計畫編號：NSC91-2118-M032-001研究期間：200208~200307研究經費：434,000[[sponsorship]]行政院國家科學委員

陳主智

計畫編號：NSC91-2118-M032-001研究期間：200208~200307研究經費：434,000行政院國家科學委員

Tamkang University Institutional Repository

????????????????  ????   ????????????????????   ?????????? ?????NSC91-2118-M-032-001- ?????91?08?01??92?07?31? ????????????     ?????????      ?????????  ?????????????     ? ? ? ? 92?11?27? ???? ??????????physiologically-based pharmaco-kinetic model, or PBPK model??????????????????????????????????????????????????????? PBPK?????????????????????????????????????????????????????????????????????????????? Bois????????????????????(MCMC)???? PBPK?????????????????????????????????Bois, Jackson, Pekari, & Smith, 1996;Gelman, Bois, & Jiang, 1996????????????MCMC????? PBPK?????????????????????????????????Bois ?????????????(measurement error)?????????????????????????????????????????????????????(process error)???????????????????????????????????????????? Bois ????????????????????????? PBPK model ????????????????????????????????????? state-space model??? Kalman filter?????????????????????????????????MCMC????????????????????????  ?????????????????????????????????????         ????       Physiologically-based pharmacokinetic models (PBPK models) have been playing an important role in assessing the dynamic relationship between internal dose and environmental exposure to hazardous chemicals. However, numerous model parameters with often very limited study subjects make statistical analysis difficult to justify. Bois et al (1996) and Gelman, Bois, and Jiang (1996) successfully applied the MCMC simulation technique in PBPK model parameter estimation problem. However, their approach attributed the random variations in internal concentrations to measurement error only, which may not be the case in real life. Due to the complexity of body metabolic mechanism, stochastic variations (process error) other than the deterministic model should be taken into account. In addition, this source of variation may surpass measurement error owing to advances in laboratory techniques nowadays. This report considers both process error and measurement error simultaneously. First, by adding random variations to the deterministic model and simplified to linear case, we obtain system of stochastic differential equations. Integrating through Ito integral, the resultant equations of internal concentrations become familiar state-space model with measurement error. The Kalman filter technique is then employed to obtain the posterior distributions of concentrations and parameters given concentrations. The MCMC technique is finally applied to generate the required samples after convergence.         Key words: Markov Chain Monte Carlo, process error, measurement error,  environmental exposure concentration, stochastic differential  equation       ???? 1. ?? The assessment of biological concentrations of a person exposed to hazardous chemicals helps industrial hygienists in determining how much risk one may develop certain diseases more precisely than environmental exposure monitoring. To find the relationship between environmental exposure and the corresponding internal concentrations, physiologically based pharmacokinetic (PBPK) models that describe the dynamic internal concentrations within body tissues have become popular in recent years (Leung, 1992; Thomas, et al, 1996; Reitz, et al, 1996). Specifically, PBPK models not only helps industrial hygienists to propose biological exposure indexes (BEIs) in more biological media and at different sampling times, it can also be used to simulate a wide variety of exposure conditions (Leung, 1992).       By dividing the whole body of a person into 4 (or 5) different groups of tissues: fat, muscle (slowly perfused), richly perfused, liver, and assuming that all metabolism takes place in the liver, dynamic concentrations within each compartment at time t can be determined by solving the mass-balance equations for each of the compartments. Specifically, let the fat, muscle, richly perfused, and liver compartments be denoted by 4,3,2,1=k , respectively. The concentrations for the non-liver compartments satisfy the equation                 [ ] 3,2,1,/)()()( =−= ktCtCQtCdtdV kkartkkk λ ,           (1) and the concentration for the liver compartment satisfies the equation        [ ] 44444max44444 /)()()(/)()()( λλ VtCKtCKtCVtCtCQtCdtdV fmart −+−−= ,    (2) where               ( ) ( )balvcinhalvvencart QQtCQtCQtC λ//)()()( ++= ,            (3) and                       ( )∑==41//)()(kckkkven QtCQtC λ                    (4) are the concentrations in the arterial blood and venous blood at time t, respectively, )(tCinh  is the chemical concentration in inhaled air, )(tCk  is the chemical concentration in the k -th compartment at time t, alvQ  is the alveolar ventilation, kQ , kV , and kλ  are the blood flow rate, volume, and tissue-blood partition coefficient of the k -th compartment, respectively. Also, bλ  is blood-air partition coefficient, cQ  is the cardiac output, maxV  is the maximum rate of metabolism, mK  is the Michaelis-Menten kinetic constant, and fK  is the first order metabolic constant.      To fit in the unknown parameters of the PBPK model for internal concentrations in practice, researchers of the field often obtain the partition coefficients kλ  and bλ  from literature, kV  and alvQ  are proportional to individual body weight, and kQ  are proportional to alvQ . And the metabolic constants maxV , mK , and fK  are obtained by scaling allometrically the in vivo value determined in laboratory animals with gas uptake techniques (see, e.g., Leung, 1992). Variations of the internal concentrations are then obtained by assigning statistical distributions to the parameters (Thomas, et al, 1996; Wu, et al, 2002). However, since these unknown model parameters are different for each individual, and the best fitting values are only empirical based on often very limited measurements, the results may not be legitimate for statistical inferences.  2. ???? In this report, to estimate the PBPK model parameters more adequately, we consider the stochastic variations of internal concentrations together with measurement error. The most important direct impact would be statistical inferences based on the estimated model parameters, especially body burden calculated from accumulated predicted compartment (such as liver or fatty tissue) concentrations. Another important application is retrospectively from sample internal concentrations to assess environmental exposure concentration, which is difficult to assess in most cases. However, the latter case is of direct concern to public population. The latter serves as two major goals in our follow-up projects.   3. ???? To estimate the PBPK model parameters with formal statistical approach, Bois et al (1996) and Gelman, Bois, and Jiang (1996) employed the Markov Chain Monte Carlo (MCMC) simulation technique for population modeling with informative prior distributions for the parameters. Given the observations and the prior information, random samples of the parameters are generated from their posterior distributions. These values are then fitted into the PBPK model to obtain the internal concentrations at time t using the Runge-Kutta numerical integration method. Differences of the expectations and the observations are attributed to measurement error. Updated observations can be obtained from the posterior distribution of the measurement error, from which new set of parameters are again generated. The procedure is repeated until large enough samples are generated from their stationary distributions after convergence.       Though the approach using MCMC simulations successfully reflects population parameters distributions upon convergence. Due to complexity of human body metabolic mechanism, the deterministic models represented by the mass-balance equations (1) and (2) are subject to stochastic variations, which should also be taken into account. That is, successive observations in time may depart from the trajectory predicted by the deterministic model. And special attention should be dealt with this source of error other than measurement error as well.  4. ???? In this report, we approximate equation (2) into a linear form, and rewrite equations (1) and (2) as stochastic differential equations by adding stochastic variations. The solutions together with equations (3) and (4) are then transformed into a dynamic linear state-space model, the established Bayesian methodology using the Kalman filter is then employed for the unobserved compartment concentrations. Together with the MCMC simulations, samples of the unknown population parameters are then generated for statistical inferences.  5. ????? The methodology of the proposed approach has already been completed. Numerical results such as real example analysis and simulations are working in progress. It is expected that the complete manuscript for submission for publication is no later than the end of February. The results are expected to be the mainstream in PBPK model statistical research. Not only the results correct Bois et al’s main framework, which is oversimplified. But it also successfully brings in state-space model structure naturally together with MCMC simulation technique. The major breakthrough of the work was completed while the project host was financially supported for a 3-month study abroad. He deeply thanks the NSC to have this opportunity to be able to work this problem in concentration within a period of time.  ???????????????????????? ???? Anderson, T. W. (1984) An Introduction to Multivariate Statistical Analysis, 2nd ed.,  John Wiley & Sons, Inc. Bois, F. Y., Jackson, E. T., Pekari, K., and Smith, M. T. (1996) Population  toxicokinetics of Benzene. Environmental Health Perspectives, 104, supplement 6, 1405-1411. Carter, C. K., and Kohn, R. (1994) On Gibbs sampling for state space models.  Biometrika, 81, 541-553. Frühwirth-Schnatter, S. (1994) Data augmentation and dynamic linear models.  Journal of Time Series Analysis, 15, 183-202. Gamerman, D. (1997) Markov Chain Monte Carlo, Chapman & Hall, London. Gelman, A., Bois, F. Y., and Jiang, J. (1996) Physiological pharmacokinetic analysis  using population modeling and informative prior distributions. Journal of the  American Statistical Association, 91(436), 1400-1412. Klebaner, F. C. (1998) Introduction to Stochastic Calculus with Applications, Imperial  College Press. Leung, H.-W. (1992) Use of physiologically based pharmacokinetic models to  establish biological exposure indexes. American Industrial Hygiene Association  Journal, 53(6), 369-374. Reitz, R. H., Gargas, M. E., Andersen, M. E., Provan, W. M., and Green, T. L. (1996)    Predicting cancer risk from vinyl chloride exposure with a physiologically based    pharmacokinetic model. Toxicology and Applied Pharmacology, 137, 253-267.  Rubin, D. B. (1987) A noniterative sampling/important resampling alternative to the  data augmentation algorithm for creating a few imputations when fractions of  missing information are modest: the SIR algorithm, comment to a paper by Tanner  and Wong. Journal of the American Statistical Association, 82, 543-546. Thomas, R. S., Bigelow, P. L., Keefe, T. J., and Yang, R. S. H. (1996) Variability in  biological exposure indices using physiologically based pharmacokinetic  modeling and Monte Carlo simulation. American Industrial Hygiene Association  Journal, 57, 23-32. West, M., and Harrison, J. (1999) Bayesian Forecasting and Dynamic Models, 2nd ed.,  Springer-Verlag, New York, Inc.  

Statistical Analysis on Environmental Risk Assessment Using Pharmacokinetic Physiologically-based Model

http://tkuir.lib.tku.edu.tw:8080/dspace/bitstream/987654321/5802/2/912118M032001.pdf

[[alternative]]Statistical Analysis on Environmental Risk Assessment Using Pharmacokinetic Physiologically-based Model

Abstract

Similar works

Full text

Available Versions

Tamkang University Institutional Repository