Modeling the transient response to volume perturbations in the beating heart by the difference equation method.

Discrete theoretical methods, compatible with the discrete features of the beating heart, are used together with experimental study to attain a quantitative understanding of the transient response to a volume perturbation and of sustained mechanical alternans (SMA) in the beating heart. This is done in three stages. In stage A, a first-order difference equation describes the stroke volume (SV) response due to the Frank-Starling mechanism. It is shown that the value of gamma, the slope of the SV-end-diastolic volume curve, determines the type of response obtained because of a perturbation: 1) nonoscillatory decay for gamma < 1,2) oscillatory decay for 1 < gamma < 2, 3) SMA for gamma = 2, and 4) chaotic response for gamma > 2. In stage B, when the effect of each SV change on the successive end-diastolic aortic pressure (P) is considered, SV response to a perturbation is determined by a second-order difference equation. The solution of this equation shows that the response is determined by gamma and by the afterload factor lambda 1 = alpha 1.delta, where alpha 1 = delta Pj + 1/delta SVj and delta = delta SVj + 1/delta Pj + 1. The responses are a nonoscillatory decay for lambda 1 < 1 - gamma (type 1), oscillatory decay for 1 - (gamma/2) > lambda 1 > 1 - gamma (type 2), SMA for lambda 1 = 1 - gamma/2 (type 3), and 2:1 electrical-mechanical response for lambda 1 > 1 - gamma/2 (type 4). In stage C, a single volume perturbation, delta SVj, will directly affect not only Pj + 1 but also the subsequent values of P. Filling volume perturbations performed with a mitral valve occluder in eight anesthetized dogs led only to type 1 and 2 responses. The responses predicted by the model (using the experimental values of gamma and lambda 1) in each of the eight open-chest dogs are compatible with the experimental responses, suggesting that it is unlikely that SMA is initiated and maintained by variations in preload and afterload.