Microdialysis is an extensively used technique for the study of solutes in brain interstitial space. The method is based on collection of substances by diffusion across a dialysis membrane positioned in the brain. The outflow concentration reflects the interstitial concentration of the substance of interest, but the relationship between these two entities is at present unclear. So far, most evaluations have been based solely on calibrations in saline. This procedure is misleading, because the ease by which molecules in saline diffuse into the probe is different from that of tissue. We describe here a mathematical analysis of mass transport into the dialysis probe in tissue based on diffusion equations in complex media. The main finding is that diffusion characteristics of a given substance have to be included in the formula. These include the tortuosity factor (lambda) and the extracellular volume fraction (alpha). We have substantiated this by studies in a well-defined complex medium (red blood cell suspensions) as well as in brain. We conclude that the traditional calculation procedure results in interstitial concentrations that are too low by a factor of lambda 2/alpha for a given compound.
SUMMARYIt is proposed to capture localized plastic deformation via the inclusion of regularized displacement discontinuities at element boundaries (interfaces) of the finite element subdivision. The regularization is based on a kinematic assumption for an interface that resembles that which is pertinent to the classical shear band concept. As a by-product of the regularization, an intrinsic band width is introduced as a 'constitutive' property rather than a geometric feature of the finite element mesh. In this way the spurious mesh sensitivity, which is obtained when the displacement approximation is continuous, can be avoided. Another consequence is that the interfacial relation between the elements is derived directly from the conventional constitutive properties of the continuously deforming material. An interesting feature is that the acoustic tensor will not only play a role for diagnosing discontinuous bifurcation but will also serve as the tangent stiffness tensor of the interface (up to within a scalar factor). An analytical investigation of the behaviour of the interface is carried out and it is shown that dilatation may indeed accompany slip within a 'shear' band for a general plasticity model. The significance of proper mesh alignment is demonstrated for a simple problem in plane strain and plane stress. It is shown that a unique structural post-peak response (in accordance with non-linear fracture mechanics) can be achieved when the plastic softening modulus is properly related to the bandwidth. The paper concludes with a numerical simulation of the gradual development of a shear band in a soil slope.
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