Cross Cutting Skills>Unit Consistency Checks Practice Test

Clinicians assessing limb perfusion sometimes estimate the volumetric blood flow rate, Q, through a stenosed artery from pressure measurements and vessel caliber. A research team proposes a simplified scaling, claiming that laminar flow under physiologic conditions follows Q ∝ ΔP r^2/η, where ΔP is the pressure drop, r the arterial radius, and η the blood viscosity. They argue that radius dominates clinically and that length, L, can be ignored. However, when comparing limbs with different lesion lengths, a coordinator asks for a unit check: Q must have units of volume per time. In contrast, ΔP has units of pressure, η has viscosity (Pa·s), and r is length. The classic Poiseuille relation is Q = (π r^4 ΔP)/(8 η L), which includes L in the denominator and a fourth-power radius dependence, ensuring correct units for volumetric flow.

To quantify a hepatic enzyme's activity in vivo, investigators compare initial velocities, v0, measured at varying substrate concentrations. One analyst proposes v0 = kcat E_tS, with turnover number kcat (s^-1), total enzyme concentration E_t, and substrate concentration S. A colleague counters that the Michaelis–Menten form v0 = kcat E_t S/(K_M + S) should be used to maintain proper saturation behavior. She asks the team to check dimensions: v0 must have concentration per time. In the first proposal, units multiply to concentration^2 per time, violating unit consistency. In the standard form, S/(K_M + S) is a dimensionless ratio of concentrations, preserving v0 as concentration per time.