This paper presents a modification of the Monte Carlo method which permits the solution of nonlinear partial differential equations. A hybrid computer implementa tion of the method is used to solve a nonlinear mathematical model of oxygen transport in the capillaries and tissue of the human brain. The tech nique is an alternative to digital computer methods and can greatly reduce the time required to produce a solution.
Chuang, K.Kazda, L.F.Windeknecht, T.A Stochastic Method of Solving Partial Differential Equations Using an Electronic Analog Computer Project Michigan Report 2900-91-T Willow Run LaboratoriesUniversity of MichiganAnn Arbor196048 pp.
2.
Curtiss, J.H.Sampling Methods Applied to Differential and Difference Equations Proceedings of Seminar on Scientific ComputationIBM CorporationNew York1949 pp. 87-109
3.
Edwards, K.H.Khana, R.R.Feasibility Study of Monte Carlo Modelling Techniques for Distributed-Parameter SystemsIEE Proceedings vol. 1171970 parts 1 and 2 pp. 2277-2293
4.
Ehrlich, L.W.Monte Carlo Solutions of Boundary Value Problems Involving the Difference Analogue of ∂2u/∂x2 + K/y ∂u/∂y = 0ACM Journal vol. 6 1959 pp. 204-218
5.
Grunewald, W.Method for Comparison of Calculated and Measured Oxygen Distribution In M Kessler, D.F. Bruley, et al., editors, Oxygen Supply ( Urban and Schwarzenberg, Munich, Germany, 1971), pp. 5-17
6.
Halberg, M.Bruley, D.F.Knisely, M.H.Simulating Oxygen Transport in the Microcirculation by Monte Carlo MethodsSimulation vol. 151970 pp. 206-212
7.
Handler, H.Monte Carlo Solution of Partial Differential Equations Using a Hybrid ComputerIEEE Transactions on Electronic Computers vol. EC-161967 pp. 603-610
8.
Hunt, D.H.Oxygen Transport in the Brain Microcirculation by a Hybrid Computer Nonlinear Monte Carlo Method PhD dissertation Clemson UniversityClemson, South Carolina1971139 pp.
9.
Hutten, H.Thews, G.Vaupel, P.Some Special Problems concerning the Oxygen Supply to Tissue as Studied by an Analogue Computer In M. Kessler, D.F. Bruley, et al., editors, Oxygen Supply (Urban and Schwarzenberg, Munich , Germany, 1971), pp. 25-31
10.
Jaede, E.K.An Application of the Monte Carlo Method to the Solution of Elliptic Partial Differential Equations Grumman Aerospace Corporation Bethpage, New York1964Grumman Research Department Memorandum RM-238 43 pp.
11.
Johnson, E.L.A Variance Reduction Technique for Hybrid Computer Generated Random Walk Solutions of Partial Differential EquationsAFIPS Proceedings Spring 1970Thomson Book Company Washington, D.C.1970 pp. 19-29
Korn, G.A.Hybrid Computer Monte Carlo TechniquesSimulation vol. 51965 pp. 234-245
14.
15.
Krogh, A.The Number and Distribution of Capillaries in Muscles with Calculations of the oxygen Pressure Head Necessary for Supplying the TissueJournal of Physiology vol. 521918 -1919 pp. 409-410
16.
Little, W.D.Hybrid Computer Solutions of Partial Differential Equations by Monte Carlo Methods AFIPS Proceedings Fall 1966Thomson Book CompanyWashington, D.C.1966 pp. 181-190
17.
McCracken, T.A.Bruley, D.F.Reneau , D.D.Knisely, M.H.Bicher, H.I.Systems Analysis of Transport Processes in Human Brain. Part 2: Systems Studies of the Simultaneous Transport of Oxygen, Glucose, and Carbon Dioxide in Human BrainProceedings First Pacific Chemical Engineering CongressKyoto, Japan 1972 pp. 137-143
18.
Metropolis, N.Ulam, S.The Monte Carlo MethodJournal of the American Statistical Association vol. 441949 pp. 335-341
19.
Metzger, H.PO2 Histograms of Three Dimensional Systems with Homogeneous and Inhomogeneous Microcirculation- a Digital Computer Study In M. Kessler, D.F. Bruley et al., editors, Oxygen Supply (Urban and Schwarzenberg , Munich , Germany, 1971) pp. 18-24
20.
Opitz, E.Schneider, M.The Oxygen Supply of the Brain and the Mechanism of Deficiency Effects Ergbnisse der Physiologie, BiologischemChemie und Experimentallen Pharmakologie vol. 461950 pp. 126-260
21.
Reneau, D.D.Bruley, D.F.Knisely , M.H.A Mathematical Simulation of Oxygen Release, Diffusion, and Consumption in the Capillaries and Tissue of the Human Brain Chemical Engineering in Medicine and BiologyPlenum PressNew York1967 pp. 135-241
22.
Reneau, D.D.Bruley, D.F.Knisely , M.H.A Digital Simulation of Transient Oxygen Transport in Capillaries - Tissue Systems ( Cerebral Grey Matter AIChE Journal vol. 151969 pp. 916-920
23.
Thews, G.Oxygen Diffusion in the Brain. A Contribution to the Questions of the Oxygen Supply of the OrgansPflugers Archives vol. 2711960 pp. 197-226
