A Stochastic Optimal Control Theory to Model Spontaneous Breathing

Kyongyob Min

Respiratory Division, Department of Internal Medicine, Itami City Hospital, Itami, Japan.

DOI: 10.4236/am.2013.411208                

Abstract

Respiratory variables, including tidal volume and respiratory rate, display significant variability. The probability density function (PDF) of respiratory variables has been shown to contain clinical information and can predict the risk for exacerbation in asthma. However, it is uncertain why this PDF plays a major role in predicting the dynamic conditions of the respiratory system. This paper introduces a stochastic optimal control model for noisy spontaneous breathing, and obtains a Shrödinger’s wave equation as the motion equation that can produce a PDF as a solution. Based on the lobules-bronchial tree model of the lung system, the tidal volume variable was expressed by a polar coordinate, by use of which the Shrödinger’s wave equation of inter-breath intervals (IBIs) was obtained. Through the wave equation of IBIs, the respiratory rhythm generator was characterized by the potential function including the PDF and the parameter concerning the topographical distribution of regional pulmonary ventilations. The stochastic model in this study was assumed to have a common variance parameter in the state variables, which would originate from the variability in metabolic energy at the cell level. As a conclusion, the PDF of IBIs would become a marker of neuroplasticity in the respiratory rhythm generator through Shr?dinger’s wave equation for IBIs.

Keywords

Biological Variability; Stochastic Processes; Optimal Stochastic Control Theory; Probability Density Function; Shrödinger’s Wave Equation

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	A. L. Goldberger, “Heartbeats, Hormones, and Health: Is Variability the Spice of Life?” American Journal of Respiratory and Critical Care Medicine, Vol. 163, No. 6, 2001, pp. 1289-1290. http://dx.doi.org/10.1164/ajrccm.163.6.ed1801a
[2]	N. S. Cherniack, G. S. Longobardo, O. R. Levine, R. Mellins and A. P. Fishman, “Periodic Breathing in Dogs,” Journal of Applied Physiology, Vol. 21, No. 6, 1966, pp. 1847-1854.
[3]	U. Frey, M. Silverman, A. L. Barabási and B. Suki, “Irregularities and Power Law Distributions in the Breathing Pattern in Preterm and Term Infants,” Journal of Applied Physiology, Vol. 85, No. 3, 1998, pp. 789-797.
[4]	B. Suki, “Fluctuations and Power Laws in Pulmonary Physiology,” American Journal of Respiratory and Critical Care Medicine, Vol. 166, No. 2, 2002, pp. 133-137. http://dx.doi.org/10.1164/rccm.200202-152PP
[5]	K. J. Astroem, “Introduction to Stochastic Control Theory,” Dover Books on Electrical Engineering, New York, 1970.
[6]	K. Min, K. Hosoi, Y. Kinoshita, S. Hara, H. Degami, T. Takada and T. Nakamura, “Use of Fractal Geometry to Propose a New Mechanism of Airway-Parenchymal Interdependence,” Open Journal of Molecular and Integrated Physiology, Vol. 2, No. 1, 2012, pp. 14-20. http://dx.doi.org/10.4236/ojmip.2012.21003
[7]	K. Yasue, “Quantum Mechanics and Optimal Stochastic Control Theory,” Kaimei-shya, Tokyo, 2007 (in Japanese).
[8]	B. Suki, N. Martinez, H. Parameswaran, A. Majumdar, R. Dellaca, C. Berry, J. J. Pillow and E. Bartolak-Suki, “Variability in the Respiratory System: Possible Origins And Implications,” B29, The Lung on the Border between Order and Chaos, 2012, p. A2682.
[9]	J. G. Venegas, T. Winkler, G. Musch, M. F. Vidal Melo, D. Layfield, N. Tgavalekos, A. J. Fischman, R. J. Callahan, G. Bellani and R. S. Harris, “Self-Organized Patchiness in Asthma as a Prelude to Catastrophic Shifts,” Nature, Vol. 434, No. 7034, 2005, pp. 777-782. http://dx.doi.org/10.1038/nature03490
[10]	G. S. Mitchell and S. M. Johnson, “Invited Review: Neuroplasticity in Respiratory Motor Control,” Journal of Applied Physiology, Vol. 94, No. 1, 2003, pp. 358-374.
[11]	P. J. Fadel, S. M. Barman, S. W. Phillips and G. L. Gebber, “Fractal Fluctuations in Human Respiration,” Journal of Applied Physiology, Vol. 97, No. 6, 2004, pp. 20562064. http://dx.doi.org/10.1152/japplphysiol.00657.2004