Uses a single capillary convection diffusion model to generate functions of linear stationary systems: h(t), H(t), R(t), eta(t).

Description

This partial differential equation models a "tissue cylinder" consisting of a single capillary, and is mathematically like BTEX10. The model has been parameterized to produce the operator functions from the Terminology paper of Bassingthwaighte et al.(1986). wIth a Dirac delta function input (unity area, input at t = 0) the response functions are: h(t) is the transport function (or transfer function) the probability density function of arrival times at the outflow, fraction of dose per second. H(t) is the cumulative residence time distribution function, and equals the fraction of the dose accumulated in a bucket at the outflow. R(t) is the residue function, the fraction of the injected mass retained in the system at time t. eta(t) is the emergence function or the fractional escape rate, the fraction of the retainied mass emerging from the outflow per second. It is equivalent to the risk function of the life insur=ance industry, the risk of dying at age t. Three situations are illustrated: 1. A general situation described above. Plot TERM, Parameter set Term. This uses a very short pulse injection. A modest diffusion coefficient illustrates the standard. If D is small the output is a delayed pulse, but numerical inaccuracies distort a very narrow pulse; for D > 0 the outflow, h(t), is nearly a Gaussian pdf, as exemplified by the closeness of fit of the spatial distribution by a Gaussian density funnction, on the spatial plot, SPACE. 2. A stirred tank. Use plot TANK and parameter set Tank. The ridiculously high diffusion coefficient disperses the matrial almost uniformly throughout the lngth of the tube, approximiating a stirred tank within the axially distributed region. The downslope of h(t) and of R(t) are almost monoexponenetial with time constabt = V/F. 3. a dispersed input function. USe plot DISP and parameter set Disp. Becasue the input is not a DIrac delta function the responses are not formally the "transport function", h(t), H(t), R(t), and eta(t), but are the delayed equivaalents. Using a lagged normal density function as input avoids the sharp discontinuities so that the numerical methods work better. (The equations are not so "stiff".)

Equations

Differential Equations

Left Boundary Conditions

Right Boundary Conditions

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Initial Conditions

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The equations for this model may be viewed by running the JSim model applet and clicking on the Source tab at the bottom left of JSim's Run Time graphical user interface. The equations are written in JSim's Mathematical Modeling Language (MML). See the Introduction to MML and the MML Reference Manual. Additional documentation for MML can be found by using the search option at the Physiome home page.