SUMMARYA ÿnite element description of uid ow through a deforming porous solid, with a hierarchical structure of pores, has been developed and implemented in the ÿnite element software package DIANA.1 Several standard element types can be used for 2-D, axisymmetric and 3-D ÿnite deformation analysis. The hierarchy is dealt with as an extra dimension, quantiÿed by a parameter x0. Both spatial and hierarchical uid ow is described by a Darcy equation. Fluid pressure and hydrostatic solid pressure are related via an elastic uid-solid interface. The state of the uid, the Darcy permeability tensor and the elastic interface depend on both spatial position and hierarchical level. Discretization and integration of uid related quantities are split into a spatial and a hierarchical part. The degrees of freedom of the ÿnite element model are the displacements of the solid, the hydrostatic pressure and a number of uid pressures on di erent hierarchical levels.Blood-perfused biological tissue can be regarded as a hierarchical porous solid, where the uid represents the blood and the hierarchy corresponds to the tree-like vascular structure. As an example, a simulation of a contracting, blood-perfused skeletal muscle is presented.
Abslract--A description of finite deformation of, and fluid flow through, a hierarchically arranged porous solid has been developed using the theory of mixtures. This hierarchical mixture consists of one solid constituent and a fluid constituent that is subdivided into a continuous series of intercommunicating compartments. Conservation laws for mass and momentum have been derived and appropriate formulations for the constitutive behaviour of the constituents are proposed. A finite element description of the hierarchical mixture model has been implemented in the software package DIANA. qrwo-dimensional, axisymmetric and three-dimensional elements can be used in finite deformation analysis. An example of application is blood perfused biological tissue. A simulation of a blood perfused contracting skeletal muscle is presented.
Mechanical interaction between tissue stress and blood perfusion in skeletal muscles plays an important role in blood flow impediment during sustained contraction. The exact mechanism of this interaction is not clear, and experimental investigation of this mechanism is difficult. We developed a finite-element model of the mechanical behavior of blood-perfused muscle tissue, which accounts for mechanical blood-tissue interaction in maximally vasodilated vasculature. Verification of the model was performed by comparing finite-element results of blood pressure and flow with experimental measurements in a muscle that is subject to well-controlled mechanical loading conditions. In addition, we performed simulations of blood perfusion during tetanic, isometric contraction and maximal vasodilation in a simplified, two-dimensional finite-element model of a rat calf muscle. A vascular waterfall in the venous compartment was identified as the main cause for blood flow impediment both in the experiment and in the finite-element simulations. The validated finite-element model offers possibilities for detailed analysis of blood perfusion in three-dimensional muscle models under complicated loading conditions.
A finite element (FE) model of blood perfused biological tissue has been developed. Blood perfusion is described by fluid flow through a series of 5 intercommunicating vascular compartments that are embedded in the tissue. Each compartment is characterized by a blood flow permeability tensor, blood volume fraction and vessel compliance. Local non-linear relationships between intra-extra vascular pressure difference and blood volume fraction, and between blood volume fraction and the permeability tensor, are included in the FE model. To test the implementation of these non-linear relations, FE results of blood perfusion in a piece of tissue that is subject to increased intramuscular pressure, are compared to results that are calculated with a lumped parameter (LP) model of blood perfusion. FE simulation of blood flow through a contracting rat calf muscle is performed. The FE model used in this simulation contains a transversely isotropic, non-linearly elastic description of deforming muscle tissue, in which local contraction stress is prescribed as a function of time. FE results of muscle tension, total arterial inflow and total venous outflow of the muscle during contraction, correspond to experimental results of an isometrically and tetanically contracting rat calf muscle.
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