Advertisement

Journal of Pharmacokinetics and Biopharmaceutics

, Volume 21, Issue 5, pp 609–636 | Cite as

Two constrained deconvolution methods using spline functions

  • Davide Verotta
Pharmacometrics

Abstract

This paper describes two new methods to solve the following estimation problem. Given n1noisy measurements (yi1,i,i=1,..., n1)of the response of a system to a knowninput [A1(t) where tindicates time], and n2noisy measurements yi,2,i=l,..., n2of the response of a system to an unknowninput [A2(t)], obtain an estimate of A2(t) and K(t) (the unit impulse response function of the system) under the model:
$$y_{ij} = \int_0^{t_{ij} } {A_j (s)} K(t_{ij} - s)ds + \varepsilon _{ij} $$
wereεi,jare independent identically distributed random variables. Both methods use spline functions to represent the unknown functions, and they automatically select the spline functions representing the unknown input and unit impulse response functions. The first method estimates separately the unit impulse response function and the input, recasting the problem in terms of inequality-constrained linear regression. The second method jointly estimates the unit impulse response function and the input function, recasting the problem in terms of inequality-constrained nonlinear regression. Simulated and real data analysis are reported.

Key words

linear regression nonlinear regression linear systems inequality constrains model selection 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    C. Chen.Linear System Theory and Design, Holt, Rinehart and Winston, New York, 1984.Google Scholar
  2. 2.
    J. J. Stephenson. Theory and transport in linear biological systems: I. Fundamental integral equation.Bull. Math. Biophys. 22:1–17 (1960).CrossRefGoogle Scholar
  3. 3.
    J. J. Stephenson. Theory and transport in linear biological systems: II. Multiflux problems.Bull. Math. Biophys. 22:113–138 (1960).CrossRefGoogle Scholar
  4. 4.
    C. Cobelli, A. Mari, S. Del-Prato, S. De-Kreutzenberg, R. Nosadini, and I. Jensen. Reconstructing the rate of appearance of subcutaneous insulin by deconvolution.Amt. J. Physiol. 253 (Regulatory Integrative Comp Physiol 16):E584-E590 (1987).Google Scholar
  5. 5.
    F. O'Sullivan and J. O'Sullivan. Deconvolution of episodic hormone data: an analysis of the role of season on the onset of puberty in cows.Biometrics 44:339–353 (1988).PubMedCrossRefGoogle Scholar
  6. 6.
    F. Langenbucher. Numerical convolution/deconvolution as a tool for correlating in vitro with in vivo drug availability.Pharm. Ind. 4:1166–1172 (1982).Google Scholar
  7. 7.
    V. F. Smolen. Theoretical and computational basis for drug bioavailability determina-tions using pharmacological data. I. General considerations and procedures.J. Pharmacokin. Biopharm. 4:337–353 (1976).CrossRefGoogle Scholar
  8. 8.
    P. Veng-Pedersen. An algorithm and computer program for deconvolution in linear pharmacokinetics.J. Pharmacokin. Biopharm. 8:463–481 (1980).CrossRefGoogle Scholar
  9. 9.
    S. Vajda, K. R. Godfrey, and P. Valko. Numerical deconvolution using system identification methods.J. Pharmacokin. Biopharm. 16:85–107 (1988).CrossRefGoogle Scholar
  10. 10.
    D. Verotta, S. Beal, and L. B. Sheiner. Semiparametric approach to pharmacokinetic-pharmacodynamic data,Am. J. Physiol. 256 (Regulatory Integrative Comp Physiol 25):R1005-R1010 (1989).PubMedGoogle Scholar
  11. 11.
    B. R. Hunt. The inverse problem of radiography.Math. Biosci. 8:161–179, 1970.CrossRefGoogle Scholar
  12. 12.
    S. V. Huffel, J. Vandewalle, M. C. Roo, and J. L. Willems. Reliable and efficient deconvolution technique based on total linear least squares for calculating the renal retention function.Med. Biol. Eng. Comput. 25:26–33, 1987.PubMedCrossRefGoogle Scholar
  13. 13.
    P. A. Jansson, R. H. Hunt, and E. K. Plyler. Resolution enhancement of spectra.J. Opt. Soc. Am. 60:596–599 (1970).CrossRefGoogle Scholar
  14. 14.
    S. W. Provencher and R. H. Vogel. Regularization techniques for inverse problems in molecular biology. In P. Deuflhard and E. Hairer (eds),Numerical Treatment of Inverse Problems in Differential and Integral Equations, Birkhauser, 1983.Google Scholar
  15. 15.
    G. Wahba. Constrained regularization for ill posed linear operator equations, with applications in meteorology and medicine. In S. S. Gupta and J. O. Berger (eds),Statistical Decision Theory and Related Topics III, Vol. 2, Academic Press, New York, 1982, pp. 383–418.Google Scholar
  16. 16.
    J. Mendelsohn and J. Rice. Deconvolution of microfluorometric histograms with B-splines.J. Am. Statist. Assoc. 77:748–753 (1985).Google Scholar
  17. 17.
    J. A. Rice. Choice of smoothing parameter in deconvolution problemsContemp. Math.,59:137–151 (1986).CrossRefGoogle Scholar
  18. 18.
    F. O'Sullivan. A statistical perspective on ill-posed inverse problems.Statist. Sci. 1:502–527 (1986).CrossRefGoogle Scholar
  19. 19.
    D. Verotta, L. B. Sheiner, W. F. Ebling, and D. Stanski. A Semiparametric approach to physiological flow models.J. Pharmacokin. Biopharm,17:463–491, (1989).CrossRefGoogle Scholar
  20. 20.
    D. G. Luenberger.Optimization by Vector Space Methods, Wiley, New York, 1969.Google Scholar
  21. 21.
    D. G. Luenberger.Linear and Nonlinear Programming, 2nd ed., Addison-Wesley, Reading, MA, 1984.Google Scholar
  22. 22.
    R. J. Carroll and D. Rupport,Transformation and Weighting in Regression, Chapman and Hall, New York, 1988.CrossRefGoogle Scholar
  23. 23.
    P. J. Bickel and K. A. Doksum.Mathematical Statistics, Holden-Day, Oakland, CA, 1977.Google Scholar
  24. 24.
    C. J. Stone. Additive regression and other nonparatnetric models,Ann. Statist. 13:689–706 (1985).CrossRefGoogle Scholar
  25. 25.
    L. Breiman. Fitting additive models to regression data. Technical Report No. 209, Department of Statistics, University of California at Berkeley, Berkeley, California, July 1989.Google Scholar
  26. 26.
    H. Akaike. A new look at the statistical model identification.IEEE Trans. Automat. Contr. 19:716–723 (1974).CrossRefGoogle Scholar
  27. 27.
    P. Craven and G. Wahba. Smoothing noisy data with spline functions.Numer. Math. 31:377–403 (1979).CrossRefGoogle Scholar
  28. 28.
    D. Verotta. Estimation and model selection using splines in constrained deconvolution.Ann. Bioeng. in press (1993).Google Scholar
  29. 29.
    D. Allen. The relationship between variable selection and data augmentation and a method for prediction.Technometrics 16:125–127 (1974).CrossRefGoogle Scholar
  30. 30.
    G. Wahba.Spline Models for Observational Data. Society for Industrial and Applied Mathematics, Philadelphia, 1990.CrossRefGoogle Scholar
  31. 31.
    D. P. Vaughan and M. Dennis. Mathematical basis of point-area deconvolution method for determining In Vivo input functions.J. Pharm. Sci. 67:663–665 (1978).PubMedCrossRefGoogle Scholar
  32. 32.
    D. Verotta. An inequality-constrained least-squares deconvolution method.J. Pharmacokin. Biopharm. 17:269–289 (1989).CrossRefGoogle Scholar
  33. 33.
    J. H. Friedman and R. Tibshirani. The monotone smoothing of scatterplots.Technometrics 26:243–250 (1984).CrossRefGoogle Scholar
  34. 34.
    J. O. Ramsay. Monotone regression splines in action.Statist. Sci. 3:425–461 (1988).CrossRefGoogle Scholar
  35. 35.
    C. Kelly and J. Rice. Monotone smoothing with application to dose-response curves and the assessment of synergism.Biometrics 46:1071–1085 (1990).PubMedCrossRefGoogle Scholar
  36. 36.
    P. Veng-Pedersen. Drug absorption evaluation in presence of changes in clearance. An algorithm and computer program for deconvolution with exact clearance correction.Biopharm. Drug Dispos. 8: 185–203, 1987.PubMedCrossRefGoogle Scholar
  37. 37.
    B. Efron.The Jackknife, the Bootstrap and Other Resampling Planes, Society for Industrial and Applied Mathematics, Philadelphia, 1982.CrossRefGoogle Scholar
  38. 38.
    B. Efron and R. Tibshirani. Bootstrap methods for standard errors, confidence intervals, and other measures of statistical accuracy.Statist. Sci. 68:54–77 (1986).CrossRefGoogle Scholar
  39. 39.
    D. Verotta. Bootstrap confidence bands in constrained linear regression. Ph.D. dissertation, Dept. of Biostatistics, University California Berkeley, Chapter3, University Microfilms International, 1992.Google Scholar
  40. 40.
    D. Verotta. Simultaneous bootstrap confidence bands in nonparametric constrained regression. Technical report #34, Department of Biostatistics, UCSF, 1993.Google Scholar
  41. 41.
    S-PLUS. Statistical Sciences Inc., Seattle, WA, 1989.Google Scholar
  42. 42.
    C. DeBoor.A Practical Guide to Splines, Springer-Verlag, New York, 1978.CrossRefGoogle Scholar
  43. 43.
    L. Breiman and P. Peters. Comparing automatic smoothers (A public service enterprise). Technical Report No. 161, Department of Statistics, University of California at Berkeley, Berkeley, California, June 1988, Revised July 1990.Google Scholar
  44. 44.
    W. R. Gillespie. CVPEX: Programs for the convolution of piecewise polynomial and polyexponential functions. Submitted.Google Scholar
  45. 45.
    QPROG. IMSL, City West Boulevard, Houston, TX 77042-3020.Google Scholar
  46. 46.
    J. M. Chambers, W. S. Cleveland, B. Kleiner, and P. A. Tukey.Graphical Methods for Data Analysis, Wadsworth, Belmont, CA, 1983.Google Scholar

Copyright information

© Plenum Publishing Corporation 1993

Authors and Affiliations

  • Davide Verotta
    • 1
    • 2
  1. 1.Department of Pharmacy and Pharmaceutical ChemistryUniversity of California San Francisco
  2. 2.Department of Epidemiology and BiostatisticsUniversity of California San Francisco

Personalised recommendations