Abstract
The ballistocardiogram is a record of movement. In the indirect method, the movement is that of the table; in the direct method, that of the body itself. In either case, the relation between such records and the force exerted upon the body by the heart and vascular system is complicated. If, however, certain simplifying assumptions are made, the relation is expressible as a linear differential equation. These assumptions are that in its elastic behavior tissue obeys Hooke's Law and that frictional forces are due to simple viscosity alone. The linear differential equation expresses a linear relationship among the various quantities: displacement, velocity, acceleration, force and higher derivatives of these quantities.
Such a linear differential equation can be “solved” electronically. Although setting up the equation to reflect accurately the mechanical situation is very difficult, perhaps impossible, certain approximations simplify matters. For the case of direct recording, we assume that approximately the body acts as a rigid mass held in position by damped elastic constraints. Under these conditions, the differential equation is very simple:
Mα + Rv + Ed = F
where F is the force acting upon the body, M the mass of the body, α the acceleration, v the the velocity, and d the displacement. R and E are coefficients related respectively to the damping viscous resistance and the elastic constants.
It is a relatively simple matter to obtain voltages proportional to α, v and d. This can be done in a variety of ways, for example, with the use of separate transducers for each, or by picking up velocity with a magnetic pick-up, integrating for d and differentiating for α by well-known electronic means(1) With 3 such voltages available, they may be combined in the proportions determined by M, R and E and the sum applied to the recording instrument to produce a record of F.
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