Annals of Biomedical Engineering

, Volume 21, Issue 6, pp 621–631

SENSOP: A derivative-free solver for nonlinear least squares with sensitivity scaling

  • I. S. Chan
  • A. A. Goldstein
  • J. B. Bassingthwaighte
Article

Abstract

Nonlinear least squares optimization is used most often in fitting a complex model to a set of data. An ordinary nonlinear least squares optimizer assumes a constant variance for all the data points. This paper presents SENSOP, a weighted nonlinear least squares optimizer, which is designed for fitting a model to a set of data where the variance may or may not be constant. It uses a variant of the Levenberg-Marquardt method to calculate the direction and the length of the step change in the parameter vector. The method for estimating appropriate weighting functions applies generally to 1-dimensional signals and can be used for higher dimensional signals. Sets of multiple tracer outflow dilution curves present special problems because the data encompass three to four orders of magnitude; a fractional power function provides appropriate weighting giving success in parameter estimation despite the wide range.

Keywords

Optimization Curve fitting Indicator dilution Probability density function Sensitivity function Model identifiability Weighting function 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bassingthwaighte, J.B.; Ackerman, F.H.; Wood, E.H. Applications of the lagged normal density curve as a model for arterial dilution curves. Circ. Res. 18:398–415; 1966.PubMedGoogle Scholar
  2. 2.
    Bassingthwaighte, J.B.; Wang, C.Y.; Chan, I.S. Blood-tissue exchange via transport and transformation by endothelial cells. Circ. Res. 65:997–1020; 1989.PubMedGoogle Scholar
  3. 3.
    Bassingthwaighte, J.B.; Chan, I.S.; Wang, C.Y. Computationally efficient algorithms for capillary convection-permeation-diffusion models for blood-tissue exchange. Ann. Biomed. Eng. 20:687–725; 1992.CrossRefPubMedGoogle Scholar
  4. 4.
    Box, G.E.P.; Hill, W.J. Correcting inhomogeneity of variance with power transformation weighting. Technometrics 16:385–389; 1974.Google Scholar
  5. 5.
    Bronikowski, T.A.; Dawson, C.A.; Linehan, J.H. Modelfree deconvolution techniques for estimating vascular transport functions. Int. J. Biomed. Comput. 14:411–429; 1983.CrossRefPubMedGoogle Scholar
  6. 6.
    Dennis, J.E.; Gay, D.M.; Welsch, R.E. Algorithm 573: NL2SOL—An adaptive nonlinear least-squares algorithm. ACM TOMS 7:369–383; 1981.Google Scholar
  7. 7.
    Donaldson, J.R.; Schnabel, R.B. Computational experience with confidence regions and confidence intervals for nonlinear least squares. Technometrics 29:67–82; 1987.Google Scholar
  8. 8.
    Duncan, G.T. An empirical study of jackknife-constructed confidence regions in nonlinear regression. Technometrics 20:123–129; 1978.Google Scholar
  9. 9.
    Hiebert, K.L. An evaluation of mathematical software that solves nonlinear least squares problems. ACM TOMS 7:1–16; 1981.Google Scholar
  10. 10.
    Kroll, K.; Bukowski, T.R.; Schwartz, L.M.; Knoepfler, D.; Bassingthwaighte, J.B. Capillary endothelial transport of uric acid in the guinea pig heart. Am. J. Physiol. 262 (Heart Circ. Physiol. 31):H420-H431; 1992.PubMedGoogle Scholar
  11. 11.
    Kuikka, J.; Levin, M.; Bassingthwaighte, J.B. Multiple tracer dilution estimates ofd- and 2-deoxy-d-glucose uptake by the heart. Am. J. Physiol. 250 (Heart Circ. Physiol. 19):H29-H42; 1986.PubMedGoogle Scholar
  12. 12.
    Lanczos, C. Applied analysis. Englewood Cliffs, NJ: Prentice-Hall; 1956.Google Scholar
  13. 13.
    Levenberg, K. A method for the solution of certain problems in least squares. Quart. Appl. Math. 2:164–168; 1944.Google Scholar
  14. 14.
    Marquardt, D.W. An algorithm for least-squares estimation of nonlinear parameters. J. Soc. Ind. Appl. Math. 11:431; 1963.CrossRefGoogle Scholar
  15. 15.
    Moré, J.J.; Garbow, B.S.; Hillstrom, K.E. Testing unconstrained optimizer software ACM TOMS 7:17–41; 1981.Google Scholar
  16. 16.
    Press, S.J. Applied multivariate analysis. New York: Holt, Rinehart and Winston; 1972.Google Scholar
  17. 17.
    Ratkowsky, D.A. Nonlinear regression modeling: Aunified practical approach. New York: Marcel Dekker; 1983.Google Scholar
  18. 18.
    Wangler, R.D.; Gorman, M.W.; Wang, C.Y.; DeWitt, D.F.; Chan, I.S.; Bassingthwaighte, J.B.; Sparks, H.V. Transcapillary adenosine transport and interstitial adenosine concentration in guinea pig hearts. Am. J. Physiol. 257 (Heart Circ. Physiol. 26):H89-H106; 1989.PubMedGoogle Scholar

Copyright information

© Pergamon Press Ltd. 1993

Authors and Affiliations

  • I. S. Chan
    • 1
  • A. A. Goldstein
    • 1
  • J. B. Bassingthwaighte
    • 1
  1. 1.Center for BioengineeringUniversity of WashingtonSeattle

Personalised recommendations