a bayesian approach for estimating detection times in horses:...

A Bayesian approach for estimating detection times in horses: exploring the

pharmacokinetics of a urinary acepromazine metabolite

J. M. MCGREE*

G. NOBLE�

F. SCHNEIDERS�

A. J. DUNSTAN�

A. R. MCKINNEY�

R. BOSTON§ &

M. SILLENCE*

*Mathematical Sciences, Queensland

University of Technology, Brisbane, Qld,

Australia; �School of Animal and Veterinary

Sciences, Charles Sturt University, Wagga

Wagga, NSW, Australia; �Australian

Racing Forensic Laboratory, Kensington,

NSW, Australia; §School of Veterinary

Medicine, New Bolton Center, University of

Pennsylvania, Kennett Square, PA, USA

McGree, J. M., Noble, G., Schneiders, F., Dunstan, A. J., McKinney, A. R.,

Boston, R., Sillence, M. A Bayesian approach for estimating detection times in

horses: exploring the pharmacokinetics of a urinary acepromazine metabolite.

J. vet. Pharmacol. Therap. doi: 10.1111/j.1365-2885.2012.01389.x.

We describe the population pharmacokinetics of an acepromazine (ACP)

metabolite (2-(1-hydroxyethyl)promazine) (HEPS) in horses for the estimation

of likely detection times in plasma and urine. ACP (30 mg) was administered to

12 horses, and blood and urine samples were taken at frequent intervals for

chemical analysis. A Bayesian hierarchical model was fitted to describe

concentration–time data and cumulative urine amounts for HEPS. The

metabolite HEPS was modelled separately from the parent ACP as the half-life

of the parent was considerably less than that of the metabolite. The clearance

(Cl/FPM) and volume of distribution (V/FPM), scaled by the fraction of parent

converted to metabolite, were estimated as 769 L/h and 6874 L, respectively.

For a typical horse in the study, after receiving 30 mg of ACP, the upper limit of

the detection time was 35 h in plasma and 100 h in urine, assuming an

arbitrary limit of detection of 1 lg/L and a small (�0.01) probability of

detection. The model derived allowed the probability of detection to be estimated

at the population level. This analysis was conducted on data collected from only

12 horses, but we assume that this is representative of the wider population.

(Paper received 2 August 2011; accepted for publication 30 January 2012)

James M. McGree, Mathematical Sciences, Queensland University of Technology, 2

George St, Brisbane, Qld 4000, Australia. E-mail: [email protected]

INTRODUCTION

Acepromazine (ACP) is a phenothiazine tranquilizer commonly

used in horses to reduce stress during transportation and to calm

excitable horses during veterinary procedures or in training

(Smith et al., 1996). As with most therapeutic substances, ACP

is prohibited in competition by most racing organizations (Chou

et al., 2002). If a sufficiently long washout period is not observed

before a race after ACP has been administered, the trainer runs

the risk of detection and associated penalties.

Drug screening is regulated by racing authorities who are in the

process of establishing limits of detection in blood and urine for a

range of therapeutic substances, based on either the parent drugs

or their metabolites. These limits are being established in order to

protect the integrity of the sport without adversely affecting the

welfare of the horse by restricting legitimate therapeutic treat-

ment, as discussed by Toutain (2010). Therefore, there is interest

in assembling detailed pharmacokinetic data to enable the rational

setting of analytical detection limits and corresponding with-

drawal times for particular prohibited substances.

Approaches to the detection of prohibited substances in horse

racing differ from country to country, such that estimated

withholding times may vary depending upon which country the

horse is racing in. For example, urine samples are favoured for

drug detection in Australia, whereas plasma samples are

preferred in the USA. Nevertheless, both countries consider a

drug test to be positive if the parent compound or one of its

metabolites is detected on race day. Therefore, to provide precise

estimates of withholding times, an understanding of the

metabolism and the renal excretion of the drug is required. This

understanding can be facilitated through the modelling of the

pharmacokinetics (PK) of the drug, that is, what the body does to

the drug after administration (Benet, 1984). In general, such

models describe how the drug is metabolized and then eliminated

from the body over time.

In this study, we consider ACP as the parent drug and

2-(1-hydroxyethyl)promazine (HEPS) as the principle metabolite.

This leads to the consideration of a parent–metabolite model that

describes the conversion of ACP to HEPS, and the elimination of

both substances from the body via the urine. The individual (or

horse) specific PK for ACP have been studied previously (Ballard

et al., 1982; Hashem and Keller, 1993; Marroum et al., 1994).

To our knowledge, limited research has appeared in the

literature in regard to the PK of HEPS, see Schneiders et al.

(2012), who suggest that detection of HEPS could be used as an

indicator for the use of ACP. Furthermore, detection times are

J. vet. Pharmacol. Therap. doi: 10.1111/j.1365-2885.2012.01389.x

� 2012 Blackwell Publishing Ltd 1

often reported in studies of a few (3–6) horses and in absolute

terms (i.e. number of hours or days) with no probability values

attached to them. In the present study, we sought to determine

more accurate detection times by using a larger number of

horses (12) and to explore a means to judge, not just the time of

detection, but the probability of detection at a given time.

We developed an understanding of the PK and renal excretion

of HEPS at a population level through the implementation of a

Bayesian hierarchical model (BHM). Covariates were available

and included breed, body weight (kg) and age (years), and these

were explored as potential effects to minimize unexplained

between-subject variability (BSV). Also, plasma concentrations

were subject to measurement error as two measurements were

taken for each sample, and analysed separately. Given the two

approaches to measure plasma concentration from each sample

generally gave differing results, this source of variability needed

to be considered in the model. Once a parsimonious statistical

model had been developed, simulation techniques were used to

estimate the detection times in plasma and urine.

ANALYTICAL METHODS

Acepromazine and HEPS were analysed by LC-MS following the

extraction from blood plasma and urine using a solid-phase

extraction method. To prepare the samples for analysis, aliquots

were spiked with internal standards (propionylpromazine hydro-

chloride and HEPS-d4 maleate), then diluted with either acetic

acid (urine) or acetic acid plus methanol (plasma).

The samples were centrifuged to remove proteinaceous

material before the supernatant fractions were loaded onto IST

Isolute HCX solid-phase extraction cartridges that had previously

been conditioned with methanol and acetic acid. The cartridges

were washed with acetic acid and methanol, dried briefly and

then eluted with an mixture of ethyl acetate, methanol and

ammonium hydroxide. The eluates were evaporated to dryness

at 60 �C under a stream of nitrogen and reconstituted in

isopropanol and ammonium acetate (20 mM) for analysis.

The samples were analysed using a Waters Acquity UPLC

system equipped with a Waters Acquity UPLC BEH C18

analytical column and a Phenomenex Security-Guard C18

guard column interfaced to an Applied Biosystems 4000 Q-Trap

mass spectrometer Melbourne, Vic., Australia. The mobile phase

varied from 100% aqueous ammonium acetate to 40% ammo-

nium acetate/60% acetonitrile, which was run as a gradient.

The instrument was operated in positive ion atmospheric

pressure chemical ionization (APCI) mode with ion source

conditions optimized for ACP.

Calibration standards and quality control samples were

prepared by spiking pooled blank equine plasma and urine with

ACP maleate and HEPS maleate. The specificity of the method

was assessed by analysing 12 blank equine plasma and 12 blank

equine urine samples. No significant matrix interference was

observed, and the calibration curves for both ACP and HEPS in

plasma and urine were linear over the full calibration ranges.

Correlation coefficients for ACP were 0.9889 or greater and for

HEPS 0.9996 or greater in both matrices. The limit of

quantification (LOQ) for both analytes in both matrices was

1 ng/mL. The limit of detection (LOD) based on a signal-to-noise

ratio >3 was <1 ng/mL in all cases.

Accuracy and precision data were collated from the quality

control samples prepared at low and high concentrations. The

accuracy of detection (% of nominal) for ACP in plasma ranged

from 95% to 109%, with a precision (% CV) of 3.6–5.3%. In urine,

values for accuracy and precision for ACP detection were 89% to

104% and 4.4% to 7.7%, respectively. The corresponding values

for HEPS were as follows: accuracy 88–104%, precision 6.4–7.9%

(plasma); and accuracy 103–119%, precision 3.1–4.5% (urine).

DESIGN AND DATA COLLECTION

Data were collected on twelve horses (geldings) after an

intravenous administration of ACP maleate equivalent to

30 mg dose of ACP, denoted as D. Injections were given into

the jugular vein after the insertion of an indwelling catheter on

the morning of the experiment. A summary of the 12 horses is

given in Table 1. All horses were weighed the day before drug

administration and at the end of the study, and it was found that

the weight of each horse had changed slightly over the duration

of the study, despite being fed a diet that was designed to

maintain a stable body weight. If this change was linear (this is

the only change we can estimate given these data), then there

was <6% change in weight across all horses. Hence, weight as a

time-varying covariate was not considered because of such small

changes, and the average (or median) of the two measurements

was used for covariate exploration.

ACP and HEPS concentration–time data were derived from the

chemical analysis (as discussed in the previous section) of blood

and urine samples collected at frequent intervals; 10-mL blood

samples were collected into lithium heparin vacutainers at the

following time points: 0.08, 0.17, 0.33, 0.67, 1, 1.5, 2, 2.5, 3, 4,

6, 8, 12 and 24 h after administration. Urine samples were

collected approximately 2, 4, 6, 8, 12, 24, 36 and 48 h after

administration. The horses were trained to urinate to a whistle.

Table 1. Summary of horses analysed

Horse

Median

weight (kg)

Age

(years) Sex Breed

Dose

(mg/kg)

1 570 11 Gelding Thoroughbred 0.053



4 544 9 Gelding Standardbred 0.055









2 J.M. McGree et al.

� 2012 Blackwell Publishing Ltd

The urine was collected into a 2-L plastic container that was

attached to each horse. Each horse also wore a specially designed

harness known as a ‘horse nappy’ between collection times to

ensure that all urine was collected. The midpoint between

collection times was used as the average urine amount. Hence,

the average urine amount was modelled over each of the time

spans. Parent ACP was not detected in urine and hence could

not be modelled. All samples were refrigerated directly after

collection and stored at )20 �C until analysed.

MODEL

Realistic statistical models for PK data are often complex systems

incorporating auxiliary variables in hierarchies or strata. Bayes-

ian statistical methodology provides powerful methods for

modelling such data, combining prior knowledge with informa-

tion from the data to yield a posterior distribution from which all

inference about unknowns can be made. Unfortunately, sampling

from this posterior distribution can be difficult in practice as no

closed-form solution may be available. This led to the develop-

ment of a collection of proven computational methods used

across a wide area of applied sciences and technology. The most

widely used computational method is Markov chain Monte Carlo

(MCMC), a technique that can be used to sample from a target

distribution (here, a posterior) that is difficult to sample from

directly. This facilitates posterior inference in complex settings.

A full-conditional hierarchical Bayesian analysis was under-

taken using WinBUGS (Lunn et al., 2000) where MCMC tech-

niques are used to sample from the posterior distribution of

estimable parameters. Reviews of Bayesian modelling of phar-

macokinetic data are given by Duffull et al. (2005) and Lunn

et al. (2002), and an example is given by Dansirikul et al. (2005).

In our analysis, two MCMC chains were run simultaneously,

each with different initial parameter estimates for each param-

eter. Convergence of the MCMC chains to the stationary

distributions was assessed in two ways. Firstly, the Gelman

and Rubin convergence diagnostic (Kass et al., 1998) was

calculated for each parameter. This diagnostic essentially

compares the between-chain variability with the within-chain

variability. If this variability is similar, the value of the diagnostic

will be close to 1. In our analysis, if values were <1.1 for all

parameters, then it was assumed that the stationary distribution

had been reached under this diagnostic. As a further check, the

trace history for each parameter was plotted to ensure that the

‘fuzzy caterpillar’ was observed (Lunn et al., 1999).

The two MCMC chains were run for 20 000 samples with the

first 10 000 being discarded as a ‘burn-in’. If the above

convergence criteria were met, the two chains were pooled to

provide samples from the stationary/posterior distribution. It is

this distribution upon which all inferences were drawn.

Parent and metabolite model

The full data set consisted of ACP and HEPS plasma concentra-

tions and urine amounts over time. Therefore, a multivariate

response model needed to be derived. A full parent–metabolite

model based on mass balance is given in Fig. 1.

Figure 1 shows a five-compartment parent–metabolite model.

Initially, the parent (P) enters into the first of two compart-

ments, where it can move between this compartment (P1) and

another (P2). A two-compartment model was proposed for this

part of the model as this was fitted in the analysis of ACP given

by Ballard et al. (1982), Hashem and Keller (1993) and

Marroum et al. (1994). A fraction of the parent is either

converted to the metabolite (FPM), excreted renally (FEP) or

eliminated nonrenally (1 ) FPM ) FEP). In the second case, the

triangle represents the bladder (P3), and the parent will

continue to accumulate here until the horse empties the

bladder. For the fraction of the parent that is converted to the

metabolite (M), a fraction of M is either excreted renally (FEM) or

nonrenally (1 ) FEM). In this first case, the metabolite accumu-

lates in the bladder (M2, represented by a triangle) until the

horse empties the bladder. The ks denote rate constants for

movement between compartments.

A simplification of this parent–metabolite model was made.

Given the half-life of the parent was considerably shorter than

that of the metabolite, we can conclude that the kinetics of the

metabolite were not being driven by the parent. In such cases, it

should be sufficient to model the metabolite alone, particularly

given we are only interested in estimating detection times for the

metabolite; notably, this is a key assumption in our approach,

and further research would be required to determine whether

this biased our analysis.

Metabolite model

Figure 2 shows the proposed model for plasma concentration

and cumulative urine amount for the metabolite only. Initially,

ACP appears in compartment P1 and is converted to HEPS that

appears in M1 (the molecular weights of ACP and HEPS are 326

and 345, respectively). Here, M1 represents the amount of HEPS

in plasma, with a fraction being excreted renally (FE) to the

bladder (M2) and a fraction excreted nonrenally (1 ) FE). The

metabolite will then accumulate in M2 until the horse empties

their bladder.

Fig. 1. Parent–metabolite model for acepromazine (P) and 2-(1-hy-

droxyethyl)promazine (M) in horses.

A Bayesian approach for estimating detection times in horses 3


From Fig. 2, the following is true

k22 ¼FECl

V

k23 ¼ð1� FEÞCl

V

FE ¼R1

0 k22M1dt

D;

where ClR ¼ FECl and ClNR ¼ (1)FE)Cl are renal clearance and

nonrenal clearance, respectively; Cl ¼ ClR + ClNR and V is the

volume of distribution.

As we only model HEPS, estimates of clearance and volume of

distribution are scaled by the fraction of the parent converted to

the metabolite, that is, Cl/FPM and V/FPM. To facilitate the

estimation of the parameters in the model, the assumption is that

FPM ¼ 1. We note, however, Ballard et al. (1982) show that

approximately 50% of injected ACP binds to red blood cells. Of

this 50%, an estimated fraction of 0.99 (±0.01) is converted to

HEPS, see Schneiders et al. (2012). Even though our parame-

terization may not reflect the true biological process, it does

allow an accurate mathematical representation of the data, see

Duffull et al. (2000).

The ordinary differential equations (ODEs) that describe the

kinetics of this model for the amount of the metabolite are as

follows. We note that these are linear ODEs and hence can be

solved analytically.

dP1

dt¼� k12P1

dM1

dt¼k12P1 � k22M1 � k23M1

dM2

dt¼k22M1;

where P1(0) ¼ D,M1(0) ¼ 0,M2(0) ¼ 0.

The above ODEs provide the structural model for the analysis.

When solved, they provide predicted responses for each of the

three compartments. For modelling the metabolite data, we are

only interested in the solution to compartments M1 and M2.

Denote the predicted responses in each of these compartments as

f1(.) and f2(.), respectively.

The following describes the full probabilistic model. In model

development, we consider a number of alternative models in an

attempt to derive a parsimonious description the multivariate

data. Instead of describing all models considered, we describe a

‘full’ model such that all other rival models are parametrically

nested within this full model.

The hierarchical structure of the probabilistic model can be

described in the following stages:

Stage 1.

yijkljkij; cijkl � Nðfijkl; cijklÞ; ð1Þ

where kij ¼ (Clij,FEij,Vij)¢, fijkl ¼ fl(kij,k12,tijkl,D) represents the

model prediction for the kth response on the jth measurement

type on the ith individual in compartment Ml, and the variance

of this response is given by cijkl ¼ r2add;l þ r2

prop;lf ðkij; tijkl;DÞ2 for

i ¼ 1,…,N (the total number of horses), j ¼ 1,2 (the two sources

for the chemical analysis), k ¼ 1,…,nijl (the total number of

observations from Ml on individual i on measurement type j) and

l ¼ 1,2.

Stage 2.

log kijj log hi;U � MVNðlog hi;UÞ; ð2Þ

where hi ¼ (Cli,FEi,Vi)¢ represents the ith individual’s mean PK

behaviour and U is the measurement error variance–covariance

matrix of PK parameters.

Stage 3.

log hij log l; Zi;X � MVNðlog l;XÞ; ð3Þ

where Zi represents the covariate values for the ith individual, Xthe between-horse variability and l ¼ (Cl,FE,V)¢ the population

PK parameters.

Stage 4 (priors).

r2add;l �Unið0;100Þ

r2prop;l �Unið0;100Þ

logðCl;VÞ0 �MVNð0;1000� IÞk12 �HNð0;1000Þ

log FE �Unið�300;0ÞU �Wð0:001� I;5ÞX �Wð0:001� I;5Þ;

where l ¼ 1,2, and Uni(), MVN, HN and W denote the uniform,

multivariate normal, half-normal and Wishart distributions,

respectively. These priors were chosen to be essentially uninfor-

mative, meaning that they should have minimal influence in

determining the parameter estimates when compared to the

observed data. It should be noted that, when estimating FE, it is

possible [through BSV and measurement error variability (MEV)]

that this estimate may be larger than 1. This is impossible in

Fig. 2. A simplified pharmacokinetic model for metabolite (2-(1-hy-

droxyethyl)promazine) after an intravenous administration of acepro-

mazine to horses.



practice, so the constraint min{1,FE} was imposed in the

estimation. As can be seen, k12 was assumed to be a fixed-

effects parameter, so no BSV (or MEV) was estimated.

RESULTS

The observed (log) concentration–time data and accumulated

urine amounts of HEPS for the 12 horses are shown in Fig. 3.

The demographic data collected for the 12 horses were breed,

weight (kg) and age (years). Of the 12 horses, four were

thoroughbreds and eight were standardbreds. The mean (± SD)

body weight of all horses was 544 (±48) kg, and the mean (±

SD) age was 8 (±2.8) years.

Structural and residual error model

The model specified in Fig. 2 was considered with additive,

proportional and a mixture of additive and proportional residual

errors on each of the two responses. These model fits can be

compared by considering deviance, calculated as minus twice

the log-likelihood. Given we typically wish to select parameter

estimates that maximize the log-likelihood, we would like to

select the model with the smallest deviance. However, as more

parameters are included, the model will generally fit the data

‘better’ and will therefore result in a smaller deviance (even

though this decrease may not be statistically significant).

Therefore, we would ideally like to select the model that has

small deviance and a small number of parameters.

The results of model fits are summarized in Table 2. The

smallest median deviance can be seen for the model that has a

proportional residual error term on plasma concentrations and

an additive residual error on the cumulative urine amounts.

Further, this model has less than or equal the number of

parameters to estimate when compared with other rival models.

Therefore, this residual error structure was assumed for further

analysis.

Measurement variability

Plasma concentrations for each horse were chemically analysed

in duplicate at each time point. We therefore explored whether

significant measurement error existed in the data. Measurement

variability was accounted for by including a random effect on

each individual parameter (i.e. Cli, FEi and Vi). Table 2 shows

the median deviance values for models fitted with and without

MEV. As can be seen, there was a reduction in the median

deviance when MEV was included. However, it is unclear

whether this is significant given the extra number of estimable

parameters introduced into the model. Hence, the deviance

information criterion (DIC) was used to compare the two models

(Spiegelhalter et al., 2002). The information criterion is as

follows.

DIC ¼ pD þ �D;

where pD ¼ �D� Dð�hÞ is the effective number of parameters in

the model, �D ¼ Eh½DðhÞ� and �h is the expectation of h.

From the above formula, the DIC penalizes large values of both�D and pD such that the model with the smallest DIC value is

preferred. For the model with MEV, �D ¼ 1978:26,

Dð�hÞ ¼ 1915:280, pD ¼ 62.981 and DIC ¼ 2041.240, com-

pared with �D ¼ 2048:94, Dð�hÞ ¼ 2012:48 , pD ¼ 36.457 and

DIC ¼ 2085.390 for the model without MEV. The results show

that the model with MEV gave the smallest DIC value, and

hence, this model was retained for further investigation.

Covariate modelling

Covariate relationships were initially investigated through the

use of scatter plots against individual parameters. If any of the

relationships warranted further investigation, then models with

these covariate relationships were fitted. Limited data were

available, so results from this part of the analysis should be

treated with caution.

0 5 10 15 20 25–1.5

–1

–0.5

0

0.5

1

1.5

2

Time (h)

log

Con

cent

ratio

n (µ

g/L)

HEPS plasma

0 10 20 30 40 500

1000

2000

3000

4000

Time (h)

Cum

ulat

ive

amou

nt (µ

g)

HEPS urine

Fig. 3. Observed plasma concentrations and

urine amounts for the acepromazine (ACP)

metabolite HEPS after intravenous dose of

30 mg of ACP to 12 horses.

Table 2. Median deviance and 95% credible interval of deviance

Model

Res.

plasma

Res.

urine

Median

deviance

95% CI for

deviance

Fig. 2 Add. Add. 2115 2100, 2137

Fig. 2 Prop. Prop. 2212 2197, 2233

Fig. 2 Add. Prop. 2312 2296, 2334

Fig. 2 Prop. Add. 2049 2034, 2069

Fig. 2 Mix. Mix. 2120 2104, 2148

Fig. 2 + MEV Prop. Add. 1977 1951, 2007

Fig. 2 + MEV + wt)V Prop. Add. 1977 1951, 2008



The covariate plots can be seen for HEPS in Fig. 4. There did

not appear to be a relationship between any covariates and the

estimates of individual parameters except for log Vi and median

horse weight (Fig. 4c). That is, it appeared that as the horse

weight increased, log Vi also increased. The plot also suggests

that the relationship (if it exists) could be linear. As such, the

400 500 600 7006

6.5

7

7.5(a)

Median weight (kg)

Log

Cl i

400 500 600 7000

0.05

0.1

0.15

0.2(b)

Median weight (kg)

FEi

400 500 600 7008

8.5

9

9.5(c)

Median weight (kg)

Log

Vi

4 6 8 10 12 14 166

6.5

7

7.5(d)

Age (years)

Log

Cl i

4 6 8 10 12 14 160

0.05

0.1

0.15

0.2(e)

Age (years)FE

i

4 6 8 10 12 14 168

8.5

9

9.5(f)

Age (years)

Log

Vi

1 26

6.5

7

7.5(g)

Breed

Log

Cl i

1 20

0.05

0.1

0.15

0.2(h)

Breed

FEi

1 28

8.5

9

9.5(i)

Breed

Log

Vi

Fig. 4. Plots of posterior means of individual parameter estimates vs. covariates.

0 5 10 15 20 25

0

2Horse 1

Time (h)

log

Con

c (µ

g/L)

log

Con

c (µ

g/L)

log

Con

c (µ

g/L)

log

Con

c (µ

g/L)

log

Con

c (µ

g/L)

log

Con

c (µ

g/L)

log

Con

c (µ

g/L)

log

Con

c (µ

g/L)

log

Con

c (µ

g/L)

log

Con

c (µ

g/L)

log

Con

c (µ

g/L)

log

Con

c (µ

g/L)

0 5 10 15 20 25−1

0

1

2Horse 2

Time (h)0 5 10 15 20 25

−2

−1

0

1

2Horse 3

Time (h)0 5 10 15 20 25

−2

−1

0

1

2Horse 4

Time (h)

0 5 10 15 20 25−1

0

1

2Horse 5

Time (h)0 5 10 15 20 25

−2

−1

0

1

2Horse 6

Time (h)0 5 10 15 20 25

−1

0

1

2Horse 7

Time (h)0 5 10 15 20 25

−2

−1

0

1

2Horse 8

Time (h)

0 5 10 15 20 25−2

−1

0

1

2Horse 9

Time (h)0 5 10 15 20 25

−2

−1

0

1

2Horse 10

Time (h)0 5 10 15 20 25

−2

−1

0

1

2Horse 11

Time (h)0 5 10 15 20 25

−2

−1

0

1

2Horse 12

Time (h)

Fig. 5. Individual fitted values for plasma concentrations of HEPS. The dashed and solid lines (-, - -) represent the duplicate measurements on the same

blood sample which the dots represent the observed data points.



(median) weight of each horse was included in the model as

follows:

log bi ¼ b1 þ b2

wi

548

� �;

where wi is the weight of the ith horse and 548 is the median

weight over all horses.

The results from the fit where weight was included as a

covariate on log V are given in Table 2. As can be seen, there

was no reduction in the median deviance of the model with

weight, compared with the model without weight. However, this

is not the full picture as it is possible for a covariate to be

significant without there being a significant reduction in

deviance. Therefore, we explored other indicators such as

whether the BSV on log V had been reduced and whether the

credible interval for b2 (from above) included zero or not.

The posterior distribution of b2 was examined, and it was

found that zero was a highly probable value. Further, the BSV of

log V without weight in the model was 0.0729, while with

weight, this reduced to 0.0709, reduction of about 3%. Given

this, it was concluded that weight was not significant, and

therefore, it was not included into the final model.

Model fit

The individual fits to the data can be seen for plasma

concentrations in Fig. 5 and cumulative urine amounts in

Fig. 6. The individual fits for plasma concentrations are repre-

sented by two lines corresponding to duplicate measurements

taken on each blood sample as the MEV was found to be

significant. In some horses, this is difficult to see, but is

particularly visible for horses 4 and 5. All individual specific

models seemed to fit the data well. The same was true for the

individual cumulative urine fits.

A summary of parameter estimates (posterior medians) for the

final model is given in Table 3. The large estimate of k12 shows a

rapid rate of conversion from ACP to HEPS. The clearance rate of

HEPS was 769 L/h with a volume of distribution of 6874 L (both

scaled by the fraction of parent converted to metabolite). The BSV

of Cl, FE and V is small, at around 30%. No estimate for the BSV of

k12 is given as this was considered as a fixed effect in the model.

Model checking and validation

Figure 7 shows the residual error plots for the final model. The

standardized residuals are given for both responses. For plasma

Table 3. Summary of parameter estimates (posterior medians) for final

model

Parameter

Cl/FPM

(L/h) FE

V/FPM

(L) k12 rprop,1 radd,2

Estimate 769 0.0856 6874 35.87 0.16 109

BSV (%) 0.09 (30) 0.11 (32) 0.07 (27) – – –

0 20 400

1000

2000

3000Horse 1

Time (h) Time (h) Time (h) Time (h)



Cum

. am

ount

(µg)

Cum

. am

ount

(µg)

Cum

. am

ount

(µg)

Cum

. am

ount

(µg)

Cum

. am

ount

(µg)

Cum

. am

ount

(µg)

Cum

. am

ount

(µg)

Cum

. am

ount

(µg)

Cum

. am

ount

(µg)

Cum

. am

ount

(µg)

Cum

. am

ount

(µg)

Cum

. am

ount

(µg)

0 20 400

2000

4000Horse 2

0 20 400

1000

2000

3000Horse 3

0 20 400

1000

2000

3000Horse 4

0 20 400

2000

4000Horse 5

0 20 400

500

1000

1500Horse 6

0 20 400

2000

4000

6000Horse 7

0 20 400

1000

2000Horse 8

0 20 400

1000

2000

3000Horse 9

0 20 400

1000

2000

3000Horse 10

0 20 400

1000

2000

3000Horse 11

0 20 400

1000

2000Horse 12

Fig. 6. Individual fitted values for cumulative urine amounts of HEPS. The solid lines represent the predicted values, and the dots represent the observed

data points.



concentrations, all plots appear reasonable given the assumption

of normally distributed residuals. The QQ plot shows discrepancy

between the standardized residuals and the fitted normal density

at the lower tail, but this was not deemed significant. Similarly,

the residual plots for cumulative urine amounts generally do not

violate the assumption for normality. Again, discrepancies

appear in the QQ plot, but at both tails.

Posterior predictive checks were used to determine whether the

model developed was consistent with the data. The checks were

performed in the following way. After fitting the final model, each

parameter has a posterior distribution. This distribution summa-

rizes the uncertainty about the estimate (given the prior and the

observed data). Ten thousand random samples were drawn from

the posterior distribution, and the concentration (lg/L) at time

t hours was simulated (based on each of the 10 000 random

samples). This produced a distribution of concentrations (lg/L) at

each time point considered. The 5th, 50th and 95th percentiles of

these distributions can be seen in Fig. 8, with the observed data as

dots. The two extreme percentiles reflect the uncertainty about the

estimated concentrations.

The plots show that some of the observations fall outside the

percentiles. This is to be expected, as in theory approximately

10% of the data points should fall outside the limits given by the

90% credible interval. For the plasma concentrations, the

uncertainty below the median seems to increase with time. This

is not surprising given that less data were available at later time

points as the concentrations of HEPS in some samples fell below

the LOD. Further, the plot is on the log-scale, so these actually

relate to very low concentrations.

For the urine data, the upper bound of the 90% credible intervals

extends noticeably higher than the observed data. This suggests

that BSV and/or MEV may be inflating estimates of FE leading to a

larger than expected amount of HEPS being excreted renally.

Thus, detection times may be longer than expected. Overall, both

posterior predictive checks seem consistent with the observed data.

Withholding and detection times

The detection time for a therapeutic substance represents the

period after administration that the laboratories can identify the

substance or its metabolite in either blood or urine. The

withholding time is usually judged by a veterinarian and

represents the period between drug administration and a race,

taking into account the published detection time and a number

of other factors.

Toutain (2010) used Monte Carlo methods to estimate

withholding times by extrapolating the detection times published

by the European Horserace Scientific Liaison Committee. The

methodology is based on the terminal phase or single exponential

model for elimination and relies on estimating a plasma-to-urine

0 10 20 30–2

–1

0

1

2(a)

Time (h)

log

Con

cent

ratio

n (µ

g/L)

0 10 20 30 40 500

1000

2000

3000

4000

5000(b)

Time (h)

Cum

ulat

ive

amou

nt (µ

g)

Fig. 8. Posterior predictive check for plasma

concentrations and urine amounts of the

acepromazine metabolite HEPS in 12 horses.

The dots represent actual observations. The

solid line represents the model, and the dashed

lines represent a 90% credible interval for the

model predictions.

0 2 4–4

–2

0

2

4Residuals vs. predicted

–2 0 2–4

–2

0

2

4QQ−plot

–4 –2 0 2 40

20

40

60Histogram of residuals

–1 0 1 2–1

0

1

2

Observed. vs. predicted

0 2000 4000–4

–2

0

2


–2 0 2–5

0

5

10QQ−plot

–5 0 50

20

40

60

80Histogram of residuals

0 2000 40000

1000

2000

3000

4000

Observed. vs. predicted

Fig. 7. Residual plots for HEPS [top row refers to plasma concentration (lg/L), and bottom row refers to cumulative urine amount (lg)]

including standardized residuals vs. predicted values, normal QQ plot of the standardized residuals, histogram of the standardized residuals and

observed vs. predicted values.



concentration ratio parameter. An important feature of this

research was the identification of the most influential parameters

through a sensitivity analysis. This revealed that the manner in

which the body processed the compound determines the

detection time rather than, for example, dose.

Given we have undertaken a full compartmental analysis

under a Bayesian framework, we provide an alternative

approach for the estimation of withholding times. This approach

will allow withholding times to be estimated from both plasma

and urine samples and allows for the practical consideration that

a horse will empty their bladder.

Probability of detection in plasma. Suppose we are interested in

determining how likely it is for a horse to have a HEPS

concentration in plasma greater than, say, the LOD of 1 lg/L at

a certain time t after being given an intravenous bolus dose of

30 mg of ACP. Given the population model developed in this

research, this can be resolved by considering the simulated data

from the posterior predictive check. In the posterior predictive

check, random samples are drawn from the posterior distribution

of parameters and data generated for some time points (t). Thus,

the probability of a horse having a HEPS concentration greater

than, say, the LOD at time t can be determined from the

simulated data. For a given time t, the number of simulated

observations that are greater than the LOD is simply counted,

then divided by the total number of simulated observations at

that particular time (t). This provides an estimate for the

probability that a concentration will be larger than the LOD. A

plot of these estimated probabilities can be seen in Fig. 9.

Figure 9 shows three plots. Figure 9a shows the estimated

probabilities of detection based on the population estimates of

parameters and therefore represents estimates for a typical horse

in the sample. Alternatively, Fig. 9b shows the estimated

probabilities of detection for the individual horses in the study.

The second plot shows how the probability of detection varies

among individual horses. This highlights the potential for

extreme detection probabilities for specific horses and the

sensitivity of the estimates to the horses studied. The last plot

is an empirical estimate of a survivor type function. The plot

shows the estimates (with 95% confidence bounds) of the

probability that a detection time is larger than some time t. For

example, it shows that at 30 h after ACP administration, while

10% of the horses would be expected to have a concentration of

HEPS in plasma greater than the LOD, in practice this percentage

may be as small as 0% or as large as 25%.

Probability of detection in urine. Assume we are interested in

determining how likely it is for a horse to have a HEPS

concentration in urine greater than, say, the LOD of 1 lg/L at a

certain time (t) after being given an intravenous bolus dose of

30 mg of ACP. The model for HEPS given in section ‘Model’

predicts the accumulated urine amount of HEPS at a given time

t. In order to determine detection times, an estimate of urine

volume or accumulated urine volume at time t needs to be made.

Figure 10 shows the observed cumulative urine volumes for

all horses against time. A strong linear relationship is evident

between cumulated volume and time, with the variability in the

response increasing with time. Therefore, a linear mixed effects

model with proportional residual error was fitted to the

cumulative urine volume with time as the independent variable.

Further details are given in Appendix A.

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

Time (h)

Pro

babi

lity

of d

etec

tion

(a)

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

Time (h)

Pro

babi

lity

of d

etec

tion

(b)

15 20 25 300

0.2

0.4

0.6

0.8

1

Time (h)

Pro

porti

on

(c)

Fig. 9. Probability plots for the detection of the acepromazine (ACP) metabolite HEPS in plasma after the intravenous dose of 30 mg of ACP; (a)

population estimate of the probability of detection, (b) individual horse estimate of the probability of detection and (c) estimated survivor type function

for detection times.

0 10 20 30 40 50 600

0.5

1

1.5

2

2.5x 104

Time

Cum

. vol

ume

Fig. 10. Observed cumulative urine volumes against time.



It is now possible to predict the cumulative amount of HEPS

and cumulative urine volume for a given time t. Given that

detection in urine is based on the concentration, these cumu-

lative predictions need to be combined and converted to

concentrations. One solution is to consider, say, 4-h windows

from the time of drug administration, and the amount of HEPS

and urine volume accumulated in these windows can be used to

predict concentration. This approach assumes the following.

• The sample is representative of the population.

• The model developed describes the population.

• A linear relationship exists between urinary excretion rate of

HEPS and the production rate of urine.

• The urinary production and excretion rate on race day (and/or

days before race day) are the same as production and

excretion rates during the study.

• The bladder is completely emptied at the beginning of each 4-h

interval.

• Urine is collected at the end of a window.

Data were simulated from the cumulative urine amount and

volume models, and the probability of detection was estimated in

the same way as plasma (described above). Results can be seen in

Fig. 11.

Again, Fig. 11 shows three plots. Figure 11a shows the

estimated probability of detection in urine given a 4-h window

0 50 100 1500

0.1

0.2

0.3

0.4

0.5

Time (h)

Pro

babi

lity

of d

etec

tion

(a)

0 50 100 1500

0.1

0.2

0.3

0.4

0.5

Time (h)

Pro

babi

lity

of d

etec

tion

(b)

45 50 55 60 65 700

0.2

0.4

0.6

0.8

1

Time (h)

Pro

porti

on

(c)

Fig. 11. Probability plots for the detection of the acepromazine (ACP) metabolite HEPS in urine after the intravenous dose of 30 mg of ACP based on

4-h windows; (a) population estimate of the probability of detection, (b) individual horse estimate of the probability of detection and (c) estimated

survivor type function for detection times.

0 10 20 30 40 500

0.5

1

1.5

2 x 104

Time

Cum

. v

Horse 1

0 10 20 30 40 500

0.5

1

1.5

2

2.5

Time

Cum

. v

Horse 2

0 10 20 30 40 500

5000

10000

15000

Time

Cum

. v

Horse 3

0 10 20 30 40 500

0.5

1

2

1.5

Time

Cum

. vHorse 4

0 10 20 30 40 500

0.5

1

1.5

2

Time

Cum

. v

Horse 5

0 10 20 30 40 500

5000

10000

15000

Time

Cum

. v

Horse 6

0 10 20 30 40 500

0.5

1

1.5

2

Time

Cum

. v

Horse 7

0 10 20 30 40 500

0.5

1

1.5

2

Time

Cum

. v

Horse 8

0 10 20 30 40 500

0.5

1

1.5

2

Time

Cum

. v

Horse 9

0 20 40 600

0.5

1

1.5

2

Time

Cum

. v

Horse 10

0 10 20 30 40 500

0.5

1

1.5

2

Time

Cum

. v

Horse 11

0 10 20 30 40 500

0.5

1

1.5

2

Time

Cum

. v

Horse 12

x 104 x 104

x 104 x 104

x 104

x 104

x 104 x 104 x 104

Fig. A1. Individual fits for cumulative urine volume model.



for a typical horse in the sample. The plot shows that there is

quite a small probability of detection past 50 h, for example. This

agrees well with the observed and predicted data shown in

Fig. 8b as it appears that very little HEPS is accumulated in urine

past this point in time. Figure 11b shows the estimated

probabilities of detection based on individual estimates of PK

parameters. These estimates relate to the individual horses in the

study. The final plot shows an empirical estimate of a survivor

type function. Detection times longer than 70 h appear rare

given our best estimate. However, when calculating appropriate

withholding times, the uncertainty around this estimate should

also be considered. Thus, while only 10% of horses might be

expected to breech the detection level at 65 h, this estimate

could be between 0% and 25% for an individual horse.

CONCLUSION

A BHM was considered for the description of the metabolism of

the parent–metabolite process for ACP. Given the half-life of ACP

is much smaller than that of the metabolite, the model was

reducible to the consideration of the metabolite only. The

structural form of this model for the description of metabolite

concentration in plasma and cumulative amount in urine was

based on mass balance with vague priors chosen for the

estimable parameters. No covariates were found to be statisti-

cally significant. This may be attributed to the small sample of

horses measured and/or relatively narrow ranges of potential

covariates, particularly for weight. Measurement variability on

plasma concentrations proved to be influential, and random

effects were included in the model to account for this.

Simulation techniques were used to show agreement between

predicted and observed data and in the estimation of detection

times for a typical horse in the sample. Detection in urine was

complicated by the need to allow the horses to empty their

bladder at various times after ACP had been administered. This

was resolved by considering urine concentration in 4-h windows

(postdose). Care should be taken when interpreting these results

as the probability profiles are likely to change if different sized

windows are considered. Nevertheless, the results shown should

provide guidance for trainers and veterinarians to estimate

appropriate withholding times to ensure that horses are not

racing with prohibited substances in their system.

All inferences in this study were based on a sample of twelve

horses. In making decisions in the model-building phase and

providing estimates of, for example, clearance and withholding

times, we assume that this sample is representative of the entire

population. Indeed, in the selection of covariates, weight seemed

somewhat linearly related to the volume of distribution. This

relationship did not prove to be statistically significant, but this

may have been due to the small sample size. Hence, care should

be taken when drawing any inferences from this work.

ACKNOWLEDGEMENT

We are very grateful to Prof. S.B. Duffull of the University of

Otago for his input into the paper. We would also like to thank

the two referees for their comments and suggestions.

REFERENCES

Ballard, S., Shults, T., Kownacki, A.A., Blake, J.W. & Tobin, T. (1982)

The pharmacokinetics, pharmacological responses and behavioral

effects of acepromazine in the horse. Journal of Veterinary Pharmacology,

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0 0.5 1 1.5 2x 104

–5

0


–4 –2 0 2 4–4

–2

0

2

4

6QQ−plot

–4 –2 0 2 4 60

10

20

30

40Histogram residuals

6 7 8 9 106

7

8

9

10Obs. vs. pred

Fig. A2. Residual plots for cumulative urine volume model.

0 10 20 30 40 50 600

0.5

1

1.5

2

2.5

3x 104

Time (h)

Cum

. V. (

mL)

Fig. A3. Posterior predictive check for cumulative urine volume model

where ‘Cum. V’ represents the cumulative amount of urine collected from

the 12 horses over the duration of the study.



rmarthur

Highlight

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APPENDIX: MODEL FOR CUMULATIVE URINE VOLUME

The individual fits can be seen in Fig. A1. From the plot, the

proposed model seems to fit the observed data well for all horses.

The residual plots and posterior predictive check can be seen

in Figs A2 & A3, respectively. The posterior predictive check

also shows the 90% credible interval for the predicted response.

The QQ plot, histogram and observed vs. predicted plots show

that the residuals generally follow a normal distribution. There

is a slight pattern in the residual vs. predicted plot, but this

was not deemed strong enough to violate our assumption

about the residuals. The posterior predictive check shows

agreement between the simulated and observed data and

shows that some observations lie outside the 90% credible

interval.



a bayesian approach for estimating detection times in horses:...

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