While maximum respiratory pressures at the mouth have been measured in numerous subjects, less data exists to characterize maximum pressures as they vary with lung volume. Maximum pressure is volume dependent because muscle tension is length dependent, because muscle tension produces higher pressure with a smaller radius of curvature, and because respiratory tissue is elastic. Rahn et al. [1] first produced static pressure-volume diagrams from a group of adult men, and later, Cook et al. [2] produced pressure-volume diagrams from a larger group of subjects including women and children. These diagrams were useful in modeling the energetics of respiration [3] and in monitoring the progress of respiratory muscle training [4]. Yet the total number of subjects tested remained small, particularly regarding females. The present paper provides additional static pressure-volume data obtained from adult volunteers, both women and men.

Data were then scrutinized in an exploratory manner to see if they could be easily and universally fit by a simple mathematical expression [5]. It was found that both men's and women's data could be described by an expression of the form:

P_e/P_max = A Ln (%VC) + B for exhalation

and

P_i/P_max = C Ln (100 - %VC) + D for inhalation

Here, P_max is the asymptotically maximum pressure that could be developed by the respiratory muscles at any lung volume and P_i is the maximum inspiratory pressure that can be developed at specific lung volumes. The average value of P_max found by determining the limit of the nonlinear P-V curve for the group of subjects was found to be 102 cmH₂O for males and 66 cmH₂O for females, and was found to be the same for both inhalation and exhalation directions. Least squares regression using Microsoft Excel yielded the following two equations:

P_e/P_max = 0.1426 Ln (%VC) + 0.3402 R² = 0.9549

and

P_i/P_max = 0.234 Ln (100% - %VC) - 0.0828 R² = 0.9642

These equations are graphed in Figure 1.

Greater (more positive) expiratory pressures were developed at higher V_L, while greater (more negative) inspiratory pressures were developed at lower V_L. In women P_e increased with volume from 39 to 61 cmH₂O and P_i decreased with volume from 66 to 28 cmH₂O. In men, P_e increased with volume from 63 to 97 cmH₂O and P₁ decreased with volume from 97 to 39 cmH₂O. These trends occur primarily for two reasons. First, respiratory muscles work both with and against respiratory tissue elastance to produce pressure. Expiratory efforts are aided by tissue elastance (lung recoil effects) at high V_L and inhibited at low V_L. Inspiratory efforts are inhibited by tissue elastance at high V_L and aided at low V_L. Second, respiratory muscles exert greater tension when they are stretched to greater lengths. Expiratory muscles are stretched when the lung is inflated, while inspiratory muscles are stretched when the lung is deflated. Both of these factors describe the general trend of the data.

The volume dependence of P_i was much more pronounced than the volume dependence of P_e in both women and men. This can be seen as a higher slope of the inspiratory equation compared to the expiratory equation in Figure 1. This may reflect a combination of strength differences between diaphragm (largely responsible for inhalation) and abdominal muscles (largely responsible for exhalation) recruitment of intercoastal muscle (largely responsible for posture maintenance), and different mechanical advantages of each type of muscle as the lung volume varies.

The Laplace equation may be relevant here. This equation states that enclosed pressure is proportional to the product of wall tension and wall thickness and inversely proportional to the radius of curvature. The Laplace equation for a sphere differs from that of a cylinder by a factor of two. Pressure in a sphere (P =

) is twice that of a cylinder (P = ), all other things being equal.

The diaphragm is positioned under the lungs and curves upward in a somewhat spherical shape. As it contracts, it becomes flatter, meaning that its radius of curvature increases. Lung volume increases as the diaphragm contracts. If the Laplace equation can be applied to the respiratory system, then it would show that inspiratory pressure should decrease as radius, and thus lung volume, increases (P_i ∝ , as long as wall tension and thickness remain steady).

The abdominal muscles are arranged differently, more like wrapping around a cylinder. The abdominal muscles flatten at smaller lung volumes instead of larger lung volumes, and the Laplace equation indicates that higher pressures should be developed at larger lung volumes (P ∝ V).

Both effects have been observed. Inspiratory pressures increase as lung volume decreases and expiratory pressure increases as lung volume increases. There is roughly a factor of two between the dependence of pressures upon lung volumes for inspiration and expiration. This could well be related to the difference in the Laplace equation for a sphere and a cylinder.

The pressure-volume data obtained in this study are of the same general magnitudes as those previously reported [1, 2, 6]. Rahn et al. [1] studied P_i in 11 men and P_e in 12 men using similar methods. The highest pressures from three efforts at each of six starting volumes were recorded. Measurements were read from a mercury manometer connected to the subjects' noses. Pressure-volume data closely match the results of the present study. Craig [6] produced pressure-volume diagrams from 10 men using methods similar to the present study. Pressures were taken from a mercury manometer connected to the subjects' mouths. These data also closely match the results of the present study. Cook et al. [2] studied 17 males and 9 females using two techniques. One technique was a conventional occlusion maneuver. The other technique involved subjects breathing into or out of large, fixed volumes. The compressibility of the air in differently sized containers provided for various ultimate lung volumes. The volumes were calculated from Boyle's Law using peak pressures that could be sustained for 1–2 seconds. Five volumes were used. It was concluded that the results of the occlusion method and the compression method were the same. In women, the compression-method P_i values were similar to those of the present study at high volumes, but slightly higher at lower volumes. The P_e values were similar at low volumes, but much higher at higher lung volumes. In men, the compression-method P_i values agree well with those of the present study. However, the P_e values are much higher than those of the present study at the higher volumes. Cook et al. [2] suggested that their P_e values might have been higher than the Rahn et al. values because of the use of mouth pressure measurements rather than nose pressure measurements. It was also hypothesized that these P_e values exceeded Craig's values due to better mouthpiece sealing.

As the results of this study are more in agreement with work of Rahn et al. [1] and Craig [6], it is more likely that there is another reason for the discrepancy. Aside from muscle strength alone, P_e and P_i are highly effort dependent. Subjects may limit their maximum pressures due to factors such as pain in the ear or general discomfort. During some maximum pressure maneuvers, researchers have observed changes in hemodynamics leading to loss of consciousness [7]. It is possible that the subjects of the Cook et al. [2] study were more highly motivated. It is also possible that these subjects were of above average strength.

Numerous authors have collected maximal pressures at a single V_L. Most recently, Wilson et al. [8] measured maximal P_e and P_i in 87 women and 48 men using partial occlusion and Bourdon gauges. The women were found to have P_e = 93 ± 17 cmH₂O and P_i = 73 ± 22 cmH₂O and the men were found to have P_e = 148 ± 34 cmH₂O and P_i = 106 ± 31 cmH₂O. It could be expected that the P_i and P_e values from a single volume study would exceed the values from a multiple volume study because more efforts are made at the optimal V_L in the single volume study, while muscle fatigue can be a factor in the multiple volume study.

Judging from inspiration values, this does not appear to be the case. In the present study, women were found to have P_i = 66 ± 32 cmH₂O at V_L = 12%VC and men were found to have P_i = 97 ± 46 cmH₂O at V_L = 14%VC. These values are virtually identical to the Wilson et al. [8] data. On the other hand, women in this study were found to have P_e = 61 ± 39 cmH₂O at V_L = 84%VC and men were found to have P_e 97 ± 42 cmH₂O at V_L = 81%VC. These values are considerably smaller than the Wilson et al. data. The P_e values of the single volume study fall in between the maximum P_e values of the present study and the maximum P_e values of the Cook et al. [2] study.

Satisfactorily describing maximum lung pressures with mathematical expressions can be helpful for respiratory mechanical modeling [3]. It is not likely that maximal pressures would be developed in young, healthy adults during quiet breathing. During exercise, and especially during expiratory flow limitation, however, maximum pressures may well be developed. For example, modeling the effects of respiratory masks during hard work could use these equations to calculate respiratory work rate. These equation forms are good because pressures and lung volumes both appear as relative rather than absolute values. That way, both men's and women's pressures could be determined with the same equations despite large differences in absolute pressures developed. Respiratory models for those conditions could well use the equations developed here.

Although we have no data to support the notion, it is possible, if their respiratory mechanics changed proportionally, that maximum pressures developed by patients with respiratory impairments could be described by the same equations as developed here. That is because these equations are in relative pressure and volume form. One would expect that P_max could be much lower in diseased patients, but P/P_max could be scaled the same. If this were so, then equations developed here could have more universal value.

Maximum pressures at the mouth have been determined to depend on lung volumes. Equations to describe these pressures have been developed, and these are in a form that may be useful for modeling and predictive purposes.