In this case, the orientation of the 2 dimensional area probe is not important. That is, the direction of its linear elements is equal in all three dimensions of space. This will be the case when the structural feature is truly isotropic. It is important to remember that this formula is applicable only when the linear structural feature has an isotropic interaction with the areal probe. Unlike the two dimensional Buffon problem, which involves the probability that a randomly thrown needle intersects a line on the floor, the space ball probe is based on the probability that a surface is hit by a line that is randomly oriented in 3D space. Briefly, it is based on a three dimensional version of the Buffon needle problem. In spite of the simplicity of the formula, its' derivation is somewhat deep and the derivation and proof are presented in the original Smith-Guttman paper. Accordingly, the number of times that a linear structural feature ( Q) passes through a probe of known area ( A) is directly related to the length per unit volume ( L V) of the structural feature. The formula that relates measurements made on images to length ( Smith and Guttman, 1953) is simple and easy to apply (Equation 1). In this article, the use and application of a virtual isotropic surface probe that readily fulfills this requirement is described. Prior to the introduction of the space ball probe, stereological estimators of length were plagued by the requirement for an isotropic interaction between the area probes and linear features such as axons and capillaries. The stereological relationship formula for estimating length density, L V ^ = 2 Structural parameters of potential interest and for which quantitative studies of length have yet to be carried out include microtubules, involved in intracellular transport, and neuropil threads, an expression of Alzheimer's disease.
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These parameters can be used to evaluate brain hemodynamics ( Kubíková et al., 2018), tissue oxygenation ( Nikolajsen et al., 2015), and tissue repair ( Lee et al., 2005 McConnell et al., 2016) and ultimately used to develop therapeutic approaches to brain disorders.
![mark west stereology mark west stereology](https://i1.rgstatic.net/publication/14185392_Unbiased_stereological_estimation_of_the_total_number_of_synapses_in_a_brain_region/links/54d2085e0cf25ba0f0423bf9/largepreview.png)
Examples using space ball probes include the length of dopaminergic axons, which can be related to the dopaminergic innervation and function of the striatum ( Li et al., 2016) the length of serotoninergic ( Liu et al., 2011) and cholinergic axons in cerebral cortex which can be related to cortical function ( Nikolajsen et al., 2011) and the length of astrocyte processes, which can be related to immune-reactivity ( McNeal et al., 2016). Estimates of the length of cellular features can provide quantitative information about various biological functions.