A quasi-one-dimensional non-linear mathematical model for the computation of the blood flow in the human systemic circulation is constructed. The morphology and physical modelling of the whole system (arteries, capillaries and veins) are completed by different methods for the different vessel generations. A hybrid method is used to solve the problem numerically, based on the governing equation (continuity, momentum and state equations), the input boundary conditions and the predetermined initial conditions. The two-step Lax-Wendroff finite-difference method is used to compute variables for each individual vessel, and the characteristic method is employed for the computation of internal boundary conditions of the vessel connection and the input and output system boundary conditions. Using this approach, blood flow, transmural pressure and blood velocity are computed at all vessel sites and for each time step. The pressure and flow waveforms obtained show reasonable agreement with clinical data and results reported in the literature. When an external conservative force field is applied to the system, the results computed from the model are intuitively correct. The term representing the external pressure added to the system by the muscle, which represents active control on the cardiovascular system, is also embodied in this model.

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A mathematical model is developed here for the impulse response of a class of systems represented by a general linear parabolic partial differential equation. Using a generalized eigenfuction expansion and the concept of a sampled-data system, a canonical set of state equations is obtained in discrete form. The impulse inputs may be any known time-varying functions on the given boundaries of the system or may be assumed to be distributed over a finite boundary region. For this computer model, no spatial discretization is required. Each discrete-time equation represents the dynamic behavior of each eigenvalue of the system and the response variable at any discrete location is given by a spatial linear combination of these state variables. Due to the discrete formulation of the system equations, numerical stability is assured. The equations are stable for any sampling time selected and the solution at any particular time desired may be simply obtained by incorporating this value directly into the model equations. The application of this method is demonstrated by solving a two-dimensional heat transfer problem subject to a single impulse input and comparisons were made to the finite element method of solution.