Page 1 of 17Bayesian Statistics (22S:138)Final Report (12/06/2010)Zhenzhen Wang, Ahmad Abu Helwa, Xiaofeng WangA Bayesian Approach for Population PharmacokineticModeling of C-PeptideIntroductionC-peptide (CP) is a byproduct of insulin production that is produced by the pancreas. CPlevel in the blood stream is an indicator of how much insulin is being produced in the body. CPis basically a protein that is made up of amino acids. When the pancreas produces insulin, it releases CP into the bloodstream in the same way that the production of heat from burning coal or wood releases smokes into the atmosphere. The amount of CP in the blood can indicate the presence or absence of disease. For example, abnormally low amounts of C-peptide in the blood suggest the insulin production is too low (or absent) because of type I diabetes, also known as juvenile or insulin-dependent diabetes. On the other hand, abnormally high amounts of C-peptide warn of the possible presence of a tumor called an insulinoma that secretes insulin. Normal levels of C-peptide may signal that all is well. However, in a person with diabetes, a normal level of C-peptide indicates the body is making plenty of insulin but the body is just not responding properly to it and this is the hallmark of type 2 diabetes (adult insulin-resistant diabetes). C-peptide, therefore, plays a crucial diagnostic role as regards to insulin. Project ObjectiveThe objective of our project is to explore a Bayesian approach for the population pharmacokinetic analysis of C-peptide in order to estimate its pharmacokinetic disposition parameters in healthy volunteers.Page 2 of 17Study Design and Data SetThe dataset that we used is obtained from an already published study. Briefly, a bolus (average mass of about 50 000 pmol) of biosynthetic CP was intravenously administered to 14 normal humans. In order to avoid the confounding effect of endogenously secreted CP, CP pancreatic secretion was suppressed through a somatostatin infusion (started two hours before the bolus administration and thereafter continued throughout the experiment). Blood samples were collected at min 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 17, 20, 25, 30, 35, 40, 45, 50, 55, 60, 70, 80, 90, 100, 110, 120, 140, 160, and 180 and CP plasma concentration was measured. Permission to use the dataset has been obtained from the Resource Facility of Population Kinetics at the University of Washington.Data Analysis and Model DesignA two-compartment model is used to fit the data. For the Bayesian population pharmacokinetic analysis, concentration-time data will be described using a three-stage hierarchical model. Concentrations will be modeled as a nonlinear function of individual-specific pharmacokinetic parameter values in the first level of the hierarchy. Distribution of these pharmacokinetic parameters around the population mean is specified in the second level of hierarchy. Priors are given in the third level of hierarchy. Analysis has been done using PKBUGS (version 1.1) and WinBUGS (version 1.3); PKBUGS is considered as an interface of the WinBUGS.The 3-stage hierarchical modelStage I: Model for the datayij = f (θi , xij ) + ε ijPage 3 of 17εij ~ N (0, τ ) , σ2 = τ -1Where;Yij: the jth observation for the ith patient.f (θi, xij): the expected value of the data from the model.θi: a vector of individual pharmacokinetic parameter values for the ith individual.xij: a sampling time. ε ij : the residual difference between the expected value and the observed value, and N represents a normal distribution with zero mean and variance σ2. Stage II: Model between subject variabilityθi ~ Np(θ, Ω-1)where;θ: vector of mean population pharmacokinetic parameters.Ω: is the variance–covariance matrix of between subject random variability.Np: represents a p dimensional multivariate normal distribution where p is the number of parameters.Stage III: Model for the priorsτ ~ Gamma(a, b)θ ~ Np(μ.mean, Σ-1)Ω ~ Wip(R, ρ)Where;μ.mean: a vector of prior population mean values of the parameters.Page 4 of 17Σ-1: the precision matrixWi: represents a Wishart distribution with parameters R and ρ.R: is the scaled prior value of the variance covariance matrix of between-subject random variability.ρ: is the degrees of freedom of the Wishart distribution.The two-compartment model is parameterized using the following disposition parameters, in the order shown:log(CL), log(Q), log(V1), and log(V2).The relationships between these disposition parameters and the rate constants are given by:Cl = k10* V1, Q= K12* V1 = k21* V2where;Cl: Clearance from the central compartment.V1: the volume of distribution of C-peptide in the central compartment.V2: the volume of distribution of the C-peptide from the peripheral compartment.Q: the distributional clearance.K12, k21: the distributional rate constantsK10: the elimination rate constantThe two compartment model can be represented by the following schemePage 5 of 17Results and DiscussionWe ran 20,000 iterations and the model converged successfully. The history plots in figure 1 shows that the three chains for all parameters of interest are well-mixed together and they indicate that all of them are converged very well. Figure 2 shows the autocorrelation plotsfor all the parameters. Cl, Q, V1, K10, K12, and K21 autocorrelation plots dropped off quickly. However, the autocorrelation for V2 dropped off a little slower relative to other parameters but it is still acceptable. Gelman-Rubin diagnostic plots in figure 3 shows that the pooled and within chains converged to stability between 6000 and 7000 iterations, so we decide to collect the statistical summary of the parameters starting at 7001 iteration after which the ratio chain is very close to 1. Table 1 shows the summary statistics of the estimated pharmacokinetic parameters.The MC error was < 5% of the standard deviation of all parameter estimates.Figure 1. History plots of the estimated parametersCl chains 1:3iteration1 5000 10000 15000 20000 150.0 200.0 250.0 300.0 350.0Page 6 of 17Q chains 1:3iteration1 5000 10000 15000 20000 100.0 200.0 300.0 400.0 500.0V1 chains 1:3iteration1 5000 10000 15000 200002.00E+33.00E+34.00E+35.00E+3V2 chains 1:3iteration1 5000 10000 15000
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