## Reassessing the Mathematical Modeling of the Contribution of Vasomotion to Vascular Resistance

*To the Editor*: This letter is motivated by the controversy surrounding the effect of vasomotion, which refers to the spontaneous oscillation in blood vessel radius, on vascular resistance. In a recent paper, Gratton et al. (2) used mathematical reasoning to imply that vascular resistance increases as the amplitude of vasomotion increases. Although this result appears consistent with experimental observations, effective vascular resistance in fact decreases with vasomotion, as shown earlier by Funk et al. (1). We address this controversy by taking a careful look at the meaning of vascular resistance and its implications for deriving a measure of vascular resistance in vessels with a time-varying radius that is physiologically meaningful.

The conclusion by Gratton et al. (2) appears consistent with experimental observations. In particular, decreased vasomotion has been correlated with low-resistance physiological states [e.g., pregnancy (2, 3)], and increased vasomotion has been correlated with high-resistance physiological states [e.g., preeclampsia and hypertension (2)]. However, these observations are paradoxical in the following sense. When the amplitude of vasomotion increases, average volume flux also increases. Because vascular resistance is defined to be a measure of the opposition or hindrance to flow, we expect the effective vascular resistance to be inversely proportional to volume flux. That is, effective vascular resistance must decrease with vasomotion. Mathematical reasoning can be used to arrive at the correct conclusion formally, but, as we shall see below, special care must be taken to calculate an effective vascular resistance when the radius of the blood vessel varies with time, as it does during vasomotion.

Volume flux (*F*) depends on the radius of the blood vessel via Poiseuille's law, and vascular resistance is defined (4) to be R_{v} = ΔP/*F*, where ΔP is the pressure difference between the ends of the vessel. When the radius varies with time *t*, both the instantaneous volume flux and the instantaneous vascular resistance also vary with time,*F* = *F*(*t*) and R_{v}= R_{v}(*t*), respectively. The appropriate quantity that reflects flow in this case is the average of the instantaneous volume flux *F̄*, computed over one period of the oscillation *T*, via_{v}, expressed in terms of instantaneous vascular resistance, is the harmonic mean of the instantaneous resistance (second to last term in *Eq. 2
*), as derived previously by Funk et al. (1), and not the time average of the instantaneous resistance (last term), as suggested more recently by Gratton et al. (2).

Using the time average of instantaneous resistance instead of the harmonic mean can easily lead to paradoxical results. If, for the sake of argument (1, 2), we model the time-varying radius as a simple sinusoidal oscillation, *r*(*t*) =*r*
_{0}(1 + λsinω*t*), where *r*
_{0} is the mean vessel radius, λ is the fractional amplitude of the oscillation (λ < 1), and ω is the frequency of the oscillation, then average volume flux evaluates to*L* is the length of the vessel and μ is the viscosity of the fluid. That is, volume flux increases with the amplitude of vasomotion. The time average of the vascular resistance evaluates to*r*(*t*).

The fact that decreased vasomotion has been correlated with low-resistance physiological states and increased vasomotion with high-resistance physiological states may seem inconsistent with the theoretical results. However, the apparent inconsistency is due to the fact that the dependence of the volume flux and vascular resistance on the mean radius, *r*
_{0}, has essentially been ignored in modeling studies because a vessel undergoing vasomotion is always compared with a static vessel with the same radius. Ultimately, changes in mean radius are very important, and it is not sufficient to consider changes in the amplitude of vasomotion alone when comparing different physiological conditions such as pregnancy and hypertension. For example, vasomotion is decreased in the circulation of mesenteric resistance arteries of pregnant rats compared with those of nonpregnant rats (2), and it alone would cause an increase in resistance, provided that mean radius is not affected. However, the mean radius of the vessels is also increased. The resulting decrease in resistance can easily overwhelm the increase in resistance due to reduced vasomotion, and the net effect is a reduction in resistance, in accordance with experimental observations.

In summary, the effective vascular resistance is equivalent to the harmonic mean of the instantaneous resistance, not the time average. Thus vasomotion contributes to flow, while decreasing vascular resistance. Use of the time average of instantaneous resistance as a measure of effective resistance leads to paradoxical conclusions and may lead to erroneous interpretations of experimental observations.

- Copyright © 2002 the American Physiological Society

## REFERENCES

The following is the abstract of the article discussed in the subsequent letter:

**Gratton, Robert J., Robin E. Gandley, John F. McCarthy, Walter K. Michaluk, Bryan K. Slinker, and Margaret K. McLaughlin.** Contribution of vasomotion to vascular resistance: a comparison of arteries from virgin and pregnant rats. *J Appl Physiol*85: 2255–2260, 1998.—Intrinsic oscillatory activity, or vasomotion, within the microcirculation has many potential functions, including modulation of vascular resistance. Alterations in oscillatory activity during pregnancy may contribute to the marked reduction in vascular resistance. The purpose of this study was *1*) to mathematically model the oscillatory changes in vessel diameter and determine the effect on vascular resistance and *2*) to characterize the vasomotion in resistance arteries of pregnant and nonpregnant (virgin) rats. Mesenteric arteries were isolated from Sprague-Dawley rats and studied in a pressurized arteriograph. Mathematical modeling demonstrated that the resistance in a vessel with vasomotion was greater than that in a static vessel with the same mean radius. During constriction with the α_{1}-adrenergic agonist phenylephrine, the amplitude of oscillation was less in the arteries from pregnant rats. We conclude that vasomotor activity may provide a mechanism to regulate vascular resistance and blood flow independent of static changes in arterial diameter. During pregnancy the decrease in vasomotor activity in resistance arteries may contribute to the reduction in peripheral vascular resistance.

- Copyright © 2002 the American Physiological Society