Abstract
Analysis of data on tissue depositions obtained by positron tomographic or NMR imaging, or of multiple tracer outflow dilution curves, requires fitting data with models composed of aggregates of capillary-tissue units. These units account for heterogeneities of flows and multisolute exchanges between longitudinally distributed regions across capillary and cell barriers within an organ. Because the analytic solutions to the partial differential equations require convolution integration, solutions are obtained relatively efficiently by a fast numerical method. Our approach centers on the use of a sliding fluid element algorithm for capillary convection, with the time step set equal to the length step divided by the fluid velocity. Radial fluxes by permeation between plasma, interstitial fluid, and cells and axial diffusion exchanges within each time step are calculated analytically. The method enforces mass conservation unless there is regional consumption. Solution for a 2-barrier, 3-region model, accurate to within 0.5%, are 100 to 1000 times faster than the corresponding, purely analytic solution, and over 10,000 times for a 4-region model. Applications include multiple indicator dilution studies of kinetics of transcapillary exchange and positron emission tomographic studies of the mechanisms of substrate transport into cells of organsin vivo.
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References
Abramowitz, M.; Stegun, I.A. Handbook of mathematical functions. New York: Dover; 1968.
Bassingthwaighte, J.B.; Ackerman, F.H.; Wood, E.H. Applications of the lagged normal density curve as a model for arterial dilution curves. Circ. Res. 18:398–415; 1966.
Bassingthwaighte, J.B.; Knopp, T.J.; Anderson, D.U. Flow estimation by indicator dilution (bolus injection): Reduction of errors due to time-averaged sampling during unsteady flow. Circ. Res. 27:277–291; 1970.
Bassingthwaighte, J.B.; Knopp, T.J.; Hazelrig, J.B. A concurrent flow model for capillary-tissue exchanges. In: Crone, C.; Lassen, N.A., eds. Capillary permeability (Alfred Benzon symp. II). Copenhagen: Munksgaard; 1970: pp. 60–80.
Bassingthwaighte, J.B. A concurrent flow model for extraction during transcapillary passage. Circ. Res. 35:483–503; 1974.
Bassingthwaighte, J.B.; Yipintsoi, T.; Knopp, T.J. Diffusional arteriovenous shunting in the heart. Microvasc. Res. 28:233–253; 1984.
Bassingthwaighte, J.B.; Goresky, C.A. Modeling in the analysis of solute and water exchange in the microvasculature. In: Renkin, E.M.; Michel, C.C.; eds. Handbook of physiology. Section 2, the cardiovascular system. Vol IV, the microcirculation. Bethesda, MD: Am. Physiol. Soc.; 1984: pp. 549–626.
Bassingthwaighte, J.B.; Kuikka, J.T.; Chan, I.S.; Arts, T.; Reneman, R.S. A comparison of ascorbate and glucose transport in the heart. Am. J. Physiol. 249 (Heart.Circ.Physiol.18):H141-H149; 1985.
Bassingthwaighte, J.B.; Chinard, F.P.; Crone, C.; Goresky, C.A.; Lassen, N.A.; Reneman, R.S.; Zierler, K.L. Terminology for mass transport and exchange. Am. J. Physiol. 250(Heart.Circ. Physiol.19):H539-H545; 1986.
Bassingthwaighte, J.B.; King, R.B.; Roger, S.A. Fractal nature of regional myocardial blood flow heterogeneity. Circ. Res. 65:578–590; 1989.
Bassingthwaighte, J.B.; Wang, C.Y. Chan, I.S. Blood-tissue exchange via transport and transformation by endothelial cells. Circ. Res. 65:997–1020; 1989.
Bassingthwaighte, J.B.; Chan, I.S.; Wang, C.Y. Blood-tissue exchange models: BTEX30 and BTEX40(UW/BIOENG-89/1). Report PB90-501396: FORTRAN Code; PB90-172578: Descriptive text. National Technical Information Services, Springfield, VA 22161; 1989.
Bohr, C. Über die spezifische Tätigkeit der Lungen bei der respiratorischen Gasaufnahme und ihr Verhalten zu der durch die Alveolarwand stattfindenden Gasdiffusion. Skand. Arch. Physiol. 22:221–280; 1909.
Bronikowski, T.A.; Linehan, J.H.; Dawson, C.A. A mathematical analysis of the influence of perfusion heterogeneity on indicator extraction. Math. Biosci. 52:27–51; 1980.
Cobelli, C.; DiStefano, III, J.J. Parameters and structural identifiability concepts and ambiguities: A critical review and analysis. Am. J. Physiol. (Regul.Integ.Comp.Physiol.8) 239:R7-R24; 1980.
Crank, J. The mathematics of diffusion. 2nd ed. Oxford: Clarendon Press; 1975.
Crone, C. The permeability of capillaries in various organs as determined by the use of ‘indicator diffusion’ method. Acta Physiol. Scand. 58:292–305; 1963.
Gayeski, T.E.J.; Honig, C.R. O2 gradients from sarcolemma to cell interior in red muscle at maximal\(\dot VO_2 \), Am. J. Physiol. 251(Heart.Circ.Physiol.20):H789-H799; 1986.
Gjedde, A.; Christensen, O. Estimates of Michaelis-Menten constants for the two membranes of the brain endothelium. J. Cereb. Blood Flow Metabol. 4:241–249; 1984.
Gonzalez-Fernandez, J.M.; Atta, S.E. Concentration of oxygen around capillaries in polygonal regions of supply. Math. Biosci. 13:55–69; 1972.
Goresky, C.A. A linear method for determining liver sinusoidal and extravascular volumes. Am. J. Physiol. 204:626–640; 1963.
Goresky, C.A.; Ziegler, W.H. Bach, G.G. Capillary exchange modeling: Barrier-limited and flow-limited distribution. Circ. Res. 27:739–764; 1970.
Guller, B.; Yipintsoi, T.; Orvis, A.L.; Bassingthwaighte, J.B. Myocardial sodium extraction at varied coronary flows in the dog: Estimation of capillary permeability by residue and outflow detection. Circ. Res. 37:359–378; 1975.
Huang, S.C.; Carson, R.E.; Hoffman, E.J.; Carson, J.; MacDonald, N.; Barrio, J.R.; Phelps, M.E. Quantitative measurement of local cerebral blood flow in humans by positron computed tomography and15O-water. J. Cereb. Blood Flow Metabol. 3:141–153; 1983.
Jacquez, J.A. Compartmental analysis in biology and medicine. Ann Arbor, MI: University of Michigan Press; 1985.
Kuikka, J.; Levin, M.; Bassingthwaighte, J.B. Multiple tracer dilution estimates of D- and 2-deoxy-D-glucose uptake by the heart. Am. J. Physiol. 250(Heart.Circ.Physiol.19):H29-H42; 1986.
Lenhoff, A.M.; Lightfoot, E.N. The effects of axial diffusion and permeability barriers on the transient response of tissue cylinders. II. Solution in time domain. J. Theor. Biol. 106:207–238; 1984.
Levin, M.; Kuikka, J.; Bassingthwaighte, J.B. Sensitivity analysis in optimization of time-distributed parameters for a coronary circulation model. Med. Prog. Technol. 7:119–124; 1980.
Lumsden, C.J.; Silverman, M. Exchange of multiple indicators across renal-like epithelia: A modeling study of six physiological regimes. Am. J. Physiol. 251(Renal.Fluid.Elect.Physiol.20):F1073-F1089; 1986.
Meier, P.; Zierler, K.L. On the theory of the indicator-dilution method for measurement of blood flow and volume. J. Appl. Physiol. 6:731–744; 1954.
Mintun, M.A.; Raichle, M.E.; Martin, W.R.W.; Herscovitch, P. Brain oxygen utilization measured with O-15 radiotracers and positron emission tomography. J. Nucl. Med. 25:177–187; 1984.
Moffett, T.C.; Chan, I.S.; Bassingthwaighte, J.B. Myocardial serotonin exchange: Negligible uptake by capillary endothelium. Am. J. Physiol. 254(Heart.Circ.Physiol.23):H570-H577; 1988.
Moler, C.; Van Loan, C. Nineteen dubious ways to compute the exponential of a matrix. SIAM Rev. 20:801–836; 1978.
Olesen, S.P.; Crone, C. Electrical resistance of muscle capillary endothelium. Biophys. J. 42:31–41; 1983.
Perl, W.; Chinard, F.P. A convection-diffusion model of indicator transport through an organ. Circ. Res. 22:273–298; 1968.
Pratt, D.T. Fast algorithms for combustion kinetics calculations. Park City, Utah International Conference on Stiff Computation; 1982.
Pratt, D.T. Exponential-fitted methods for integrating stiff systems of ordinary differential equations: Application to homogeneous, gas-phase chemical kinetics. New Orleans, LA: JANNAF Propulsion Conference; 1984.
Reid, J.G. Structural identifiability in linear time-invariant systems. IEEE Trans. Automat. Control 22:242–246, 1977.
Renkin, E.M. Transport of potassium-42 from blood to tissue in isolated mammalian skeletal muscles. Am. J. Physiol. 197:1205–1210; 1959.
Rose, C.P.; Goresky, C.A. Vasomotor control of capillary transit time heterogeneity in the canine coronary circulation. Circ. Res. 39:541–554; 1976.
Rose, C.P.; Goresky, C.A.; Bach, G.G. The capillary and sarcolemmal barriers in the heart: An exploration of labeled water permeability. Circ. Res. 41:515–533; 1977.
Rose, C.P.; Goresky, C.A. Bélanger, P.; Chen, M.J. Effect of vasodilation and flow rate on capillary permeability surface product and interstitial space size in the coronary circulation: A frequency domain technique for modeling multiple dilution data with Laguerre functions. Circ. Res. 47:312–328; 1980.
Roth, A.C.; Feigl, E.O. Diffusional shunting in the canine myocardium. Circ. Res. 48:470–480; 1981.
Sangren, W.C.; Sheppard, C.W. A mathematical derivation of the exchange of a labeled substance between a liquid flowing in a vessel and an external compartment. Bull. Math. Biophys. 15:387–394; 1953.
Sharan, M.; Jones, Jr., M.D.; Koehler, R.C.; Traystman, R.J.; Popel, A.S. A compartmental model for oxygen transport in brain microcirculation. Ann. Biomed. Eng. 17:13–38; 1989.
Sheppard, C.W. Basic principles of the tracer method. New York: Wiley; 1962.
Stephenson, J.L. Theory of the measurement of blood flow by the dilution of an indicator. Bull. Math. Biophys. 10:117–121; 1948.
Taylor, G. Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. R. Soc. Lond. A 219:186–203; 1953.
Taylor, G. The dispersion of matter in turbulent flow through a pipe. Proc. R. Soc. Lond. A 223:446–468; 1954.
Wangler, R.D.; Gorman, M.W.; Wang, C.Y.; DeWitt, D.F.; Chan, I.S.; Bassingthwaighte, J.B.; Sparks, H.V. Transcapillary adenosine transport and interstitial adenosine concentration in guinea pig hearts. Am. J. Physiol. 257(Heart.Circ.Physiol.26):H89-H106; 1989.
Zierler, K.L. Theory of the use of arteriovenous concentration differences for measuring metabolism in steady and non-steady states. J. Clin. Invest. 40:2111–2125; 1961.
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Bassingthwaighte, J.B., Chan, I.S.J. & Wang, C.Y. Computationally efficient algorithms for convection-permeation-diffusion models for blood-tissue exchange. Ann Biomed Eng 20, 687–725 (1992). https://doi.org/10.1007/BF02368613
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DOI: https://doi.org/10.1007/BF02368613