Skip to main content
SpringerLink
Log in

Modeling blood flow heterogeneity

  • Published:
Annals of Biomedical Engineering Aims and scope Submit manuscript

Abstract

It has been known for some time that regional blood flows within an organ are not uniform. Useful measures of heterogeneity of regional blood flows are the standard deviation and coefficient of variation or relative dispersion of the probability density function (PDF) of regional flows obtained from the regional concentrations of tracers that are deposited in proportion to blood flow. When a mathematical model is used to analyze dilution curves after tracer solute administration, for many solutes it is important to account for flow heterogeneity and the wide range of transit times through multiple pathways in parallel. Failure to do so leads to bias in the estimates of volumes of distribution and membrane conductances. Since in practice the number of paths used should be relatively small, the analysis is sensitive to the choice of the individual elements used to approximate the distribution of flows or transit times. Presented here is a method for modeling heterogeneous flow through an organ using a scheme that covers both the high flow and long transit time extremes of the flow distribution. With this method, numerical experiments are performed to determine the errors made in estimating parameters when flow heterogeneity is ignored, in both the absence and presence of noise. The magnitude of the errors in the estimates depends upon the system parameters, the amount of flow heterogeneity present, and whether the shape of the input function is known. In some cases, some parameters may be estimated to within 10% when heterogeneity is ignored (homogeneous model), but errors of 15–20% may result, even when the level of heterogeneity is modest. In repeated trials in the presence of 5% noise, the mean of the estimates was always closer to the true value with the heterogeneous model than when heterogeneity was ignored, but the distributions of the estimates from the homogeneous and heterogeneous models overlapped for some parameters when outflow dilution curves were analyzed. The separation between the ditributions was further reduced when tissue content curves were analyzed. It is concluded that multipath models accounting for flow heterogeneity are a vehicle for assessing the effects of flow heterogeneity under the conditions applicable to specific laboratory protocols, that efforts should be made to assess the actual level of flow heterogeneity in the organ being studied, and that the errors in parameter estimates are generally smaller when the input function is known rather than estimated by deconvolution.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abounader, R., J. Vogel, and W. Kuschinsky. Patterns of capillary plasma perfusion in brains of conscious rats during normocapnia and hypercapnia.Circ. Res. 76:120–126, 1995.

    PubMed  CAS  Google Scholar 

  2. Audi, S. H., G. S. Krenz, J. H. Linehan, D. A. Rickaby, and C. A. Dawson. Pulmonary capillary transport function from flow-limited indicators.J. Appl. Physiol. 77:332–351, 1994.

    PubMed  CAS  Google Scholar 

  3. Audi, S. H., J. H. Linehan, G. S. Krenz, C. A. Dawson, S. B. Ahlf, and D. L. Roerig. Estimation of the pulmonary capillary transport function in isolated rabbit lungs.J. Appl. Physiol. 78:1004–1014, 1995.

    Article  PubMed  CAS  Google Scholar 

  4. 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.

    PubMed  CAS  Google Scholar 

  5. Bassingthwaighte, J. B. Plasma indicator dispersion in arteries of the human leg.Circ. Res. 19:332–346, 1966.

    PubMed  CAS  Google Scholar 

  6. Bassingthwaighte, J. B., and M. Levin. Analysis of coronary outflow dilution curves for the estimation of cellular uptake rates in the presence of heterogeneous regional flows.Basic Res. Cardiol. 76:404–410, 1981.

    Article  PubMed  CAS  Google Scholar 

  7. Bassingthwaighte, J. B., and C. A. Goresky. Modeling in the analysis of solute and water exchange in the microvas-culature. In:Handbook of Physiology, Section 2, The Cardiovascular System, vol. 4, The Microcirculation, edited by E. M. Renkin and C. C. Michel, Bethesda, MA, American Physiological Society, 1984, pp. 549–626.

    Google Scholar 

  8. Bassingthwaighte, J. B., M. A. Malone, T. C. Moffett, R. B. King, S. E. Little, J. M. Link, and K. A. Krohn. Validity of microsphere depositions for regional myocardial flows.Am. J. Physiol. 253 (Heart Circ. Physiol.. 22):H184-H193, 1987.

    PubMed  CAS  Google Scholar 

  9. Bassingthwaighte, J. B., R. B. King, and S. A. Roger. Fractal nature of regional myocardial blood flow heterogeneity.Circ. Res. 65:578–590, 1989.

    PubMed  CAS  Google Scholar 

  10. 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.

    Article  PubMed  CAS  Google Scholar 

  11. Bronikowski, T. A., C. A. Dawson, J. H. Linehan, and D. A. Rickaby. A mathematical model of indicator extraction by the pulmonary endothelium via saturation kinetics.Math. Biosci. 61:237–266, 1982.

    Article  Google Scholar 

  12. Bronikowski, T. A., C. A. Dawson, and J. H. Linehan. On indicator dilution and perfusion heterogeneity: A stochastic model.Math. Biosci. 83:199–225, 1987.

    Article  Google Scholar 

  13. Chan, I. S., A. A. Goldstein, and J. B. Bassingthwaighte. SENSOP: A derivative-free solver for non-linear least squares with sensitivity scaling.Ann. Biomed. Eng. 21:621–631, 1993.

    Article  PubMed  CAS  Google Scholar 

  14. 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.

    Article  PubMed  CAS  Google Scholar 

  15. Cobelli, C., M. P. Saccomani, E. Ferranini, R. A. Defronzo, R. Gelfand, and R. Bonadonna. A compartmental model to quantitate in vivo glucose transport in the human forearm.Am. J. Physiol. 257(Endocrinol. Metal. 20): E943-E958, 1989.

    PubMed  CAS  Google Scholar 

  16. Cousineau, D. F., C. A. Goresky, J. R. Rouleau, and C. P. Rose. Microsphere and dilution measurements of flow and interstittial space in dog heart.J. Appl. Physiol. 77:113–120, 1994.

    PubMed  CAS  Google Scholar 

  17. Duling, B. R., and D. H. Damon. An examination of the measurement of flow heterogeneity in striated muscle.Circ. Res. 60:1–13, 1987.

    PubMed  CAS  Google Scholar 

  18. Glenny, R. W., and H. T. Robertson. Fractal modeling of pulmonary blood flow heterogeneity.J. Appl. Physiol. 70: 1024–1030, 1991.

    PubMed  CAS  Google Scholar 

  19. Gonzalez, F., and J. B. Bassingthwaighte. Heterogeneities in regional volumes of distribution and flows in the rabbit heart.Am. J. Physiol. 258(Heart Circ. Physiol. 27):H1012-H1024, 1990.

    PubMed  CAS  Google Scholar 

  20. Haselton, F. R., R. E. Parker, R. J. Roselli, and T. R. Harris. Analysis of lung multiple indicator data with an effective diffusivity model of capillary exchange.J. Appl. Physiol. 57:98–109, 1984.

    PubMed  CAS  Google Scholar 

  21. Hoedt-Rasmussen, K., E. Sveinsdottir, and N. A. Lassen. Regional cerebral blood flow in man determined by intraarterial injection of radioactive inert gas.Circ. Res. 18:237–247, 1966.

    PubMed  CAS  Google Scholar 

  22. 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.

    PubMed  CAS  Google Scholar 

  23. King, R. B., A. Deussen, G. R. Raymond, and J. B. Bassingthwaighte. A vascular transport operator.Am. J. Physiol. 265(Heart Circ. Physiol. 34):H2196-H2208, 1993.

    PubMed  CAS  Google Scholar 

  24. Kirkebo, A., A. Haugan, and I. Tyssebotn. Blood flow heterogeneity in the renal cortex during burn shock in dogs.Acta Physiol. Scand. 123:205–213, 1985.

    Article  PubMed  CAS  Google Scholar 

  25. Knopp, T. J., W. A. Dobbs, J. F. Greenleaf, and J. B. Bassingthwaighte. Transcoronary intravascular transport functions obtained via a stable deconvolution technique.Ann. Biomed. Eng. 4:44–59, 1976.

    Article  PubMed  CAS  Google Scholar 

  26. Kroll, K., and D. W. Stepp. Adenosine kinetics in the canine coronary circulation.Am. J. Physiol. 269(Heart Circ. Physiol. 38), 1996.

    Google Scholar 

  27. Rose, C. P., and C. A. Goresky. Vasomotor control of capillary transit time heterogeneity in the canine coronary circulation.Circ. Res. 39:541–554, 1976.

    PubMed  CAS  Google Scholar 

  28. Rowlett, R. D., and T. R. Harris. A comparative study of organs models and numerical techniques for the evaluation of capillary permeability from multiple-indicator data.Math. Biosci. 29:273–298, 1976.

    Article  Google Scholar 

  29. Stapleton, D. D., T. C. Moffett, D. G. Baskin, and J. B. Bassingthwaighte. Autoradiographic assessment of blood flow heterogeneity in the hamster heart.Microcirculation 2:277–282, 1995.

    Article  PubMed  CAS  Google Scholar 

  30. van Beek, J. H. G. M., J. B. Bassingthwaighte, and S. A. Roger. Fractal networks explain regional myocardial flow heterogeneity.Adv. Exp. Med. Biol. 248:249–257, 1989.

    PubMed  Google Scholar 

  31. Wolpers, H. G., V. Geppert, A. Hoeft, H. Korb, R. Schrader, and G. Hellige. Estimation of myocardial blood flow heterogeneity by transorgan helium transport functions.Pflugers Arch. 401:217–222, 1984.

    Article  PubMed  CAS  Google Scholar 

  32. Yipintsoi, T., W. A. Dobbs, Jr., P. D. Scanlon, T. J. Knopp, and J. B. Bassingthwaighte. Regional distribution of diffusible tracers and carbonized microspheres in the left ventricle of isolated dog hearts.Circ. Res. 33:573–587. 1973.

    PubMed  CAS  Google Scholar 

  33. Zierler, K. L.. A critique of compartmental analysis.Annu. Rev. Biophys. Bioeng. 10:531–562, 1981.

    Article  PubMed  CAS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Center for Bioengineering, University of Washington, Box 357962, 98195-7962, Seattle, WA, U.S.A.

    Richard B. King, Gary M. Raymond & James B. Bassingthwaighte M.D., Ph.D.

Authors
  1. Richard B. King
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gary M. Raymond
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. James B. Bassingthwaighte M.D., Ph.D.
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

King, R.B., Raymond, G.M. & Bassingthwaighte, J.B. Modeling blood flow heterogeneity. Ann Biomed Eng 24, 352–372 (1996). https://doi.org/10.1007/BF02660885

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02660885

Keywords

Search

Navigation