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Model-Based Analysis of Mechanisms Responsible for Sleep-Induced Carbon Dioxide Differences

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Abstract

This work describes a comprehensive mathematical model of the human respiratory control system which incorporates the central mechanisms for predicting sleep-induced changes in chemical regulation of ventilation. The model integrates four individual compartments for gas storage and exchange, namely alveolar air, pulmonary blood, tissue capillary blood, body tissues, and gas transport between them. An essential mechanism in the carbon dioxide transport is its dissociation into bicarbonate and acid, where a buffering mechanism through hemoglobin is used to prevent harmfully low pH levels. In the current model, we assume high oxygen levels and consider intracellular hydrogen ion concentration as the principal respiratory control variable. The resulting system of delayed differential equations is solved numerically. With an appropriate choice of key parameters, such as velocity of blood flow and gain of a non-linear controller function, the model provides steady-state results consistent with our experimental observations measured in subjects across sleep onset. Dynamic predictions from the model give new insights into the behaviour of the system in subjects with different buffering capacities and suggest novel hypotheses for future experimental and clinical studies.

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References

  • Aittokallio, J., Virkki, A., Aittokallio, T., Saaresranta, T., Polo-Kantola, P., Polo, O., in press. Non-invasive respiratory monitoring during wakefulness and sleep in pre- and postmenopausal women. Respir. Physiol. Neurobiol. Available online 31 May 2005 (doi:10.1016/j.resp.2005.03.012).

  • Aittokallio, T., Gyllenberg, M., Polo, O., 2002. Adjustment of the human respiratory system to increased upper airway resistance during sleep. Bull. Math. Biol. 64, 3–28.

    Article  PubMed  Google Scholar 

  • Batzel, J., Tran, H., 2000a. Stability of the human respiratory control system I. Analysis of a two-dimensional delay state-space model. Math. Biol. 41, 45–79.

    Article  MATH  MathSciNet  Google Scholar 

  • Batzel, J., Tran, H., 2000b. Stability of the human respiratory control system II. Analysis of a three-dimensional delay state-space model. Math. Biol. 41, 80–102.

    Article  MATH  MathSciNet  Google Scholar 

  • Batzel, J.J., Timischl-Teschl, S., Kappel, F., 2005. A cardiovascular-respiratory control system model including state delay with application to congestive heart failure in humans. J. Math. Biol. 50(3), 293–335.

    Article  PubMed  MATH  MathSciNet  Google Scholar 

  • Batzel, J.J., Tran, H.T., 2000c. Modeling instability in the control system for human respiration: applications to infant non-REM sleep. Appl. Math. Comput. 110(1), 1–51.

    Article  MathSciNet  MATH  Google Scholar 

  • Browne, H.A.K., Adams, L., Simonds, A., Morrell, M.J., 2003. Ageing does not influence the sleep-related decrease in the hypercapnic ventilatory response. Eur. Respir. J. 21(3), 523–529.

    Article  PubMed  Google Scholar 

  • Carley, D.W., Shannon, D.C., 1988a. A minimal mathematical model of human periodic breathing. J. Appl. Physiol. 65, 1400–1409.

    Google Scholar 

  • Carley, D.W., Shannon, D.C., 1988b. Relative stability of human respiration during progressive hypoxia. J. Appl. Physiol. 65, 1389–1399.

    Google Scholar 

  • Caruana-Montaldo, B., Gleeson, K., Zwillich, C.W., 2000. The control of breathing in clinical practice. Chest 117(1), 205–225.

    Article  PubMed  Google Scholar 

  • Chiari, L., Avanzolini, G., Ursino, M., 1997. A comprehensive simulator of the human respiratory system: Validation with experimental and simulated data. Ann. Biomed. Eng. 25, 985–999.

    PubMed  Google Scholar 

  • Clark, J.S., Votteri, B., Ariagno, R.L., Cheung, P., Eichhorn, J.H., Fallat, R.J., Lee, S.E., Newth, C.J.L., Rotman, H., Sue, D.Y., 1992. Noninvasive assessment of blood gases. Am. Rev. Respir. Dis. 145, 220–232.

    PubMed  Google Scholar 

  • Cleave, J.P., Levine, M.R., Fleming, P.J., 1984. The control of ventilation: A theoretical analysis of the response to transient disturbances. J. Theor. Biol. 108, 261–283.

    Article  PubMed  MathSciNet  Google Scholar 

  • Cleave, J.P., Levine, M.R., Fleming, P.J., Long, A.M., 1986. Hopf bifurcations and the stability of the respiratory control system. J. Theor. Biol. 119, 299–318.

    Article  PubMed  MathSciNet  Google Scholar 

  • Colrain, I.M., Trinder, J., Fraser, G., Wilson, G.V., 1987. Ventilation during sleep onset. J. Appl. Physiol. 63(5), 2067–74.

    PubMed  Google Scholar 

  • den Aardweg, J.G.V., Karemaker, J.M., 2002. Influence of chemoreflexes on respiratory variability in healthy subjects. Am. J. Respir. Crit. Care Med. 165, 1041–1947.

    PubMed  Google Scholar 

  • Despopulous, A., Silbernagl, S., 2003. Color Atlas of Physiology, 5th edn. Thieme Medical Publishers, New York, NY, USA.

  • ElHefnawy, A., Saidel, G.M., Bruce, E.N., Cherniack, N.S., 1990. Stability analysis of CO2 control of ventilation. J. Appl. Physiol. 69, 498–503.

    PubMed  Google Scholar 

  • Erdi, P., Toth, J., 1989. Mathematical Models of Chemical Reactions. Princeton University Press, Princeton, NJ.

    MATH  Google Scholar 

  • Fowler, A., Kalamangalam, G., 2000. The role of the central chemoreceptor in causing periodic breathing. IMA J. Math. Appl. Med. Biol. 17(2), 147–167.

    Article  PubMed  MATH  Google Scholar 

  • Fowler, A.C., Kalamangalam, G.P., 1993. A mathematical analysis of the grodins model of respiratory control. IMA J. Math. Appl. Med. Biol. 10, 249–280.

    Article  PubMed  MATH  Google Scholar 

  • Grodins, F.S., Buell, J., Bart, A.J., 1967. Mathematical analysis and digital simulation of the respiratory control system. J. Appl. Physiol. 22(2), 260–276.

    PubMed  Google Scholar 

  • Guglielmi, N., Hairer, E., 2000. Users' guide for the code radar5. Technical report, University of Geneve, Department of Mathematics.

  • Guglielmi, N., L'Aquila, Hairer, E., 2001. Implementing radau iia methods for stiff delay differential equations. Computing 67, 1–12.

    Article  MATH  MathSciNet  Google Scholar 

  • Janssens, J.P., Perrin, E., Bennani, I., de Muralt, B., Titelion, V., Picaud, C., 2001. Is continuous transcutaneous monitoring of P CO 2 (TcCO2) over 8 h reliable in adults? Respir. Med. 95, 331–335.

    Article  PubMed  Google Scholar 

  • Javaheri, S., 1999. A mechanism of central sleep apnea in patients with heart failure. N. Engl. J. Med. 341(13), 949–954.

    Article  PubMed  Google Scholar 

  • Khoo, M.C.K., 1990. A model-based evaluation of the single-breath CO2 ventilatory response test. J. Appl. Physiol. 68(1), 393–399.

    PubMed  Google Scholar 

  • Khoo, M.C.K., 2000. Determinants of ventilatory instability and variability. Respir. Physiol. 122(2-3), 168–182.

    Google Scholar 

  • Khoo, M.C.K., Gottschalk, A., Pack, A.I., 1991. Sleep-induced periodic breathing and apnea: A theoretical study. J. Appl. Physiol. 70(5), 2014–2024.

    PubMed  Google Scholar 

  • Khoo, M.C.K., Kronauer, R.E., Strohl, K.P., Slutsky, A.S., 1982. Factors inducing periodic breathing in humans: A general model. J. Appl. Physiol. 53, 644–659.

    PubMed  Google Scholar 

  • Kirby, T.P., Wraith, P.K., Cort, S. C.D., Airlie, M.A.A., Hill, J.E., Carson, E.R., Flenley, D.C., Warren, P.M., 1994. Modelling the dynamic ventilatory response to hypoxia in normal subjects. J. Theor. Biol. 166, 135–147.

    Article  PubMed  Google Scholar 

  • Kuboyama, T., Hori, A., Sato, T., Mikami, T., Yamaki, T., Ueda, S., 1997. Changes in cerebral blood flow velocity in healthy young men during overnight sleep and while awake. Electroencephalogr. Clin. Neurophysiol. 102(2), 125–131.

    Article  PubMed  Google Scholar 

  • Levine, M., Cleave, J.P., Dodds, C., 1995. Can periodic breathing have advantages for oxygenation? J. Theor. Biol. 172(4), 355–368.

    Article  Google Scholar 

  • Longobardo, G.S., Cherniack, N.S., Gothe, B., 1989. Factors affecting respiratory system stability. Ann. Biomed. Eng. 17, 377–396.

    Article  PubMed  Google Scholar 

  • Longobardo, G.S., Evangelisti, C.J., Cherniack, N.S., 2002. Effects of neural drives on breathing in the awake state in humans. Respir. Physiol. 129(3), 279–402.

    Article  PubMed  Google Scholar 

  • Longobardo, G.S., Goethe, B., Goldman, M.D., Cherniack, N.S., 1982. Sleep apnea considered as a control system instability. Respir. Physiol. 50, 311–333.

    Article  PubMed  Google Scholar 

  • Marcus, C.L., Glomb, W.B., Basinski, D.J., Davidson, S.L., Keens, T.G., 1994. Developmental pattern of hypercapnic and hypoxic ventilatory responses from childhood to adulthood. J. Appl. Physiol. 76(1), 314–320.

    PubMed  Google Scholar 

  • Maxima Development Team 2004. Maxima: A sophisticated computer algebra system. (Maxima was previously knows as DOE Macsyma).

  • Mitchell, G.S., Douse, M.A., Foley, K.T., 1990. Receptor interactions in modulating ventilatory activity. Am. J. Physiol. 259, 911–920.

    Google Scholar 

  • Moreau, K.L., Donato, A.J., Tanaka, H., Jones, P.P., Gates, P.E., Seals, D.R., 2003. Basal leg blood flow in healthy women is related to age and hormone replacement therary status. J. Physiol. 547, 309–316.

    Article  PubMed  Google Scholar 

  • NIST, 2004. Standard Reference Database 40. NIST, http://kinetics.nist.gov/ solution/index.php. NDRL/NIST Solution Kinetics Database on the Web.

  • Noshiro, M., Furuya, M., Linkens, D., Goode, K., 1993. Nonlinear identification of the P CO 2 control system in man. Comput. Methods Prog. Biomed. 40(3), 189–202.

    Article  Google Scholar 

  • Orr-Walker, B.J., Horne, A.M., Evans, M.C., Grey, A.B., Murray, M. A.F., McNeil, A.R., Reid, I.R., 1999. Hormone replacement therapy causes a respiratory alkalosis in normal posmenopausal women. J. Clin. Endocrinol. Metab. 84(6), 1997–2001.

    Article  PubMed  Google Scholar 

  • Pedersen, M. E.F., Fatemian, M., Robbins, P.A., 1999. Identification of fast and slow ventilatory responses to carbon dioxide under hypoxic and hyperoxic conditions in humans. J. Physiol. 521(1), 273–287.

    Article  PubMed  Google Scholar 

  • Prabhakar, N.R., Peng, Y.-J., 2004. Peripheral chemoreceptors in health and disease. J. Appl. Physiol. 96, 359–366.

    Article  PubMed  Google Scholar 

  • R Development Core Team 2004. R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-00-3.

  • Saaresranta, T., Polo, O., 2002. Hormones and breathing. Chest 122, 2165–2182.

    Article  PubMed  Google Scholar 

  • Saunders, K.B., Bali, H.N., Carson, E.R., 1980. A breathing model of the respiratory system: The controlled system. J. Theor. Biol. 84, 135–161.

    Article  PubMed  Google Scholar 

  • Saunders, K.B., Stradling, J., 1993. Chemoreceptor drives and short sleep-wake cycles during hypoxia: a simulation study. Ann. Biomed. Eng. 21, 465–474.

    Article  PubMed  Google Scholar 

  • Ursino, M., Magosso, E., Avanzolini, G., 2001a. An integrated model of the human ventilatory control system: The response to hypercapnia. Clin. Physiol. 21(4), 447–464.

    Article  Google Scholar 

  • Ursino, M., Magosso, E., Avanzolini, G., 2001b. An integrated model of the human ventilatory control system: the response to hypoxia. Clin. Physiol. 21(4), 465–477.

    Article  Google Scholar 

  • Vielle, B., 2000. A new explicit stability criterion for human periodic breathing. J. Math. Biol. 41(6), 546–558.

    Article  PubMed  MATH  MathSciNet  Google Scholar 

  • Vielle, B., Chauvet, G., 1998. Delay equation analysis of human respiratory stability. Math. Biosci. 152, 105–122.

    Article  PubMed  MATH  MathSciNet  Google Scholar 

  • West, J.B., 1977. Bioengineering Aspects of the Lung, vol. 3. Marcel Dekker.

  • Whiteley, J.P., Gavaghan, D.J., Hahn, C.E.W., 2003. Periodic breathing induced by arterial oxygen partial pressure oscillations. Math. Med. Biol. 20, 205–224.

    Article  PubMed  MATH  Google Scholar 

  • Williams, A.J., 1998. Abc of oxygen: Assessing and interpreting arterial blood gases and acid-base balance. BMJ 317, 1213–1216. clinical review.

    Google Scholar 

  • Younes, M., Ostoroski, M., Thompson, W., Colleen, L., Shewchuk, W., 2001. Chemical control stability in patients with obstructive sleep apnea. Am. J. Respir. Crit. Care Med. 163(5), 1181–1190.

    PubMed  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Turku, FIN-20014, Turku, Finland

    T. Aittokallio & A. Virkki

  2. Turku Centre for Computer Science (TUCS), Lemminkäisenkatu 14A, FIN-20520, Turku, Finland

    T. Aittokallio & A. Virkki

  3. Department of Mathematics and Statistics, Rolf Nevanlinna Institute, University of Helsinki, FIN-00014, Helsinki, Finland

    M. Gyllenberg

  4. Sleep Research Unit, Department of Physiology, University of Turku, FIN-20014, Turku, Finland

    T. Aittokallio, O. Polo & J. Toivonen

  5. Department of Pulmonary Medicine, Tampere University Hospital, FIN-33521, Tampere, Finland

    O. Polo

Authors
  1. T. Aittokallio
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  2. M. Gyllenberg
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  3. O. Polo
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  4. J. Toivonen
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  5. A. Virkki
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Correspondence to A. Virkki.

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Aittokallio, T., Gyllenberg, M., Polo, O. et al. Model-Based Analysis of Mechanisms Responsible for Sleep-Induced Carbon Dioxide Differences. Bull. Math. Biol. 68, 315–341 (2006). https://doi.org/10.1007/s11538-005-9059-3

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