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A Mathematical Model of Human Respiration at Altitude

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Abstract

We developed a mathematical model of human respiration in the awake state that can be used to predict changes in ventilation, blood gases, and other critical variables during conditions of hypocapnia, hypercapnia and these conditions combined with hypoxia. Hence, the model is capable of describing ventilation changes due to the hypocapnic-hypoxia of high altitude. The basic model is that of Grodins et al. [Grodins, F. S., J. Buell, and A. J. Bart. J. Appl. Physiol. 22:260–276, 1967]. We updated the descriptions of (1) the effects of blood gases on cardiac output and cerebral blood flow, (2) acid–base balance in blood and tissues, (3) O2 and CO2 binding to hemoglobin and most importantly, (4) the respiratory-chemostat controller. The controller consists of central and peripheral sections. The central chemoceptor-induced ventilation response is simply a linear function of brain \(P_{\rm CO_2}\) above a threshold value. The peripheral response has both a linear term similar to that for the central chemoceptors, but dependent upon carotid body \(P_{\rm CO_2}\) and with a different threshold and a complex, nonlinear term that includes multiplication of separate terms involving carotid body \(P_{\rm O_2}\) and \(P_{\rm CO_2}.\) Together, these terms produce ‘dogleg’-shaped curves of ventilation plotted against \(P_{\rm CO_2}\) which form a fan-like family for different values of \(P_{\rm CO_2}.\) With this chemical controller, our model closely describes a wide range of experimental data under conditions of solely changes in \(P_{\rm CO_2}\) and for short-term hypoxia coupled with \(P_{\rm CO_2}\) changes. This model can be used to accurately describe changes in ventilation and respiratory gases during ascent and during short-term residence at altitude. Hence, it has great applicability to studying O2-delivery systems in aircraft.

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Abbreviations

HCO3 :

Bicarbonate

C :

Content or concentration, L gas 100 L blood−1

CSF:

Cerebrospinal fluid

H :

Hematocrit, fractional

HbCO:

Carbamino hemoglobin

mM:

Concentration, millimoles L−1

MR:

Metabolic rate, mL min−1

P :

Gas partial pressure, torr

Q :

Blood flow, L min−1

RQ:

Respiratory Quotient

S :

Hemoglobin O2 saturation, fractional

SF:

Shunt fraction

T :

Respiratory threshold, torr

V :

Volume, L

\(\dot{V}\) :

Ventilation, L min−1

f :

Fraction

α:

Gas solubility, mM L blood−1

t :

time, s

A:

Alveolar

a:

Arterial

b:

Brain

bl:

Blood

cb:

Carotid body

D:

Deadspace

et:

End tidal

I:

Inspiratory (ventilation)

pl:

Plasma

rc:

Red blood cell

ti:

Tissue

T:

Tidal (volume)

v:

Venous

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Acknowledgments

We dedicate this paper to Dr. Fred Grodins first Chair of the Department of Biomedical Engineering at the University of Southern California. One of us (MBW) was a junior faculty member in this Department from 1967 to 1970. Although MBW was not involved at that time with respiratory physiology, he came away with a great appreciation for Fred’s wisdom and his ability to simplify complex problems. We hope that Fred would approve of this work and see it as a significant contribution to understanding the control of respiration.

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Authors and Affiliations

  1. Department of Pharmacology, Physiology and Neuroscience, University of South Carolina School of Medicine, 6439 Garners Ferry Road, Columbia, SC, 29209, USA

    Matthew Bernard Wolf & Robert P. Garner

  2. Odyssey II Solutions, Wichita, KS, 67208, USA

    Robert P. Garner

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  1. Matthew Bernard Wolf
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Correspondence to Matthew Bernard Wolf.

Appendix

Appendix

Figure A1
figure 15

Cardiac index data (solid circles) from Richardson et al.,39 upper panel, and the ratio of brain blood flow to its resting value (lower panel) from various sources (open symbols) are plotted against \(P_{\rm aO_2}.\) The dashed lines are sigmoidal fits to the data. In Eq. (6), the fit was scaled so that cardiac output was unchanged at resting \(P_{\rm AO_2}\)

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Figure A2
figure 16

Tidal volume is shown as a function of total ventilation. Experimental data (symbols) are from various sources.9,12–16,36,37 The curve fit (Eq. 13) is shown by the dashed line

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Wolf, M.B., Garner, R.P. A Mathematical Model of Human Respiration at Altitude. Ann Biomed Eng 35, 2003–2022 (2007). https://doi.org/10.1007/s10439-007-9361-3

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