Revisiting a physiologically based pharmacokinetic model for cocaine with a forensic scope

A whole-body permeability-rate-limited physiologically based pharmacokinetic (PBPK) model for cocaine was developed with the aim to predict the concentration–time profiles of the drug in blood and different tissues in humans.

Abstract

A whole-body permeability-rate-limited physiologically based pharmacokinetic (PBPK) model for cocaine was developed and adjusted with the pharmacokinetic data from studies with animals and reparametrized scaling to humans with the aim to predict the concentration–time profiles of the drug in blood and different tissues in humans. Estimated time course concentrations could be used as an interpretation tool by forensic toxicologists. The model estimations were compared successfully with pharmacokinetic parameters and time to peak for some effects reported in the literature. Once developed, the PBPK model was employed to predict the time course tissue concentrations reported in previous distribution studies introducing individualizing data. The heart and brain concentrations estimated by the model match adequately with the time and duration of some effects such as chronotropic and psychoactive effects, respectively. This work is the first attempt for employing PBPK modeling as a tool for forensic interpretation. Future modeling of other cocaine metabolite profiles or interaction when co-administered with other substances, such as alcohol, might be developed in the future.

Introduction

Among the tasks undertaken by forensic toxicologists, interpretation of laboratory results is the most challenging because it's supposed to give a biological meaning (i.e. impairment) to a numerical value (i.e. drug concentration in plasma). Such interpretations may be used to prove recent or chronic use of a substance, to determine the cause of death or to establish the timeframe in which a crime was committed under the influence of substances. In order to achieve such interpretations, some mathematical models have been developed based on drug elimination profiles.1 A first step towards establishing a relationship between the pharmacological effects and plasmatic concentrations relies on ensuring that the drug is available to its target,2 thus a model that can represent drug distribution to target organs could help in interpretations.

Several compartmental pharmacokinetic (PK) models have been developed to describe drug distribution in the body, and physiologically-based pharmacokinetic (PBPK) models integrate a wide variety of information, such as physicochemical drug properties, physiological parameters like blood flow, and biochemical processes like biotransformation pathways to better represent the pharmacokinetics of drugs within the body.3 PBPK models allow the extrapolation to different species, and the incorporation of variables such as age, sex, ethnicity or health status to explore the variability of pharmacokinetic patterns, which has been useful in drug research and development,4,5 as well as in health risk assessment.6

Given the current uses of PBPK models in the pharmaceutical and regulatory fields, and their capability for prediction, we propose that they could provide a robust interpretation tool for forensic toxicologists. In order to explore the interpretative potential of PBPK models with a forensic scope, a diffusion limited PBPK model for cocaine was developed with two main objectives: first, proposing PBPK modeling as a useful tool in the forensic field exposing some of its advantages to predict and interpret concentrations; and second, constructing a PBPK model for cocaine from the data reported in the literature. The resulting model and comparisons with reported data in the literature are discussed in this work.

Recently, cocaine use appears to be increasing in the two largest markets: North America and Europe.7 Cocaine, like amphetamines, is classified as a stimulant. These drugs act via dopaminergic receptors to produce neurochemical magnification of the pleasure response, and enhance alertness and a sense of wellbeing. Cocaine interferes with presynaptic catecholamine reuptake, which results in accumulation at the postsynaptic site. The subsequent activation of the sympathetic nervous system produces many of the toxic effects of the drug.8 Cocaine also blocks the re-uptake of dopamine in the central nervous system, accounting for the feeling of euphoria experienced shortly after drug intake9,10 Additionally, cocaine produces powerful positive reinforcing qualities, probably because of its effect on cortical systems during repeated cocaine reinforcement, leading to maladaptive changes that contribute to the risks of drug use for exacerbation of other psychiatric disorders.9,10 Chronic use of the drug can result in abnormalities of the dopaminergic pathways, leading to psychiatric complications such as paranoid psychosis and mood disorders.11

Cocaine is very lipophilic and therefore, rapidly traverses most biologic membranes. It readily crosses the blood-brain barrier and accumulates within the central nervous system, resulting in brain concentrations that are four times the usual peak plasma concentrations.12 Tachycardia and an intense euphoria occur within 5 to 11 minutes after intravenous (i.v.) administration.12,13 Cocaine is rapidly metabolized by serum and hepatic cholinesterases to inactive compounds, benzoylecgonine and ecgonine methyl ester,14,15 which are renally excreted.16

The plasma elimination half-life varies markedly, perhaps because of the differences in cholinesterase activity.17 The half-life is also dependent upon the route of administration, average values being 0.6 hours after intravenous administration, 0.9 hours after oral intake, and 1.3 hours after intranasal use.18,19 Several pharmacokinetic and effect studies of cocaine in humans are reported in the literature, however, there is continuing controversy about which is the best pharmacokinetic model to fit plasma concentrations and explain pharmacodynamic effects. Additionally, the significant inter-individual variation in the rate of metabolism makes it difficult to compare human studies or to extrapolate those performed in animals. On the other hand, pharmacodynamic effects do not correlate with cocaine plasmatic concentrations after i.v. administration. There has been some discussion in the literature regarding the complexity of direct interpretation of cocaine concentration in plasma because plasma concentrations vary widely and often do not correlate with those in the brain.20 Thus, these concentrations cannot be used to predict reliably the severity of intake, to guide clinical management and, from a forensic scope, to interpret the influence of cocaine on a legal controversy. PBPK models represent a good tool to predict drug concentrations in different tissues, and, consequently, help in the interpretation of plasma concentrations and its relationship with concentrations in target tissues leading to a better understanding of the physiological effects, and including metabolite formation modeling. Previous effort in developing a PBPK model for cocaine in rats was performed by Bonate et al.21 based on the Well Stirred Tank approach; however, fat and muscle predictions showed poor fits compared with the other tissues for all time points and were discarded from the sampling matrix during optimization. Additionally, their model scaled to humans was a poor fit following intravenous administration and coupling with metabolite pharmacokinetics was not considered. In this paper we present a PBPK model developed with a permeability-rate-limited approach, which is indicated to use when tissue–drug concentrations do not decline in a parallel way to drug concentrations in the blood, just as is the cocaine behavior, resulting in an improved fit of all tissues and a better estimation of plasmatic concentrations after an intravenous administration. The model's usefulness from a forensic scope is also discussed.

Experimental

Model development

Preliminary PBPK model

The model used to simulate cocaine PK in rats is presented in Fig. 1. A whole body permeability-rate-limited PBPK model was selected, this type of model is indicated when tissue–drug concentrations do not decline in a parallel way to drug concentrations in blood/plasma,3 and it is especially useful to model organs with limited permeability like the brain. Since we are interested in predicting the time course of a substance for which the main target organ is the brain, we selected this type of PBPK model. The model consisted of 13 compartments, representing the tissues of interest: lungs, brain, fatty tissue, bones, heart, kidneys, muscle, skin, spleen, intestine, liver, venous blood and arterial blood (Fig. 1A).22–25 Each one but venous and arterial blood consisted of two well-stirred sub-compartments (Fig. 1B) with a permeability rate-limited transfer between them described by a permeability surface-area product (PS_T). Each sub-compartment is described with a mass balance ordinary differential equation (ODE).

Fig. 1. (A) Thirteen compartment PBPK model for an intravenous administration of cocaine in a rat. Q is blood flow for lungs (Lu), brain (Br), fatty tissue (Fa), bone (Bo), heart (He), muscle (Mu), kidney (Ki), skin (Sk), spleen (Sp), intestine (In), and liver (Li). Hv is the hepatic vein and Ha is the hepatic artery. CL_K represents renal clearance and CL_H represents hepatic clearance. (B) Sub-compartment representation of tissues in the PBPK model. C_arterial is the inflowing arterial concentration, C_venous is the outflowing venous concentration, C₁ is the concentration in the vascular sub-compartment of the tissue, C₂ is the concentration in the extravascular sub-compartment, V₁ is the vascular volume, V₂ is the extravascular volume and PS is the permeability-surface area product. Adapted from Thompson and Beard.26.

Metabolic clearance by esterases was considered from the liver cellular matrix, kidney cellular matrix and plasma as the main mechanism of elimination. The compartments were catalogued into two types: eliminating and non-eliminating tissues or organs. For non-eliminating organs, the general differential eqn (1) and (2) were employed to describe the rate of change in the amount of drug in the blood tissue and cellular matrix. The mass balance of eliminating tissues (liver and kidney) is described by differential eqn (3). Finally, the resulting “mixed” venous blood concentration (C_v) of cocaine is described by summation of all venous blood concentrations leaving the organs considered in the model, considering also in this mass balance the metabolic clearance from this compartment by esterases, the mass balance differential equation is described as follows (eqn (4)).

1

2

3

4

For eqn (1)–(4), C_bT = concentration in blood tissue at a given time (t), C_v = venous blood at a given time (t). V_bT = volume; Q_lu = blood flow; Q_Ti = blood flow for tissue i; C_in = input concentration, PS_T = permeability surface-area product, K_p:T = plasma tissue partition coefficient, Cl_p = plasmatic clearance.

PBPK rat model parametrization

The physiological parameters obtained for rats from a literature review are shown in Table 1. Experimental estimates of the plasma tissue partition coefficients (K_p:Ti) for each tissue compartment were unavailable and were calculated by Rodgers, Leahy and Rowland.27 This method is based on tissue composition and describes the expected solubility of a basic chemical in each tissue compartment. The tissue compositions and physiological blood data used for partition coefficient calculation are reported in the ESI.†

Table 1. Specific physiological parameters for rats.

Tissue	Cardiac output (Qc = 0.055 L min–1) d		Tissue volume		Vascular tissue volume
	% Qc	Q (L min–1)	% BW	Density b (kg L–1)	%tissue volume
Brain	2.82 a	0.0016	0.6 c	1.04	3 b
Kidneys	14.1 b	0.0078	0.98 d	1.05	16 b
Liver	19.94 b	0.0110	4.9 d	1.04	21 b
Hepatic artery	2.4 b	0.0013	—	—	—
Hepatic vein	17.54	0.0096	—	—	—
Spleen	2 c	0.0011	0.26 e	1.05	22 b
Intestine	15.54 a	0.0085	3.8 e	1.04	3.5 b
Heart	5.8 d	0.0032	0.33 b	1.03	26 b
Lungs	100 c	0.0550	0.67 d	1.05	36 b
Muscle	27.8 b	0.0153	40.43 b	1.04	4 b
Bone	5.4 d	0.0030	7.7 d	1.43	4 b
Skin	10.5 e	0.0058	16.9 d	1.11	2 b
Fat	5.1 d	0.0028	8.2 d	0.916	1 b
Arterial blood	100 c	0.0550	2.72 f	1.054	—
Venous blood	100 c	0.0550	5.44 f	1.054	—

^aSakaeda, 1998.28

^bBrown, 1997.29

^cThompson, 2012.26

^dDelp, 1998.30

^eDavies, 1993.31

^fPoulin, 2002.32

Diffusion-limited model optimization

Data from Nayak et al.33 were employed to fit the PBPK model in rats. The simultaneous solving of every ODE proposed for each compartment was programmed using the multi-paradigm numerical computing environment and programming language MATLAB R2017a, version 7.8 developed by MathWorks. For modeling and simulation ode15s solver was employed. Model fitting was performed using the function ‘lsqcurvefit’ with default options and the sum of least squares deviation as the criterion for model discrimination. Initially, fourteen model parameters were optimized which included PS_T for each organ and clearance from the liver cellular matrix, kidney cellular matrix and plasma. After a preliminary adjustment, the model sub estimated the brain concentrations; since it is the target organ, it was imperative to improve this estimation. There are several methods to estimate partition coefficients, the methods result in predictions with different accuracies depending on the tissue and the drug class.25,34 With the aim to improve predictions, in a second attempt, K_p:brain, and other three partition coefficients for poorly simulated tissues (K_p:f, K_p:spleen, K_p:L) were included as variables to optimize and were allowed to vary during the optimization, using the previous diffusion-limited structure.

Statistical index for goodness of fit

In order to compare experimental data with simulations and select the model which better fits the Nayak et al. data, the index proposed by Krishnan et al. was employed.35 The index was calculated for each available time of collection data for all three PBPK models (eqn (5))

5

Benzoylecgonine sub-model simulation

A PBPK well stirred tank sub-model for metabolite benzoylecgonine was also developed. The model consisted of the same 13 compartments and the same physiological parameters Q_T and V_T were employed_. Benzoylecgonine is a zwitterion, so K_p:Ti was calculated according to the Rodgers and Rowland equations for zwitterions,36,37 K_p:Ti data are reported in Table 2. Assuming 89% of biotransformation of cocaine into benzoylecgonine38 in a non-saturable process, the differential equations for plasma, liver and kidney were connected with the core PBPK, renal clearance for benzoylecgonine was considered equal to 0.020 L min^–1 kg^–1 (ref. 39) and then concentrations were modelled with ode15s solver in MATLAB2017a.

Table 2. Model PBPK parameters in humans.

	Reference value male 70 kg	Range a	Cf a	Reference
Regional blood flow distribution (percent cardiac output) in humans
Tissue
Adipose (pQf)	5	4.16–6	1.2	Brown et al.29
Bone (pQbone)	5	4.16–6	1.2	Brown et al.29
Brain (pQbrain)	12.0	10–14.4	1.2	Brown et al.29
Heart (pQh)	4	3.3–4.8	1.2	Brown et al.29
Kidneys (pQk)	19.0	15.8–22.8	1.2	Brown et al.29
Liver (pQL)	25	20.8–30.0	1.2	Brown et al.29
Hepatic artery(pQha)	6.9	5.75–8.28	1.2	Calculated by difference
Lung (pQlung)	100	—	—	Brown et al.29
Muscle (pQm)	17.0	14.16–20.4	1.2	Brown et al.29
Spleen(pQspleen)	2	1.6–2.4	1.2	Pilari et al.41
Skin (pQskin)	5	4.16–6	1.2	Brown et al.29
Rest of the body (pQrob)	4.5	3.75–5.4	1.2	Calculated by difference to mass balance
Reference values for masses of organs and tissues (percent body weight)
Arterial (pVa)	2.57	2.14–3.08	1.2
Adipose (pVf)	21.42	17.8–25.7	1.2
Bone (pVbone)	14.29	11.9–17.14	1.2
Brain (pVbrain)	2.0	1.66–2.4	1.2
Heart (pVh)	1.2	1–1.44	1.2
Intestine(pVint)	1.44	1.2–1.72	1.2
Kidneys (pVk)	0.44	0.36–0.52	1.2
Liver (pVL)	2.57	2.14–3.08	1.2
Lung (pVlung)	1.71	1.42–2.05	1.2
Muscle (pVm)	40.00	33.33–48	1.2
Spleen (pVspleen)	0.26	0.21–0.31	1.2
Skin (pVs)	3.71	3.09–4.45	1.2
Venose (pVv)	5.14	4.28–6.16	1.2
Rest of the body (pVrob)	3.25			Calculated by difference
Density (g mL–1)
Arterial	1.054
Adipose	0.916
Bone	1.04
Brain	1.04
Heart	1.03
Intestine	1.04
Kidneys	1.05
Liver	1.04
Lung	1.05
Muscle	1.04
Spleen	1.05
Skin	1.11
Venose	1.054
Rest of the body	—
Fraction of blood in tissues (dimensionless)
Adipose (pVfb)	0.020	0.016–0.024	1.2	Values calculated from ICRP reference values40
Bone (pVbbone)	0.038	0.032–0.046	1.2
Brain (pVbrainv)	0.046	0.038–0.055	1.2
Heart (pVhv)	0.583	0.486–0.699	1.2
Intestine (pVintv)	0.321	0.267–0.385	1.2
Kidneys (pVkv)	0.346	0.833–0.4152	1.2
Liver (pVLv)	0.299	0.249–0.359	1.2
Lung (pVLv)	0.468	0.390–0.561	1.2
Muscle (pVmv)	0.269	0.224–0.322	1.2
Spleen (pVspleenv)	0.053	0.044–0.064	1.2
Skin (pVsv)	0.058	0.048–0.696	1.2
Permeability surface area product (LL–1 min–1)
Adipose (PSf)	0.9877	0.823–1.185	1.2	Adjusted b
Bone (PSbone)	0.0607	0.050–0.073	1.2	Adjusted b
Brain (PSbrain)	59.340	49.45–71.20	1.2	Adjusted b
Heart (PSh)	2.1793	1.816–2.615	1.2	Adjusted b
Intestine (PSint)	0.3749	0.312–0.449	1.2	Adjusted b
Kidneys (PSk)	0.5039	0.419–0.605	1.2	Adjusted b
Liver (PSL)	2.1802	1.817–2.616	1.2	Adjusted b
Lung (PSlung)	0.2106	0.175–0.253	1.2	Adjusted b
Muscle (PSm)	0.1350	0.112–0.162	1.2	Adjusted b
Spleen (PSspleen)	0.4070	0.339–0.488	1.2	Adjusted b
Skin (PSs)	0.0395	0.033–0.047	1.2	Adjusted b
Cocaine partition coefficient plasma: tissue (dimensionless)
Adipose (Kp:f)	3.2458	2.70–3.88	1.2	Adjusted b
Bone (Kp:bone)	1.2509	1.04–1.50	1.2	Calculated c
Brain (Kp:brain)	1.3191	1.10–1.58	1.2	Adjusted b
Heart (Kp:h)	3.3789	2.82–4.05	1.2	Calculated c
Intestine (Kp:int)	3.7487	3.12–4.50	1.2	Calculated c
Kidneys (Kp:k)	6.7441	5.62–8.09	1.2	Calculated d
Liver (Kp:L)	9.1711	7.64–11.0	1.2	Adjusted b
Lung (Kp:lung)	5.4719	4.56–6.57	1.2	Calculated c
Muscle (Kp:m)	2.4497	2.04–2.94	1.2	Calculated c
Spleen (Kp:spleen)	8.4717	7.06–10.16	1.2	Adjusted b
Skin (Kp:s)	2.5328	2.11–3.04	1.2	Calculated c
Total clearance cocaine	0.03 L min–1 kg–1	0.022–0.35	1.16	Chow et al.13 Range from Barnett et al.42
Benzoylecgonine partition coefficient plasma: tissue (dimensionless)
Adipose (Kp:f)	0.2347	0.195–0.282	1.2	Calculated d
Bone (Kp:bone)	0.4126	0.344–0.495	1.2	Calculated d
Brain (Kp:brain)	0.4512	0.376–0.541	1.2	Calculated d
Heart (Kp:h)	1.1428	0.952–1.563	1.2	Calculated d
Intestine (Kp:int)	1.1787	0.982–1.414	1.2	Calculated d
Kidneys (Kp:k)	2.0804	1.734–2.496	1.2	Calculated d
Liver (Kp:L)	1.8723	1.560–2.247	1.2	Calculated d
Lung (Kp:lung)	1.7237	1.436–2.068	1.2	Calculated d
Muscle (Kp:m)	0.8152	0.679–0.978	1.2	Calculated d
Spleen (Kp:spleen)	1.4284	1.190–1.714	1.2	Calculated d
Skin (Kp:s)	0.8089	0.674–0.970	1.2	Calculated d
Total clearance benzoylecgonine	0.00375 L min–1 kg–1	0.0031–0.0045	1.2	Adjusted

^aSince variability of reference values for masses of organs and tissues is not completely reported in the literature, a Cf of 1.2 (20% of variation) was assumed for all tissues and parameters. Ranges were calculated according to McLeod et al., (Confidence Factor (Cf) = 1.2; Probability{μ/Cf < X < Cf*μ} = 0.95).43

^bAdjusted during fitting our PBPK model with Nayak et al. data.

^cCalculated according to the Rodgers approach for moderate to strong bases.44

^dCalculated according to the Rodgers approach for zwitterions.36

Extrapolation from rat to human PBPK model

The rat PBPK model was scaled up to humans by incorporating data from the literature on human physiological parameters for cardiac output, blood flows to individual tissues, volume of individual tissues, and blood volume fractions for individual tissues assuming a 70 kg male.40K_p:t and PS_t values were fixed according to the previous results with a rat model, the value of total clearance in humans was taken from the literature13 and then separated in renal, hepatic and plasmatic clearance according to the proportions estimated in the rat model. Parameters considered in the human model are reported in Table 2.

In order to extend the model to another common route of administration in humans, we employed data from a controlled study in humans with the intranasal route of administration (snorting) published by Jeffcoat et al. Intranasal absorption was modeled as a linear second order process that is governed by 2 rates K_IN1 and K_IN2 and an absorbed fraction F. Conceptually, absorption can be thought of as involving 2 sequential steps, with first order absorption from the uptake compartment (where the dose is administered) to an intermediate compartment with rate K₁ followed by absorption from the intermediate compartment to the target organ (venous blood) with rate K₂.

6

7

For eqn (6) and (7), F is the absorbed fraction or fraction bioavailable, K_IN1 and K_IN2 are the values for intranasal uptake. AIN1 is the amount of cocaine (mg min^–1) in the compartment where the dose is deposited and A_IN2 is the amount in the intermediate compartment from which the compound enters the circulation.

Sensitivity analysis

The PBPK human model contains a total of 74 parameters for predicting the distribution of cocaine and benzoylecgonine in humans. Sensitivity analysis was conducted with Monte Carlo uncertainty analysis programmed in MATLAB (MATLAB R2017a). Model parameters were chosen from normal distributions for body weight and dose, and lognormal for everyone else considering a variability of 20% of the mean value (Confidence factor (Cf) = 1.2, see Table 2 for details). The normalized sensitivity coefficient (NSC) values were determined by the following equation (eqn (8)):

8

where m is the response variable (e.g., venous concentration), ΔM is the change in response variable, p is the value of the parameter, and Δp is the change in the parameter value. Variation in each parameter was considered according to Table 2, and the NSCs were computed for a simulation of 32 mg kg^–1 iv dose for a subject with a body weight equal to 70 kg.

Results and discussion

PBPK model for rats

The permeability-limited PBPK model simulations for cocaine kinetics in the rat are presented in Fig. 2. The initial PBPK model underpredicted the time course of brain concentrations, however, after allowing to fit K_p:Ti for brain, spleen, liver and adipose tissue, the model improved considerably (index for goodness: 0.4722 and 0.2665, respectively). Plasma cocaine concentration–time course after intravenous injection resulting from our model shows a biphasic curve on a semilog scale like the one obtained in a two-compartment model.

Fig. 2. PBPK model fitted to Nayak et al.33 data.

The model predicted pharmacokinetic constants in rats close to the ones reported in the literature by other authors; when the initial blood samples are drawn prior to 1 min after injection, a t_1/2α of approximately 1 min or less is calculated,45 our model estimates a t_1/2α = 0.335 min. Regarding t_1/2β, the value calculated for rats with this model was t_1/2β = 22.28 min which agrees with the values reported in the literature for similar doses (19 ± 3 min (n = 9, 2 mg kg^–1 dose)46 and 18 min (n = 3, [3H]-8 mg kg^–1 dose).33 The model predicts a total clearance of 334.07 mL min^–1 kg^–1 which exceeds the value reported in the literature (range 51–204 mL min^–1 kg^–1).45,47–49 Distribution volume (V_d) is another parameter which might be useful as it's been reported previously;50 however, attempts at experimental estimation of this parameter have shown differences due to the method of estimation and the use of arterial, or venous samples.38,44,47,48,51 The values for V_d range from 0.9 to 6.11; our model estimated a V_d of 2.73 L kg^–1 which is consistent with previous results.

Regarding benzoylecgonine pharmacokinetics in rats, concentrations time-course predicted for benzoylecgonine in plasma and the brain show a brain/plasma ratio <1 after the first minute post-administration. These findings are in agreement with previous reports in the literature.38 In contrast to cocaine, benzoylecgonine is a much more polar substance and is distributed less in tissues with a high lipid content, such as the brain, which explains the difference with respect to cocaine in which the brain/plasma ratio is greater than 1 after the initial distribution phase.

PBPK in humans

Pharmacokinetics of cocaine in humans has been reported and discussed by several authors,52–58 and distribution of intravenous cocaine has been modeled by one17 or two compartment systems,59 however the number of individuals studied is insufficient to cover a wide spectrum of associated biological variability, which would explain the difference between reports. One of the advantages of the use of PBPKs is that they allow adjusting the model according to individual characteristics, for example body weight, which allows a more adequate approximation of the variation of pharmacokinetic parameters to an individual, which is of particular interest in the forensic field. Much is even discussed about whether these pharmacokinetic parameters are dose-dependent. The PBPK models allow the simulation of different doses, which consents to clarify the dependence or not of the estimated parameters.

The main results from sensitivity analysis are shown in Table 3, a detailed summary can be consulted in the ESI.† In summary, a small NCS value means that the variation in the parameter in question does not result in an important change in the estimations by the PBPK model, it means that the estimation is not sensitive to this parameter and therefore is not related importantly to the concentrations in that tissue. The parameters with values of NSC >0.1 were considered relevant to simulation. The parameters that have the greatest influence on the concentration peaks in the brain and the heart are the volume of the organs and the cardiac output they receive, as well as the distribution coefficients plasma tissue (K_p:Ti). It is remarkable that, according to the results of analysis, the permeability surface-area products (PS) do not have an important influence on these values.

Table 3. Normalized sensitivity coefficient (NSC) values calculated at the cocaine peak concentration in the brain (T = 4.24) and in the heart (T = 3.41 min) from sensitivity analysis by Monte Carlo from 1000 virtual subjects with a body weight of 70 kg and a total dose of cocaine of 32 mg. Values of NSC >0.1 were considered relevant to simulation, this values are presented in bold.

	Maximum brain peak concentration of cocaine (T = 4.24 min)	Maximum heart peak concentration of cocaine (T = 3.41 min)
V brain	0.3461	0.0072
V h	0.0067	0.3119
V hv	0.0042	0.2648
Q brain	0.3137	0.0297
Q h	0.0055	0.3088
K p:brain	0.1062	0.0077
K p:h	0.0047	0.2143
Total clearance	0.1329	0.1189

In order to evaluate our model, some pharmacokinetic parameters are compared and discussed below. The beta half-life estimated by Chow et al. is 48 ± 13 minutes (± SD),13 Javaid et al. reported a range from 16 to 87 minutes,17 and subsequently 43 min,18 while Barnett et al. reported values ranging from 40–91 min (ref. 42) and Perez-Reyes et al. around 64 ± 5 min.60 Our model scaled to humans under considerations made in Monte Carlo simulation (BW = 70 kg, total dose = 32 mg of cocaine) predicts a half time t_1/2β = 56.77 min (p = 0.955, 47.7–85.9 min), which is consistent with previous experimental findings by other authors. Regarding total clearance, Chow et al. reported 2.10 L min^–1,13 Javaid et al. 2.45 L min^–1,18 and Barnett et al. reported a range of 1.2–1.9 L min kg^–1;42 our calculation with PBPK is 2.45 L min^–1 kg^–1 (p = 0.955, 2.06–3.7 L min^–1 kg^–1) which is consistent with previous reports.

One of the advantages of PBPK models is that they can predict the behavior over time of the concentrations in different organs and tissues, which favors a more adequate interpretation of the pharmacodynamic effects. The relationship between cocaine concentrations in the blood and pharmacological effects is undoubtedly complex; immediately after dosing, blood concentrations are high and tissue concentrations are low, whereas later, in the post-absorptive stage, tissue concentrations may be higher than blood concentrations. Cocaine presents hysteresis, which implies that the plasma concentration is not representative of the concentrations in the target organs, for example the brain and the cardiovascular system. This phenomenon prevents in practice adequately interpreting plasma values and estimating the magnitude of the effects and their duration, these values are key answers in legal cases like driving under the influence of drugs, alcohol or psychoactive treatment. Establishing impairment or estimating the last time of intake is important to know when clinical or forensic analysis is intended. Time to peak “high” subjective effects of cocaine after intravenous administration is reported at 5 ± 1 minutes;60 due to the rapid distribution of cocaine, the plasmatic concentration decreases considerably within 5 minutes. The concentration of cocaine in the target organ (brain) increases during the first minutes after administration instead. Our model predicts a maximum concentration in the brain at 4.07 min (p = 0.95, range 3–4.78 min) which is consistent with the time of presentation of psychoactive effects; the subjective effects last for about 30 minutes to return to the baseline in agreement with an important decay of concentration in the brain. These findings are illustrated in Fig. 3.

Fig. 3. Left: Time course concentration of cocaine in plasma and brain, estimated by PBPK scaled to humans. Solid line is the mean of 1000 Monte Carlo simulations for a subject of 70 kg and dose administered of 32 mg of cocaine, scattered lines represent ± one and two standard deviations (p = 0.995). Right: Cocaine brain to plasma ratio calculated from PBPK simulation.

Regarding cardiovascular effects, the time to the maximum peak of mean heart rate increase was reported to manifest at 4.7 ± 0.7 min by Perez-Reyes et al.,60 and within 5–15 min by Chow et al.13 with a duration of about 30 min and 60 min, respectively. Chow et al. proposed in their paper a bicompartmental model for cocaine; in their approach, the concentration in the secondary compartment or BioPhase is related to the chronotropic effect. PBPK model prediction for time to present a heart maximum concentration ranges from 2.80 to 6.36 min (Fig. 4), and it has a considerable decay around 40–60 min, which is consistent with the time of presentation and the duration of pharmacodynamic effects. The beta-1-adrenergic receptor is responsible for heart rate and it predominates in this tissue heart, several studies support the idea about a peripheral site of action for cocaine with a local cardiotoxic effect on the heart.61–63 Although there is the presence of this type of receptor in other tissues, because they have a specific role in each one,64 it is reasonable to think that the proposed BioPhase in the pharmacokinetic study by Chow et al. is probably the concentration in the heart if the effect measured is the chronothropic effect on the heart.

Fig. 4. Time course cocaine concentration in the heart estimated by PBPK scaled to humans. Solid line is the mean of 1000 Monte Carlo simulations for a subject of 70 kg and dose administered of 32 mg of cocaine, scattered lines represent ± one and two standard deviations (p = 0.995).

Plasma concentration data from a study by Barnet et al.42 were employed to explore the performance of the model to predict the time course of cocaine concentration in plasma. One of the advantages of PBPK modeling is their capability to adjust predictions to individual cases modifying parameters like body weight or dosing scheme, leading to individualized predictions. This feature is of particular interest when interpreting the results of a blood sample collected in forensic investigations to determine whether an individual was under the influence of a substance at a given time, or if the intake dose was sufficient to cause impairment. We adjust body weight and dose to individual values of the subjects studied by Barnet et al., the resulting simulation is shown in Fig. 5. From the resulting graphs, our model makes predictions adequately for the first two hours in all cases, the behavior of higher doses (200 mg) is estimated better than lower ones (100 mg). During the first two hours for 100 mg dose, as the plasma concentration becomes smaller, the variation predicted by the model increases, this phenomenon continues to increase after 2 hours where the data progressively move away from the predictions of the model. However, the model is accurate when concentrations are not too low, as it can be seen in individuals who received higher doses (200 mg), the model performed a good estimation even after 2 hours. The fall in accuracy after two hours for low doses is possibly related to the magnitude of concentration more than with the time after dosing since the values move away from the prediction when the concentration is close to or below 10^–1 mg L^–1; the same behavior is shown by mean prediction (Fig. 5, line) and the Monte Carlo prediction at a probability of 0.95 and 20% of variability (Fig. 5, green scattered lines), leading one to think that the influence of interindividual variability becomes more important to accuracy when the concentration of cocaine is low. Future optimization with more human pharmacokinetics could improve predictions.

Fig. 5. PBPK estimations of the concentration (mg L^–1) of cocaine in plasma in subjects studied by Barnet et al.42 Solid line is the mean of 1000 Monte Carlo simulations for each subject, scattered lines represent ± two standard deviations (p = 0.995).

The most common routes of administration of cocaine are injection, smoking and snorting (intranasal). Among routes of administration, the most used for the first consumption of cocaine is intranasal, so with the purpose of exploring the behavior of the model employing other routes of administration different from intravenous injection, the PBPK model for humans was fitted with Jeffcoat et al.65 data, resulting constants for absorption and bioavailability are K₁ = 0.2745 min^–1, K₂ = 0.0233 min^–1 and F = 0.38000, the bioavailability (F) calculated by our model for the intranasal route is in the range reported by other authors from 0.25 to 0.60.17,42,66 Time courses concentration for plasma, brain and heart were estimated with Monte Carlo simulation as the previous intravenous model (Fig. 6). The model predicts plasma concentration adequately and; as it was expected, change to the intranasal route of administration delays the maximum brain or heart peak concentration of cocaine compared with the intravenous administration, in agreement with other reports where plasma concentrations after nasal insufflations peak in 15 to 60 minutes. Some authors propose that the delay resulting from a decrease in the rate of absorption is caused by vasoconstriction in the nasal mucosa.17,18,66

Fig. 6. From left to right, PBPK estimations of plasma, brain and heart concentrations for cocaine and plasma concentrations reported by Jeffcoat et al.65 for administration of an intranasal dose of 94.3 ± 0.5 (SD) mg of cocaine base to six masculine subjects of weight 75.7 ± 2.1 kg. Solid line is the mean of 1000 Monte Carlo simulations for the average weight of group, scattered lines represent ± two standard deviations (p = 0.995).

For interpretation purposes, common practice among forensic toxicologists is the quantification of metabolites, even if they are inactive, and the use of ratios of the parent drug and its metabolite concentrations for either estimation of the time interval between the last intake and the time of sample collection or to distinguish acute from chronic intake.1 In order to evaluate our PBPK model for prediction of metabolite benzoyilecgonine concentrations we performed estimations for administration of two intravenous doses of 32 mg at 0 and 15 minutes and the results were compared with the experimental data reported by Isenschmid et al.56 who analyzed for cocaine and its metabolites the plasma from 10 human subjects at various intervals after administration of two rapid doses of 32 mg of cocaine; since the body weight was not reported in this study, modelling was performed for a masculine subject of 70 kg. Fig. 7 shows the Monte Carlo estimations for 1000 subjects. Data from Isenschmid et al. for concentration in plasma of both cocaine and metabolite benzoylecgonine fit well in the interval of prediction of our model despite the lack of data in relation to the mean body weight and its standard deviation to calculate Cf for Monte Carlo. The fact that the model can also adequately predict the time course of metabolite concentrations favors its use in the forensic context to support estimations of the last intake.

Fig. 7. PBPK estimations of plasmatic concentration for cocaine and metabolite benzoylecgonine and data reported by Isenschmid et al.56 for administration of two doses intravenous of 32 mg at 0 and 15 minutes modeled for a masculine subject of 70 kg. Solid line is the mean of 1000 Monte Carlo simulations for a subject of 70 kg, scattered lines represent ± two standard deviations (p = 0.995).

Model limitations

Although the model adequately matches the time and duration of some effects and pharmacokinetic parameters, some additional efforts should be made regarding chronic administration and tolerance developed because of both acute and chronic administration, in order to improve predictions. On the other hand, the relative importance of serum cholinesterase and other liver esterases in the disposition of cocaine in vivo could influence importantly the risk for toxicity or fatal overdoses following cocaine use. Homozygous or heterozygous inheritance of the atypical esterase may cause failure to hydrolyze cocaine or to hydrolyze it at an intermediate rate, respectively;67 chronic liver disease and pregnancy also may reduce cholinesterase activity. These particularities are not contemplated in our current PBPK model; however, due to the importance of serum cholinesterase in the biotransformation of cocaine and its clinical and forensic implications, it would be convenient in future models to include interindividual variations associated with heredity, as well as interactions with other drugs, both therapeutic and drugs of abuse.

On the other hand, the chronotropic effect associated with cocaine may also alter pharmacokinetic behavior. Cardiac output is the volume of blood pumped by the heart per minute, it is the product of the heart rate (HR) and the stroke volume (SV), which is the volume of blood pumped per beat. Consequently, we consider that changes in heart beat (chronotropic effect) have necessarily an influence on the cardiac output and, thus, the pharmacokinetic distribution of cocaine while the chronotropic effect lasts. Previous studies have reported that cocaine produces a tachycardic effect without a significant increase in cardiac output, which endorses the decision of not considering it within our model in agreement with parsimony criteria.60 Though our model predicts adequately pharmacokinetic parameters, this phenomenon should be considered in the future to improve predictions.

Finally, one factor that potentially increases the risk of toxicity in drug abusers is the concomitant use of ethanol. Ethanol was detected in a nearly half of the studied cases (48.91%) by García-Repetto et al.,68 their findings suggest that the concomitant use of cocaine and ethanol is widespread among the population studied. This interaction is not considered by the actual PBPK model, however it can be easily extended in future modellings if the experimental time course data of concentrations during co-administration are available to fit and validate the model.

Conclusions

The need for understanding the relationship between plasma drug concentrations and pharmacological or toxic effects is important for many reasons, including understanding the mechanism of drug action, treatment of overdose cases and interpretation of concentration and psychoactive effect related to consumption, whether for clinical or forensic purposes. The proposal that the activity of a drug depends on blood concentration and not on dose was made in the 1940s,69 and it became the basis of therapeutic drug monitoring to preserve concentrations in therapeutic ranges and to understand the individual variability, among other uses. This concept has been extended to drugs of abuse in order to understand pharmacological and toxic responses observed, however, plasmatic concentrations of drugs of abuse are not often directly related to psychoactive effects. This circumstance constitutes the biggest challenge in forensic toxicology test interpretation as it is difficult to make inferences about the influence of the plasmatic concentration of a substance on the behavior or perception of the subject under investigation. Since the traditional pharmacokinetic approach does not allow one to estimate concentrations in target organs, when the free concentration in the target organ is not equal to that in plasma (i.e. the brain/plasma simulated ratios of cocaine (Fig. X)), PBPK models can offer a significant advantage to support the interpretation of concentrations due to the capability of the model to predict concentrations in the biological compartment of interest.

A diffusion limited PBPK model for cocaine was developed in order to propose an interpretative tool with clinical or forensic scope. The model scaled up to humans considers the coupling with the pharmacokinetics with the main metabolite, benzoyilecgonine; this characteristic is important in order to perform an interpretation with forensic scope since the parent drug and metabolite concentrations are usual evidence in legal controversies.

The literature offers considerable toxicological data about cocaine-related deaths, and it is clear that the correlation of blood concentration with toxicity is not generally possible due to the fact that acute cardiac effects of cocaine are independent of the cocaine concentration in the blood.70 Attribution of driving under influence is another difficult task although cocaine is particularly dangerous for road traffic because of the discrepancy between the subjective feeling of increased performance on the one hand and the actual fitness which objectively has not increased to the same extent on the other hand.71 The estimated time course of heart and brain concentrations by the PBPK model developed match adequately with the time of presentation and duration of some effects such as chronothropic and psychoactive effects, respectively. Additionally, the PBPK models allow the adjustment of values such as weight, dose and sex of the individual, among other variables, which leads to the individualization of the estimates and improves the interpretation of the findings without generalizing. Characteristics that have not been considered in this model, such as the interindividual variability of the cholinesterase activity involved in the biotransformation of cocaine, can be taken into account by simply adjusting the corresponding value in equations; also, the interaction with other substances present may be considered.

This work represents a novel proposal to employ PBPK modeling as a tool for forensic interpretation, since giving a meaning with legal consequences to a concentration found in a biological sample is no simple task. As such, being able to correlate plasmatic drug concentrations with organ tissue concentration is a reliable way to establish relationships with some effects. This incipient work, while modeling a well-studied drug of forensic interest, has shown predictions in accordance to clinical reports, which implies that these models have enormous potential in the future for forensic toxicologists; PBPK models are usually employed in the study of drugs with therapeutic value and are supported by posterior pre-clinical and clinical studies; however such assays are not usually done or authorized for substances of forensic interest in standard pharmaceutical practice as their value relies on their harmful effects. Accordingly, these models can provide useful information on substances which cannot or should be administered in a very limited way to humans under controlled conditions, such as the named New Psychoactive Substances or other drugs and poisons of interest in the forensic field. Prediction for potentially harmful substances with these models can also optimize pre-clinical assays in animals and their posterior extrapolation to humans can be used to understand their effects and PK in order to detect their use, abuse, fatal intoxication or influence in legal and medico-legal cases.

Conflicts of interest

There are no conflicts to declare.