How to maximize pulmonary oxygen uptake: a systematic analysis of mathematical models representing ventilation‐perfusion matching

Abstract

A major determinant of how efficient the lungs can transport oxygen into the bloodstream is the regional matching of airflow and blood flow – referred to as ventilation-perfusion (V/Q) matching. It is appreciated that the mechanisms governing V/Q matching play pathophysiological roles in the etiology of disease states such as acute pulmonary embolism and atelectasis. Yet how physiological V/Q matching is achieved remains poorly understood. This study aims to investigate how V and Q together govern oxygen transport and theoretically determine the optimal ratio of air and blood flow to maximize oxygen uptake. Clinical V/Q ratios are regionally measured by the ratio of 2 distinct contrast agents that are inhaled and infused. In other words, clinical V/Q ratio measurements are not a ratio of flow rates—they are ratio of contrast agent signals. To understand how the ratio of flow rates affect oxygen uptake, we derive and analyze series of progressively more sophisticated models of V/Q matching: Model A is a simple linear compartmental formulation; Model B accounts for the effect of nonlinear hemoglobin on oxygen solubility; Model C accounts for the spatial profile of a pulmonary capillary; and Model D accounts for both hemoglobin and spatial transport. Each model is analyzed using linear stability analysis. The results of our analysis conflict with the intuition that a V/Q ratio (of flow rates) equal to 1 maximizes oxygen uptake. We find that for each of the four models analyzed, optimal oxygen delivery is typically associated with a V/Q ratio greater than 1. Moreover, we observe that oxygen uptake is not only dependent upon the V/Q ratio, but the magnitude of air and blood flow – with increasing blood flow, even larger increases in air flow are required to maintain a given level of oxygen in the bloodstream. Thus our analysis reveals a non-linear relationship between optimal air and blood flow rates, V/Q ratios, and oxygenation. This non-linearity is consistent with cardiac output and ventilation data measured in humans over a range of cardiovascular exercise intensities.