Analog and digital simulation of the radocardiogram Page: 3 of 25
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THE MATHEMATICAL MODEL AND ITS USE IN DIGITAL COMPUTER SIMULATIONS
The model consists of four heart chambers, each ejecting a fixed fraction of its contained
blood volume (and hence of its retained tracer) with each systole, and a function describing the
lung delay (Fig. 1). By restricting the computation to points at end-systole and end-diastole, no
integration is required in the program, and the calculation is simple and rapid.
Each of the four heart chambers behaves as follows:
At end-diastole, q - q + I
At end-systole, r - q - e and q - q - (1 - e)
where q is total quantity of tracer in the heart chamber, I is the total quantity of tracer input to
the chamber during its diastole, r is the quantity of tracer output from the chamber during systole,
and e is the constant ejection fraction of the chamber.
The mechanism is written here in computer-language replacement statements instead of
ordinary equations, because the recursion relations are very simple when so expressed and are
unnecessarily complicated by subscripting to indicate the phase of each cycle and the chamber
involved. The programming sequence shown in the sample programs below guarantees the
required reciprocal behavior of the heart chambers and its relationship to the lung transit-time
Lung transit-time distribution
The transit-time distribution chosen to represent the delay of tracer in the lung is most
simply represented for digital computation by its "residue function":
and H*' - H
where i has values 0, 1, 2 ... This generates a curve of the type shown in Fig. 2.
H* is the residue function in the terminology of Bassingthwaighte [15J], that is 1-H, where
H is the familiar cumulative frequency function, the integral of h, the frequency function of
tracer transit times. Here, however, we are dealing with a discrete or discontinuous frequency
function in contrast to the continuous frequency function usually employed in circulatory
dynamics. Justification for the use of the residue function-and of this particular one-is given
in reference (3]. a is the turnover constant of any single compartment in the series of equal
size compartments that make up the lung delay. n is the number of such compartments. The
mean time of tracer in the lung is given by n/X. Alternatively, the distribution ma'y be regarded
simply as a convenient and reasonable two parameter curve for the lung delay, with parameters
a and n. i is the dummy index used to number each cardiac cycle, H* is the value of the residue
function at ventricular end-systole and H* *is used to indicate the corresponding value at vcntric-
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Parker, H.G.; Van Dyke, D.C.; Upham, F.T. & Windsor, A.A. Analog and digital simulation of the radocardiogram, article, June 1, 1974; Berkeley, California. (https://digital.library.unt.edu/ark:/67531/metadc1026005/m1/3/: accessed March 19, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.