### Mathematical Biosciences and Engineering

2019, Issue 6: 7299-7326. doi: 10.3934/mbe.2019365
Research article

# Estimation of probability distributions of parameters using aggregate population data: analysis of a CAR T-cell cancer model

• Received: 28 March 2019 Accepted: 01 August 2019 Published: 09 August 2019
• In this effort we explain fundamental formulations for aggregate data inverse problems requiring estimation of probability distribution parameters. We use as a motivating example a class of CAR T-call cancer models in mice. After ascertaining results on model stability and sensitivity with respect to parameters, we carry out first elementary computations on the question how much data is needed for successful estimation of probability distributions.

Citation: Celia Schacht, Annabel Meade, H.T. Banks, Heiko Enderling, Daniel Abate-Daga. Estimation of probability distributions of parameters using aggregate population data: analysis of a CAR T-cell cancer model[J]. Mathematical Biosciences and Engineering, 2019, 16(6): 7299-7326. doi: 10.3934/mbe.2019365

### Related Papers:

• In this effort we explain fundamental formulations for aggregate data inverse problems requiring estimation of probability distribution parameters. We use as a motivating example a class of CAR T-call cancer models in mice. After ascertaining results on model stability and sensitivity with respect to parameters, we carry out first elementary computations on the question how much data is needed for successful estimation of probability distributions.

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Ernstberger, et al., Experimental design and estimation of growth rate distributions in size-structured shrimp populations, CRSC TR08-20, Center for Research in Scientific Computation, N. C. State University, Raleigh, NC, November, 2008; Inverse Probl., 25 (2009), 095003 (28 pp). [6] H. T. Banks, L. W. Botsford, F. Kappel, et al., Estimation of growth and survival in size-structured cohort data: An application to larval striped bass (Morone saxatilis), CAMS Tech. Rep. 89-10, University of Southern California, 1989; J. Math. Biol., 30 (1991), 125–150. [7] H. T. Banks and B. G. Fitzpatrick, Estimation of growth rate distributions in size-structured popu-lation models, CAMS Tech. Rep. 90-2, University of Southern California, January, 1990; Q. Appl. Math., 49 (1991), 215–235. [8] D. Abate-Daga and M. L. Davila, CAR models: next-generation CAR modifications for enhanced T-cell function, Mol. Ther-Oncolytics, 3 (2016), 16014. [9] E. K. Moon, C. Carpenito, J. Sun, et al., Expression of a functional CCR2 receptor enhances tumor localization and tumor eradication by retargeted human T cells expressing a mesothelin-specific chimeric antibody receptor, Clin. Cancer Res., 17 (2011), 4719–4730. [10] H. T. Banks and N. L. Gibson, Well-posedness in Maxwell systems with distributions of polariza-tion relaxation parameters, CRSC-TR04-01, Center for Research in Scientific Computation, N. C. State University, Raleigh, NC, January, 2004; Appl. Math. Lett., 18 (2005), 423–430. [11] H. T. Banks and N. L. Gibson, Electromagnetic inverse problems involving distributions of di-electric mechanisms and parameters, CRSC-TR05-29, August, 2005; Q. Appl. Math., 64 (2006), 749–795. [12] H. T. Banks and D. M. Bortz, Inverse problems for a class of measure dependent dynamical systems, J. Inverse Ill-pose. Probl., 13 (2005), 103–121. [13] H. T. Banks, D. M. Bortz and S. E. Holte, Incorporation of variability into the mathematical modeling of viral delays in HIV infection dynamics, Math. Biosci., 183 (2003), 63–91. [14] H. T. Banks, D. M. Bortz, G. A. Pinter, et al., Modeling and imaging techniques with potential for application in bioterrorism, CRSC TR03-02, January 2003; Chapter 6 in Bioterrorism: Math-ematical Modeling Applications in Homeland Security (H.T. Banks and C. Castillo-Chavez, eds.), Front. Appl. Math., FR28, SIAM, Philadelphia, 2003, 129–154. [15] H. T. Banks, J. H. Barnes, A. Eberhardt, et al., Modeling and computation of propagating waves from coronary stenosis, Comput. Appl. Math., 21 (2002), 767–788. [16] H. T. Banks, S. Hu, Z. R. Kenz, et al., Material parameter estimation and hypothesis testing on a 1D viscoelastic stenosis model: methodology, CRSC-TR12-09, April, 2012; J. Inverse Ill-pose. Probl., 21 (2013), 25–57. [17] H. T. Banks, S. Hu, Z. R. Kenz, et al., Model validation for a noninvasive arterial stenosis detection problem, CRSC-TR12-22, December, 2012; Math. Biosci. Eng., 11 (2013), 427–448. [18] H. T. Banks and G. A. Pinter, A probabilistic multiscale approach to hysteresis in shear wave propagation in biotissue, CRSC-TR04-03, January, 2004; SIAM J. Multiscale Model. Sim., 3 (2005), 395–412. [19] H. T. Banks, Chapter 14.4 of A Functional Analysis Framework for Modeling, Estimation and Control in Science and Engineering, Taylor and Frances Publishing, 2012. [20] G. de Vries, T. Hillen, M. Lewis, et al., A Course in Mathematical Biology: Quantitative Modelling with Mathematical and Computational Methods, SIAM, Philadephia, 2006. [21] S. I. Rubinow, Introduction to Mathematical Biology, John Wiley & Sons, New York, 1975. [22] L. J. Allen, An Introduction to Mathematical Biology. Pearson Education, Inc., Pearson Prentice Hall, Upper Saddle River, NJ, 2007. [23] M. Braun and M. Golubitsky, Differential Equations and their Applications. Vol. 4. Springer-Verlag New York, Inc., New York, NY, 1983. [24] G. E. Collins and A. G. Akritas, Polynomial real root isolation using Descarte's rule of signs, Proceedings of the third ACM symposium on Symbolic and algebraic computation, ACM, 1976. [25] H. T. Banks, L. Bekele-Maxwell, L. Bociu, et al., The complex-step method for sensitivity analysis of non-smooth problems arising in biology, CRSC-TR15-11, Center for Research in Scientific Computation, N. C. State University, Raleigh, NC, October, 2015; Eurasia. J. Math. Comput. Appl., 3 (2015), 16–68. [26] H. T. Banks, K. Bekele-Maxwell, R. A. Everett, et al., Dynamic modeling of problem drinkers undergoing behavioral treatment, CRSC-TR16-12, Center for Research in Scientific Computation, N. C. State University, Raleigh, NC, October, October, 2016; Bull. Math. Biol., 79 (2017), 1254–1273. [27] H. T. Banks and K. L. Bihari, Modeling and estimating uncertainty in parameter estimation, CRSC-TR99-40, Center for Research in Scientific Computation, N. C. State University, Raleigh, NC, October, 2016; December, 1999; Inverse Probl., 17 (2001), 95–111. [28] H. T. Banks, K. B. Flores, I. G. Rosen, et al., The Prohorov Metric Framework and aggregate data inverse problems for random PDEs, CRSC-TR18-05, Center for Research in Scientific Computa- tion, N. C. State University, Raleigh, NC, June, 2018; Commun. Appl. Anal., 22 (2018), 415–446. [29] H. T. Banks, J. Catenacci and S. Hu, Use of difference-based methods to explore statistical and mathematical model discrepancy in inverse problems, CRSC-TR15-05, Center for Research in Scientific Computation, N. C. State University, Raleigh, NC, May, 2015. J. Inverse Ill-pose. P., 24 (2016), 413-–433. [30] H. T. Banks, J. E. Banks, N. G. Cody, et al., Population model for the decline of Homalodisca vitripennis (HEMIPTERA: CICADELLIDAE) over a ten-year period, CRSC-TR18-06, Center for Research in Scientific Computation, N. C. State University, Raleigh, NC, June, 2018; J. Biol. Dyn., 13 (2019), 422–446. [31] H. T. Banks and K. Kunisch, Estimation Techniques for Distributed Parameter Systems, Birkhas-ser, Boston, 1989. [32] H. T. Banks and P. Kareiva, Parameter estimation techniques for transport equations with applica-tion to population dispersal and tissue bulk flow models, J. Math. Biol., 17 (1983), 253–273. [33] A. C. Atkinson and R. A. Bailey, One hundred years of the design of experiments on and off the pages of Biometrika, Biometrika, 88 (2001), 53–97. [34] A. C. Atkinson and A. N. Donev, Optimum Experimental Designs, Oxford University Press, New York, 1992. [35] M. P. F. Berger and W. K. Wong (Editors), Applied Optimal Designs, John Wiley & Sons, Chich-ester, UK, 2005. [36] V. V. Fedorov, Theory of Optimal Experiments, Academic Press, New York and London, 1972. [37] V. V. Fedorov and P. Hackel, Model-Oriented Design of Experiments, Springer-Verlag, New York, 1997. [38] W. Mueller and M. Stehlik, Issues in the optimal design of computer simulation experiments, Appl. Stoch. Model. Bus., 25 (2009), 163–177. [39] M. Patan and B. Bogacka, Optimum experimental designs for dynamic systems in the presence of correlated errors, Comput. Stat. Data An., 51 (2007), 5644–5661. [40] D. Ucinski and A. C. Atkinson, Experimental design for time-dependent models with correlated observations, Stud. Nonlinear Dyn. E., 8 (2004), Article 13: The Berkeley Electronic Press. [41] H. T. Banks, K. Holm and F. Kappel, Comparison of optimal design methods in inverse problems, CRSC-TR10-11, Center for Research in Scientific Computation, N. C. State University, Raleigh, NC, July, 2010; Inverse Probl., 27 (2011), 075002. [42] H. T. Banks, A. Cintr'on-Arias and F. Kappel, Parameter selection methods in inverse problem for-mulation, CRSC-TR10-03, Center for Research in Scientific Computation, N. C. State University, Raleigh, NC, revised November 2010; Mathematical Model Development and Validation in Physi-ology: Application to the Cardiovascular and Respiratory Systems, Lecture Notes in Mathematics, Vol. 2064, Mathematical Biosciences Subseries; Springer-Verlag, Berlin, 2013. [43] M. Avery, H. T. Banks, K. Basu, et al., Experimental design and inverse problems in plant bi-ological modeling, CRSC-TR11-12, Center for Research in Scientific Computation, N. C. State University, Raleigh, NC, October, 2011; J. Inverse Ill-pose. P., 20 (2012), 169–191. [44] H. T. Banks and K. L. Rehm, Experimental design for vector output systems, CRSC-TR12-11, Center for Research in Scientific Computation, N. C. State University, Raleigh, NC, April, 2012; Inverse Probl. Sci. En., 22 (2014), 557–590. [45] H. T. Banks and K. L. Rehm, Experimental design for distributed parameter vector systems, CRSC-TR12-17, Center for Research in Scientific Computation, N. C. State University, Raleigh, NC, August, 2012; Appl. Math. Lett., 26 (2013), 10–14. [46] H. T. Banks, S. Dediu, S. L. Ernstberger, et al., Generalized sensitivities and optimal experimental design, CRSC-TR08-12, Center for Research in Scientific Computation, N. C. State University, Raleigh, NC, September, 2008, revised November, 2009; J. Inverse Ill-pose. P., 18 (2010), 25–83. [47] B. M. Adams, H. T. Banks, M. Davidian, et al., Model fitting and prediction with HIV treatment interruption data, CRSC TR05-40, Center for Research in Scientific Computation, N. C. State University, Raleigh, NC, October, 2005; Bull. Math. Biol., 69 (2007), 563–584.
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