# Lumped Parameter Mathematical Models

10.1 Introduction

Since the pioneering work of Otto Frank in 1899, there have been many types of mathematical models of blood flow. The aim of these models is a better understanding of the biofluid mechanics in cardiovascular systems. Mathematical computer models aim to facilitate the understanding of the cardiovascular system in an expensive and noninvasive way. One example of the motivation for such a model occurred when Raines et al. in 1972 observed that patients with severe vascular disease have pressure waveforms that are markedly different from those in healthy persons. By developing a model that would predict which changes in parameters affect those pressure waveforms in what way, the scientists might provide a means for diagnosing vascular disease before it becomes severe.

Lumped-parameter models are in common use for studying the factors that affect pressure and flow waveforms. A lumped-parameter model is one in which the continuous variation of the system's state variables in space is represented by a finite number of variables, defined at special points called nodes. The model would be less computationally expensive, with a correspondingly lower spatial resolution, while still providing useful information at important points within the model. Lumped-parameter models are good for helping to study the relationship of cardiac output to peripheral loads, for example, but because of the finite number of lumped elements, they cannot model the higher spatial-resolution aspects of the system without adding many, many more elements.

Chapter 10 will begin by addressing the general electrical analog model of blood flow, based on electrical transmission line equations.

Later in Chap. 10 a specific model of flow through a human mitral valve will be presented in some detail.

### 10.2 Electrical Analog Model of Flow in a Tube

In Chap. 7, Sec. 7.7, a solution was developed, which was published by Greenfield and Fry in 1965, that shows the relationship between flow and pressure for axisymmetric, uniform, fully developed, horizontal, Newtonian, pulsatile flow. The Fry solution is particularly useful when modeling the relationship of the pressure gradient in a tube or blood vessel to the instantaneous blood velocity. In Chap. 8 we used the Fry solution to develop an electrical analog of a typical pressure-measuring catheter. Using well-known solutions to RLC circuits, it became possible to characterize parameters like the natural frequency and dimen-sionless damping ratio of the system.

I describe, here in Chapter 10, an alternative development of the modeling of flow through any vessel or tube based on that analog. Let the subscript V represent properties of the blood vessel that we are modeling and the subscript L represent properties of the terminating load. In Chap. 7, the terminating load was a transducer, but in the more general case of the isolated blood vessel within a cardiovascular system, the terminating load represents the effects of a group of distal blood vessels, often capillaries.

Figure 10.1 shows the electrical schematic of a circuit representing a length of artery, terminating in a capillary bed. The values for resistance, capacitance, and inductance for each resistor, capacitor, and inductor, respectively, are calculated from the blood vessel properties on a per unit length basis.

Recall that the hydrodynamic resistance of a blood vessel depends on its radius and length as well as the viscosity of the fluid flowing in the

LV1 RV1 LV2

qc 1

RV2 LV 3

9c 2

RV 3

Figure 10.1 The electrical schematic of a model of blood flowing through a vessel or tube with the V transcript representing the characteristics of the vessel and L representing the terminal load.

n in

tube. For this type of discretized model, the resistance in each discrete resistor can be written as

pr m

The inductance in the vessel for each discrete inductor element becomes lv = /L = Cu ^ (10.2)

pr m

The capacitance of the vessel is the compliance of the vessel, or dA/dP, and depends on the pressure at the point where the capacitor is located.