Respiration Physiology (1967) 3, 130-l 35 ; North-Holland Publishing Company, Amsterdam

COMPARATIVE QUANTITATIVE MORPHOLOGY OF THE MAMMALIAN LUNG: TRACHEA1

S. M. TENNEY AND D. BARTLETT, JR.

Department of Physiology, Dartmouth Medical School, Hanover, New Hampshire, 03755, U.S.A.

Abstract. Measurements of tracheal dimensions in a variety of mammals, covering the extremes of body sire, indicate that tracheal length is nearly proportional to M2.27, and tracheal radius is pro- portional to Mo28. The ratio of ventilation to cross-sectional area of the trachea is a constant for all species, hence average linear flow velocity in the trachea is the same in all mammals. Assuming dead space volume is proportional to tracheal volume, the dead space volume was found to be proportional to M1e06, in other words, it is proportional to both tidal volume and total pulmonary capacity; and since respiratory frequency is proportional to M- o.22, dead space ventilation is com- puted to be proportional to MO.‘*, and hence, is nearly proportional to expired ventilation. From this it may be inferred that alveolar ventilation is proportional to metabolic CO2 production in all mammals and thus alveolar Pco, is the same in all mammals. Similarly, since alveolar ventilation is proportional to oxygen consumption, alveolar PO, will also be nearly the same in all mammals at sea level.

Alveolar gas Dead space Alveolar ventilation Lung Comparative morphometry Trachea

Before oxygen is taken up by blood in the lungs of mammals it must be transported from the atmosphere, first by convection (ventilation) into the lungs, and second by physical diffusion across the alveolar wall and pulmonary capillary membrane. For each of these two processes there are relationships between structure and function which may be expected to follow a principle of most useful design. Predictions which follow from elementary considerations of physiological requirements can be tested simply and directly by analysis of dimensions of relevant units in the lungs, with attention focused on the problems posed by variation of body size (allometry).

The great range of size of mammals, from mouse to elephant, poses problems of structural design for the lungs which originate with the fact that all mammals have approximately the same body temperature. From the time that the physical laws governing flow of heat were first formulated they have been applied to the problem

Accepted for publication 9 May 1967. 1 This work was supported by a grant from the National Heart Institute, National Institutes of

Health, U.S.P.H.S. (HE-02888436).

130

COMPARATIVE MORPHOMETRY OF THE MAMMALIAN TRACHEA 131

of regulation of body temperature in living animals. Since, in the simplest model, heat production will be proportional to body mass (or volume), but heat loss, other things being equal, will be proportional to body surface area, thermal balance at the same temperature in all species will be difficult unless body surface area : body volume ratio can be held constant. However, it is clear that the smaller an animal the greater will be its area: volume ratio, and hence, to maintain the same body temperature, heat production (metabolic rate) must remain more nearly proportional to surface area than to body mass. Numerous measurements have verified the general validity of this view, although the prediction that resting oxygen consumption should be proportional to the square of the cube root of body mass (Vo2ccM0.(j6), as the straight application of the surface area law would require, is not seen. Rather, \t,,a~M’.‘~ (BRODY, 1945), the deviation of experiment from prediction originating in geometric “shape factors” and the fact that, in interspecific comparisons, “other things” are never equal.

Given then, the law relating metabolic rate (oxygen consumption) and body size, it is not surprising to find that minute volume of ventilation is very nearly proportional to rate of oxygen consumption. Experimental data illustrate the fact that the slope of the line of best fit for ventilation data is 0.75 (GUYTON, 1947a), very nearly the same as for the line relating oxygen consumption and body size (0.74). The interspecific constancy of the ratio of oxygen flow to air flow has been noted by others (GUNTHER and GUERRA, 1955; STAHL, 1963); and in an earlier morphometric study of the lung we have shown that the diffusing surface area of the lung is proportional to resting oxygen flow requirements (TENNEY and REMMERS, 1963). From this it may be inferred that the ratio of membrane resistance to oxygen flow is also an interspecific con- stant.

The present study considers convective transport, in which case the dimensions of the trachea are of primary concern. In the most general terms, the principal of similitude would lead us to predict that both tracheal radius and tracheal length would be proportional to the cube root of body mass. In terms of function, there are two minima which might be sought, one to conserve matter, and the other, to conserve energy. The former is the dead space problem, and if we assume that the dead space is proportional to tracheal volume, then tracheal volume should be as small as possible in order that each breath will be maximally efficient with respect to gas exchange. Clearly, though, by minimizing tracheal radius, airflow resistance wiJ1 increase, and this will add to the work of breathing. By direct measurement of tracheal dimensions nature’s solution to the compromise should emerge. There are only a very few data in the literature on which to base any conclusion. GUYTON (1947b) measured the tracheal diameter of ten mice, and this value, taken together with an assumed value in man concluded that the 1/3rd power law was exactly obeyed. There are also a few measurements to suggest that dead space volume is roughly one third of tidal volume (DEVOURS, 1966; MBAD, 1960). We have sought to extend the range of measurements with particular interest in increasing the accuracy of predicting alveolar gas composition.

132 S. M. TENHEY AND D. BARTLETT, JR

Methods and results

In a wide variety of animals, covering the extremes of body size, tracheas were measured in situ in freshly killed animals. Tracheal length was taken as the distance from the epiglottis to the carina, and internal diameter was measured just below the larynx.

TRACHEAL LENGTH

Since the lungs must be more or less centrally placed, tracheal length may be expected to vary as a linear dimension of the animal. In other words, LaM0*33. In fig. 1, the anatomical measurements do not confirm the prediction exactly, and what is found is LaM’.“.

TRACHEAL DIAMETER

This is the crucial dimension, for upon it the decision between minimizing volume or resistance rests, and the resistance alternative indudes more than one possibility. For example, if laminar flow is the central concern, then the Poiseuille formula applies, and we should expect a constant ratio between ventilation (VE) and tracheal radius (R) to the fourth power (iiEaR4); or, since vEaM”*75, RaM’.“. Actually, this is not a likely possibility, since the major resistance to air flow occurs, not in the trachea, but in the smaller airways, and they would provide a more useful site for an optimizing design based on consideration of laminar flow resistance. A more reasonable design criterion for the trachea might be to fix its dimensions so that flow remained laminar, hence avoiding the wasteful expenditure of energy with turbulent flow. Since the components of Reynold’s No. include linear flow velocity and pipe radius in the numerator, but velocity varies inversely as radius squared (cross-sectional area), the invariant ratio we seek is between ventilation and tracheal radius; or, RaM’.“.

Neither prediction is satisfied by the measurements. The data on tracheal radius, shown in fig. 1, indicate that RaM”*3g, which means that tracheal cross-sectional area is proportional to ventilation, and from this we deduce that average linear flow velocity is an interspecific constant. GUYTON (1947b) has also pointed out that, since pressure drop, at constant velocity, varies inversely as the square of the radius, but proportionately with length, it is possible that the total difference, from mouth to alveolus, would be the same in all species. Most measurements do, in fact, indicate that intrapleural pressure is about the same in all species (CROSFILL and WIDDI- COMBE, 1961). Further, if velocity is invariant then as tracheal size increases the likelihood of turbulent flow increases directly. The critical value of Reynold’s No. is exceeded, even at resting ventilatory volumes, in tracheas larger than man’s.

DEAD SPACE VOLUME

Assuming that dead space volume is proportional to tracheal volume, and computing the latter as if it were a cylinder, we find that dead space volume is proportional to M1*05, and since lung volume - actually, total lung capacity (TENNEY and REAMERS,

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134 S. hi. TENNEY AND D. BARTLETT, JR.

1963) - is very nearly proportional to body size (VL~CM ’ *02) it emerges that dead space volume is very nearly a constant fraction of lung volume.

ALVEOLAR VENTILATION

Knowing minute volume of ventilation and dead space volume we can compute the relationship of alveolar ventilation to body size if the variation in respiratory frequency is known. It would be good pump design to make the tidal volume pro- portional to lung volume, and since VE=f VT, then M0.75ccf Ml.‘, and faM-“*25. The experimentally determined value of the exponent relating respiratory frequency and body mass is -0.28. The value found for the exponent, based on different series of observations has varied from -0.25 (GUYTON, 1947b) to -0.29 (GUNTHER and GUERRA, 1955). We find that the slope of the best fit of all the data on respiratory frequency in the Handbook of Respiration (1958) is -0.28.

VD=fVD. Hence, VDaM’.“. And since VA=%-VD, VA=M’.‘~ [l-M’*“]. It is problematical whether the residual mass exponent in the parenthetical expression of this equation is anything other than a small error in the measurements. If that is the case, then, since ~~~~ = k(Vco2/VA), we see that alveolar CO2 tension is independ- ent of body size, since V&VA a(M0.74/M0.7s) = const. In other words, in all mammals, dead space volume is a constant fraction of lung volume and since tidal volume is proportional to lung volume, it is a constant fraction of tidal volume, and dead space ventilation is a constant fraction of total ventilation. This was assumed by DEJOURS who calculated the necessary dead space, alveolar, and tidal volume relationships consequent to that assumption (1966). The few alveolar CO2 values in the literature (DITTMER and GREBE, 1958) support the deductions of this work and the validity of Dejours’ assumption.

Discussion

There are small differences between the predicted values of the exponents in the allometric equations and those which were actually found. Although the measure- ments are extremely simple, the range is very wide, and for some computed values, like dead space, terms are squared and multiplied, which would lead to a magnification of small errors. Further, one must have reserved confidence in the accuracy of some of the physiological measurements, particularly minute volume of ventilation, in all but a very few spacies, and this is a required value to calculate alveolar ventilation. Still, with all of the problems, the relationships emerge clearly consistent with the principle of most useful design.

A fundamental question arises over the suitability of relating the dimensions and derived functional capacities of the lung to resting conditions of the animal. A much more likely and meaningful relationship might be to the exercising animal-to conditions of maximal oxygen consumption and high levels of ventilation-if the design of the lung has evolved through natural selection, since the conditions of survival are probably dependent upon great effort, and the ability to sustain it. It may be that maximal values are proportional to resting values, in which case the

COMPARATlVE MORPHOMBTRY OF THJJ MAMMALIAN TRACHEA 135

analysis given is directly applicable. The data are i~u~cient to know with certainty if that is so, and whether, for example, maximal ventilation and maximal oxygen consumption are more nearly proportional to body weight, rather than to body surface area. The interpretation of pulmonary dimensions will depend on several, as yet unknown, general interspecific characteristics.

Probably the most far-reaching conclusion of these measurements is with regard to the alveolar gas composition. The data strongly suggest that alveolar Pcoz is constant for all mammals, and since R varies only slightly between species, alveolar PO2 must also be constant.

References

BRODY, S. (1945). Bioenergetics and Growth. New York, remold Publ. Corp. CROSFZLL, M. L. and J. G. WIDDICXMBE (1961). Physical characteristics of the chest and lungs and

the work of breathing in different mammalian species. J. Physiol. (LOT&.) 158: 1-14. DJOURS, P. (1966). Respiration. New York, Oxford University Press. GUNTHER, B. and E. GUEXRA (1955). Biological similarities. Rdaphysiol. lutinoumericanu5 : 169-186, GWTON, A. C. (1947a). Measurement of the respiratory volumes of laboratory animals. Am. J.

Physiol. 150: 70-77. GUYTON, A. C. (1947b). Analysis of respiratory patterns in laboratory animals. Am. J. Physiol.

150: 78-83. Handbook of Respiration (1958), edited by D. S. Dittmer and R. M. Grebe. Philadelphia, W. B.

Saunders Co. MEAD, J. (1960). Control of respiratory frequency. J. Appi. Physiol. 15: 325-336. STAHL, W. R. (1963). Similarity analysis of physiological systems. Perspectives in Bioi. and Med.

6: 291-321. PENNEY, S. M. and J. E. REMMEas (1963). Comparative quantitative morphology of the mammalian

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