Modeling of Pharmacokinetics of Cocaine in Human Reveals the Feasibility for Development of Enzyme Therapies for Drugs of Abuse

A promising strategy for drug abuse treatment is to accelerate the drug metabolism by administration of a drug-metabolizing enzyme. The question is how effectively an enzyme can actually prevent the drug from entering brain and producing physiological effects. In the present study, we have developed a pharmacokinetic model through a combined use of in vitro kinetic parameters and positron emission tomography data in human to examine the effects of a cocaine-metabolizing enzyme in plasma on the time course of cocaine in plasma and brain of human. Without an exogenous enzyme, cocaine half-lives in both brain and plasma are almost linearly dependent on the initial cocaine concentration in plasma. The threshold concentration of cocaine in brain required to produce physiological effects has been estimated to be 0.22±0.07 µM, and the threshold area under the cocaine concentration versus time curve (AUC) value in brain (denoted by AUC2∞) required to produce physiological effects has been estimated to be 7.9±2.7 µM·min. It has been demonstrated that administration of a cocaine hydrolase/esterase (CocH/CocE) can considerably decrease the cocaine half-lives in both brain and plasma, the peak cocaine concentration in brain, and the AUC2∞. The estimated maximum cocaine plasma concentration which a given concentration of drug-metabolizing enzyme can effectively prevent from entering brain and producing physiological effects can be used to guide future preclinical/clinical studies on cocaine-metabolizing enzymes. Understanding of drug-metabolizing enzymes is key to the science of pharmacokinetics. The general insights into the effects of a drug-metabolizing enzyme on drug kinetics in human should be valuable also in future development of enzyme therapies for other drugs of abuse.


Analysis of the structural identifiability of the model
The system-experiment model is By using Eqs.(S1) to (S4) and the initial conditions, we have in Eqs.(S8) to (S11) are all observable parameters. From Eq.(8), we obtain Substitution of Eq.(S12) into Eq.(S9) gives (S13) Combination of Eqs.(S10) and (S11) gives Substitution of Eqs.(S12) and (S14) into Eq.(S15) gives Substitution of Eq.(S16) into Eq.(S13) gives Equation (S17) can be rewritten in the following standard form: in which coefficients A, B, and C can be determined by known constants: As well known, Eq.(S18) mathematically can have two solutions, denoted as 1 s and 2 s here for convenience:  Table S1, although their exact values are dependent on the specific functions used for the data fitting. Nevertheless, our numerical tests always qualitatively indicate that A < 0, B < 0, and C > 0, which gives (S24) When pb K is known, Eq.(S12) provides p V uniquely, Eq.(S14) provides b V uniquely, and Eq.(S16) provides bp K uniquely.
In summary, although the model mathematically may have two solutions associated with 1 s and 2 s , there is only one physically meaningful solution for the values of parameters p V , b V , pb K , and bp K used in the model. So, under the condition that p V , b V , pb K , and bp K all must be positive values, all unknown parameters of the model are uniquely identifiable and, therefore, the model is structurally identifiable.

Impacts of the catalytic parameters of enzyme on cocaine concentration in brain
With all of the model parameters calibrated, we can further discuss possible impacts of the catalytic parameters (k cat and K M ) of enzyme on cocaine concentration in brain, i.e. ) , when a typical addiction dose of cocaine is administered. It has been known that for a typical addiction dose of cocaine, the peak cocaine concentration in plasma is expected to be about 1 to 5 µM 1,2 Hence, the model was first used to determine the curves of ) Depicted in Figure S2 are the predicted curves of cocaine uptake and clearance in brain when  Figure S2). As expected, when the enzyme activity increases, the enzymatic hydrolysis of cocaine is always faster so that the ) ( 2 t x value (cocaine concentration in brain) in the presence of a more active enzyme is always smaller than the corresponding ) ( 2 t x value in the presence of only endogenous BChE at any given time (t) after cocaine is administered. The difference between the ) values corresponding to the two enzymes becomes larger and larger with the time (t), as seen in Figure S2. Hence, when the enzyme activity increases, the cocaine half-life in brain (i.e. t b1/2 ), AUC2 ∞ , and the peak concentration of cocaine in brain all decrease. Due to the decrease of the cocaine peak concentration and half-life in brain, the time to reach the decreased cocaine peak concentration in brain also decreases with increasing the catalytic activity of the enzyme, particularly in increasing k cat , as seen in Tables S2 and S3.
As seen in Table S2 and Figure S2 (upper panel), for a given initial cocaine concentration in plasma, when K M decreases from 4.5 µM to 0.45 µM, the t b1/2 and AUC2 ∞ values in brain decrease significantly, while the peak concentration of cocaine in brain also decreases slightly.
When K M decreases further from 0.45 µM, the t b1/2 , AUC2 ∞ , and peak concentration of cocaine in brain decrease little. This is because when K M is sufficiently small, the enzyme has already been saturated and the catalytic reaction has reached the maximum rate such that further decrease in K M no longer can increase the reaction rate.
As seen in Table S3 and Figure S2 (lower panel), for a given initial cocaine concentration in plasma, when k cat increases for each order of magnitude, the AUC2 ∞ value in brain decreases for about a order of magnitude while the t b1/2 and peak concentration of cocaine in brain also decrease significantly. Thus, so long as the k cat value is sufficiently large, the t b1/2 and AUC2 ∞ values and the peak concentration of cocaine in brain all can be neglected.
In addition, because V max  k cat ·[E], the same impact of increasing k cat may also be achieved

Additional information about the evaluation of available high-activity cocaine-metabolizing enzymes for their effects on the cocaine concentration in brain
As AUC2 ∞ and cocaine peak concentration in brain are the primary determinants of the overall cocaine reward/stimulation effects in brain, cocaine-metabolizing enzymes CocHs and CocE all can considerably decrease the overall cocaine reward/stimulation effects in brain. To simplify further discussion below, we will first focus on the AUC2 ∞ values. As the increase in the initial cocaine concentration in plasma increases both the peak concentration and half-life of cocaine in brain, AUC2 ∞ is a quadratic polynomial function of faster than the simpler linear correction, as the data in Table 1 show.
As seen in  has a larger catalytic rate constant (k cat , which is the dominant factor affecting the enzyme activity in the condition of a very high substrate concentration), whereas CocH2 (k cat = 1,730 min -1 and K M = 1.1 µM) has a higher catalytic efficiency (k cat /K M , which is the dominant factor affecting the enzyme activity in the condition of a low substrate concentration).Therefore, it is significant for improving the potency of the enzyme to decrease K M when cocaine concentration is low, whereas increasing k cat is always the most effective.   Table S3. Effect of k cat on cocaine peak concentration (Peak in μM) and the area under curve (AUC2 ∞ in μM·min) in the brain when K M = 4.5 μМ. Physiological concentration (0.035 μM) of the enzyme in plasma is used in the modeling.  Figure S1. Plots of cocaine half-life in brain (t b1/2 ) versus the initial cocaine concentration in plasma ( ) 0 (