Optimal structural and functional organization of the circulatory and respiratory systems

I. B. Bukharov

Matematičeskoe modelirovanie, Tome 17 (2005) no. 9, pp. 3-26.

Résumé

In this article, mathematical models dealing with the structure and steady functional states of the circulatory and respiratory systems, based on the solving of optimization problems with different optimality criteria are reviewed. As a result of mathematical modeling, the principal structural and functional parameters of the circulatory (vessel diameters of the arterial tree, end-diastolic and stroke volume of the isolated left ventricle, left ventricular ejection as a function of time, hemodynamic vascular resistance for the isolated left ventricle) and respiratory (branch diameters of the bronchial tree, breathing frequency, inspiratory and expiratory phase duration, air flow as a function of time) systems are determined. The best agreement between the optimal and observed parameters is obtained by using mathematical models based on the minimum energy consumption criterion.

Bibliographie

[1] Hess W. R., “Das Prinzip des kleinstein Kraftverbrauches im Dienste hämodynamischer Forschung”, Arch. Anat. Physiol. (Physiol. Abteil.), 1914, no. 1–2, 1–62 | MR

[2] Peters R. H., The ecological implications of body size, Cambridge University Press, Cambridge, 1983, 329 pp.

[3] Brody S., Bioenergetics and growth, Reinhold Publishing Company, New York, 1945, 1023 pp.

[4] Murray C. D., “The physiological principle of minimum work”, Proc. Natl. Acad. Sci., USA, 12:2 (1926), 207–214 | DOI

[5] Bukharov I. B., Khanin M. A., “Optimal structure of the microcirculatory bed”, J. Theor. Biol., 169:3 (1994), 267–273 | DOI | MR

[6] Cohn D., “Optimal systems. II: The vascular system”, Bull. Math. Biophys., 17:3 (1955), 219–227 | DOI | MR

[7] Chernousko F. L., “Optimalnaya struktura vetvyaschikhsya truboprovodov”, Prikl. mat. mekh., 41:2 (1977), 376–383 | MR

[8] Melkumyants A. M., “Optimalnaya struktura arterialnoi seti skeletnykh myshts”, Byull. eksp. biol. med., 1978, no. 9, 259–262 | MR

[9] Grigoryan S. S., Gotsiridze N. Sh., “Ob optimalnoi organizatsii organnoi arterialnoi seti”, Mekhanika biologicheskikh sploshnykh sred, Izd-vo MGU, M., 1986, 119–124

[10] Grigoryan S. S., Gotsiridze N. Sh., “Optimalnaya organizatsiya vetvyascheisya sosudistoi seti pri pulsiruyuschem potoke”, Mekhanika biologicheskikh sploshnykh sred, Izd-vo MGU, M., 1986, 125–131

[11] Singhal S., Henderson R., Horsfield K., “Morphometry of the human pulmonary arterial tree”, Circulat. Res., 33:2 (1973), 190–197 | MR

[12] Noldus E. J., “Optimal control aspects of left ventricular ejection dynamics”, J. Theor. Biol., 63:3 (1976), 275–309 | DOI | MR

[13] Pfeiffer K. P., Kenner T., “On the optimal strategy of cardiac ejection”, Cardiovascular system dynamics, eds. Kenner T. et al., Plenum, NY, 1982, 133–136 | MR

[14] Hämäläinen R. P., Hämäläinen J. J., “On the minimum work criterion in optimal control models of left-ventricular ejecton”, IEEE Trans. Biomed. Eng., 32:10 (1985), 951–956 | DOI

[15] Hämäläinen J. J., Hämäläinen R. P., “Energy cost minimization in left-ventricular ejection: an optimal control model”, J. Appl. Physiol., 61:6 (1986), 1972–1979 | MR

[16] Hämäläinen J. J., “Optimal stroke volume in left-ventricular ejection”, IEEE Trans. Biomed. Eng., 36:2 (1989), 172–182 | DOI | MR

[17] Van der Horn G. J., Westerhof N., Elzinga G., “Optimal power generation by the left ventricle: a study in the anesthetized open thorax cat”, Circulat. Res., 56:3 (1985), 252–261 | MR

[18] Sunagawa K., Manghan W. L., Sagawa K., “Optimal arterial resistance for the maximum stroke volume studied in isolated canine left ventricle”, Circulat. Res., 56:6 (1985), 586–595 | MR

[19] Myhre E. S. P., Johansen A., Bjorustad J., Piene H., “The effect of contractility, and preload on matching between the canine left ventricle and afterload”, Circulat., 73:2 (1986), 161–171

[20] Burkhoff O., Sagawa K., “Ventricular efficiency predicted by an analytical model”, Amer. J. Physiol., 250:5 (1986), R1021–R1027 | MR

[21] Yamashiro S. M., Daubenspeck J. A., Bennet F. M., “Optimal regulation of left-ventricular ejection pattern”, Appl. Math. Comput., 5:1 (1979), 41–54 | DOI | MR | Zbl

[22] Westerhof N., Elzinga G., Sipkema P., “An artificial arterial system for pumping hearts”, J. Appl. Physiol., 31:8 (1971), 776–781

[23] Livnat A., Yamashiro S. M., “Optimal control evaluation of left-ventricular systolic dynamics”, Amer. J. Physiol., 240:3 (1981), R370–R383 | MR

[24] Bloomfield M. E., Glod L. D., Reddy R. V., Katz A. I., Moreno A. H., “Thermodynamic characterization of the contractile state of the myocardium”, Circulat. Res., 30:5 (1972), 520–534

[25] Buancristiani J. F., Liedtke A. J., Strong R. M., Urschel C. W., “Parameter estimates of left ventricular model during ejection”, IEEE Trans. Biomed. Eng., 20:1 (1973), 110–114 | DOI

[26] Elzinga G., Westerhof N., “Pressure and flow generated by the left ventricle against different impedances”, Circulat. Res., 32:2 (1973), 178–186

[27] Randall O. S., Van den Bos G. G., Westerhof N., “Systemic compliance: does it play a role in the genesis of essential hypertension?”, Cardiovasc. Res., 18:8 (1984), 455–462 | DOI

[28] Sunagawa K., Manghan W. L., Burkhoff D., Sagawa K., “Left-ventricular interaction with arterial load studied in isolated canine ventricle”, Amer. J. Physiol., 245:4 (1983), H773–H780

[29] Deswysen B., Charlier A. A., Gevers M., “Quantitative evaluation of the systemic vascular bed by parameter estimation of a simple model”, Med. Biol. End. Comput., 18:2 (1980), 153–166 | DOI | MR

[30] Sunagawa K., Manghan W. L., Sagawa K., “Stroke volume effect of changing input impedance over selected frequency ranges”, Amer. J. Physiol., 248:4 (1985), H477–H484

[31] Suga H., Hayashi T., Shirahata M., “Ventricular systolic pressure-volume area as predictor of cardiac oxygen consumption”, Amer. J. Physiol., 240:1 (1981), H39–H44

[32] Manghan W. L., Sunagawa K., Burkhoff D., Sagawa K., “Effect of arterial impedance changes on the end-systolic pressure-volume relation”, Circulat. Res., 54:5 (1984), 595–602

[33] Suga H., Igarashi Y., Yamada O., Gito Y., “Mechanical efficiency of the left ventricle as a function of preload, after-load and contractility”, Heart and Vessels, 1:1 (1985), 3–8 | DOI

[34] Weibel E. R., Morphometry of the Human Lung, Academic Press, New York, 1963, 175 pp. | MR

[35] Wilson T. A., “Design of the bronchial tree”, Nature, 213:5077 (1967), 668–669 | DOI

[36] Campbell E. J. M., Westlake E. K., Cherniack R. M., “Simple methods of estimating oxygen consumption and efficiency of the muscles of breathing”, J. Appl. Physiol., 11:3 (1957), 303–308

[37] Rohrer F., “Der Stromungswiderstand in dem menschlichen Atemwegen”, Pflüg. Arch. Ges. Physiol., 162 (1915), 225–299 | DOI

[38] Otis A. B., Fenn W. D., Rahn H., “Mechanics of breathing in man”, J. Appl. Physiol., 2:5 (1950), 592–607

[39] Rahn H., Otis A. B., Chadwick L. E., “The pressure-volume diagram of the thorax and lung”, Amer. J. Physiol., 146:2 (1946), 161–178

[40] Otis A. B., Benbower W. C., “Effect of gas density on resistance to respiratory gas flow in man”, J. Appl. Physiol., 2:3 (1949), 300–306 | MR

[41] Barcroft J., Margaria R., “Some effects of carbonic acid on the character of human respiration”, J. Physiol., 72:2 (1931), 175–185

[42] Lindhard J., “Über das Minutenvolum des Herzens bei Ruhe und in Muskelarbeit”, Pflüg. Arch. Ges. Physiol., 161:3 (1915), 233–383 | DOI

[43] Liljestrand G., “Untersuchungen über die Atmungsarbeit”, Skand. Arch. Physiol., 35:3 (1918), 199–293

[44] Yamashiro S. M., Grodins F. S., “Optimal regulation of respiratory airflow”, J. Appl. Physiol., 30:5 (1971), 597–602 | MR

[45] Morrow P. E., Vosteen R. E, “Pneumotachographic studies in man and dog incorporating a portable wireless transducer”, J. Appl. Physiol., 5:3 (1953), 348–360 | MR

[46] Patterson J. L., Goetz R. H., Doyle J. T., “Cardiorespirtory dynamics in the ox and giraffe, with comparative observations on man and other mammals”, Ann. N.Y. Acad. Sci., 127 (1965), 393–413 | DOI | MR

[47] Proctor D. F., Hardy J. B., “Studies on respiratory airflow”, Bull. Johns Hopkins Hosp., 85:2 (1949), 253–280 | MR

[48] Silverman L., Lee G., Plotkin T., “Airflow measurements on human subjects with and without respiratory resistance at several work rates”, Arch. Ind. Hyg. Occupat. Med., 3:4 (1951), 461–478 | MR

[49] Yamashiro S. M., Daubenspeck J. A., Lauritsen T. N., Grodins F. S., “Total work rate of breathing optimization in $\mathrm{CO}_2$ inhalation and exercise”, J. Appl. Physiol., 38:4 (1975), 702–709 | MR

[50] Gray J. S., Grodins F. S., Carter E. T., “Alveolar and total ventilation and the dead space problem”, J. Appl. Physiol., 9:3 (1956), 307–320

[51] Kuester J. L., Mize S. H., Optimization techniques with Fortran, McGraw, New York, 1973, 500 pp. | Zbl

[52] Asmussen E., Christensen E. H., “Die Mittelkapazitat der Lungen bei erhohten $\mathrm{O}_2$-Bedarf”, Skand. Arch. Physiol., 82:3–4 (1939), 201–211 | MR

[53] Johnson A. T., Masaitis C., “Prediction of Inhalation Time/Exhalation Time ratio during exercise”, IEEE Trans. Biomed. Eng., 23:3 (1976), 376–382 | DOI

[54] Johnson A. T., Berlin H. M., “Exhalation time characterizing exhaustion while wearing respiratory protective masks”, J. Amer. Ind. Hyg. Ass., 35:4 (1974), 463–467 | MR

[55] Mead J., “Control of respiratory frequency”, J. Appl. Physiol., 15:3 (1960), 325–336 | MR

[56] Obraztsov I. F., Khanin M. A., Optimalnye biomekhanicheskie sistemy, Meditsina, M., 1989, 271 pp. | MR

[57] Hill A. V., “The heat of shortening and the dynamic constants of muscle”, Proc. Roy. Soc. Biol., 126:2 (1938), 136–195 | DOI

[58] Dempsey J. A. Rankin J., “Physiologic adaptations of gas transport systems to muscular work in health and disease”, Amer. J. Phys. Med., 46:6 (1967), 582–647 | MR

[59] Homscher E., Rall J. A., “Energetics of shortening muscles in twitches and titanic contractions”, J. Gen. Physiol., 62:6 (1973), 663–676 | DOI

[60] Hill A. V., Trails and trials in physiology, Arnold, London, 1965, 374 pp.

[61] Bartlett R. G., Brubach H. F., Specht T., “Oxygen cost of breathing”, J. Appl. Physiol., 12:4 (1958), 413–426 | MR

[62] Hämäläinen R. P., “Adaptive control of respiratory mechanics”, Trans. ASME, 95:3 (1973), 327–331 | MR

[63] Clark F. J., Von Euler C., “On the regulation of depth and rate of breathing”, J. Physiol. (London), 222:2 (1968), 267–295

[64] Mikhailov V. V., “Effektivnost chastogo i redkogo dykhaniya u sportsmenov pri myshechnoi deyatelnosti tsiklicheskogo tipa”, Teor. prakt. fiz. kult., 23:3 (1960), 203–208

[65] Zamir M., “The role of shear forces in arterial branching”, J. Gen. Physiol., 67:2 (1975), 213–222 | DOI | MR

[66] Ruttimann U. E., Yamamoto W. S., “Respiratory airflow patterns that satisfy power and force criteria of optimality”, Ann. Biomed. Eng., 1:2 (1972), 146–159 | DOI | MR

[67] Bukharov I. B., Khanin M. A., “A mathematical model of the functional state of the oxygen transport system”, Bull. Math. Biol., 42:6 (1980), 627–645 | MR | Zbl

[68] Bukharov I. B., Khanin M. A., “A mathematical model of the pathological function state of the oxygen transport system”, Bull. Math. Biol., 46:2 (1984), 115–125 | MR | Zbl

[69] Bukharov I. B., Khanin M. A., “A mathematical model of the exercise functional state of the oxygen transport system”, J. Theor. Biol., 137:3 (1989), 191–201 | MR

[70] Bukharov I. B., Khanin M. A., “Optimal arterial blood pressure”, J. Theor. Biol., 148:3 (1991), 289–294 | DOI | MR

[71] Bukharov I. B., Khanin M. A., “Optimal pulmonary arterial blood pressure”, J. Theor. Biol., 162:4 (1993), 431–445 | DOI | MR