# 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.

Heart chambers

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

distribution.

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":

k-0 k

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-

ular end-diastole.-3-

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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.