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A two-compartment in vivo pharmacokinetic model (Figure 1) with Michaelis-Menten elimination in plasma was built to describe the cocaine concentration-time profile in the human body and characterize the relationship between the profile and the observed pharmacological activity of cocaine reported by Fowler et al. [37] Assuming that all compartments are homogeneous, the model is based on the following differential equations:(1)(2)where and represent the cocaine concentrations in plasma (compartment 1) and brain (compartment 2), respectively. KM is the Michaelis-Menten constant of the enzyme for cocaine. Vmax is the maximum rate of the enzymatic reaction when the enzyme is saturated by the substrate (cocaine): Vmax≡kcat·[E] in which kcat is the catalytic rate constant and [E] is the concentration of the enzyme in plasma. Kpb is the constant for cocaine diffusion from plasma compartment to brain compartment, and Kbp is the constant for cocaine diffusion from brain compartment to plasma compartment. Vp and Vb refer to the effective volumes of compartments 1 (plasma) and 2 (brain), respectively. Standard Michaelis-Menten equation of BChE-catalyzed cocaine hydrolysis was used for a couple of reasons: (1) BChE-catalyzed hydrolysis of cocaine is the dominant cocaine-metabolizing pathway in plasma and the other cocaine-metabolizing pathways may be neglected [10]; (2) the products of BChE-catalyzed hydrolysis of cocaine do not significantly inhibit BChE [38]. Further, in the presence of an exogenous cocaine-metabolizing enzyme with a >1,000-fold improved catalytic efficiency against cocaine, the catalytic activity of the endogenous enzyme is negligible.
Figure data removed from full text. Figure identifier and caption: 10.1371/journal.pcbi.1002610.g001 A two-compartment model.The model has one compartment representing brain (striatum) tissue, which exchanges radiotracer with plasma compartment (volume Vp) with two diffusion constants Kpb and Kbp. In the plasma compartment, cocaine molecules experience an enzymatic Michaelis-Menten elimination, where KM is the Michaelis-Menten constant and Vmax is the maximum velocity of cocaine conversion to the metabolites.
Concerning the structural identifiability [39], [40], [41] of the model, there are two model outputs, denoted by and for convenience, and we have and in which as discussed below. Vb, Vp, Kpb, and Kbp are unknown parameters/variables, whereas c, Vmax, and KM are the known constants (see below). It should be noted that Vb, Vp, Kpb, and Kbp all must be positive values in order to be physically meaningful. An analysis using the Taylor series expansion method revealed that there is only one physically meaningful solution for the values of parameters , , , and used in the model. So, under the condition that , , , and all must be positive values, all unknown parameters of the model associated with Eqs. (1) and (2) are uniquely identifiable and, therefore, the model is structurally identifiable.
Model fitting and parameter estimation: PET data were selected to fit the model as PET is superior over other techniques, such as microdialysis, that have been used to determine the cocaine distribution and its time course in living subject (human). In addition, PET imaging analysis can reveal the variation of cocaine concentration in brain starting from seconds after cocaine is injected into a living subject. Therefore, the theoretical model was fitted to the experimentally observed data reported by Fowler et al. [37] These PET data were chosen based on several reasons. First, the PET data were obtained for human subjects, which is consistent with kcat and KM of human BChE that we have been studying in our lab [28], [29], [30], [31], [32]. Second, the cocaine concentrations in plasma at various time points were also measured for the same subject(s) along with the PET measurement. In addition, the time course of cocaine appearing in the striatum of brain [37] is consistent with the time course of the mean subjective “high” in human subjects reported by Cook et al., [42] a well-documented euphoria experienced after the intravenous (i.v.) administration of cocaine. The experimental data – uptake and clearance of the [11C]cocaine radioactivity in the human corpus striatum over a 35-minute period after the injection of [11C]cocaine depicted in Figure 2 of the previous report [37], and the relative concentration of [11C]cocaine in human plasma depicted in Figure 4A of the previous report [37] – were digitized for fitting with the results obtained from the theoretical pharmacokinetic simulation. Although the experimental data for uptake and clearance of the [11C]cocaine radioactivity in the human cerebellum were also available, we chose not to directly model these data in our finally generated model because cocaine always had the highest concentration in the striatum [37], [43]. To include the uptake and clearance of radioactivity in the human cerebellum, one more compartment and three additional parameters would need to be included in the pharmacokinetic model. The extra parameters would add some additional flexibility (and also uncertainty) to the model during the calibration of the model parameters. The difference for the distribution of (−)-cocaine in brain other than striatum can be corrected by adjusting the volume parameter Vb during the model fitting.
Figure data removed from full text. Figure identifier and caption: 10.1371/journal.pcbi.1002610.g002 Modeling of available experimental data (dots) for the concentrations of cocaine in brain (black) and in plasma (red) of human subject.The experimental data came from reference [37]. Our pharmacokinetic modeling and simulation were performed by use of ADAPT II program [44] and a MATLAB program developed in our own lab for numerical solution of differential equations defined in Eqs. (1) and (2) [45], [46], [47]. Curves of the cocaine concentrations versus time in both compartments generated by numerical integration were evaluated for the closest to the observed PET data. The fitting was judged by using the root-mean-squared error (RMSE). All points were given the equal weight during the least-squares fitting process.
Area under the curve (AUC) analysis: The cocaine exposure in brain can be characterized by the area under the cocaine concentration versus time curve (AUC) in brain. For convenience, the AUC values in plasma and brain within time t after the cocaine administration are denoted by AUC1t and AUC2t, respectively, and can be evaluated via(3)in which i = 1 and 2 that refer to plasma and brain, respectively. In practice, the numerical integration with Eq. (3) was carried out by using the well-known linear trapezoidal rule. AUCit = AUCi∞ when t = ∞.
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