Any network of resistances, however complex, can always be reduced to a single “equivalent” resistor that relates the total flow through the network to the pressure difference across the network. Of course, one way of finding the overall resistance of a network is to perform an experiment to see how much flow goes through it for a given pressure difference between its inlet and outlet. Another approach to finding the overall resistance of a network is to calculate it from knowledge of the resistances of the individual elements in the network and how they are connected. When one looks at the overall design of the body’s vascular system, one sees two patterns: 1.the arterial, arteriolar, capillary, and venous segments are connected in series; and 2.within each segment, there are many vessels arranged in parallel.
Vessels in Series
When vessels with individual resistances R1, R2, …, Rn are connected in series, the overall resistance of the network is simply the sum of the individual resistances, as indicated by the following formula:
Rs=R1 + R2 +…+ Rn
Figure 6-3A shows an example of three vessels connected in series between some region where the pressure is Pi and another region with a lower pressure Po, so that the total pressure difference across the network, ΔP, is equal to Pi – Po. By the series resistance equation, the total resistance across this network (Rs) is equal to R1 + R2 + R3. By the basic flow equation, the flow through the network (Q) is equal to ΔP/Rs. It should be intuitively obvious that Q is also the flow (volume/time) through each of the elements in the series, as indicated in Figure 6-3B. Fluid particles may move with different velocities (distance/time) in different elements of a series network, but the volume that passes through each element in a minute must be identical.
As shown in Figure 6-3C, a portion of the total pressure drop across the network occurs within each element of the series. The pressure drop across any element in the series can be calculated by apply the basic flow equation to that element, for example, ΔP1=QR1. Note that the largest portion of the overall pressure drop will occur across the element in the series with the largest resistance to flow (R2 in Figure 6-3).
One implication of the series resistance equation is that elements with the highest relative resistance to flow contribute more to the network‘s overall resistance than do elements with relatively low resistance. Therefore, high-resistance elements are inherently in an advantageous position to be able to control the overall resistance of the network and therefore the flow through it.
Vessels in Parallel
As indicated in Figure 6-4, when several tubes with individual resistances R1, R2, …, Rn are brought together to form a parallel network of vessels, one can calculate a single overall resistance for the parallel network Rp according to the following formula:
1/Rp=1/R1 + 1/R2 +…+ 1/Rn
The total flow through a parallel network is determined by ΔP/Rp. As the preceding equation implies, the overall effective resistance of any parallel network will always be less than that of any of the elements in the network. In general, the more parallel elements that occur in the network, the lower the overall resistance of the network. Thus, for example, a capillary bed that consists of many individual capillary vessels in parallel can have a very low overall resistance to flow even through the resistance of a single capillary is relatively high.