Parameter Identification for Population Equations Modeling Erythropoiesis

Conference paper

Abstract

Physiological models explaining the anemia of chronic kidney disease have become more complicated over the last years. Identification of model parameters poses difficulties as measurements are very limited. A model for erythropoiesis, consisting of coupled partial differential equations, is adapted to individual patients. The numerical approximations make use of evolution operators and are based on the theory of abstract Cauchy problems. The abstract Cauchy problems corresponding to the model equations are approximated by Cauchy problems on finite-dimensional subspaces of the state space of the original problem. A low approximation dimension suffices to obtain accurate numerical solutions and estimates for the parameters. An example of (locally) well identifiable parameters expressing numerical convergence for increasing dimensions of the finite dimensional approximating system is discussed. Moreover, it is demonstrated that a clever choice of cost-functionals can reduce the observation time needed for parameter identification from 150 to 90 days.

Keywords

Abstract cauchy problems Anemia Chronic kidney disease Erythropoiesis Parameter estimation Structured population equations Weighted cost functionals 

References

  1. 1.
    M. Abramowits, I. Stegun (eds.), Handbook of Mathematical Functions, 10th edn. National Bureau of Standards—Applied Mathematics Series (1972)Google Scholar
  2. 2.
    A. Attarian, J. Batzel, B. Matzuka, H.T. Tran, in Application of the unscented Kalman filtering to parameter estimation, ed. by J.J. Batzel, M. Bachar, F. Kappel. Mathematical Modeling and Validation in Physiology: Application to the Cardiovascular and Respiratory Systems, Lecture Notes in Mathematics (Mathematical Biosciences Subseries), vol 2064 (Springer, New York, 2013) pp. 75–88Google Scholar
  3. 3.
    A. Besarab, W. Bolton, J. Browne, J. Egrie, A. Nissenson, D. Okamoto, S. Schwab, D. Goodkin, The effects of normal as compared with low hematocrit values in patients with cardiac disease who are receiving hemodialysis and epoetin. N. Engl. J. Med. 339, 584–590 (1998)CrossRefGoogle Scholar
  4. 4.
    A. Collins, R. Brenner, J. Ofman, E. Chi, N. Stuccio-White, M. Krishnan, C. Solid, N. Ofsthun, J. Lazarus, Epoetin alfa use in patients with ESRD: an analysis of recent US prescribing patterns and hemoglobin outcomes. Am. J. Kidney Dis. 46, 481–488 (2005)CrossRefGoogle Scholar
  5. 5.
    E. Davies, One-Parameter Semigroups. (Academic Press, London, 1980)Google Scholar
  6. 6.
    K.J. Engel, R. Nagel, One Parameter-Semigroups for Linear Evolution Equations. (Springer, Berlin, 2000)Google Scholar
  7. 7.
    S. Fishbane, J. Berns, Hemoglobin cycling in hemodialysis patients treated with recombinant human erythropoietin. Kidney Int. 68, 1337–1343 (2005)CrossRefGoogle Scholar
  8. 8.
    D. Fuertinger, F. Kappel, A numerical method for structured population equations modeling control of erythropoiesis. in 1st IFAC Workshop on the Control of Systems Modeled by Partial Differential Equations (CPDE 2013), September 25–27, Paris (France) (2013)Google Scholar
  9. 9.
    D. Fuertinger, F. Kappel, A parameter identification technique for structured population equations modeling erythropoiesis in dialysis patients, Lecture Notes in Engineering and Computer Science. in Proceedings of the World Congress on Engineering and Computer Science 2013, WCECS 2013, 23–25 October, 2013, San Francisco, USA, pp. 940–944Google Scholar
  10. 10.
    D. Fuertinger, F. Kappel, S. Thijssen, N. Levin, P. Kotanko, A model of erythropoiesis in adults with sufficient iron availability. J. Math. Biol. 66(6), 1209–1240 (2013)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    D.H. Fuertinger, A model for erythropoiesis. Ph.D. thesis, University of Graz, Austria, 2012Google Scholar
  12. 12.
    A. Go, G. Chertow, D. Fan, C. McCulloch, C. Hsu, Chronic kidney disease and the risks of death, cardiovascular events, and hospitalization. N. Engl. J. Med. 351, 1296–1305 (2004)CrossRefGoogle Scholar
  13. 13.
    K. Ito, F. Kappel, Evolution Equations and Approximations (World Scientific, Singapore, 2002)MATHGoogle Scholar
  14. 14.
    E. Jones, T. Oliphant, P. Peterson et al., SciPy: Open source scientific tools for Python. http://www.scipy.org/ (2001)
  15. 15.
    F. Kappel, K. Zhang, Approximation of linear age-structured population model using Legendre polynomials. J. Math. Anal. Appl. 180, 518–549 (1993)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    M. Lichtman, E. Beutler, T. Kipps, U. Seligsohn, K. Kaushansky, J. Prchal (eds.), Williams Hematology, 7th edn. (McGraw-Hill, New York, 2005)Google Scholar
  17. 17.
    S. Nadler, J. Hidalgo, T. Bloch, Prediction of blood volume in normal human adults. Surgery 51, 224–232 (1962)Google Scholar
  18. 18.
    J. Nelder, R. Mead, A simplex method for function minimization. Comput. J. 7, 308–313 (1965)CrossRefMATHGoogle Scholar
  19. 19.
    J. Nocedal, S. Wright, Numerical Approximation, 2nd edn. Operations Research and Financial Engineering. (Springer, NewYork, 2006)Google Scholar
  20. 20.
    A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. (Springer, Berlin, 1983)Google Scholar
  21. 21.
    G. van Rossum, F.L. Drake (eds.), Python Reference Manual. Python Software Foundation, http://docs.python.org/ref/ref.html (2012)
  22. 22.
    G. Strippoli, J. Craig, C. Manno, F. Schena, Hemoglobin targets for the anemia of chronic kidney disease: a meta-analysis of randomized, controlled trials. JASN 15, 3154–3165 (2004)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Renal Research InstituteNew YorkUSA
  2. 2.Institute for Mathematics and Scientific ComputingUniversity of GrazGrazAustria

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