title: Modeling the transient response to volume perturbations in the beating heart by the difference equation method.
abstract: 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.
authors.affiliation: Department of Biomedical Engineering, Hadassah University Hospital, Ein Kerem, Jerusalem, Israel.
authors.forename: D
authors.initials: D
authors.lastname: Adler
authors.forename: S D
authors.initials: SD
authors.lastname: Nikolic
authors.forename: E H
authors.initials: EH
authors.lastname: Sonnenblick
authors.forename: E L
authors.initials: EL
authors.lastname: Yellin
mesh_terms: D000818:Animals; D001810:Blood Volume; D003198:Computer Simulation; D004285:Dogs; D006339:Heart Rate; D006439:Hemodynamics; D006801:Humans; D008955:Models, Cardiovascular