By Roustem N. Miftahof, Hong Gil Nam
Mathematical modelling of physiological platforms supplies to increase our realizing of complicated organic phenomena and pathophysiology of illnesses. during this e-book, the authors undertake a mathematical method of symbolize and clarify the functioning of the gastrointestinal approach. utilizing the mathematical foundations of skinny shell idea, the authors patiently and comprehensively advisor the reader throughout the basic theoretical recommendations, through step by step derivations and mathematical workouts, from easy idea to complicated physiological types. functions to nonlinear difficulties relating to the biomechanics of belly viscera and the theoretical boundaries are mentioned. distinct consciousness is given to questions of complicated geometry of organs, results of boundary stipulations on pellet propulsion, in addition to to medical stipulations, e.g. useful dyspepsia, intestinal dysrhythmias and the influence of gear to regard motility issues. With finish of bankruptcy difficulties, this e-book is perfect for bioengineers and utilized mathematicians.
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Extra info for Mathematical Foundations and Biomechanics of the Digestive System
30). Introduce polar coordinates α1 and α2 on S, such that α1 is the axial and α2 is the polar angular coordinate. They are related to the global Cartesian coordinates by 40 Shells of complex geometry À Á rðαi Þ ¼ xi þ yj þ z k ¼ R0 i sin α2 þ k cos α2 þ α1 j: (2:59) The Lamé parameters Ai and curvatures kij are given by A1 ¼ 1; k11 ¼ 1=R1 ¼ 0; A2 ¼ R0 ; k12 ¼ 0; k22 ¼ 1=R2 ¼ 1=R0 : (2:60) For the coefﬁcients θi ¼ 1 þ Hðαi Þ=Ri we have θ1 ¼ 1; θ2 ¼ 1 þ Hðαi Þ=R0 : (2:61) Hence, from Eq. 51) for yi, we obtain y1 ¼ @Hðαi Þ ; @α1 y2 ¼ 1 : R0 þ Hðαj Þ (2:62) On substituting Eqs.
However, in the modelling of the small and large intestine cylindrical coordinates become natural, whereas for working with the pregnant uterus, eyeball and urinary bladder spherical coordinates are more practical. All of the exercises which follow are aimed at obtaining speciﬁc relations of interest in cylindrical and spherical coordinates, respectively. g and reciprocal fr1 ; r2 ; m g bases on the surface S in cylindrical 1. Find the natural fr1 ; r2 ; m and spherical coordinates. 2. Find the unit base vectors ei and ei (i = 1, 2) and the Lamé parameters of the surface.
1 Fictitious deformations Most biological shells are of complex geometry. This is a result of the considerable anatomical variability of the organs they model. For example, the human stomach resembles a horn or a hook, the pregnant uterus, a pear, and the urine-ﬁlled bladder, a prolate or oblate spheroid. Convenient parameterization of such shells is a difﬁcult analytical task and sometimes is even unfeasible. Almost all numerical methods, on the other hand, are based on discretization of the computational domain and hence may appear to be secluded from the problem.