Development of a Compartment Model to Study the Pharmacokinetics of Medical THC after Oral Administration

: The therapeutic potential of delta9-tetrahydrocannabinol (THC), a primary cannabinoid in the cannabis plant, has led to its development into oral medical products for treating various conditions. However, THC, being a psychoactive substance, can lead to addiction if taken in inappropriate amounts. Thus, studying the pharmacokinetics of THC is crucial for understanding how the drug behaves in the body after administration. This study aims to develop a multi-compartmental model to investigate the pharmacokinetics of medical THC and its metabolites after oral administration. Using the law of mass action, the model was converted into ordinary differential equations (ODEs) to describe the rate of concentration changes of THC and its metabolites in each compartment. The nonstandard finite difference (NSFD) method was then applied to construct numerical solution schemes, which were implemented in MATLAB along with estimated pharmacokinetic rate constants. The results demonstrate that the simulation curves depicting the plasma concentration–time profiles of THC and 11-hydroxy-THC (THC-OH) closely resemble actual data samples, indicating the model’s accuracy. Moreover, the model predicts the pharmacokinetics of THC and its metabolites in various tissues. Consequently, this model serves as a valuable tool for enhancing our understanding of the pharmacokinetics of THC and its metabolites, guiding dosage adjustments, and determining administration durations for oral medical THC.


Introduction
Delta9-tetrahydrocannabinol (THC), one of the cannabinoids found in the cannabis plant, exhibits several clinically useful pharmacological characteristics [1].As a result, THC is extracted and developed into a medical product, providing an alternative treatment for patients who do not respond to conventional therapies.This includes individuals experiencing chemotherapy-induced nausea and vomiting, medication-resistant epilepsy, anorexia in AIDS patients, and those seeking to enhance their quality of life in palliative care.Numerous studies have demonstrated the positive effects of medical THC in treating various conditions [2,3].However, THC, as a psychoactive compound, can lead to addiction if used inappropriately.
Various forms of medical THC products are currently available to address diverse treatment needs, including oral, inhalation, smoking, and injection methods.Among these, oral dosage forms such as capsules or tablets are widely preferred due to their convenience and the ability to accurately determine the administration dose.Upon oral administration, THC is absorbed through the gastrointestinal tract, including the stomach and small intestine, with a bioavailability ranging from 0.1% to 0.2% [4].Subsequently, it undergoes first-pass metabolism in the liver after entering via the hepatic portal vein.In the liver, enzymes such as CYP2C and CYP3A metabolize THC into its primary psychoactive metabolite, 11-hydroxy-THC (THC-OH), and a secondary metabolite, 11-nor-9carboxy-THC (THCCOOH) [5].THC-OH serves as the primary psychoactive metabolite, whereas THCCOOH is non-psychoactive.The remaining THC and its metabolites enter the bloodstream after passing through the heart, with both THC and THC-OH simultaneously reaching the brain.Approximately 20% of THC and other cannabinoids are excreted in urine, while over 65% are eliminated in feces within five days [6].
Compartmental modeling is another method for studying pharmacokinetics, encompassing processes like absorption, distribution, metabolism, and excretion.It is a mathematical technique that divides the body into compartments to represent its entirety.This approach helps overcome challenges and reduces costs associated with clinical studies.Previous research has proposed compartment models to describe THC's complex pharmacokinetics.For example, in 2015, Jules et al. [7] introduced a four-compartment pharmacokinetic model for THC after oral, intravenous, and pulmonary dosing.In 2017, Rakesh [8] explored THC and its active metabolite's pharmacokinetics and pharmacodynamics using effect compartment modeling-based approaches.In 2020, Cristina et al. [9] developed a population multi-compartment pharmacokinetic model for THC and its active and inactive metabolites following controlled smoked cannabis delivery.Additionally, in 2023, our team created a compartment model to investigate THC and its metabolites' pharmacokinetics after smoking [10].
Typically, analytical solutions suffice for compartment models with one or two compartments.However, when dealing with three or more compartments, we encounter a sizable system of differential equations, some of which yield non-real solutions.These analytical solutions were detailed in our previous work [10].However, leveraging non-real solutions for simulations proves challenging.Hence, numerical methods are essential to mitigate the complexity of solving analytical solutions for multi-compartment models.One such numerical method used to solve the system of ordinary differential equations (ODEs) in compartment pharmacokinetic models is the nonstandard finite difference (NSFD) method for constructing numerical solution schemes.Originating from the standard finite difference (SFD) method developed by Mickens [11,12], the NSFD method has been shown in several previous studies to yield more stable numerical results and better consistency with differential equations compared with the original SFD method, irrespective of the chosen step size [13][14][15][16].
As discussed earlier, THC possesses psychoactive properties and carries the risk of addiction.Therefore, administering medical THC requires caution, as the precise initial dosage for different patient groups remains undetermined.It is generally advisable to commence treatment with a small dosage, such as once daily, and gradually increase it if the patient tolerates it well.Consequently, having a tool to investigate and explain the pharmacokinetics of medical THC becomes crucial, as it enhances our comprehension and aids in determining the dosage and administration duration, thus optimizing therapeutic efficacy and safety.In this study, we introduce a compartmental model for the pharmacokinetics of medical THC post-oral administration.The aim of this study was to develop a comprehensive model for thoroughly investigating the pharmacokinetics of medical THC and its metabolites in the body after oral administration.The model was converted into ordinary differential equations (ODEs) to portray the rate of concentration change of THC and its metabolites across bodily compartments, employing the law of mass action.For simulating the results, we employed the NSFD method to devise numerical solution schemes.These schemes were then implemented in MATLAB, incorporating estimated pharmacokinetic rate constants for THC and its metabolites, facilitating the generation of simulation curves for comparison with actual concentration samples.

Methodology
In this section, we establish a multi-compartment model to elucidate the pharmacokinetics of THC post-oral administration.The compartmental model was created to represent the concentration of THC and its metabolites in different parts of the body.The model was then converted into ordinary differential equations (ODEs) to describe the rate of changes in THC and its metabolite concentrations within each compartment, employing the law of mass action.Subsequently, we employed the nonstandard finite difference method (NSFD) to transform the ODEs into numerical solution schemes.Additionally, we provided the estimated pharmacokinetic rate constants used to simulate the results, including examples of actual plasma concentrations of THC and THC-OH that are compared with our simulation results.

Compartmental Modeling
We divided the compartment model representing the body into four phases: the first phase involved the absorption compartment, representing the gastrointestinal tract, to elucidate THC absorption; the second phase addressed the pharmacokinetics of THC; the third phase focused on the pharmacokinetics of THC-OH; and the fourth phase examined the pharmacokinetics of THCCOOH in plasma.In this model, we considered the pharmacokinetics of THC and THC-OH in both blood and tissues by dividing the body into three physiologically significant compartments: the central compartment, representing the blood (plasma) or systemic circulation; the rapidly equilibrating tissue compartment, representing organs like the liver, lungs, and kidneys; and the slowly equilibrating tissue compartment, representing tissues such as muscle, fat, and bone.Additionally, we created effect compartments for THC and THC-OH to describe their concentrations at the sites of effect.This representation is illustrated in Figure 1.We defined the variables and rate constants for the absorption, distribution, metabolism, and excretion of THC and its metabolites in each compartment.The concentrations were represented as follows: C a (t) for THC in the absorption compartment (gastrointestinal tract), C 1 (t) for THC in the central compartment (plasma), C 2 (t) for THC in the rapidly equilibrating tissue compartment, C 3 (t) for THC in the slowly equilibrating tissue compartment, and C e1 (t) for THC in the effect compartment.C 4 (t) for THC-OH in the central compartment, C 5 (t) for THC-OH in the rapidly equilibrating tissue compartment, C 6 (t) for THC-OH in the slowly equilibrating tissue compartment, C e2 (t) for THC-OH in the effect compartment, and C 7 (t) for THCCOOH in the central compartment.Also, we denoted the absorption rate constant of THC as k a and the distribution rate constants of THC in each compartment as k 12 , k 21 , k 13 , k 31 , and k 1e .The metabolic rate constants of THC to THC-OH and THCCOOH were represented by k 14 and k 17 , respectively.For THC-OH, the distribution rate constants in each compartment were k 45 , k 54 , k 46 , k 64 , and k 4e , while k 47 represented the metabolic rate constant of THC-OH to THCCOOH.The elimination rate constants of THC-OH and THCCOOH from the central compartment were k 40 and k 70 , respectively.Additionally, k e10 and k e20 denoted the elimination rate constants of THC and THC-OH from the effect compartment.
By applying the law of mass action to the compartment model illustrated in Figure 1, we derived a set of ordinary differential equations representing the rate of change of THC and its metabolite concentrations in each compartment, as follows: where C 0 is the initial concentration of THC (ng/mL).Next, we applied the NSFD method to Equation (1) to construct numerical solution schemes, thus mitigating the challenge of solving analytical solutions for large systems of equations.

NSFD Schemes
The nonstandard finite difference method (NSFD), a numerical approach, has been utilized in continuous models.It was developed from the standard finite difference method (SFD) by Mickens [11,12].It serves as an alternative method to ensure the stability of numerical solutions, regardless of the chosen step size.The NSFD procedures are based on the following rules: (1) The following represents the discrete first derivative: where ψ and ϕ depends on step-size △t = h and satisfy the conditions: The ψ and ϕ can have different functions from one another.While specific forms for a given equation can be readily found, there are currently no general guidelines for choosing the functions ψ(h) and ϕ(h).Frequently employed functional representations for ψ(h) and ϕ(h) are where λ is some parameter appearing in the differential equations.
(2) Both linear and nonlinear terms may require a nonlocal representation on the discrete computational; for example: C =⇒ C k+1 .
Preliminary rules exist for constructing denominator functions for a system of coupled first-order ordinary differential equations: (1) Form an initial, finite difference model by replacing all first-derivatives by discrete forward-Euler terms: (2) For a particular discrete equation, in general, its dependent variable will occur linearly in its evaluation at the (k + 1)-th time step.Solve for this dependent variable at the (k + 1)-th time step, in terms of all other dependent variables evaluated at the k-th time step.(3) The denominator function can be chosen as follows If a factor in each discrete equation has an expression of the form (1 + λh), where λ is made up of one or more parameters that occur in the original differential equations.(4) In the discrete finite-difference schemes constructed in point 1, replace h by the appropriate ϕ(h, λ).(5) For the case where λ = 0, use the denominator function ϕ(h) = h.
Hence, from the ordinary differential equations in Equation ( 1), we construct the NSFD schemes for the first derivative and the nonlocal representations for other terms, thereby transforming Equation (1) into: It can be observed that if we already possess the scheme at the (k + 1)-th time step from the preceding equation, we will construct the scheme with the (k + 1)-th time step instead of the k-th time step in the subsequent equation.This approach ensures a more continuous, accurate, and efficient calculation by leveraging the latest computed values.
From Equation (2), for solving C a,k+1 , we get Given the occurrence of 1 + k a h, according to number 3 in the preliminary rules, the denominator function should be chosen to have the following form: Solving similarly for the other C i,k+1 (i = 1, 2, ..., 7, e1, e2), we obtain the denominator functions as follows: Therefore, based on the preliminary rules for constructing the denominator function, we obtain the NSFD schemes of Equation (1) as follows: By rearranging Equation ( 4), we obtain the solution schemes as follows: where the various ϕ depend on step-size (h) as shown in Equation (3).

Pharmacokinetic Parameters
To simulate the results, we needed to know the pharmacokinetic rate constants of THC and its metabolites.In this study, we derived these rate constants from our previous research on the pharmacokinetics of THC and its metabolites after smoking [10], adjusting them to obtain simulation results as close to the actual data as possible.Therefore, we obtained the estimated pharmacokinetic rate constants for THC and THC-OH, as shown in Table 1.
Table 1.The estimated pharmacokinetic rate constants of THC and THC-OH after 10 mg oral administration [10].

Actual Data
In this study, we compared our simulation results with actual data samples obtained from a clinical study conducted by Guy and Robson [17].The actual data consisted of plasma concentrations of THC and THC-OH after the oral administration of 10 mg of THC.Experimental data from the clinical study are presented in Table 2.We will compare the actual data sample, as presented in Table 2, with the simulated results of THC and THC-OH concentrations, which will be discussed in the conclusion and discussion section.

Simulation Results
For the simulations, we utilized the NSFD scheme specified in System (5) and implemented it in the MATLAB program.We employed the estimated pharmacokinetic rate constants from Table 2 and selected a step size (h) of 0.05.Consequently, we obtained simulated concentrations of THC and THC-OH in plasma and other tissues following the oral administration of 10 mg of THC, replicating the experimental conditions of the clinical studies conducted by Guy and Robson [17].These results are presented in Figures 2 and 3 for THC and THC-OH, respectively.
In Figure 2, the plasma THC concentration, denoted as C 1 (t), experienced a rapid increase, peaking at 4.52 ng/mL between 50 and 55 min.Subsequently, it declined swiftly within the 55-360 min interval, followed by a gradual decrease over time.In the absorption compartment (C a (t)), the THC concentration decreased rapidly from an initial value of 24.85 ng/mL within the 0-200 min interval, gradually reaching 0 ng/mL after 200 min.Concurrently, the THC concentration in other tissues varied, contingent upon the distribution rate constants.The maximum concentrations of THC in the rapidly equilibrating tissue, slowly equilibrating tissue, and the effect site were 3.36 ng/mL at 50-55 min, 1.68 ng/mL at 50-55 min, and 0.043 ng/mL at 160-235 min, respectively.Following peak concentrations, the levels gradually declined to 0 ng/mL over time.The simulated THC concentration following oral administration of 10 mg in each compartment is illustrated in Figure 4.
In Figure 3, the plasma concentration of THC-OH (C 4 (t)) surged rapidly, peaking at 6.76 ng/mL at 105 min.Concurrently, THC-OH concentration in other tissues reached their respective maximum levels before gradually declining over time.The peak concentrations of THC-OH in the rapidly equilibrating tissue, slowly equilibrating tissue, and the effect site were 0.042 ng/mL at 105 min, 0.022 ng/mL at 105 min, and 1.60 ng/mL at 250-260 min, respectively.The simulated concentration of THC-OH following oral administration of 10 mg THC in each compartment is depicted in Figure 5.

Conclusions and Discussion
In this study, we developed a compartment model to describe the pharmacokinetics of delta9-tetrahydrocannabinol (THC) and its metabolites following oral administration.The compartmental model was converted into ordinary differential equations (ODEs), which represent the rate of change in the concentration of THC and its metabolites in each compartment using the law of mass action.We then applied the nonstandard finite difference method (NSFD) to create numerical solution schemes.These schemes were implemented in a MATLAB program, incorporating the estimated pharmacokinetic rate constants to generate the simulation results.The rate constants were adjusted based on our previous study on the pharmacokinetics of THC and its metabolites following smoking [10].Consequently, we obtained the simulation results of THC and THC-OH concentrations after the oral administration of 10 mg THC, under the clinical study conditions by Guy and Robson [17].The NSFD scheme, as depicted in Equation ( 5), was used with a chosen step size (h) of 0.05, as shown in Figures 4 and 5.An explanation of the behavior of these result graphs is provided in the Results section.
For the simulated THC concentration in plasma, C 1 (t), when compared to the actual data from the study by Guy and Robson, the simulated plasma THC concentration curves closely resembled the actual data, as shown in Figure 6a, with an R-squared value of 0.8167 and a root mean square error (RMSE) of 0.6626.Similarly, for the simulated THC-OH concentration in plasma, C 4 (t), the simulated plasma THC-OH concentration curve closely matched the actual data, despite a discrepancy in the peak time, as shown in Figure 6b.The R-squared value was 1.0189 and the RMSE was 0.7848.This discrepancy may be related to the estimated pharmacokinetic parameters.Furthermore, when comparing the simulation results with another clinical study that administered the same initial dosage of 10 mg oral medical THC, such as the study by Nadulski et al. [18], we observed differences in the mean maximum concentrations of THC and THC-OH in plasma (C 1 and C 4 , respectively).In Nadulski's study, the mean maximum concentration of THC was 3.20 ng/mL at a mean maximum time of 63 min, while the mean maximum concentration of THC-OH was 4.48 ng/mL at a mean maximum time of 90 min.These slight discrepancies between our simulation results and Nadulski's study may be attributed to variations in the clinical study conditions.Based on our study, the model accurately produces pharmacokinetic curves for THC and its metabolites in the plasma after oral administration, closely matching the actual data from the study conducted by Guy and Robson.Furthermore, our model can predict the concentration of THC and its metabolites in other bodily compartments, which provides an advantage over most clinical studies that typically focus only on plasma concentrations.Consequently, our model serves as a valuable tool for advancing our understanding of the pharmacokinetics of medical THC, aiding in determining appropriate dosages and administration durations for oral medical THC products.Additionally, the model demonstrates potential for further refinement and development in the future.

Figure 1 .
Figure 1.A compartment pharmacokinetic model of THC and its metabolites after oral administration.

Figure 2 .Figure 3 .
Figure 2. The simulated THC concentration after the administration of 10 mg of THC orally in each compartment.(a) in the absorption compartment; (b) in the central compartment (plasma); (c) in the rapidly equilibrating tissue compartment; (d) in the slowly equilibrating tissue compartment; and (e) in the effect site compartment.

Figure 4 .
Figure 4.The simulated THC concentration after the oral administration of 10 mg of THC in each compartment.

Figure 5 .
Figure 5.The simulated THC-OH concentration after the oral administration of 10 mg of THC in each compartment.

Figure 6 .
Figure 6.The actual and simulated concentration in plasma after the oral administration of 10 mg THC: (a) the simulated THC concentration; (b) the simulated THC-OH concentration.