A nonlinear biphasic model of flow-controlled infusions in brain: Mass transport analyses

A biphasic nonlinear mathematical model is proposed for the mass transport that occurs during constant flow-rate infusions into brain tissue. The model takes into account geometric and material nonlinearities and a hydraulic conductivity dependent upon strain. The biphasic and convective–diffusive transport 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 the inclusion of these nonlinearities produced modest changes in the interstitial concentration but important variations in drug penetration and bulk concentration. Increased penetration of the drug but smaller bulk concentrations were obtained at smaller strains caused by combination of parameters such as increased Young’s modulus and initial hydraulic conductivity. This indicates that simulations of constant flow-rate infusions under the assumption of infinitesimal deformations or rigidity of the tissue may yield lower bulk concentrations near the infusion cavity and over-predictions of the penetration of the infused agent. The analyses also showed that decrease in the infusion flow rate of a fixed amount of drug results in increased penetration of the infused agent. From the clinical point-of-view, this may promote a safer infusion that delivers the therapeutic range over the desired volume while avoiding damage to the tissue by minimizing deformation and strain.     

A nonlinear biphasic viscohyperelastic model for articular cartilage

Experiments on articular cartilage have shown nonlinear stress-strain curves under finite deformations as well as intrinsic viscous effects of the solid phase. The aim of this study was to propose a nonlinear biphasic viscohyperelastic model that combines the intrinsic viscous effects of the proteoglycan matrix with a nonlinear hyperelastic constitutive equation. The proposed equation satisfies objectivity and reduces for uniaxial loading to a solid type viscous model in which the actions of the springs are represented by the hyperelastic function proposed by Holmes and Mow [1990. J. Biomechanics 23, 1145-1156.]. Results of the model, that were efficiently implemented in an updated Lagrangian algorithm, were compared with experimental infinitesimal data reported by DiSilverstro and Suh [2001. J. Biomechanics 34, 519-525.] and showed acceptable fitting for the axial force (R(2)=0.991) and lateral displacement (R(2)=0.914) curves in unconfined compression as well as a good fitting of the axial indentation force curve (R(2)=0.982). In addition, the model showed an excellent fitting of finite-deformation confined compression stress relaxation data reported by Ateshian et al. [1997. J. Biomechanics 30, 1157-1164.] and Huang et al. [2005. J. Biomechanics 38, 799-809.] (R(2)=0.993 and R(2)=0.995, respectively). The constitutive equation may be used to represent the mechanical behavior of the proteoglycan matrix in a fiber reinforced model of articular cartilage.     

A fully nonlinear viscohyperelastic model for the brain tissue applicable to dynamic rates

Understanding the mechanical response of the brain to external loadings is of critical importance in investigating the pathological conditions of this tissue during injurious conditions. Such injurious loadings may occur at high rates, for example among others, during road traffic or sport accidents, falls, or due to explosions. Hence, investigating the injury mechanism and design of protective devices for the brain requires constitutive modeling of this tissue at such rates. Accordingly, this paper is aimed at critically investigating the physical background for viscohyperelastic modeling of the brain tissue with scrutinizing the elastic fields pertinent to large, time dependent deformations, and developing a fully nonlinear multimode Maxwell model that can mathematically explain such deformations. The proposed model can be calibrated using the simple monotonic uniaxial deformation of the sample extracted from the tissue, and does not require additional information from relaxation or creep experiments. The performance of the proposed model is examined using the experimental results of two different studies, which reveals a desirable agreement. The usefulness, limitations, and future developments of the proposed model are discussed in this paper.     

A nonlinear biphasic fiber-reinforced porohyperviscoelastic model of articular cartilage incorporating fiber reorientation and dispersion

A nonlinear biphasic fiber-reinforced porohyperviscoelastic (BFPHVE) model of articular cartilage incorporating fiber reorientation effects during applied load was used to predict the response of ovine articular cartilage at relatively high strains (20%). The constitutive material parameters were determined using a coupled finite element-optimization algorithm that utilized stress relaxation indentation tests at relatively high strains. The proposed model incorporates the strain-hardening, tension-compression, permeability, and finite deformation nonlinearities that inherently exist in cartilage, and accounts for effects associated with fiber dispersion and reorientation and intrinsic viscoelasticity at relatively high strains. A new optimization cost function was used to overcome problems associated with large peak-to-peak differences between the predicted finite element and experimental loads that were due to the large strain levels utilized in the experiments. The optimized material parameters were found to be insensitive to the initial guesses. Using experimental data from the literature, the model was also able to predict both the lateral displacement and reaction force in unconfined compression, and the reaction force in an indentation test with a single set of material parameters. Finally, it was demonstrated that neglecting the effects of fiber reorientation and dispersion resulted in poorer agreement with experiments than when they were considered. There was an indication that the proposed BFPHVE model, which includes the intrinsic viscoelasticity of the nonfibrillar matrix (proteoglycan), might be used to model the behavior of cartilage up to relatively high strains (20%). The maximum percentage error between the indentation force predicted by the FE model using the optimized material parameters and that measured experimentally was 3%.     

Data-guide for brain deformation in surgery: comparison of linear and nonlinear models

Background  

Pre-operative imaging devices generate high-resolution images but intra-operative imaging devices generate low-resolution images. To use high-resolution pre-operative images during surgery, they must be deformed to reflect intra-operative geometry of brain.     

A distributed nonlinear model of lung tissue elasticity

Maksym, Geoffrey N., and Jason H. T. Bates. Adistributed nonlinear model of lung tissue elasticity.J. Appl. Physiol. 82(1): 32-41, 1997.- We present a theory relating the static stress-strainproperties of lung tissue strips to the stress-bearing constituents,collagen and elastin. The fiber pair is modeled as a Hookean spring(elastin) in parallel with a nonlinear string element (collagen), whichextends to a maximum stop length. Based on a series of fiber pairs, wedevelop both analytical and numerical models with distributedconstituent properties that account for nonlinear tissue elasticity.The models were fit to measured stretched stress-strain curves of fiveuniaxially stretched tissue strips, each from a different dog lung. Wefound that the distributions of stop length and spring stiffness followinverse power laws, and we hypothesize that this results from thecomplex fractal-like structure of the constituent fiber matrices inlung tissue. We applied the models to representative pressure-volume(PV) curves from patients with normal, emphysematous,and fibrotic lungs. The PV curves were fit to theequation V = A  Bexp(KP),where V is volume, P is transpulmonary pressure, andA, B, andK are constants. Our models lead to apossible mechanistic explanation of the shape factorK in terms of the structuralorganization of collagen and elastin fibers.

     

A nonlinear viscoelastic model of lung tissue mechanics.

There have been a number of attempts recently to use linear models to describe the low-frequency (0-2 Hz) dependence of lung tissue resistance (Rti) and elastance (Eti). Only a few attempts, however, have been made to account for the volume dependence of these quantities, all of which require the tissues to be plastoelastic. In this paper we specifically avoid invoking plastoelasticity and develop a nonlinear viscoelastic model that is also capable of accounting for the nonlinear and frequency-dependent features of lung tissue mechanics. The model parameters were identified by fitting the model to data obtained in a previous study from dogs during sinusoidal ventilation. The model was then used to simulate pressure and flow data by use of various types of ventilation patterns similar to those that have been employed experimentally. Rti and Eti were estimated from the simulated data by use of four different estimation techniques commonly applied in respiratory mechanics studies. We found that the estimated volume dependence of Rti and Eti is sensitive to both the ventilation pattern and the estimation technique, being in error by as much as 217 and 22%, respectively.     

Unconfined compression of articular cartilage: nonlinear behavior and comparison with a fibril-reinforced biphasic model

Mechanical behavior of articular cartilage was characterized in unconfined compression to delineate regimes of linear and nonlinear behavior, to investigate the ability of a fibril-reinforced biphasic model to describe measurements, and to test the prediction of biphasic and poroelastic models that tissue dimensions alter tissue stiffness through a specific scaling law for time and frequency. Disks of full-thickness adult articular cartilage from bovine humeral heads were subjected to successive applications of small-amplitude ramp compressions cumulating to a 10 percent compression offset where a series of sinusoidal and ramp compression and ramp release displacements were superposed. We found all equilibrium behavior (up to 10 percent axial compression offset) to be linear, while most nonequilibrium behavior was nonlinear, with the exception of small-amplitude ramp compressions applied from the same compression offset. Observed nonlinear behavior included compression-offset-dependent stiffening of the transient response to ramp compression, nonlinear maintenance of compressive stress during release from a prescribed offset, and a nonlinear reduction in dynamic stiffness with increasing amplitudes of sinusoidal compression. The fibril-reinforced biphasic model was able to describe stress relaxation response to ramp compression, including the high ratio of peak to equilibrium load. However, compression offset-dependent stiffening appeared to suggest strain-dependent parameters involving strain-dependent fibril network stiffness and strain-dependent hydraulic permeability. Finally, testing of disks of different diameters and rescaling of the frequency according to the rule prescribed by current biphasic and poroelastic models (rescaling with respect to the sample's radius squared) reasonably confirmed the validity of that scaling rule. The overall results of this study support several aspects of current theoretical models of articular cartilage mechanical behavior, motivate further experimental characterization, and suggest the inclusion of specific nonlinear behaviors to models.     

A novel three-phase model of brain tissue microstructure

We propose a novel biologically constrained three-phase model of the brain microstructure. Designing a realistic model is tantamount to a packing problem, and for this reason, a number of techniques from the theory of random heterogeneous materials can be brought to bear on this problem. Our analysis strongly suggests that previously developed two-phase models in which cells are packed in the extracellular space are insufficient representations of the brain microstructure. These models either do not preserve realistic geometric and topological features of brain tissue or preserve these properties while overestimating the brain's effective diffusivity, an average measure of the underlying microstructure. In light of the highly connected nature of three-dimensional space, which limits the minimum diffusivity of biologically constrained two-phase models, we explore the previously proposed hypothesis that the extracellular matrix is an important factor that contributes to the diffusivity of brain tissue. Using accurate first-passage-time techniques, we support this hypothesis by showing that the incorporation of the extracellular matrix as the third phase of a biologically constrained model gives the reduction in the diffusion coefficient necessary for the three-phase model to be a valid representation of the brain microstructure.     

A stochastic model of brain cell differentiation in tissue culture

 The timing of cell differentiation can be controlled both by cell-intrinsic mechanisms and by cell-extrinsic signals. Oligodendrocyte type-2 astrocyte progenitor cells are known to be the precursor cells that give rise to oligodendrocytes. When stimulated to divide by purifed cortical astrocytes or by platelet-derived growth factor, these progenitor cells generate oligodendrocytes in vitro with a timing like that observed in vivo. The most widely accepted model of this process assumes a cell-intrinsic biological clock that resides in the progenitor cell. The intrinsic clock model originally proposed in 1986 remains as the dominant theoretical concept for the analysis of timed differentiation in this cell lineage. However, the results of a recent experimental study (Ibarrola et al., Developmental Biology, vol. 180, 1–21, 1996) are most consistent with the hypothesis that the propensity of a clone of dividing O-2A progenitor cells initially to generate at least one oligodendrocyte may be regulated by cell-intrinsic mechanisms, but that environmental signals regulate the extent of further oligodendrocyte generation. We propose a stochastic model of cell differentiation in culture to accommodate the most recent experimental findings. Our model is an age-dependent branching stochastic process with two types of cells. The model makes it possible to derive analytical expressions for the expected number of progenitor cells and of oligodendrocytes as functions of time. The model parameters were estimated by fitting these functions through data on the average (sample mean) number of both types of cells per colony at different time intervals from start of experiment. Using this method we provide a biologically meaningful interpretation of the observed pattern of oligodendrocyte generation in vitro and its modification in the presence of thyroid hormone. Received: 18 April 1997 / Revised version: 30 November 1997     

A permeation-diffusion-reaction model of gas transport in cellular tissue of plant materials

Gas transport in fruit tissue is governed by both diffusion and permeation. The latter phenomenon is caused by overall pressure gradients which may develop due to the large difference in O(2) and CO(2) diffusivity during controlled atmosphere storage of the fruit. A measurement set-up for tissue permeation based on unsteady-state gas exchange was developed. The gas permeability of pear tissue was determined based on an analytical gas transport model. The overall gas transport in pear tissue samples was validated using a finite element model describing simultaneous O(2), CO(2), and N(2) gas transport, taking into account O(2) consumption and CO(2) production due to respiration. The results showed that the model described the experimentally determined permeability of N(2) very well. The average experimentally determined values for permeation of skin, cortex samples, and the vascular bundle samples were (2.17+/-1.71)x10(-19) m(2), (2.35+/-1.96)x10(-19) m(2), and (4.51+/-3.12)x10(-17) m(2), respectively. The permeation-diffusion-reaction model can be applied to study gas transport in intact pears in relation to product quality.     

Patient-specific model of brain deformation: application to medical image registration

This contribution presents finite element computation of the deformation field within the brain during craniotomy-induced brain shift. The results were used to illustrate the capabilities of non-linear (i.e. accounting for both geometric and material non-linearities) finite element analysis in non-rigid registration of pre- and intra-operative magnetic resonance images of the brain. We used patient-specific hexahedron-dominant finite element mesh, together with realistic material properties for the brain tissue and appropriate contact conditions at boundaries. The model was loaded by the enforced motion of nodes (i.e. through prescribed motion of a boundary) at the brain surface in the craniotomy area. We suggest using explicit time-integration scheme for discretised equations of motion, as the computational times are much shorter and accuracy, for practical purposes, the same as in the case of implicit integration schemes. Application of the computed deformation field to register (i.e. align) the pre-operative images with the intra-operative ones indicated that the model very accurately predicts the displacements of the tumour and the lateral ventricles even for limited information about the brain surface deformation. The prediction accuracy improves when information about deformation of not only exposed (during craniotomy) but also unexposed parts of the brain surface is used when prescribing loading. However, it appears that the accuracy achieved using information only about the deformation of the exposed surface, that can be determined without intra-operative imaging, is acceptable. The presented results show that non-linear biomechanical models can complement medical image processing techniques when conducting non-rigid registration. Important advantage of such models over the previously used linear ones is that they do not require unrealistic assumptions that brain deformations are infinitesimally small and brain stress-strain relationship is linear.     

A stochastic model for facilitated transport of oxygen through tissue slices

Longmuir and co-workers have reported that respiration of certain tissue slices is approximated by Michaelis-Menten kinetics. From this and other experimental findings, Longmuir proposed that a carrier is involved in tissue oxygen transport. Gold developed a deterministic model to examine this hypothesis. This report presents a stochastic model for a fixed site carrier in a more general framework that includes the stochastic counter-part to Gold's deterministic model as a special case. The kinetics of tissue oxygen consumption predicted by the model are examined for various cases.     

A biphasic multilayer computational model of human skin

The present study investigates the layer-specific mechanical behavior of human skin. Motivated by skin’s histology, a biphasic model is proposed which differentiates between epidermis, papillary and reticular dermis, and hypodermis. Inverse analysis of ex vivo tensile and in vivo suction experiments yields mechanical parameters for each layer and predicts a stiff reticular dermis and successively softer papillary dermis, epidermis and hypodermis. Layer-specific analysis of simulations underlines the dominating role of the reticular dermis in tensile loading. Furthermore, it shows that the observed out-of-plane deflection in ex vivo tensile tests is a direct consequence of the layered structure of skin. In in vivo suction experiments, the softer upper layers strongly influence the mechanical response, whose dissipative part is determined by interstitial fluid redistribution within the tissue. Magnetic resonance imaging-based visualization of skin deformation in suction experiments confirms the deformation pattern predicted by the multilayer model, showing a consistent decrease in dermal thickness for large probe opening diameters.

     

Finite deformation of soft tissue: analysis of a mixture model in uni-axial compression

The dynamic finite deformational behavior of a biphasic model for soft hydrated tissue is examined. In the case of uni-axial confined compression the displacement and stress fields are derived for steady-state permeation, creep, and stress-relaxation. It is shown how to use the results of this analysis to obtain the constitutive relations, as well as the associated material parameters, from the corresponding experiments. It is also shown that the solutions from the theory go much farther, giving a detailed account of the deformation and interaction of the fluid and solid phases in the tissue.