# Oxygen Boundary Crossing Probabilities

• N. A. Busch
• I. A. Silver
Part of the Advances in Experimental Medicine and Biology book series (AEMB, volume 215)

## Summary

The probability that an oxygen particle will reach a time dependent boundary is required in oxygen transport studies involving solution methods based on probability considerations. A Volterra integral equation is presented, the solution of which gives directly the boundary crossing probability density function. The boundary crossing probability is the probability that the oxygen particle will reach a boundary within a specified time interval. When the motion of the oxygen particle may be described as strongly Markovian, then the Volterra integral equation can be rewritten as a generalized Abel equation, the solution of which has been widely studied.

## Keywords

Probability Density Function Oxygen Transport Volterra Integral Equation Monte Carlo Simulation Study Current Time Interval
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. Anderssen, R.S., De Hoog, F.R. and Weiss, R. (1973). On the numerical solution of Brownian motion processes. J. Appl. Probability, 10, 409–418.Google Scholar
2. Bownds, J.M. (1982). Theory and performance of a sub-routine for solving Volterra integral equations. Computing, 28, 317–332.
3. Bownds, J.M. and Applebaum, L. (1985). Algorithm 627, a Fortran subroutine for solving Volterra integral equations. ACM Trans. Math. Software, 11, 58–65.Google Scholar
4. Bownds, J.M. and Wood, B. (1976). On numerically solving nonlinear Volterra integral equations with fewer computations. SIAM J. Numerical Analysis, 13, 705–519.Google Scholar
5. Bownds, J.M. and Wood, B. (1979). A smoothed projection method for singular nonlinear Volterra equations. J. Approxim. Theory, 25, 120–141.Google Scholar
6. Chandrasekhar, S. (1943). Stochastic problems in physics and astronomy. Reviews of Modern Physics, 15, 1–89.
7. Cuzik, J. (1981). Boundary crossing probabilities for stationary gaussian processes and Brownian motion. Trans. Am. Math. Soc. 263, 469–492.Google Scholar
8. Durbin, J. (1971). Boundary-crossing probabilities for the Brownian motion and poisson processes and techniques for computing the power of the Kolmogorov-Smirnov test. J. Appl. Probability, 8, 431–453.Google Scholar
9. Durbin J. (1985). The first passage density of a continuous Gaussian process to a general boundary. J. Appl. Probability, 22, 99–122.
10. Fortet, R. (1943). Les fonctions aleatoires du type de Markoff associees a certaines equations lineaires aux derivees partielles du type parabolique. J. Math. Pures Appl. 22, 177–243.Google Scholar
11. Lakshminarayanaiah, N. (1969). Transport Phenomena in Membranes.Academic Press, New York.Google Scholar
12. Lakshminarayanaiah, N. (1984). Equations of Membrane Biophysics. Academic Press, New York.Google Scholar
13. Lightfoot, E.N., Jr (1974). Transport Phenomena and Living Systems. New York.Google Scholar
14. Ricciardi, L.M., Sacerdote, L. and Sato, S. (1984). On an integral equation for first passage time probability densities. J. Appl. Probability, 21, 302–314.Google Scholar
15. Ricciardi, L.M. and Sato, S. (1983). A note on the evaluation of first passage time probability densities. J. Appl. Probability, 20, 197–201.Google Scholar
16. Smith, C.S. (1972). A note on boundary-crossing probabilities for the Brownian motion. J. Appl. Probability, 9, 857–861.Google Scholar
17. Uhlenbeck, G.E. and Ornstein, L.S. (1930). On the theory of Brownian motion. Physical Review, 36, 823–841.
18. Uhlenbeck, G.E. and Wang, M.C. (1945). On the theory of the Brownian motion II. Reviews of Modern Physics, 17, 323–342.