 0.203 ] mm4 /(N s). With these parameters, IGA simulations showed peak stresses due to a 1-mL injection to reach 48.8 kPa at the site of injection, decaying after drug volume disperses into the tissue. The poro-hyper-viscoelastic neo-Hookean model captures the confined compression response of subcutaneous tissue and gelatin hydrogels. IGA implementation enables predictive simulations of drug delivery.

Ion-Induced Swelling Behavior of Articular Cartilage due to Non-Newtonian Flow and Its Effects on Fluid Pressure and Solid Displacement

In this study, we examine the behavior of articular cartilage equilibrated in a salt (NaCl) solution during non-Newtonian fluid flow that follows an Ostwald-de Waele model. A linearly elastic and isotropic rectangular strip of cartilage is considered for analysis. A continuum theory of mixtures has been employed to develop a coupled system of partial differential equations for the solid displacement and the fluid pressure by considering the important factor of the ion concentration by assuming the cartilage as a deformable porous media. The coupled system of partial differential equations is solved using the numerical method named method of lines. In most cases, shear-thinning fluid is compared to the shear-thickening fluid to magnify the difference. Graphical results show that shear-thickening fluids bring more solid deformation and shows less fluid pressure in comparison to the shear-thinning fluids.

Solute Transport across the Lymphatic Vasculature in a Soft Skin Tissue

Convective transport of drug solutes in biological tissues is regulated by the interstitial fluid pressure, which plays a crucial role in drug absorption into the lymphatic system through the subcutaneous (SC) injection. In this paper, an approximate continuum poroelasticity model is developed to simulate the pressure evolution in the soft porous tissue during an SC injection. This poroelastic model mimics the deformation of the tissue by introducing the time variation of the interstitial fluid pressure. The advantage of this method lies in its computational time efficiency and simplicity, and it can accurately model the relaxation of pressure. The interstitial fluid pressure obtained using the proposed model is validated against both the analytical and the numerical solution of the poroelastic tissue model. The decreasing elasticity elongates the relaxation time of pressure, and the sensitivity of pressure relaxation to elasticity decreases with the hydraulic permeability, while the increasing porosity and permeability due to deformation alleviate the high pressure. An improved Kedem-Katchalsky model is developed to study solute transport across the lymphatic vessel network, including convection and diffusion in the multi-layered poroelastic tissue with a hybrid discrete-continuum vessel network embedded inside. At last, the effect of different structures of the lymphatic vessel network, such as fractal trees and Voronoi structure, on the lymphatic uptake is investigated. In this paper, we provide a novel and time-efficient computational model for solute transport across the lymphatic vasculature connecting the microscopic properties of the lymphatic vessel membrane to the macroscopic drug absorption.

Compressive stress relaxation behavior of articular cartilage and its effects on fluid pressure and solid displacement due to non-Newtonian flow

In this study, we investigate the effects of the power-law index and permeability parameter on the deformation of soft tissue (articular cartilage) which is bathed in the non-Newtonian fluid under stress-relaxation in compression. Ramp displacement is imposed on the surface of hydrated soft tissue. Deformation of the tissue and the fluid pressure is examined for the fast and slow rate of compression. We have employed a linear biphasic mixture theory to develop a mathematical model for compressive stress-relaxation behavior of articular cartilage for non-Newtonian flow. Numerical results indicate that shear-thinning fluids induce less solid deformation and exhibit more fluid pressure as compared to shear-thickening fluids for fast and slow rate of compression. The results also show that linear permeability induces more deformation as compared to strain-dependent nonlinear permeability due to viscoelastic nature of articular cartilage.

Tumescent Injections in Subcutaneous Pig Tissue Disperse Fluids Volumetrically and Maintain Elevated Local Concentrations of Additives for Several Hours, Suggesting a Treatment for Drug Resistant Wounds

PURPOSE

Bolus injection of fluid into subcutaneous tissue results in accumulation of fluid at the injection site. The fluid does not form a pool. Rather, the injection pressure forces the interstitial matrix to expand to accommodate the excess fluid in its volume, and the fluid becomes bound similar to that in a hydrogel. We seek to understand the properties and dynamics of externally tumesced (swollen) subcutaneous tissue as a first step in assessing whether tumescent antibiotic injections into wounds may provide a novel method of treatment.

METHODS

Subcutaneous injections of saline are performed in live and dead pigs and the physical properties (volume, expansion ratio, residence time, apparent diffusion constant) of the resulting fluid deposits are observed with diffusion-weighted magnetic resonance imaging, computed tomography, and 3D scanning.

RESULTS

Subcutaneous tissue can expand to a few times its initial volume to accommodate the injected fluid, which is dispersed thoroughly throughout the tumescent volume. The fluid spreads to peripheral unexpanded regions over the course of a few minutes, after which it remains in place for several hours. Eventually the circulation absorbs the excess fluid and the tissue returns to its original state.

CONCLUSIONS

Given the evidence for dense fluid dispersal and several-hour residence time, a procedure is proposed whereby tumescent antibiotic injections are used to treat drug-resistant skin infections and chronic wounds that extend into the subcutaneous tissue. The procedure has the potential to effectively treat otherwise untreatable wounds by keeping drug concentrations above minimum inhibitory levels for extended lengths of time.

Foundations of modeling in cryobiology-II: Heat and mass transport in bulk and at cell membrane and ice-liquid interfaces

Modeling coupled heat and mass transport in biological systems is critical to the understanding of cryobiology. In Part I of this series we derived the transport equation and presented a general thermodynamic derivation of the critical components needed to use the transport equation in cryobiology. Here we refine to more cryobiologically relevant instances of a double free-boundary problem with multiple species. In particular, we present the derivation of appropriate mass and heat transport constitutive equations for a system consisting of a cell or tissue with a free external boundary, surrounded by liquid media with an encroaching free solidification front. This model consists of two parts-namely, transport in the "bulk phases" away from boundaries, and interfacial transport. Here we derive the bulk and interfacial mass, energy, and momentum balance equations and present a simplification of transport within membranes to jump conditions across them. We establish the governing equations for this cell/liquid/solid system whose solution in the case of a ternary mixture is explored in Part III of this series.

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.

Radial flow through deformable porous shells

The problem of radially directed fluid flow through a deformable porous shell is considered. General nonlinear diffusion equations are developed for spherical, cylindrical and planar geometries. Solutions for steady flow are found in terms of an exact integral and perturbation solutions are also developed. For unsteady flow, perturbation methods are used to find approximate small-time solutions and a solution valid for slow compression rates. These solutions are used to investigate the deformation of the porous material with comparisons made between the planar and the cylindrical geometries.

TRANSPORT PROPERTIES OF CARTILAGINOUS TISSUES

Cartilaginous tissues, such as articular cartilage and intervertebral disc, are avascular tissues which rely on transport for cellular nutrition. Comprehensive knowledge of transport properties in such tissues is therefore necessary in the understanding of nutritional supply to cells. Furthermore, poor cellular nutrition in cartilaginous tissues is believed to be a primary source of tissue degeneration, which may result in osteoarthritis (OA) or disc degeneration. In this mini-review, we present an overview of the current status of the study of transport properties and behavior in cartilaginous tissues. The mechanisms of transport in these tissues, as well as experimental approaches to measuring transport properties and results obtained are discussed. The current status of bioreactors used in cartilage tissue engineering is also presented.

Nonlinear dependence of hydraulic conductivity on tissue deformation during intratumoral infusion

Efficiency of intratumoral infusion for drug and gene delivery depends on intrinsic tissue structures as well as infusion-induced changes in these structures. To this end, we investigated effects of infusion pressure (P(inf)) and infusion-induced tissue deformation on infusion rate (Q) in three mouse tumor models (B16.F10, 4T1, and U87) and developed a poroelastic model for interpreting data and understanding mechanisms of fluid transport in tumors. The collagen concentrations in these tumors were 2.9+/-1.2, 12.2+/-0.9, and 18.1+/-3.5 microg/mg wet wt. of tissues, respectively. During the infusion, there existed a threshold infusion pressure (P(t)), below which fluid flow could not be initiated. The values of P(t) for these tumors were 7.36, 36.8, and 29.4 mmHg, respectively. Q was a bell-shaped function of P(inf) in 4T1 tumors but increased monotonically with increasing P(inf) in other tumors. These observations were consistent with results from numerical simulations based on the poroelastic model, suggesting that both the existence of P(t) and the nonlinear relationships between Q and P(inf) could be explained by infusion-induced tissue deformation that anisotropically affected the hydraulic conductivity of tissues. These results may be useful for further investigations of intratumoral infusion of drugs and genes.

Anisotropic hydraulic permeability in compressed articular cartilage

The extent to which articular cartilage hydraulic permeability is anisotropic is largely unknown, despite its importance for understanding mechanisms of joint lubrication, load bearing, transport phenomena, and mechanotransduction. We developed and applied new techniques for the direct measurement of hydraulic permeability within statically compressed adult bovine cartilage explant disks, dissected such that disk axes were perpendicular to the articular surface. Applied pressure gradients were kept small to minimize flow-induced matrix compaction, and fluid outflows were measured by observation of a meniscus in a glass capillary under a microscope. Explant disk geometry under radially unconfined axial compression was measured by direct microscopic observation. Pressure, flow, and geometry data were input to a finite element model where hydraulic permeabilities in the disk axial and radial directions were determined. At less than 10% static compression, near free-swelling conditions, hydraulic permeability was nearly isotropic, with values corresponding to those of previous studies. With increasing static compression, hydraulic permeability decreased, but the radially directed permeability decreased more dramatically than the axially directed permeability such that strong anisotropy (a 10-fold difference between axial and radial directions) in the hydraulic permeability tensor was evident for static compression of 20-40%. Results correspond well with predictions of a previous microstructurally-based model for effects of tissue mechanical deformations on glycosaminoglycan architecture and cartilage hydraulic permeability. Findings inform understanding of structure-function relationships in cartilage matrix, and suggest several biomechanical roles for compression-induced anisotropic hydraulic permeability in articular cartilage.

Perfusion studies of steady flow in poroelastic myocardium tissue

The behaviour of the heart has always elicited interest and particularly the study of its myocardium, as 5-10% of the blood pumped by the heart is passed through the coronary arteries to the myocardium itself. An in-depth investigation of the myocardium behaviour is useful. The present work aims to investigate how myocardium perfusion is influenced by myocardial stress and diseased states, and in general by LV pumping abnormalities. LV myocardial perfusion can then serve as a possible index of the capacity of the LV to respond to its work demand, and thus of the risk of heart failure. The poroelastic analysis of the myocardium based on finite element method (FEM) for regional perfusion through a rectangular element with various physiological ranges of loading conditions was studied.

Quantitative three-dimensional analysis and diffusion modeling of oligonucleotide concentrations after direct intraparenchymal brain infusion

We compared quantitative experimental results on the diffusion of 35S-labeled phosphorothioate oligonucleotide (PS-ODN) after intraparenchymal infusion in rat brain, with the distributions predicted by Fick's second law of diffusion. Fischer 344 rats underwent identical intracerebral infusions of 36S-PS-ODN. After 0, 5, 11, 23, and 47 h, groups of animals were sacrificed and sequential brain cryosections subjected to autoradiography. The resulting experimental data were compared to the predicted distributions, for estimation of the apparent free diffusion coefficient, D*. Volumes of distribution and total content of 36 S-PS-ODN in the parenchyma were also computed, to monitor loss of total material. The values for D* were within the expected range for the 21-mer PS-ODN used, but a progressive decrease in D* over time was noted. The model requires D* to remain constant and, thus, does not adequately explain the spread of 35S-PS-ODN following infusion. The progressive slowing of spread over time suggests that at later time points, 35S-PS-ODN may be fixed by tissue binding or cellular uptake in the brain. Loss of material via vascular and CSF clearance may also contribute to the lack of fit. Our results highlight issues to be addressed in the modeling and experimental design of the intraparenchymal infusion process.

A Porous Media Approach to Finite Deformation Behaviour in Soft Tissues

The present work presents a porous medium formulation for the biomechanical analysis of soft tissues. An updated Lagrangian approach is developed to study the coupled effects of low speed flows of fluid phases, in partially or fully saturated conditions, and the finite deformation occurring in the solid matrix. The procedure developed allows both for the evaluation of coupled geometric and material non-linearities. The main theoretical and computational aspects of this multiphase formulation are discussed. The finite element method is used for the numerical solution of the resulting coupled system of equations. A reference case is reported with regard to healthy and degenerative phases of intervertebral segment. Results reported allow for a detailed interpretation of the formulation reliability, also by comparison with existing experimental data. In particular, the role played by the fluid on the load carrying mechanism is pointed out, thus stressing the importance of a multiphase approach to the overall behaviour of the spinal motion segment in time.

Interstitial hydraulic conductivity in a fibrosarcoma

Convective transport of therapeutic agents in solid tumors can be improved through intratumoral infusion. To optimize the convection, we investigated the dependence of the hydraulic conductivity on tissue deformation induced by interstitial fluid pressure gradient during the infusion. Two experimental systems were used in the investigation: 1) one-dimensional perfusion through tumor slices and 2) intratumoral infusion using a needle. With these systems, we found that the apparent hydraulic conductivity (K(app)) could be altered by several orders of magnitude in fibrosarcomas through changes in perfusion conditions. When the perfusion pressure was less than a threshold level, fluid flow in tissues could not be detected. When the perfusion pressure was increased above the threshold level, K(app) depended on perfusion system and pressure. The maximum variation in K(app) in fibrosarcomas reached 80,260-fold in our experiments. The large variation in K(app) could be explained by perfusion pressure-induced tissue deformation. These experimental data suggest that the hydraulic conductivity is very sensitive to tissue deformation and imply that it is possible to improve intratumoral infusion of therapeutic agents through optimization of infusion conditions.

Mechanics of interstitial-lymphatic fluid transport: theoretical foundation and experimental validation

Interstitial fluid movement is intrinsically linked to lymphatic drainage. However, their relationship is poorly understood, and associated pathologies are mostly untreatable. In this work we test the hypothesis that bulk tissue fluid movement can be evaluated in situ and described by a linear biphasic theory which integrates the regulatory function of the lymphatics with the mechanical stresses of the tissue. To accomplish this, we develop a novel experimental and theoretical model using the skin of the mouse tail. We then use the model to demonstrate how interstitial-lymphatic fluid movement depends on a balance between the elasticity, hydraulic conductivity, and lymphatic conductance as well as to demonstrate how chronic swelling (edema) alters the equipoise between tissue fluid balance parameters. Specifically, tissue fluid equilibrium is perturbed with a continuous interstitial infusion of saline into the tip of the tail. The resulting gradients in tissue stress are measured in terms of interstitial fluid pressure using a servo-null system. These measurements are then fit to the theory to provide in vivo estimates of the tissue hydraulic conductivity, elastic modulus, and overall resistance to lymphatic drainage. Additional experiments are performed on edematous tails to show that although chronic swelling causes an increase in the hydraulic conductivity, its greatly increased distensibility (due to matrix remodeling) dampens the driving forces for fluid movement and leads to fluid stagnation. This model is useful for examining potential treatments for edema and lymphatic disorders as well as substances which may alter tissue fluid balance and/or lymphatic drainage.

Dynamics of human milk extraction: a comparative study of breast feeding and breast pumping

We describe a mathematical model of the flow and deformation in a human teat. Our aim is to compare the theoretical milk yield during infant breast feeding with that obtained through the use of a breast pump. Infants use a peristaltic motion of the tongue, along with some suction, to extract milk, whereas breast pumps use a cyclic pattern of suction only. Our model is based on quasi-linear poroelasticity whereby the teat is modelled as a cylindrical porous elastic material saturated with fluid. We impose a cyclic axial suction pressure difference across the teat and impose a radial compressive force moving along the teat which mimics infant suckling. This is compared to the case of cyclic and steady pumping only which models the action of breast pumps. The results illustrate that there is an optimal time to apply the compressive force during the suction cycle that will increase the flow rate in our theoretical teat. The model and results may be of use in the future design of effective breast pumps.

Transport of fluid and ions through a porous-permeable charged-hydrated tissue, and streaming potential data on normal bovine articular cartilage

Using the triphasic mechano-electrochemical theory [Lai et al., J. biomech. Engng 113, 245-258 (1991)], we analyzed the transport of water and ions through a finite-thickness layer of charged, hydrated soft tissue (e.g. articular cartilage) in a one-dimensional steady permeation experiment. For this problem, we obtained numerically the concentrations of the ions, the strain field and the fluid and ion velocities inside when the specimen is subject to an applied mechanical pressure and/or osmotic pressure across the layer. The relationships giving the dependence of streaming potential and permeability on the negative fixed charge density (FCD) of the tissue were derived analytically for the linear case, and calculated for the nonlinear case. Among the results obtained were: (1) at a fluid pressure difference of 0.1 MPa across the specimen layer, there is a 10% flow-induced compaction at the downstream boundary; (2) the flow-induced compaction causes the FCD to increase and the neutral salt concentration to decrease in the downstream direction; (3) while both ions move downstream, relative to the solvent (water), the anions (Cl-) move with the flow whereas cations (Na+) move against the flow. The difference in ion velocities depends on the FCD, and this difference attained a maximum at a physiological FCD of around 0.2 meq ml-1; (4) the apparent permeability decreases nonlinearly with FCD, and the apparent stiffness of the tissue increases with FCD; and (5) the streaming potential is not a monotonic function of the FCD but rather it has a maximum value within the physiological range of FCD for articular cartilage. Finally, experimental data on streaming potential were obtained from bovine femoral cartilage. These data support the triphasic theoretical prediction of non-monotonicity of streaming potential as a function of the FCD.

Flow-induced deformation from pressurized cavities in absorbing porous tissues

The behaviour of a cavity during an injection of fluid into biological tissue is considered. High cavity pressure drives fluid into the neighbouring tissue where it is absorbed by capillaries and lymphatics. The tissue is modelled as a nonlinear deformable porous medium with the injected fluid absorbed by the tissue at a rate proportional to the local pressure. A model with a spherical cavity in an infinite medium is used to find the pressure and displacement of the tissue as a function of time and radial distance. Analytical and numerical solutions for a step change in cavity pressure show that the flow induces a radial compression in the medium together with an annular expansion, the net result being an overall expansion of the medium. Thus any flow induced deformation of the material will aid in the absorption of fluid.

A microstructurally-based finite element model of the incised human cornea

A mechanical model of the human cornea is proposed and employed in a finite element formulation for simulating the effects of surgical procedures, such as radial keratotomy, on the cornea. The model assumes that the structural behavior of the cornea is governed by the properties of the stroma. Arguments based on the microstructural organization and properties of the stroma lead to the conclusion that the human cornea exhibits flexural and shear rigidities which are negligible compared to its membrane rigidity. Accordingly, it is proposed that to a first approximation, the structural behavior of the cornea is that of a thick membrane shell. The tensile forces in the cornea are resisted by very fine collagen fibrils embedded in the ground substance of the stromal lamellae. When the collagen fibrils are cut, as in radial keratotomy, it is argued that they become relaxed since there is negligible transfer of load between adjacent fibrils due to the low shear modulus of the ground substance. The forces in the cornea are then resisted only by the remaining uncut fibrils. The cutting of fibrils induces an anisotropy and inhomogeneity in the membrane rigidity. By assuming a uniform angular distribution of stromal lamellae through the corneal thickness, geometric arguments lead to a quantitative representation for the anisotropy and inhomogeneity. All material behavior is assumed to be in the linear elastic regime and with no time-dependency. The resulting constitutive model for the incised cornea has been employed in a geometrically non-linear finite element membrane shell formulation for small strains with moderate rotations. A number of numerical examples are presented to illustrate the effectiveness of the proposed constitutive model and finite element formulation. The dependence of the outcome of radial keratotomy, measured in terms of the immediate postoperative shift in corneal power, on a number of important factors is investigated. These factors include the value of the elastic moduli of the stromal lamellae (dependent on the patient's age), the incision depth, the optic zone size, the number of incisions and their positions, and the intraocular pressure. Results have also been compared with expected surgical corrections predicted by three expert surgeons and show an excellent correspondence.