Floyd B. Hanson

Professor Emeritus
Laboratory for Advanced Computing
Laboratory for Control and Information
Department of Mathematics, Statistics, and Computer Science
University of Illinois at Chicago

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Annotated Publications in
Stochastic Modeling in Biology and Medicine

(Most recent papers have preprint links.)

13. F. B. Hanson and H. C. Tuckwell, Persistence Times of Populations with Large Random Fluctuations, Theor. Population Biol., vol. 14, pp. 46-61, 1978.

16. C. Tier and F. B. Hanson, Persistence of Density Dependent Stochastic Populations, Math. Biosci., vol. 53, pp. 89-117, 1981.

17. F. B. Hanson and H. C. Tuckwell, Logistic Growth with Random Density Independent Disasters, Theor. Population Biol., vol. 19, pp. 1-18, 1981.

18. F. B. Hanson and C. Tier, A Stochastic Model of Tumor Growth, Math. Biosci., vol. 61, pp. 73-100, 1982.

20. F. B. Hanson and C. Tier, Harvesting Under Small Demographic Uncertainty, Lecture Notes in Biomathematics 52, pp. 401-406, 1983.

21. F. B. Hanson and H. C. Tuckwell, Diffusion Approximation for Neuronal Activity Including Reversal Potentials, J. Theor. Neurobiol., vol. 2, pp. 127-153, 1983.

22. D. Ryan and F. B. Hanson, Optimal Harvesting in Disastrous Environments, in Mathematical Models of Renewable Resources, vol. II, edited by R. Lamberson, The Humboldt State University Mathematical Modeling Group, Arcata, Cal., pp. 171-177, 1984.

23. D. Ryan and F. B. Hanson, Poisson search and harvesting, in Mathematical Models of Renewable Resources, vol. III, edited by R. Lamberson, The Humboldt State University Mathematical Modeling Group, Arcata, Cal., pp. 109-112, 1984.

24. F. B. Hanson and D. Ryan, Optimal Harvesting in a Stochastic Environment with Density Dependent Jumps, in Mathematical Models of Renewable Resources, vol. III, edited by R. Lamberson, The Humboldt State University Mathematical Modeling Group, Arcata, Cal., pp. 117-123, 1984.

25. D. Ryan and F. B. Hanson, Optimal Harvesting with Exponential Growth in an Environment with Random Disasters and Bonanzas, Math. Biosci., vol. 74, pp. 37-57, 1985.

26. D. Ryan and F. B. Hanson, Optimal Harvesting of a Logistic Population in an Environment with Stochastic Jumps, J. Math. Biol., vol. 24, pp. 259-277, 1986.

29. F. B. Hanson, Bioeconomic model of the Lake Michigan alewife fishery, Can. J. Fish. Aquat. Sci., vol. 44, suppl. 2, pp. 298-305, 1987.

Abstract: This multidimensional fisheries model was developed for the Great Lakes Fisheries Commission's Assessment and Prediction of Yields (ASPY) Workshop at Quetico. The model was the motivation for developing high performance computational algorithms for multidimensional stochastic dynamic programming solutions.

30[NRM88]. F. B. Hanson and D. Ryan, Optimal harvesting with density dependent random effects, Natural Resource Modeling, vol. 2, pp. 439-455, 1988.

Abstract: This paper was presented at the Third Pacific Coast Resource Modeling Conference, University of California, Davis, CA, June 1984. The effect of sudden large, randomly occurring density dependent disasters on the optimal harvest policy and optimal expected return for an exploited population. The population grows logistically with disasters occurring on a time scale very short compared to the natural growth scale. The case of density dependent frequency is also treated. Stochastic dynamic programming is used in the optimization and a refined numerical method is successful in computing the solutions. For a set of realistic field data it is found that random effects typically have a significant effect on both optimal return and optimal effort levels. The effect of density dependence is far more pronounced for optimal return than for optimal effort levels.

35[MBS89]. F. B. Hanson and D. Ryan, Mean and Quasi-Deterministic Equivalence for Linear Stochastic Dynamics, Math. Biosci., vol. 93, pp. 1-14, 1989.

Abstract: In this paper, Hanson and Ryan show an equivalence principle that the size of the quasi-deterministic population (i.e., using the infinitesimal mean and variance) is the same as the mean population governed by a linear, Markov stochastic differential equation. A new, nonlinear, exactly analyzable, stochastic differential equation with nontrivial Poisson noise is presented as a counter-example to show that the linear result is a sharp result. The importance of this result is that it shows the circumstances under which the simpler quasi-deterministic approximation can be used properly in applications.

63[JMB97]. F. B. Hanson and H. C. Tuckwell, Population Growth with Randomly Distributed Jumps, J. Math. Biology, vol. 36(2), pp. 169-187, December 1997.

Abstract: The growth of populations with continuous deterministic and random jump components is treated. Three special models in which random jumps occur at the time of events of a Poisson process and admit formal explicit solutions are considered: A) Logistic growth with random disasters having exponentially distributed amplitudes; B) Logistic growth with random disasters causing the removal of a uniformly distributed fraction of the population size; and C) Exponential decay with sudden increases (bonanzas) in the population and with each increase being an exponentially distributed fraction of the current population. Asymptotic and numerical methods are employed to determine the mean extinction time for the population, qualitatively and quantitatively. For Model~A, this time becomes exponentially large as the carrying capacity becomes much larger than the mean disaster size. Implications for colonizing species for Model~A are discussed. For Model~B and C, the practical notion of a small, but positive, effective extinction level is chosen, and in these cases the expected extinction time rises rapidly with population size, yet at less than an exponentially large order.

64[MBS98]. F. B. Hanson and D. Ryan, Optimal Harvesting with Both Population and Price Dynamics, Mathematical BioSciences, vol. 148, issue 2, pp. 129-146, March 1998.

Abstract: We consider the effects of large inflationary price fluctuations on the computed optimal harvest strategy for a randomized Schaefer model. Both prices and population sizes are assumed random with both background (Wiener) and jump (Poisson) components. Population fluctuations are assumed to be density independent, i.e., relative changes are independent of population size. Stochastic dynamic programming is employed to find the optimal harvesting effort and economic return for a realistic set of bioeconomic data for Pacific halibut. It is found that inflationary effects have a pronounced influence on the optimal return, even in a hazardous or disastrous environment. However, optimal harvesting effort levels are much less sensitive to inflationary effects.

81[KU02cancer]. J. J. Westman, B. R. Fabijonas, D. L. Kern, and F. B. Hanson, Probabilistic Rate Compartment Cancer Model: Alternate versus Traditional Chemotherapy Scheduling, in Stochastic Theory and Control, Proceedings of a Workshop held in Lawrence, Kansas, October 18-20, 2001, Lecture Notes in Control and Information Sciences, B.Pasik-Duncan (Editor), Springer-Verlag, New York, pp. 491-506, 24 July 2002 (invited paper).

Abstract: A four-compartment model for the evolution of cancer based on the characteristics of the cells is presented. The model is expanded to account for intrinsic and acquired drug resistance. This model can be explored to see the evolution of drug resistance starting from a single cell. Numerical studies are performed illustrating the palliative nature of chemotherapeutic treatments. Computational results are given for traditional treatment schedules. An alternate schedule for treatments is developed increasing the life expectancy and quality of life for the patient. A framework for the alternate scheduling is presented that addresses life expectancy, quality of life, and risk of metastasis. A key feature of the alternate schedule is that information for a particular patient can be used resulting in a personalized schedule of treatments. Alternate scheduling is compared to traditional treatment scheduling.

82[MTNS02]. J. J. Westman, t. R. Fabijonas, D. L. Kern, and F. B. Hanson, Cancer Treatment Using Multiple Chemotherapeutic Agents Subject to Drug Resistance, Proceedings of the 15th International Symposium on Mathematical Theory of Networks and Systems, 6 CD pages, 12 August 2002.

Abstract: A compartment model for the evolution of cancer subject to multiple chemotherapeutic agents is presented. The formulation accounts for the heterogeneous nature of cancer and drug resistance, leading to a system that can be used as a tool for optimizing treatments. Chemotherapeutic or cytotoxic agents have limited effectiveness due to of toxicity and the development of drug resistance. Toxicity limits the amount of the agent that can be used and acts as an upper bound on the dosage. Drug resistance causes the clinical value of the cytotoxic agent to become nominal after a small number of treatments. Therefore, several non-cross resistant cytotoxic agents are used to extend the number of treatments that can be adminis tered.

93[ACC05] Siddhartha P. Chakrabarty and Floyd B. Hanson, Optimal Control of Drug Delivery to Brain Tumors for a Distributed Parameters Model, Proceedings of 2005 American Control Conference, pp. 973-978, 08 June 2005.

Abstract: The growth and treatment of brain tumors is mathematically examined using a distributed parameters model. The model is a system of three coupled reaction diffusion equations involving the tumor cells, normal tissue and the drug concentration. An optimal control problem is designed, with the drug delivery rate as the control and solved to obtain the state and co-state equations as well as the regular control using a modified double shot forward-backward method. This gives rise to a coupled system of equations with a forward state equation and a backward co-state equation, which is solved using a double shot method. A numerical procedure based upon the Crank-Nicolson method is used to solve the coupled system of two three-dimension partial differential equations.

94[CDC05] Floyd B. Hanson and Siddhartha P. Chakrabarty, Optimal Control of Drug Delivery to Brain Tumors for a PDE Driven Model Using the Galerkin Finite Element Method , Proceedings of Joint 44th IEEE Conference on Decision and Control and European Control Conference, pp. 1613-1618, 12 December 2005.

Abstract: The Galerkin finite element method is used to examine the optimal drug delivery to brain tumors. The PDE driven mathematical model is a system of three coupled reaction diffusion equations involving the tumor cells, the normal tissue and the drug concentration. An optimal control problem is formulated keeping in mind the primary goals of the treatment, i.e., minimizing the tumor cell density and reducing the side effects of drugs. A distributed parameter method based on application of variational calculus to a pseudo-Hamiltonian, is used to obtain a coupled system of forward state equations and backward co-state equations. The Galerkin form of the finite element method is used due to its greater facility in numerically representing complex structures such as those in the brain. Finally, a two-dimensional circular disk test case is considered and partitioned into a set of rectangular finite elements in polar coordinates, with bilinear basis functions in the interior, but linear-quadratic basis function for elements adjacent to the boundary to exactly satisfy the no-flux boundary conditions.

98[CDC06] Siddhartha P. Chakrabarty and Floyd B. Hanson, Cancer Drug Delivery in Three Dimensions For a Distributed Parameter Control Model Using Finite Elements , Proceedings of 45th Conference on Decision and Control , pp. 2088-2093, San Diego, CA, December 2006.

Abstract: Optimal distributed parameter control and a Galerkin finite element method are used to develop procedures for an ideal model of optimal drug delivery to brain tumors. The mathematical model comprises of a system of three coupled reaction diffusion models, involving the density of tumor cells, normal tissue and also the drug concentration. An optimal control problem is formulated with the goal of minimizing the tumor cell density and reducing the side effects of the drug. A distributed parameter method based on a simple application of variational calculus is used on a pseudo-Hamiltonian, which is then used to obtain a coupled system of forward state equations and backward co-state equations. The Galerkin finite element method is used to realistically represent the brain structure. Finally, an ideal three dimensional test case is considered and partitioned into a set of brick finite elements in spherical coordinates, with tri-linear basis functions. Non-uniqueness of nodes in spherical coordinates is removed by combining like nodes, such as at the origin, at the poles and at the polar angle discontinuity. The Galerkin ODEs are solved by a combination of Crank-Nicolson and predictor-corrector methods.

101[MBS09]. Siddhartha P. Chakrabarty and Floyd B. Hanson, "Distributed Parameters Deterministic Model for Treatment of Brain Tumors Using Galerkin Finite Element Method ," Mathematical Biosciences, pp. 1-54, accepted for publication, 27 March 2009.

Abstract: In this paper we present a distributed parameters deterministic model for treatment of brain tumors using Galerkin finite element method. The dynamic model comprises of system of three coupled reaction diffusion models, involving the tumor cells, the normal tissues and the drug concentration. An optimal control problem is formulated with the goal of minimizing the tumor cell density and reducing the side effects of the drug. A distributed parameters method based on the application of variational calculus is used on an integral-Hamiltonian, which is then used to obtain an optimal coupled system of forward state equations and backward co-state equations. The Galerkin finite element method is used to realistically represent the brain structure as well as to facilitate computation. Finally a three dimensional test case is considered and partitioned into a set of spherical finite elements, using tri-linear basis functions, except for the elements effected by singularities of polar and azimuthal angles, as well the origin.

• This material is based upon work supported by awards from the National Science Foundation under Grants Numbers NSF 880699, 9102343, 9301107, 9626692, 9973231.

• Supercomputing research support was provided by
  • San Diego Supercomputing Center NPACI Account UIL203;
  • National Center for Supercomputing Applications Grant Numbers DCR860001N, DMS890009N, DMS900016N, DMS920003N, DMS960002N;
  • Pittsburgh Supercomputing Center Grant Numbers DMS940001P and DMS9400011P;
  • Argonne National Laboratory's Advanced Computing Research Facility;
  • Los Alamos National Laboratory's Advanced Computing Laboratory ACL-Accounts Z803835 and gwqfbh-d.

• Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation or other agency.

June 1, 1998