Abstract
Oxygen consumption in small tissue regions cannot be measured directly, but assessment of oxygen transport and metabolism at the regional level is possible with imaging techniques using tracer15O-oxygen for positron emission tomography. On the premise that mathematical modeling of tracer kinetics is the key to the interpretation of regional concentration-time curves, an axially-distributed capillary-tissue model was developed that accounts for oxygen convection in red blood cells and plasma, nonlinear binding to hemoglobin and myoglobin transmembrane transport among red blood cells, plasma, interstitial fluid and parenchymal cells, axial dispersion, transformation to water in the tissue, and carriage of the reaction product into venous effluent. Computational speed was maximized to make the model useful for routine analysis of experimental data. The steady-state solution of a parent model for tracer oxygen and tracer water. The set of models provides estimates of oxygen consumption, extraction, and venouspO2 by fitting model solutions to experimental tracer curves of the regional tissue content or venous outflow. The estimate myocardial oxygen consumption for the whole heart was in good agreement with that measured directly by the Fick method and was relatively insensitive to noise. General features incorporated in the model make it widely applicable to estimating oxygen consumption in other organs from data obtained by external detection methods such as positron emission tomography.
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Bassingthwaighte, J. B., F. H. Ackerman, and E. H. Wood. Applications of the lagged normal density curve as a model for arterial dilution curves.Circ. Res. 18:398–415, 1966.
Bassingthwaighte, J. B.. Plasma indicator dispersion in arteries of the human leg.Cir. Res. 19:332–346, 1966.
Bassingthwaighte, J. B., T. J. Knopp, and J. B. Hazelrig. A. concurrent flow model for capillary-tissue exchange. In: Capillary permeability (Alfred Benzon Symp. II), edited by C. Crone and N. A. Lassen. Copenhagen Munksgaard, 1970, pp. 60–80.
Bassingthwaighte, J. B., T. Yipintsoi, and R. B. Harvey. Microvasculature of the dog left ventricular myocardium.Microvasc. Res. 7:229–249, 1974.
Bassingthwaighte, J. B.. Physiology and theory of tracer washout techniques for the estimation of myocardial blood flow: flow estimation from tracer washout.Progr. Cardiovasc. Dis. 20:165–189, 1977.
Bassingthwaighte, J. B., and C. A. Goresky. Modeling in the analysis of solute and water exchange in the microvasculature. In: Handbook of physiology, sect. 2. The cardiovascular system, vol. IV. The microcirculation, edited by E. M. Renkin and C. C. Michel. Bethesda, MD: American Physiological Society, 1984, pp. 549–626.
Bassingthwaighte, J. B., R. B. King, and S. A. Roger. Fractal nature of regional myocardial blood flow heterogeneity.Circ. Res. 65:578–590, 1989.
Bassingthwaighte, J. B., C. Y. Wang, and J. S. Chan. Blood-tissue exchange via transport and transformation by endothelial cells.Circ. Res. 65:997–1020, 1989.
Bassingthwaighte, J. B., M. A. Malone, T. C. Moffett, R. B. King, I. S. Chan, J. M. Link, and K. A. Krohn. Molecular and particulate depositions for regional myocardial flows in sheep.Circ. Res. 66:1328–1344, 1990.
Bassingthwaighte, J. B., I. S. Chan, and C. Y. Wang. Computationally efficient algorithms for capillary convection-permeation-diffusion models for blood-tissue exchange.Ann. Biomed. Eng. 20:687–725, 1992.
Bassingthwaighte, J. B., and D. A. Beard. Fractal15O-water washout from the heart.Circ Res. 77:1212–1221, 1995.
Bergmann, S. R., P. Herrero, J. Markham, C. J. Weinheimer, and M. N. Walsh. Noninvasive quantitation of myocardial blood flow in human subjects with oxygen-15-labeled water and positron emission tomography.J. Am. Coll. Cardiol. 14:639–652, 1989.
Bohr, C. Über-die spezifische tatigkeit der lungen bei der respiratori schen gasaufnahme und ihr verhalten zu der durch die alveolarwand staffindenden gasdiffusion.Skand. Arch. Physiol. 22:221–280, 1909.
Buerk, D. G., and Bridges, E. A simplified algoprithm for computing the variation in oxyhemoglobin saturation with pH, pCO2, T, and DPG.Chem. Eng. Commun. 47:113, 1986.
Clark, A., Jr., W. J. Federspiel, P. A. A. Clark, and G. R. Cokelet. Oxygen delivery from red cells.Biophys. J. 47: 171–181, 1985.
Clough, A. V., A. Al-Tinawi, J. H. Linehan, and C. A. Dawson. Regional transit time estimation from image residue curves.Ann. Biomed. Eng. 22:128–143, 1994.
de Koming, L., J. C. Hoofd, and E. Kreuzer. Oxygen transport and the function of myoglobin. Theoretical model and experiments in chicken gizzard smooth muscle.Pflügers Arch. 389:211–217, 1981.
Desjardins, C., and B. R. Duling. Microvessel hematocrit: measurement and implications for capillary oxygen transport.Am. J. Physiol. 252:H494-H503, 1987.
Deussen, A., and J. B. Bassingthwaighte. Modeling15O-oxygen tracer data for estimating oxygen consumption.Am. J. Physiol. 270 (Heart. Circ. Physiol. 39):H1115-H1130, 1996.
Federspiel, W. J., and I. H. Sarelius. An examination of the contribution of red cell spacing to the uniformity of oxygen flux at the capillary wall.Microvasc. Res. 27:273–285, 1984.
Goldstick, T. K., V. T. Ciuryla, and L. Zuckerman. Diffusion of oxygen in plasma and blood.Adv. Exp. Med. Biol. 75:183–190, 1976.
Gonzalez, F., and J. B. Bassingthwaighte. Hererogeneities in regional volumes of distribution and flows in the rabbit heat.Am. J. Physiol. 258 (Heart. Circ. Physiol. 27):H1012-H1024, 1990.
Grieb, P., R. E. Forster, D. Strome, C. W. Goodwin, and P. C. Pape. O2 exchange between blood and brain tissues studied with18O2 indicator dilution technique.J. Appl. Physiol. 58:1929–1941, 1985.
Groebe, K., and G. Thews. Theoretical analysis of oxygen supply to contracted skeletal muscle.Adv. Exp. Med. Biol. 200:495–514, 1986.
Groebe, K., and G. Thews. Effects of red cell spacing and red cell movement upon oxygen release under conditions of maximally working skeletal muscle. In: Oxygen transport to tissue XI, edited by K. Rakusan, G. P. Biro, T. K. Goldstick, and Z. Turek, New York: Plenum Press, 1989, pp. 175–185.
Grunewald, W. A., and W. Sowa. Capillary structures and O2 supply to tissue. An analysis with a digital diffusion model as applied to the skeletal muscle.Rev. Physiol. Biochem. Pharmacol. 77:149–209, 1977.
Hellums, J. D. The resistance to oxygen transport in the capillaries relative to that in the surrounding tissue.Microvasc. Res. 13:131–136, 1977.
hellums, J.D., P.K. Nair, N.S. Huang, and N. Ohshima. Simulation of intraluminal gas transport processes in the microcirculation.Am. Biomed. Eng. 24:1–24, 1996.
Homer, L. D., J. B. Shelton, C. H. Dorsey, and T. J. Williams. Anisotropic diffusion of oxygen in slices of rat muscle.Am. J. Physiol. 246 (Reg. Int. Comp. Physiol. 15):R107-R113, 1984.
Huang, S. C., D. G. Feng, and M. E. Phelps. Model dependency and estimation reliability in measurement of cerebral oxygen utilization rate with oxygen-15 and dynamic positron emission tomography.J. Cereb. Blood Flow Metab. 6:105–119, 1986.
Hudson, J. A., and D. B. Cater. An analysis of factors affecting tissue oxygen tension.Proc. R. Soc. Lond. B161:247, 1964.
King, R. B., J. B. Bassingthwaighte, J. R. S. Hales, and L. B. Rowell. Stability of heterogeneity of myocardial blood flow in normal awake baboons.Circ. Res. 57:285–295, 1985.
King, R. B., G. M. Raymond, and J. B. Bassingthwaighte. Modeling blood flow heterogeneity.Ann. Biomed. Eng. 24: 352–372, 1996.
Krogh, A. The number and distribution of capillaries in muscles with calculations of the oxygen pressure head necessary for supplying the tissue.J. Physiol. 52:409–415, 1919.
Napper, S. A., and R. W. Schubert. Mathematical evidence for flow-induced changes in myocardial oxygen consumption.Ann. Biomed. Eng. 16:349–365, 1988.
Page, E., L. P. McCallister, and B. Power. Stereological measurements of cardiac ultrastructures implicated in excitation-contraction coupling.Proc. Natl. Acad. Sci. U.S.A. 68:1465–1466, 1971.
Page, E., and L. P. McCallister. Quantitative electron microscopic description of heart muscle cells.Am. J. Cardiol. 31:172–181, 1973.
Popel, A. S. Analysis of capillary-tissue diffusion in multicapillary systems.Math. Biosci. 39:187–211, 1978.
Popel, A. S. Theory of oxygen transport to tissue.Crit. Rev. Biomed. Eng. 17:257–321, 1989.
Raichle, M. E., R. L. Grubb, Jr., J. O. Eichling, and M. M. Ter-Pogossian. Measurement of brain oxygen utilization with radioactive oxygen-15: experimental verification.J. Appl. Physiol. 40:638–640, 1976.
Reneau, D. D., D. F. Bruley, and M. H. Knisely. A mathematical simulation of oxygen release, diffusion, and consumption in the capillaries and tissue of the human brain. In: Chemical engineering in medicine and biology. New York: Plenum Press, 1967, pp. 135–241.
Riveros-Moreno, V., and J. B. Wittenberg. The self-diffusion coefficients of myoglobin and hemoglobin in concentrated solutions.J. Biol. Chem. 247:895–901, 1972.
Rose, C. P., C. A. Goresky, and G. G. Bach. The capillary and sarcolemmal barriers in the heart: an exploration of labeled water permeability.Circ. Res. 41:515–533, 1977.
Rose, C. P., and C. A. Goresky. Limitations of tracer oxygen uptake in the canine coronary circulation.Circ. Res. 56:57–71, 1985.
Singh, M. P., M. Sharan, and A. Aminataei. Development of mathematical formulae for O2 and CO2 dissociation curves in the blood.IMA J. Math. Appl. Med. Biol. 6:25–46, 1989.
Taegtmeyer, H. Carbohydrate interconversions and energy production.Circulation 72(Suppl. IV):1–8, 1985.
Tangelder, G. J., D. W. Slaaf, A. M. M. Muijtjens, T. Arts, M. G. A. oude Egbrink, and R. S. Reneman. Velocity profiles of blood platelets and red blood cells flowing in arterioles of the rabbit mesentery.Circ. Res. 59:505–514, 1986.
Ter-Pogossian, M. M., J. O. Eichling, D. O. Davis, and M. J. Welch. The measurein vivo of regional oxygen utilization by means of oxyhemoglobin labeled with radioactive oxygen-15.J. Clin. Invest. 49:381–391, 1970.
Ter-Pogossian, M. M., and P. Herscovith. Radioactive oxygen-15 in the study of cerebral blood flow, blood volume, and oxygen metabolism.Semin. Nucl. Med. 15:377–394, 1985.
Thews, G. Die Sauerstoffdiffusion im Gehirn. Ein Betrag lur Frage der Sauerstroffversorgung der Organe.Pflügers Arch. 271:197, 1960.
Wittenberg, B. A., and J. B. Wittenberg. Transport of oxygen in muscle.Am. Rev. Physiol. 51:857–878, 1989.
Zierler, K. L. A critique of compartmental analysis.Ann. Rev. Biophys. Bioeng. 10:531–562, 1981.
Adair, G. S. The hemoglobin system. VI. The oxygen dissociation curve of hemoglobin.J. Biol. Chem. 63:529, 1925.
Yipintsoi, T., P. D. Scanlon, and J. B. Bassingthwaighte. Density and water content of dog ventricular myocardium.Proc. Soc. Exp. Biol. Med. 141:1032–1035, 1972.
Safford, R. E., E. A. Bassingthwaighte, and J. B. Bassingthwaighte. Diffusion of water in cat ventricular myocardium.J. Gen. Physiol. 72:513–538, 1978.
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Li, Z., Yipintsoi, T. & Bassingthwaighte, J.B. Nonlinear model for capillary-tissue oxygen transport and metabolism. Ann Biomed Eng 25, 604–619 (1997). https://doi.org/10.1007/BF02684839
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DOI: https://doi.org/10.1007/BF02684839