Abstract.
This paper considers a model of the human cardiovascular-respiratory control system with one and two transport delays in the state equations describing the respiratory system. The effectiveness of the control of the ventilation rate is influenced by such transport delays because blood gases must be transported a physical distance from the lungs to the sensory sites where these gases are measured. The short term cardiovascular control system does not involve such transport delays although delays do arise in other contexts such as the baroreflex loop (see [46]) for example. This baroreflex delay is not considered here. The interaction between heart rate, blood pressure, cardiac output, and blood vessel resistance is quite complex and given the limited knowledge available of this interaction, we will model the cardiovascular control mechanism via an optimal control derived from control theory. This control will be stabilizing and is a reasonable approach based on mathematical considerations as well as being further motivated by the observation that many physiologists cite optimization as a potential influence in the evolution of biological systems (see, e.g., Kenner [29] or Swan [62]). In this paper we adapt a model, previously considered (Timischl [63] and Timischl et al. [64]), to include the effects of one and two transport delays. We will first implement an optimal control for the combined cardiovascular-respiratory model with one state space delay. We will then consider the effects of a second delay in the state space by modeling the respiratory control via an empirical formula with delay while the the complex relationships in the cardiovascular control will still be modeled by optimal control. This second transport delay associated with the sensory system of the respiratory control plays an important role in respiratory stability. As an application of this model we will consider congestive heart failure where this transport delay is larger than normal and the transition from the quiet awake state to stage 4 (NREM) sleep. The model can be used to study the interaction between cardiovascular and respiratory function in various situations as well as to consider the influence of optimal function in physiological control system performance.
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References
Batzel, J.J., Tran, H.T.: Modeling Instability in the Control System for Human Respiration: applications to infant non-REM sleep. Appl. Math. Comput. 110, 1–51 (2000)
Batzel, J.J., Tran, H.T.: Stability of the human respiratory control system. Part1: Analysis of a two dimensional delay state-space delay model. J. Math. Biol. 41 (1), 45–79 (2000)
Batzel, J.J., Tran, H.T.: Stability of the human respiratory control system. Part2: Analysis of a three dimensional delay state-space model. J. Math. Biol. 41 (1), 80–102 (2000)
Bennett, L.S., Langford, B.A., Stradling, J.R., Davies, R.J.: Sleep fragmentation indices as predictors of daytime sleepiness and nCPAP response in obstructive sleep apnea. Am. J. Resp. Crit. Care Med. 158 (3), 778–786 (1998)
Bevier, W.C., Bunnell, D.E., Horvath, S.M.: Cardiovascular function during sleep of active older adults and the effects of exercise. Exp. Gerontol. 22 (5), 329–37 (1987)
Bocchi, E.A., Vilella de Moraes, A.V., Esteves-Filho, A., Bacal, F., Auler, J.O., Carmona, M.J., Bellotti, G., Ramires, A.F.: L-arginine reduces heart rate and improves hemodynamics in severe congestive heart failure. Clin. Cardiol. 23 (3), 205–210 (2000)
Bruschi, C., Fanfulla, F., Traversi, E., Patruno, V., Callegari, G., Tavazzi, L., Rampulla, C.: Identification of chronic heart failure patients at risk of Cheyne-Stokes respiration. Monaldi Arch. Chest Dis. 54 (4), 319–324 (1999)
Burgess, H.J., Kleiman, J., Trinder, J.: Cardiac activity during sleep onset. Psychophysiology 36 (3), 298–306 (1999)
Cherniack, N.S.: Apnea and periodic breathing during sleep. New Engl. J. Med. 341 (13), 985–987 (1999)
Chiariello, M., Perrone-Filardi, P.: Pathophysiology of heart failure. Miner Electrolyte Metab. 25 (1–2), 6–10 (1999)
Crystal, R.G., West, J.B., et al. (eds.): The Lung: Scientific foundations. Lippincott-Raven Press, Philadelphia, 1997
Fincham, W.F., Tehrani, F.T.: A mathematical model of the human respiratory system. J. Biomed. Eng. 5 (2), 125–133 (1983)
Fishman, A.P., Cherniack, N.S., Widdicombe, J.G., Geiger, S.R., (eds.): Handbook of Physiology: The Respiratory System, Volume II, Control of Breathing, Part 2. Am. Phys. Soc. Bethesda, Maryland, 1986
Grodins, F.S., Buell, J., Bart, A.J.: Mathematical analysis and digital simulation of the respiratory control system. J. Appl. Physiol. 22 (2), 260–276 (1967)
Grodins, F.S.: Integrative cardiovascular physiology: a mathematical synthesis of cardiac and blood vessel hemodynamics. Quart. Rev. Biol. 34 (2), 93–116 (1959)
Grodins, F.S.: Control Theory and Biological Systems. Columbia University Press, New York, 1963
Guyton, A.C.: Textbook of Medical Physiology. 7th ed. W. B. Saunders Company, Philadelphia, 1986
Hall, M.J., Xie, A., Rutherford, R., Ando, S., Floras, J.S., Bradley, T.D.: Cycle length of periodic breathing in patients with and without heart failure. Am. J. Respir. Crit. Care. Med. 154 (2 pt. 1), 376–381 (1996)
Hambrecht, R., Gielen, S., Linke, A., Fiehn, E., Yu, J., Walther, C., Schoene, N., Schuler, G.: Effects of exercise training on left ventricular function and peripheral resistance in patients with chronic heart failure: A randomized trial. Jama 283 (23), 3095–3101 (2000)
Hanly, P., Zuberi, N., Gray, R.: Pathogenesis of Cheyne-Stokes respiration in patients with congestive heart failure: relationship to arterial PCO2. Chest 104 (4), 1079–1084 (1993)
Huntsman, L.L., Noordergraaf, A., Attinger, E.O.: Metabolic autoregulation of blood flow in skeletal muscle: a model. In: J. Baan, A. Noordergraaf, J. Raines (eds.), Cardiovascular System Dynamics, MIT Press, Cambridge, 1978, pp. 400–414
Javaheri, S.: A mechanism of central sleep apnea in patients with heart failure. New Engl. J. Med. 341 (13), 949–54 (1999)
Kappel, F., Peer, R.O.: A mathematical model for fundamental regulation processes in the cardiovascular system. J. Math. Biol. 31 (6), 611–631 (1993)
Kappel, F., Peer, R.O.: Implementation of a cardiovascular model and algorithms for parameter identification. SFB Optimierung und Kontrolle, Karl-Franzens-Universität Graz, technical report Nr. 26, 1995
Kappel, F., Propst, G.: Approximation of feedback control for delay systems using Legendre polynomials. Conf. Sem. Mat. Univ. Bari. Nr. 201, 1984
Kappel, F., Salamon, D.: Spline approximation for retarded systems and the Riccati equation. Siam J. Control Optim. 25 (4), 1082–1117 (1987)
Kappel, F., Lafer, S., Peer, R.O.: A model for the cardiovascular system under an ergometric workload. Surv. Math. Ind. 7, 239–250 (1997)
Katz, A.M.: Physiology of the heart: 2nd edition. Raven Press, NY, 1992
Kenner, T.: Physical and mathematical modeling in cardiovascular systems. In: N.H.C. Hwang, E.R. Gross, D.J. Patel (eds.) Clinical and Research Applications of Engineering Principles, University Park Press, Baltimore, 1979, pp. 41–109
Khoo, M.C.K., Kronauer, R.E., Strohl, K.P., Slutsky, A.S.: Factors inducing periodic breathing in humans: a general model. J. Appl. Physiol. 53 (3), 644–659 (1982)
Khoo, M.C.K.: A model of respiratory variability during non-REM sleep. In: G.D. Swanson, F.S. Grodins, R.L. Hughson, (Eds.) Respiratory Control: A Modeling Perspective, Plenum Press, New York, 1989
Khoo, M.C.K.: Modeling the effect of sleep-state on respiratory stability. In: M.C.K. Khoo (eds.) Modeling and Parameter Estimation in Respiratory Control, Plenum Press, New York, 1989, pp. 193–204
Khoo, M.C.K., Yamashiro, S.M.: Models of control of breathing. In: H.A. Chang, M. Paiva (eds.) Respiratory Physiology: an analytical approach, Marcel Dekker, New York, 1989, pp. 799–829
Khoo, M.C.K., Gottschalk, A., Pack, A.I.: Sleep-induced periodic breathing and apnea: a theoretical study. J. Appl. Physiol. 70 (5), 2014–2024 (1991)
Klinke, R., Silbernagl, S. (Eds.): Lehrbuch der Physiologie. Georg Thieme Verlag, Stuttgart, 1994
Koo, K.W., Sax, D.S., Snider, G.L.: Arterial blood gases and pH during sleep in chronic obstructive pulmonary disease. Am. J. Med. 58 (5), 663–670 (1975)
Krieger, J., Maglasiu, N., Sforza, E., Kurtz, D.: Breathing during sleep in normal middle age subjects. Sleep 13 (2), 143–154 (1990)
Kugler, J., Maskin, C., Frishman, W.H., Sonnenblick, E.H., LeJemtel, T.H.: Regional and systemic metabolic effects of angiotensin-converting enzyme inhibition during exercise in patients with severe heart failure. Circulation 66 (6), 1256–1261 (1982)
Lorenzi-Filho, G., Rankin, F., Bies, I., Douglas Bradley, T.: Effects of inhaled carbon dioxide and oxygen on Cheyne-Stokes respiration in patients with heart failure. Am. J. Respir. Crit. Care Med. 159 (5 pt. 1), 1490–1498 (1999)
Mancia, G.: Autonomic modulation of the cardiovascular system during sleep. New Engl. J. Med. 328 (5), 347–349 (1993)
Martin, L.: Pulmonary physiology in clinical practice : the essentials for patient care and evaluation. Mosby Press, St. Louis, 1987
Mateika, J.H., Mateika, S., Slutsky, A.S., Hoffstein, V.: The effect of snoring on mean arterial blood pressure during non-REM sleep. Am. Rev. Respir. Dis. 145 (1), 141–146 (1992)
Moraes, D.L., Colucci, W.S., Givertz, M.M.: Secondary pulmonary hypertension in chronic heart failure: the role of the endothelium in pathophysiology and management. Circulation 102 (14), 1718–1723 (2000)
Niebauer, J., Clark, A.L., Anker, S.D., Coats, A.J.S.: Three year mortality in heart failure patients with very low left ventricular ejection fractions. Int. J. Cardiol. 70 (3), 245–247 (1999)
Noordergraaf, A., Melbin, J.: Introducing the pump equation. In: T. Kenner, R. Busse, H. Hinghofer-Szalkay (eds.) Cardiovascular System Dynamics: Models and Measurements, Plenum Press, New York, 1982, pp. 19–35
Ottesen, J.T.: Modeling of the baroreflex-feedback mechanism with time-delay. J. Math. Biol. 36 (1), 41–63 (1997)
Parmley, W.W.: Pathophysiology of congestive heart failure. Am. J. Cardiol. 56 (2), 7A–11A (1985)
Peer, R.O.: Mathematische Modellierung von grundlegenden Regelungsvorgängen im Herzkreislauf-System. Technical Report, Karl-Franzens-Universität und Technische Universität Graz, 1989
Peskin, C.S.: Lectures on mathematical aspects of physiology. In: F.C. Hoppensteadt (eds.) Mathematical Aspects of Physiology, Lect. Appl. Math. 19, Am. Math. Soc, Providence, Rhode Island, 1981, pp. 1–107
Phillipson, E.A., Bowes, G.: Control of breathing during sleep. In: A.P. Fishman, N.S. Cherniack, J.G. Widdicombe, S.R. Geiger (eds.) Handbook of Physiology, Section 3: The Respiratory System, Volume II, Control of Breathing, Part 2, Am. Phys. Soc. Bethesda, Maryland, 1986, pp. 649–690
Pinna, G.D., Maestri, R., Mortara, A., La Rovere, M.T., Fanfulla, F., Sleight, P.: Periodic breathing in heart failure patients: testing the hypothesis of instability of the chemoreflex loop. J. Appl. Physiol. 89 (6), 2147–2157 (2000)
Podszus, T.: Kreislauf und Schlaf. In: H. Schulz (ed.) Kompendium Schlafmedizin für Ausbildung, Klinik und Praxis, Deutsche Gesllschaft für Schlafforschung und Schlafmedizin, Landsberg, Lech: Ecomed-Verl.-Ges., 1997
Quaranta, A.J., D’Alonso, G.E., Krachman, S.L.: Cheyne-Stokes respiration during sleep in congestive heart failure. Chest 111 (2), 467–473 (1997)
Richardson, D.W., Wasserman, A.J., Patterson, J.L., Jr.: General and regional circulatory responses to change in blood pH and carbon dioxide tension. J. Clin. Invest. 40, 31–43 (1961)
Rowell, L.B.: Human Cardiovascular Control. Oxford University Press, 1993
Russell, D.L.: Mathematics of finite-dimensional control systems: theory and design. Marcel Dekker, New York, 1979
Schneider, H., Schaub, C.D., Andreoni, K.A., Schwartz, A.R., Smith, P.L., Robotham, J.L., O’Donnell, C.P.: Systemic and pulmonary hemodynamic responses to normal and obstructed breathing during sleep. J. Appl. Physiol. 83 (5), 1671–1680 (1997)
Schoen, F.J.: The heart-a pathophysiological perspective, Ch. 13. In: R.S. Cotran et al. (eds) Robbins Pathologic Basis of Disease, Sixth Edition, W. B. Saunders, Philadelphia, 1999
Shepard, J.W., Jr: Gas exchange and hemodynamics during sleep. Med. Clin. North Am. 69 (6), 1243–1264 (1985)
Somers, V.K., Dyken, M.E., Mark, A.L., Abboud, F.M.: Sympathetic-nerve activity during sleep in normal subjects. New Engl. J. Med. 328 (5), 303–307 (1993)
Sullivan, M.J., Higginbotham, M.B., Cobb, F.R.: Increased exercise ventilation in patients with chronic heart failure: intact ventilatory control despite hemodynamic and pulmonary abnormalities. Circulation 77 (3), 552–559 (1988)
Swan, G.W.: Applications of optimal control theory in biomedicine. Marcel Dekker, Inc. New York, 1984
Timischl, S.: A Global Model of the Cardiovascular and Respiratory System. PhD thesis, Karl-Franzens-Universität Graz, 1998
Timischl, S., Batzel, J.J., Kappel, F.: Modeling the human cardiovascular-respiratory control system: an optimal control application to the transition to non-REM sleep. Spezialforschungsbereich F-003 Technical Report 190, Karl-Franzens-Universität, 2000
Topor, Z.L., Johannson, L., Kasprzyk, J., Remmers, J.E.: Dynamic ventilatory response to CO2 in congestive heart failure patients with and without central sleep apnea. J. Appl. Physiol. 91 (1), 408–416 (2001)
Tsuruta, H., Sato, T., Shirataka, M., Ikeda, N.: Mathematical model of the cardiovascular mechanics for diagnostic analysis and treatment of heart failure: part 1 model description and theoretical analysis. Med. Biol. Eng. Comp. 32 (1), 3–11 (1994)
Wasserman, K., Whipp, B.J., Casaburi, R.: Respiratory control during exercise. In: A.P. Fishman, N.S. Cherniack, J.G. Widdicombe, S.R. Geiger (eds.) Handbook of Physiology, Section 3: The Respiratory System, Volume II, Control of Breathing, Part 2, Am. Phys. Soc. Bethesda, Maryland, 1986
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Supported by FWF (Austria) under grant F310 as a subproject of the Special Research Center F003 “Optimization and Control”
Mathematics Subject Classification (2000): 92C30, 49J15
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Batzel, J., Kappel, F. & Timischl-Teschl, S. A cardiovascular-respiratory control system model including state delay with application to congestive heart failure in humans. J. Math. Biol. 50, 293–335 (2005). https://doi.org/10.1007/s00285-004-0293-3
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DOI: https://doi.org/10.1007/s00285-004-0293-3