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Tortuosity of the extracellular space describes hindrance posed to the diffusion process by a geometrically complex medium in comparison to an environment free of any obstacles. Calculating tortuosity in biologically relevant geometries is difficult. Yet this parameter has proved very important for many processes in the brain, ranging from ischemia and osmotic stress to delivery of nutrients and drugs. It is also significant for interpretation of the diffusion-weighted magnetic resonance data. We use a volume-averaging procedure to obtain a general expression for tortuosity in a complex environment. A simple approximation then leads to tortuosity estimates in a number of two-dimensional (2D) and three-dimensional (3D) geometries characterized by narrow pathways between the cellular elements. It also explains the counterintuitive fact of lower diffusion hindrance in a 3D environment. Comparison with Monte Carlo numerical simulations shows that the model gives reasonable tortuosity estimates for a number of regular and randomized 2D and 3D geometries. Importantly, it is shown that addition of dead-end pores increases tortuosity in proportion to the square root of enlarged total extracellular volume fraction. This conclusion is further supported by the previously described tortuosity decrease in ischemic brain slices where dead-end pores were partially occluded by large macromolecules introduced into the extracellular space.

Magnetic data from archaeological sites have traditionally been displayed by contour, isometric, and dotdensity plotting, or by simulated gray-scale techniques using symbol overprinting. These methods do not show fine linear structures in the data which are of great interest to archaeologists. If true gray-scale methods using a modern video display, followed by film recording for hard copy are employed, image processing techniques can be applied to enhance the geometric structures of archaeological interest. Interpolation techniques for enlarging data to full screen size, along with compression methods to keep data within gray-scale capabilities, are needed. Such techniques would introduce minimum distortion and allow faint details to be seen in the vicinity of strong anomalies. Postprocessing methods based on rapid image spatial filtering and enhancement algorithms could then be applied in an interactive environment.

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