Mathematical models of diffusion-limited gas bubble dynamics in tissue

Abstract

Mathematical models of bubble evolution in tissue have recently been incorporated into risk functions for predicting the incidence of decompression sickness (DCS) in human subjects after diving and/or flying exposures. Bubble dynamics models suitable for these applications assume the bubble to be either contained in an unstirred tissue (two-region model) or surrounded by a boundary layer within a well-stirred tissue (three-region model). The contrasting premises regarding the bubble-tissue system lead to different expressions for bubble dynamics described in terms of ordinary differential equations. However, the expressions are shown to be structurally similar with differences only in the definitions of certain parameters that can be transformed to make the models equivalent at large tissue volumes. It is also shown that the two-region model is applicable only to bubble evolution in tissues of infinite extent and cannot be readily applied to bubble evolution in finite tissue volumes to simulate how such evolution is influenced by interactions among multiple bubbles in a given tissue. Two-region models that are incorrectly applied in such cases yield results that may be reinterpreted in terms of their three-region model equivalents but only if the parameters in the two-region model transform into consistent values in the three-region model. When such transforms yield inconsistent parameter values for the three-region model, results may be qualitatively correct but are in substantial quantitative error. Obviation of these errors through appropriate use of the different models may improve performance of probabilistic models of DCS occurrence that express DCS risk in terms of simulated in vivo gas and bubble dynamics.