Advertisement

Spline functions in convolutional modeling of verapamil bioavailability and bioequivalence. I: conceptual and numerical issues

  • J. Popović
Article

Summary

A cubic spline function for describing the verapamil concentration profile, resulting from the verapamil absorption input to be evaluated, has been used. With this method, the knots are taken to be the data points, which has the advantage of being computationally less complex. Because of its inherently low algorhythmic errors, the spline method is less distorted and more suitable for further data analysis than others. The method has been evaluated using simulated verapamil delayed release tablet concentration data containing various degrees of random noise. The accuracy of the method was determined by how well the estimates of input rate and extent represented the true values. It was found that the accuracy of the method was of the same order of magnitude as the noise level of the data. Spline functions in convolutional modeling of verapamil formulation bioavailability and bioequivalence, as shown in the numerical simulation investigation, are very powerful additional tools for assessing the quality of new verapamil formulations in order to ensure that they are of the same quality as already registered formulations of the drug. The development of such models provides the possibility to avoid additional or larger bioequivalence and/or clinical trials and to thus help shorten the investigation time and registration period.

Keywords

Spline convolution input rate simulation verapamil 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Dedík, L., Durišová, M. (1995): CXT — A programme for analysis of linear, dynamic systems in the frequency domain. Int. S. Bio-Med. Comput., 39, 231–241.CrossRefGoogle Scholar
  2. 2.
    Dedík, L., Durišová, M. (1996): CXT-MAIN: A software package for determination of the analytical form of the pharmacokinetic system weighting function. Comput. Methods, Programs, Biomed.,51, 183–192.CrossRefGoogle Scholar
  3. 3.
    Dedík, L., Durišová, M. (1999): System approach in technical, environmental and biomedical studies. Publishing House of Slovak University of Technology, Bratislava.Google Scholar
  4. 4.
    Durišová, M., Dedík, L. (1994): Comparative study of human pentacaine pharmacokinetics in time and frequency domains. Methods Find. Exp. Clin. Pharmacol., 16, 219–232.PubMedGoogle Scholar
  5. 5.
    Dedík, L., Durišová, M. (1994): Frequency response method in pharmacokinetics. J. Pharmacokinet. Biopharmacy., 22, 293–307.CrossRefGoogle Scholar
  6. 6.
    Durišová, M., Dedík, L., Balan, M. (1995): Building a structured model of acomplex pharmacokinetic system with time delays. Bull. Math. Biol., 57, 787–808.PubMedGoogle Scholar
  7. 7.
    Durišová, M., Dedík, L. (1997): Modeling in frequency domain used for assessment of in vivo dissolution profile. Pharm. Res., 14, 860–864.CrossRefPubMedGoogle Scholar
  8. 8.
    Dedík, L., Durišová, M. (2001): Modeling drug absorption from enteric coated granules. Methods Find. Exp. Clin. Pharmacol., 23, 213–217.CrossRefPubMedGoogle Scholar
  9. 9.
    Dedík, L., Durišová, M. (2002): System-approach methods for modeling and testing similarity of in vitro dissolutions of drug dosage formulations. Comput. Methods Programs Biomed., 69, 49–55.CrossRefPubMedGoogle Scholar
  10. 10.
    Dedík, L., Durišová, M. (2002): System-approach to modeling metabolite formation from parent drug: a working example with methotrexate. Methods Find. Exp. Clin. Pharmacol., 24, 481–486.CrossRefPubMedGoogle Scholar
  11. 11.
    Popović J. (2004): Classical Michaelis-Menten and system theory approach to modeling metabolite formation kinetics. Eur. J. Drug Metab. Pharmacokinet., 29, 205–214.CrossRefPubMedGoogle Scholar
  12. 12.
    Durišová, M., Dedík, L. (2005): New mathematical methods in pharmacokinetic modeling. Basic Clin. Pharmacol. Toxicol., 96, 335–342.CrossRefPubMedGoogle Scholar
  13. 13.
    Agatonović-Kustrin S., Beresford R. (2000): Basic concepts of artificial neural network (ANN) modeling and its application in pharmaceutical research. J. Pharm. Biomed. Anal., 22, 717–727.CrossRefPubMedGoogle Scholar
  14. 14.
    Yamamura S., Nishizawa K., Hirano M., Momose Y., Kimura A. (1998): Prediction of plasma levels of aminoglycoside antibiotic in patients with severe illness by means of an artificial neural network simulator. J. Pharm. Pharm. Sci., 1, 95–101.PubMedGoogle Scholar
  15. 15.
    Yamamura S., Takehira R., Kawada K., Nishizawa K., Katayama M., Hirano M., Momose Y. (2003): Application of artificial neural network modeling to identify severely ill patients whose aminoglycoside concentrations are likely to fall below therapeutic concentrations. J. Clin. Pharm. Ther., 28, 425–432.CrossRefPubMedGoogle Scholar
  16. 16.
    Yamamura S. (2003): Clinical application of artificial neural network (ANN) modeling to predict pharmacokinetic parameters of severely ill patients. Adv. Drug. Deliv. Rev., 55, 1233–1251.CrossRefPubMedGoogle Scholar
  17. 17.
    Chow H.H., Tolle K.M., Roe DJ. Elsberry V., Chen H. (1997): Application of neural networks to population pharmacokinetic data analysis. J. Pharm. Sci., 86, 840–845.CrossRefPubMedGoogle Scholar
  18. 18.
    Sun Y., Peng Y., Chen Y., Shukla A.J. (2003): Application of artificial neural networks in the design of controlled release drug delivery systems. Adv. Drug. Del. Rev., 55, 1201–1215.CrossRefGoogle Scholar
  19. 19.
    Turner J.V., Maddalena D.J., Cutler D.J. (2004): Pharmacokinetic parameter prediction from drug structure using artificial neural networks. Int. J. Pharm., 270, 209–219.CrossRefPubMedGoogle Scholar
  20. 20.
    Babuška R. (1998): Fuzzy modeling for control. Kluwer Academic Publishers, Boston, USA.Google Scholar
  21. 21.
    Sproule B.A., Naranjo C.A., Türksen LB. (2002): Fuzzy pharmacology: theory and applications. Trends Pharmacol. Sci., 23, 412–417.CrossRefPubMedGoogle Scholar
  22. 22.
    Sproule B.A., Bazoon M., Shulman K.I., Türksen I.B., Naranjo C.A. (1997): Fuzzy logic pharmacokinetic modeling: application to lithium concentration prediction. Clin. Pharmacol. Ther., 62, 29–40.CrossRefPubMedGoogle Scholar
  23. 23.
    Kilic K., Sproule B.A., Türksen I.B., Naranjo C.A. (2002): Fuzzy system modeling in pharmacology: an improved algorithm. Fuzzy Sets Syst., 130, 253–264.CrossRefGoogle Scholar
  24. 24.
    Nestorov I. (2001): Modelling and simulation of variability and uncertainly in toxicokinetics and pharmacokinetics. Toxicol. Lett., 120, 411–420.CrossRefPubMedGoogle Scholar
  25. 25.
    Pannier A.K., Brand R.M., Jones D.D. (2003): Fuzzy modeling of skin permeability coefficients. Pharm. Res., 20, 143–148.CrossRefPubMedGoogle Scholar
  26. 26.
    Hirono S., Nakagome I., Hrano H., Matsushita Y., Yoshii F., Moriguchi I. (1994): Non-congeneric structure-pharmacokinetic property correlation studies using fuzzy adaptive least-squeres: oral bioavailability. Biol. Pharm. Bull., 17, 306–309.PubMedGoogle Scholar
  27. 27.
    Pintore M., van de Waterbeemd H., Piclin N., Chrétien J.R. (2003): Prediction of oral bioavailability by adaptive fuzzy partitioning. Eur. J. Med.Chem., 38, 427–431.CrossRefPubMedGoogle Scholar
  28. 28.
    Nauck D., Kruse R. (1999): Obtaining interpretable fuzzy classification rules from medical data. Artif. Intell. Med., 16, 149–169.CrossRefPubMedGoogle Scholar
  29. 29.
    Catto J.W., Linkens D.A., Abbod M.F., Chen M., Burton J.L., Feeley K.M., Hamdy F.C. (2003): Artificial intelligence in predicting bladder cancer outcome: a comparison of neuro-fuzzy modeling and artificial neural networks. Clin. Cancer Res., 9, 4172–4177.PubMedGoogle Scholar
  30. 30.
    Mandelbrot B.B: (1976): Fractal geometry of turbulence — Housdorff dimension, dispersion and nature of singularities of fluid motion. Cr. Acad. Sci. A Math., 282, 119–120.Google Scholar
  31. 31.
    Mandelbrot B.B. (1982): The fractal geometry of nature. W.H. Freeman and Co., San Francisco, USA.Google Scholar
  32. 32.
    Ben-Avraham D., Havlin S. (2000): Diffusion and reactions in fractals and disordered systems. Cambridge University Press, Cambridge, UK.CrossRefGoogle Scholar
  33. 33.
    Koch H.P. (1933): The concept of fractals in the pharmaceutical sciences. Pharmazie, 48, 643–659.Google Scholar
  34. 34.
    Macheras P., Argyrakis P., Polymilis C. (1996): Fractal geometry, fractal kinetics and chaos en route to biopharmaceutical sciences. Eur. J. Drug Metab. Pharmacokin., 21, 77–86.CrossRefGoogle Scholar
  35. 35.
    Weiss M. (1999): The anomalous pharmacokinetics of amiodarone explained by nonexponential tissue trapping. J. Pharmacokin. Biopharm., 27, 383–396.CrossRefGoogle Scholar
  36. 36.
    Fuite J., Marsh R., Tuszyòski J. (2002): Fractal pharmacokinetics of the drug mibefradil in the liver. Phys. Rev., 66, No. 021904, Part 1.Google Scholar
  37. 37.
    Dokoumetzidis A., Macheras P. (2003): A model for transport and dispersion in the circulatory system based on the vascular fractal tree. Ann. Biomed. Eng., 31, 284–293.CrossRefPubMedGoogle Scholar
  38. 38.
    Karalis V., Dokoumetzidis A., Macheras P. (2004): A physiologically based approach for the estimation of recirculatory parameters. J. Pharmacol. Exp. Ther., 308, 198–205.CrossRefPubMedGoogle Scholar
  39. 39.
    Macheras P., Argyrakis P. (1997): Gastrointestinal drug absorption: is it time to consider heterogeneity as well as homogeneity? Pharm. Res., 14, 842–847.CrossRefPubMedGoogle Scholar
  40. 40.
    Dokoumetzidis A., Karalis V., Ilaidis A., Macheras P. (2004): The heterogeneous course of drug transit through the body. Trends Pharmacol. Sci., 25, 140–146.CrossRefPubMedGoogle Scholar
  41. 41.
    Veng Pedersen P. (1980): model-independent method of analyzing input in linear pharmacokinetic systems having polyexponential impulse response. I: Theoretical analysis. J. Pharm. Sci., 69, 298–305.CrossRefGoogle Scholar
  42. 42.
    Veng Pedersen P. (1980): Model — independent method of analyzing input in linear pharmacokinetic systems having polyexponential impulse response. II: Numerical evaluation. J. Pharm. Sci., 69, 305–312.CrossRefGoogle Scholar
  43. 43.
    Veng Pedersen P. (1980): Novel deconvolution method for linear pharmacokinetic systems with polyexponential impulse response. J. Pharm. Sci., 69, 312–318.CrossRefGoogle Scholar
  44. 44.
    Veng Pedersen P. (1980): Novel approach to bioavailablity testing: statistical method for comparing drug input calculated by a least-squares deconvolution technique. J. Pharm. Sci., 69, 318–324.CrossRefGoogle Scholar
  45. 45.
    Popović J. (1998): Cubic spline functions and polynomials for calculation of absorption rate. Eur. J. Drug Metab. Pharmacokin., 23, 469–473.CrossRefGoogle Scholar
  46. 46.
    Schoenberg J.I. (1946): Contributions to the problem of approximation of equidistant data by analytic functions. Quart. Appl. Math., 4, Part A, 45–99, Part B 112–114.Google Scholar
  47. 47.
    Curry B.H., Schoenberg J.I. (1996): On Polya frequency functions IV: The fundamental spline functions and their limits. J. Anal. Math., 17, 71–107.CrossRefGoogle Scholar
  48. 48.
    Reinsch H.C. (1967): Smoothing by spline functions. Numerische Mathematik, 10, 177–183.CrossRefGoogle Scholar
  49. 49.
    Ahlberg HJ., Nilson N.E., Walsh L.J. (1967): The theory of splines and their applications. Academic Press, New York, San Francisco, London.Google Scholar
  50. 50.
    Greville E.N.T. (1969): Theory and applications of spline functions. Academic Press, New York, San Francisco, London.Google Scholar
  51. 51.
    Dunfield G.I., Read F.J. (1972): Determination of reaction rates by the use of cubic spline interpolation. J. Phys. Chem., 57, 2178–2183.CrossRefGoogle Scholar
  52. 52.
    Chou S.F., Sirisena R.H. (1978): Computation of optimal controls for non-linear distributed — parameter systems using multivariate spline functions. Int. J. Systems Sci., 9, 1387–1395.CrossRefGoogle Scholar
  53. 53.
    Muth M.M.A., Willsky S.A. (1978): A sequential method for spline approximation with variable knots. Int. J. Systems Sci., 9, 1055–1067.CrossRefGoogle Scholar
  54. 54.
    Wold S. (1971): Analysis of kinetic data by means of spline functions. Chem. Scripta, 1, 97–102.Google Scholar
  55. 55.
    Wold S. (1974): Spline functions in data analysis. Technometrics, 16, 1–11.CrossRefGoogle Scholar
  56. 56.
    Yeh C.K., Kwan C.K. (1978): A Comparison of numerical integrating algorithms by trapezoidal, Lagrange and spline approximation. J. Pharmacokin. Biopharm., 6, 79–98.CrossRefGoogle Scholar
  57. 57.
    Popović J., Popović V. (1985): Cubic spline functions in pharmacokinetic data analysis. Period. Biol., 87, 293–296.Google Scholar
  58. 58.
    Popović J., Popović V. (1993): Analysis of toxicokinetic data by means of spline functions. Arch. Toxicol. Kin. Xen. Metab., 1, 79–93.Google Scholar
  59. 59.
    Popović J. (1997): Polynomials vs cubic spline functions for model independent deconvolution calculations of absorption rate. Eur. J. Clin. Pharmacol., 52 (Suppl.), 446.Google Scholar
  60. 60.
    Popović J. (1998): Cubic spline function and polynomials for deconvolution calculations of absorption rate — numerical evaluation. Arch. Toxicol. Kinet. Xenobiot. Metab., 6, 99–107.Google Scholar
  61. 61.
    Anderson S., Hauck W.W. (1990): Consideration of individual bioequivalence. J. Pharmacokin. Biopharm., 18, 259–273.CrossRefGoogle Scholar
  62. 62.
    Hauck W.W., Anderson S. (1992): Types of bioequivalence and related statistical considerations. Int. J. Clin. Pharmacol. Ther. Toxicol., 30, 181–187.PubMedGoogle Scholar
  63. 63.
    Enderenyi L., Amidon G.L., Midha K.K., Skelly J.P. (1998): Individual bioequivalence: attractive in principle, difficult in practice. Pharm. Res., 15, 1321–1325.CrossRefGoogle Scholar
  64. 64.
    Williams R.L., Patnaik R.N., Chen M.L. (2000): The basis for individual bioequivalence. Eur. J. Drug. Metab. Pharmacokin., 25, 13–17.CrossRefGoogle Scholar
  65. 65.
    Ekbohm G., Melander H. (1989): The subject-by-formulation interaction as a criterion of interchangeability of drugs. Biometrics, 45, 1249–1254.CrossRefGoogle Scholar
  66. 66.
    Patnaik R., Williams R.L. (2000): Subject-by-formulation interaction in bioequivalence: conceptual and statistical issues. Pharm. Res., 17, 375–380.CrossRefPubMedGoogle Scholar
  67. 67.
    Endrenyi L., Taback N., Tothfalusi L. (2000): Properties of the estimated variance component for subject-by-formulation interaction in studies of individual bioequivalence. Stat. Med., 19, 2867–2878.CrossRefPubMedGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • J. Popović
    • 1
  1. 1.Faculty of MedicinePharmacology DepartmentNovi SadRepublic of Serbia

Personalised recommendations