Dt

Note that the resistance, inductance, and capacitance of each vessel segment can vary, for example, in a tapering vessel.

10.2.1 Nodes and the equations at each node

In electrical circuit analysis, a node is defined as a point in a circuit where two or more circuit elements join. In Fig. 10.1 we will take the combination of a resistor, inductor, and capacitor as being a single element of our model, and number the nodes at the center of each element at the points where the resistor, capacitor, and inductor are joined. The nodes in Fig. 10.1 are labeled 1, 2, 3,. . ., n.

In the model, the input flow to the first element is labeled q1. The input pressure is labeled Pin. The flow leaving node 1 toward the capacitor is labeled qc1. The pressure at node 1 is P1. Note that the model does not contain any information about the pressure at points between the input point and node 1.

Considering node 1, we can now write an equation for the pressure drop across the first resistor and inductor as shown in Eq. (10.4) as the first-order differential equation .

In Eq. (10.4), PIN represents the input pressure to the vessel. P1 is the pressure at node 1 and q is the flow through the vessel. The time rate of change of flow is designated dq/dt. Note that LV1 and RV1 can vary from node to node, and can even vary with pressure, which would cause our model to be nonlinear.

A second differential equation, which is also first order, can be written at node 1. This equation describes the flow into the capacitor at node 1. The flow q1 is the vessel compliance at this point, C1 multiplied by dP/dt as shown in Eq. (10.5) . Conceptually, at a specific point in time, it may be useful at this point to think about pressure at each of the nodes, P1, P2, . . ., Pn, as being the independent variables and the flows, q1; q2, . . ., qn, as being the dependent variables, which depend on pressure. Mathematically, the P's and q's are dependent on time and location and time is the only independent variable. We will use the model to try to understand the relationship between flow and pressure in the vessel.

This system of two first-order differential equations can be written once for each node in the model. At the end, we will end up with a system of 2n first-order, ordinary differential equations, with two first-order equations written for each node: node 1 through node n. Although a large model can be computationally complex, the equations for a single node are relatively straightforward.

If we repeat the two equations for node 2 we end up with the following:

dt dP

The general equations for a general node i, between node 2 and node n will be

10.2.2 Terminal load

If we are modeling the aorta, for example, it is possible to divide the vessel into segments and write a set of equations for each finite segment. If we continue to add the downstream details of every branch of the aorta then the model will become larger and larger, more and more complicated, and more and more computationally expensive. Finally, it becomes impractical to individually model each one of the tens of billions of capillaries in the circulatory system. Instead, a more practical solution is that we will lump together all of the elements downstream of, or distal to, the main vessel that we are trying to understand with our model.

Because the capillaries are primarily resistance vessels we could begin estimating a load resistance that is based on the pressure at node n, and the flow moving through the entire capillary bed. The total, terminal load resistance, RTis equal to the pressure at node n divide by the total flow qn as shown in Eq. (10.10).

Although Rt would be a good first-order estimate of the terminal load in our model, we also know from empirical evidence that the capillary bed does not act as a pure resistance element. If we ended the model of the aorta, for example, in a single resistance we would find that pressure waves are reflected proximally because of a mismatch in impedance. In many cases, we would predict standing pressure waves by examining the model, where we know from the empirical data that no standing pressure waves truly exist. That is to say, the pressure along the vessel will not show a steady pressure gradient as one might expect, but a time varying pressure gradient that looks like periodic noise when one plots pressure gradient versus distance along the axis of the vessel. In fact, the vessels downstream of our model that make up the terminal load also exhibit a capacitive effect.

It has been suggested that a terminal load for our model could be estimated as a resistor in series with a second resistor parallel to a capacitor as shown in Fig. 10.2. For steady flow, or at very low frequencies, the total load impedance is equal to the total load resistance, which in this case is the sum of the two resistors. The total terminal resistance Rt is equal to RL1 plus RL2, the sum of the values of the two terminal load resistors.

One method of estimating the capacitance of the load, CL, is by impedance matching. For practical, biological systems, we would expect the output impedance of our model to match the input impedance of the terminal load, so that wave reflections are minimized. The ratio of RL1 and

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