Actively Circulating Volume as a Consequence of Stochasticity within Microcirculation

Author(s)
Viktor V. Kislukhin

ABSTRACT

It is well established that in the pathology of the cardio-vascular system (CVS) only a portion of the blood volume (BV) can be in active circulation. This portion of BV is named the actively circulating volume (ACV) and is evaluated from a monotone decrease of dilution curve produced by an intravascular tracer. In given paper is presented Markov chain as a math model of the flow of a tracer throughout CVS. The consideration of CVS as a set of segments with respect to an anatomical structure and assuming the existence for CVS steady-state condition; leads to the Markov chain of the finite order with constant coefficients. The conclusions of the article are 1) there are open and closed microvessels, such that the switching from open to closed and back is a stochastic process, 2) if the switching is slow then the ACV, as the volume of heart chambers and only open for circulation vessels, can be detected.

It is well established that in the pathology of the cardio-vascular system (CVS) only a portion of the blood volume (BV) can be in active circulation. This portion of BV is named the actively circulating volume (ACV) and is evaluated from a monotone decrease of dilution curve produced by an intravascular tracer. In given paper is presented Markov chain as a math model of the flow of a tracer throughout CVS. The consideration of CVS as a set of segments with respect to an anatomical structure and assuming the existence for CVS steady-state condition; leads to the Markov chain of the finite order with constant coefficients. The conclusions of the article are 1) there are open and closed microvessels, such that the switching from open to closed and back is a stochastic process, 2) if the switching is slow then the ACV, as the volume of heart chambers and only open for circulation vessels, can be detected.

KEYWORDS

Blood Volume, Actively Circulating Volume, Microcirculation, Vasomotion, Markov Chain, Math Model of Cardiovascular System

Blood Volume, Actively Circulating Volume, Microcirculation, Vasomotion, Markov Chain, Math Model of Cardiovascular System

Cite this paper

nullV. Kislukhin, "Actively Circulating Volume as a Consequence of Stochasticity within Microcirculation,"*Applied Mathematics*, Vol. 2 No. 4, 2011, pp. 508-513. doi: 10.4236/am.2011.24066.

nullV. Kislukhin, "Actively Circulating Volume as a Consequence of Stochasticity within Microcirculation,"

References

[1] H. C. Lawson, “The Volume of Blood—A Critical Examination of Methods for Its Measurement,” In: W. F. Hamilton and P. Dow, Ed., The Handbook of Physiology: Section 2, Circulation, Waerly Press, Baltimore, Vol. 1, 1962, pp. 23-49.

[2] C. J. Wiggers, “Physiology of Shock,” The Mechanisms of Peripheral Circulatory Failure, The Commonwealth Fund, New York, 1950, pp. 253-286.

[3] W. C. Shoemaker, “Measurement of Rapidly and Slowly Circulating Red Cell Volumes in Hemorrhagic Shock,” American Journal of Physiology, Vol. 202, No. 6, 1962, pp. 1179-1182.

[4] C. F. Rothe, R. H. Murray and T. D. Bennett, “Actively Circulating Blood Volume in Endotoxin Shock Measured by Indicator Dilution,” American Journal of Physiology, Vol. 236, No. 2, February 1979, pp. 291-300.

[5] A. Hoeft, B. Schorn, A. Weyland, M. Scholz, W. Buhre, E. Stepanek, S. J. Allen and H. Sonntag, “Bedside Assessment of Intravascular Volume Status in Patients Undergoing Coronary Bypass Surgery,” Anesthesiology, Vol. 81, No. 1, July 1994, pp. 76-86. doi:10.1097/00000542-199407000-00012

[6] V. I. Romanovsky, “Discrete Markov Chains,” Wolters-Noordhoff, Groningen, 1970.

[7] J. L. Stephenson, “Theory of the Measurement of Blood Flow by the Dilution of an Indicator,” Bulletin of Mathematical Biology, Vol. 10, No. 3, September 1948, pp. 117-121. doi:10.1007/BF02477486

[8] P. Meier and K. L. Zierler, “On the Theory of the Indicator-Dilution Method for Measurement of Blood Flow and Volume,” Journal of Applied Physiology, Vol. 6, No. 12, June 1954, pp. 731-744.

[9] R. Bellman, “Mathematical Methods in Medicine,” World Scientific, Singapore, 1983.

[10] W. Feller, “An Introduction to Probability Theory and Its Applications,” John Wiley & Sons Ltd., New York, Vol. 1, 1959.

[11] K. Zierler, “Indicator Dilution Methods for Measuring Blood Flow, Volume, and Other Properties of Biological Systems: A Brief History and Memoir,” Annals of Biomedical Engineering, Vol. 28, No. 8, August 2000, pp. 836-848. doi:10.1114/1.1308496

[12] A. Krogh, “The Anatomy and Physiology of Capillaries,” Hafner Publishing Co., New York, 1959.

[13] E. M. Renkin, S. D. Gray and L. R. Dodd, “Filling of Microcirculation in Skeletal Muscles during Timed India Ink Perfusion,” American Journal of Physiology, August 1981, Vol. 241, No. 2, pp. 174-86.

[14] V. V. Kislukhin, “Vasomotion Model Explanation for Urea Rebound,” ASAIO Journal, Vol. 48, No. 3, May- June 2002, pp. 296-299. doi:10.1097/00002480-200205000-00016

[15] T. Schroder, U. Rosler, I. Frerichs, G. Hahn, J. Ennker and G. Hellige, “Errors of the Backextrapolation Method in Determination of the Blood Volume,” Physics in Medicine and Biology, Vol. 44, No. 1, January 1999, pp. 121-301. doi:10.1088/0031-9155/44/1/010

[16] K. Parthasarathi and H. H. Lipowsky, “Capillary Recruitment in Response to Tissue Hypoxia and Its Dependence on red Blood Cell Deformability,” American Journal of Physiology, Vol. 277, No. 6, December 1999, pp. 2145-2157.

[17] C. H. Baker and H. D. Wycoff, “Time-Concentration Curves and Dilution Spaces of T-1824 and I-1824 and I-131-Labeled Proteins in Dogs,” American Journal of Physiology, Vol. 201, No. 6, December 1961, pp. 1159- 1163.

[1] H. C. Lawson, “The Volume of Blood—A Critical Examination of Methods for Its Measurement,” In: W. F. Hamilton and P. Dow, Ed., The Handbook of Physiology: Section 2, Circulation, Waerly Press, Baltimore, Vol. 1, 1962, pp. 23-49.

[2] C. J. Wiggers, “Physiology of Shock,” The Mechanisms of Peripheral Circulatory Failure, The Commonwealth Fund, New York, 1950, pp. 253-286.

[3] W. C. Shoemaker, “Measurement of Rapidly and Slowly Circulating Red Cell Volumes in Hemorrhagic Shock,” American Journal of Physiology, Vol. 202, No. 6, 1962, pp. 1179-1182.

[4] C. F. Rothe, R. H. Murray and T. D. Bennett, “Actively Circulating Blood Volume in Endotoxin Shock Measured by Indicator Dilution,” American Journal of Physiology, Vol. 236, No. 2, February 1979, pp. 291-300.

[5] A. Hoeft, B. Schorn, A. Weyland, M. Scholz, W. Buhre, E. Stepanek, S. J. Allen and H. Sonntag, “Bedside Assessment of Intravascular Volume Status in Patients Undergoing Coronary Bypass Surgery,” Anesthesiology, Vol. 81, No. 1, July 1994, pp. 76-86. doi:10.1097/00000542-199407000-00012

[6] V. I. Romanovsky, “Discrete Markov Chains,” Wolters-Noordhoff, Groningen, 1970.

[7] J. L. Stephenson, “Theory of the Measurement of Blood Flow by the Dilution of an Indicator,” Bulletin of Mathematical Biology, Vol. 10, No. 3, September 1948, pp. 117-121. doi:10.1007/BF02477486

[8] P. Meier and K. L. Zierler, “On the Theory of the Indicator-Dilution Method for Measurement of Blood Flow and Volume,” Journal of Applied Physiology, Vol. 6, No. 12, June 1954, pp. 731-744.

[9] R. Bellman, “Mathematical Methods in Medicine,” World Scientific, Singapore, 1983.

[10] W. Feller, “An Introduction to Probability Theory and Its Applications,” John Wiley & Sons Ltd., New York, Vol. 1, 1959.

[11] K. Zierler, “Indicator Dilution Methods for Measuring Blood Flow, Volume, and Other Properties of Biological Systems: A Brief History and Memoir,” Annals of Biomedical Engineering, Vol. 28, No. 8, August 2000, pp. 836-848. doi:10.1114/1.1308496

[12] A. Krogh, “The Anatomy and Physiology of Capillaries,” Hafner Publishing Co., New York, 1959.

[13] E. M. Renkin, S. D. Gray and L. R. Dodd, “Filling of Microcirculation in Skeletal Muscles during Timed India Ink Perfusion,” American Journal of Physiology, August 1981, Vol. 241, No. 2, pp. 174-86.

[14] V. V. Kislukhin, “Vasomotion Model Explanation for Urea Rebound,” ASAIO Journal, Vol. 48, No. 3, May- June 2002, pp. 296-299. doi:10.1097/00002480-200205000-00016

[15] T. Schroder, U. Rosler, I. Frerichs, G. Hahn, J. Ennker and G. Hellige, “Errors of the Backextrapolation Method in Determination of the Blood Volume,” Physics in Medicine and Biology, Vol. 44, No. 1, January 1999, pp. 121-301. doi:10.1088/0031-9155/44/1/010

[16] K. Parthasarathi and H. H. Lipowsky, “Capillary Recruitment in Response to Tissue Hypoxia and Its Dependence on red Blood Cell Deformability,” American Journal of Physiology, Vol. 277, No. 6, December 1999, pp. 2145-2157.

[17] C. H. Baker and H. D. Wycoff, “Time-Concentration Curves and Dilution Spaces of T-1824 and I-1824 and I-131-Labeled Proteins in Dogs,” American Journal of Physiology, Vol. 201, No. 6, December 1961, pp. 1159- 1163.