The Unit of Vascular Resistance: A Naming Opportunity

The metric system is based on three fundamental units: the kilogram (kg, mass), the meter (m, distance), and the second (s, time). Often a combination of these three is given a name (called a derived unit), usually honoring a famous scientist. For example, a newton, the unit of force named after the English physicist and mathematician Isaac Newton (1642 – 1727), is a kg m s⁻²; a joule, the unit of energy named for English physicist James Joule (1818 – 1889), is a kg m² s⁻²; a pascal, the unit of pressure named for French mathematician and physicist Blaise Pascal (1623 – 1662), is a kg m⁻¹ s⁻²; a watt, the unit of power named for Scottish engineer James Watt (1736 – 1819), is a kg m² s⁻³; and a rayl, the unit of acoustic impedance named for English physicist John Strutt (1842 – 1919) who is also known as Lord Rayleigh, is a kg m⁻² s⁻¹.

In Chapter 1 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss the human circulatory system. We talk about blood pressure, p, which is usually expressed in mmHg or torr, but in the metric system is given in pascals. We also analyze blood flow or cardiac output, i, sometimes expressed in milliliters per minute, but properly should be m³ s⁻¹. Then Russ and I introduce is the vascular resistance.

We define the vascular resistance R in a pipe or a segment of the circulatory system as the ratio of pressure difference across the pipe or segment to the flow through it:

R = Δp/i .                  (1.58)

The units are Pa m⁻³ s. Physiologists use the peripheral resistance unit (PRU), which is torr ml⁻¹ min.

What name is given to the Pa m⁻³ s, or equivalently the kg m⁻⁴ s⁻¹? Sometimes it’s called the “acoustic ohm,” stressing its analogy to the electrical unit of the ohm (a volt per amp). If the unit for electrical resistance can honor a scientist, German physicist Georg Ohm (1789 – 1854), why can’t the unit for mechanical resistance do the same? Let’s name the unit for vascular resistance!

I know what you’re thinking: we already have a name, the peripheral resistance unit. True, but I see three disadvantages with the PRU. First, it’s based on oddball units (pressure in torr? time in minutes?), so it’s not standard metric. Second, sometimes it’s defined using the second rather than the minute, so it’s confusing and you always must be on your toes to avoid making a mistake. Third, it wastes the chance to honor a scientist. We can do better.

My first inclination was to name this unit after the French physician Jean Poiseuille (1797 – 1869). He is the hero of Sec. 1.17 in IPMB. His equation relating the pressure drop and flow through a tube—often called the Poiseuille law—explains much about blood flow. However, Poiseuille already has a unit. The coefficient of dynamic viscosity has units of kg m⁻¹ s⁻¹, which is sometimes called a poiseuille. It’s not used much, but it would be confusing to adopt it for vascular resistance in addition to viscosity. Moreover, the old cgs unit for viscosity, g cm⁻¹ s⁻¹, is also named for Poiseuille; it’s called the poise, and it is commonly used. With two units already, Poiseuille is out.

Henry Darcy (1803 – 1858) was a French engineer who made important contributions to hydraulics, including Darcy’s law for flow in a porous medium. However, an older unit of hydraulic permeability is the darcy. Having another unit named after Darcy (even if it’s an mks unit instead of an oddball obsolete unit) would complicate things. So, no to Mr. Darcy.

The Irish physicist and mathematician George Stokes (1819 – 1903) helped develop the theoretical justification for the Poiseuille law. I’m a big fan of Stokes. He seems like a perfect candidate. However, the cgs unit of kinematic viscosity, the cm² s⁻¹, is called the stokes. He’s taken.

The Poiseuille law is sometimes called the Hagen-Poiseuille law, after the German scientist Gotthilf Hagen (1797 – 1884). He would be a good candidate for the unit, and some might choose to call a kg m⁻⁴ s⁻¹ a hagen. Why am I not satisfied with this choice? Hagen appears to be more of a hydraulic engineer than a biomedical scientist, and one theme in IPMB is to celebrate researchers who work at the interface between physics and physiology. Nope.

My vote goes to William Harvey (1578 – 1657), the English physician who first discovered the circulation of blood. I can find no units already named for Harvey. He doesn’t have a physics education, but he did make quantitative estimates of blood flow to help establish his hypothesis of blood circulation (such numerical calculations were uncommon in his day, but are in the spirit of IPMB). Harvey is a lot easier to pronounce than Poiseuille. Moreover, my favorite movie is Harvey.

We can name the kg m⁻⁴ s⁻¹ as the harvey (Ha), and 1 Ha = 1.25 × 10^-10 PRU (we may end up using gigaharveys when analyzing the peripheral resistance in people). One final advantage of the harvey: for those of you who disagree with me, you can claim that “Ha” actually stands for Hagen.