Experimental setup and analytical method
Six Standardbred horses (four mares and two geldings) 6–20 years old and weighing 430–584 kg were included in the study and assigned to a randomised crossover design including four treatments and four periods. Each treatment started with an intravenous bolus dose immediately followed by 3 h of constant rate infusion of dexamethasone 21-phosphate disodium salt (Dexadreson 2 mg mL−1, Intervet AB, Sollentuna, Sweden). The dose levels were (bolus + infusion) 0.1 + 0.07 μg kg−1, 1 + 0.7 μg kg−1 and 10 + 7 μg kg−1 dexamethasone. For the control level 0.9% saline was used. Before the bolus dose (time = 0) a pre-dose blood sample was drawn. Additional blood samples were drawn during and after infusion at hours 1, 2, 3, 4, 5, 6, 9, 12, 18, 24, 36 and 48. A minimum of a 1 week wash-out period was allowed between drug treatments. The study was approved by the Ethics Committee for Animal Experiments, Uppsala, Sweden (C333/11). Total plasma dexamethasone and cortisol concentrations were analysed and quantified using Ultra High Performance Liquid Chromatography-Tandem Mass Spectrometry (UHPLC-MS/MS). The analytical method was described before elsewhere [21].
Dexamethasone exposure
A two-compartment model (Eq. 1, Fig. 1a) was fitted to experimental dexamethasone-time course data.
$$\left\{{\begin{array}{ll}V_{c} \frac{{\text{d}}C_{p}}{\text{d}t} = \text{Inf}(t) - Cl \cdot C_{p} + Cl_{d} \left( C_{t} - C_{p} \right), & C_{p} (0) = \frac{\text{D}}{V_{c}} \\{V_{t} \frac{{{\text{d}}C_{t} }}{{{\text{d}}t}} = Cl_{d} \left( {C_{p} - C_{t} } \right),} & C_{t} (0) = 0\end{array}}\right.$$
(1)
Cp and Ct denote drug concentration in central (plasma) and peripheral compartments. Vc, Vt, Cl and Cld denote, respectively, the central and peripheral volumes, plasma clearance and inter-compartmental distribution parameter. Inf(t) represents the constant rate infusion regimen and D is the bolus dose administered at time t = 0.
Cortisol turnover
Cortisol was modelled by a turnover model (Eqs. 2–4, Fig. 1b).
$$\frac{{{\text{d}}R}}{{{\text{d}}t}} = k_{\text{in}} (t) \cdot I(t) - k_{\text{out}} R,\quad R(0) = R_{\text{eq}} (t = 0,C_{{p,\,{\text{eq}}}} = 0)$$
(2)
R is cortisol concentration, kout the fractional turnover rate and Req stands for the expression in Eq. 5. The oscillatory behaviour of turnover rate was modelled by Eq. 3. Note that in the following ω = 2π/24 h−1.
$$k_{\text{in}}(t) = k_{\text{avg}} \cdot
\left(1 + \alpha \cdot \cos\left(\omega\, (t - t_0)\right) \right)$$
(3)
kavg is positive and corresponds to the average turnover rate. t0 is the phase shift between − 12 and 12 h, and α, a number between 0 and 1, describes the amplitude of the oscillations as a proportion of kavg. Choosing α this way ensures positivity of the turnover rate for all choices of parameters. The period was fixed at 24 h. The inhibitory dexamethasone mechanism function was modelled as
$$I(t) = 1 - \frac{{I_{ \text{max} }\,C_{p}^{n} (t)}}{{IC_{50}^{n} + C_{p}^{n} (t)}}\,,$$
(4)
where Imax is maximum inhibitory capacity, IC50 the potency of dexamethasone and n is a Hill exponent. This model is a modified version of the single cosine model presented in [26].
Cortisol concentration under constant dexamethasone exposure
An oscillating turnover rate leads to oscillating cortisol concentration. Keeping dexamethasone exposure in Eq. 4 constant at a fixed concentration Cp, eq, the cortisol response is given by
$$R_{\text{eq}} (t,C_{{p,\,{\text{eq}}}} ) = A + B \cdot { \cos }\left( {\omega\, (t - C)} \right)\,,$$
(5)
where
$$\begin{aligned}
A & =
\frac{k_{\text{avg}}}{k_{\text{out}}}
\cdot \left(1 -
\frac{I_{\text{max}}\,C_{p,\,\text{eq}}^n}{IC_{50}^n + C_{p,\,\text{eq}}^n }
\right)\,, \\
B & =
\frac{k_{\text{avg}} \cdot \alpha}{\sqrt{k_{\text{out}}^2 + \omega^2}}
\cdot \left(1 -
\frac{I_{\text{max}}\,C_{p,\,\text{eq}}^n}{IC_{50}^n + C_{p,\,\text{eq}}^n }
\right)\,, \\
C & = \frac{1}{\omega}\,\arctan
\left(
\frac{k_{\text{out}}\,\sin(\omega t_0) + \omega\,\cos(\omega t_0)}{k_{\text{out}}\,\cos(\omega t_0) - \omega\,\sin(\omega t_0)} \right)\,.
\end{aligned}$$
(6)
A describes the average cortisol response, B the amplitude and C the phase shift of the oscillation. The model predicts only changes in the average cortisol response and amplitude due to changes in dexamethasone exposure. A derivation of Req in Eq. 5 as well as for A, B and C in Eq. 6 can be found in the Appendix. The ideas are similar to the calculations presented in Krzyzanski et al. [27].
Residual error variance model
Kinetic data was modelled on a log scale. For the drug exposure model in Eq. 1 it was assumed that
$${ \log }\left( {C_{p} \left( {t_{ij} } \right)} \right) = { \log }\left( {\widehat{{C_{p} }}\left( {t_{ij} } \right)} \right) + e_{ij} .$$
(7)
For the cortisol response model in Eq. 2 a combined error model with proportional and additive error was assumed. This was described by
$$R\left( {t_{ij} } \right) = \hat{R}\left( {t_{ij} } \right)\left( {1 + s_{ij}^{(1)} } \right) + s_{ij}^{(2)} .$$
(8)
Here, Cp(tij) and R(tij) are the jth measurement of the plasma concentration of dexamethasone in the central compartment and cortisol, respectively, measured for subject i at time point tij. \(\widehat{{C_{p} }}(t_{ij} )\) and \(\hat{R}(t_{ij} )\) are the predicted concentrations for subject i at time point tij. eij as well as \(s_{ij}^{(1)}\) and \(s_{ij}^{(2)}\) were assumed to be normally distributed with zero mean and respective standard deviations ε as well as \(\sigma_{1}\) and \(\sigma_{2}\).
Statistical parameter model
IIV was modelled by making the following assumptions about the distribution of the parameters in Eqs. 1–4. The process of deciding which parameters were modelled with correlation is described in the Supplementary.
All parameters involved in the description of dexamethasone exposure were modelled independently log-normally distributed, i.e.,
$$\begin{aligned}
\log(\theta) & \sim {\text{Log-Normal}}\,(\mu ,\tau^{2})\,, \\
\mu & \sim {\text{Normal}}\,(m_{0} ,1)\,, \\
\tau & \sim {\text{Student-t}}\,(4,s_{0} ,0.25)\,,
\end{aligned}$$
(9)
where m0 and s0 are prior parameters and θ stands for Cl, Cld, Vc and Vt. Some parameters in the cortisol turnover model were modelled with correlations as
$$\left( {\begin{array}{*{20}c} {{ \log }\left( {k_{\text{avg}} } \right)} \\ {{ \log }\left( {k_{\text{out}} } \right)} \\ {{ \log }\left( {IC_{50} } \right)} \\ {{\text{logit}}\left( \alpha \right)} \\ {{\text{logit}}\left( {\frac{{t_{0} + 12{\text{h}}}}{{24{\text{h}}}}} \right)} \\ \end{array} } \right) \sim {\text{Normal}}\left( {{\varvec{\upmu}},\,{\varvec{\Omega}}} \right),$$
(10)
where Ω = LDLT, \(\varvec{D} = \text{diag}(\tau_{1}^{2} ,\;\tau_{2}^{2} ,\;\tau_{3}^{2} ,\;\tau_{4}^{2} ,\;\tau_{5}^{2} )\) and L is a lower-triangular matrix. In this representation the matrix D contains the variances and LLT is the correlation matrix. In addition
$$\begin{aligned}
n &\sim {\text{Normal}}\,(\mu_n ,\tau_n)\,, \\
I_{\text{max}} &\sim {\text{Logit-Normal}}\,(\mu_{{I_{ \text{max} } }} ,\tau_{{I_{ \text{max} } }} )\,.
\end{aligned}$$
(11)
Hyperparameters μ in Eqs. 10, 11 and the diagonal elements of D as well as τ in Eq. 11 are distributed as
$$\begin{aligned}
\mu &\sim {\text{Normal}}\,(m_0, v)\,, \\
\tau &\sim {\text{Student-t}}\,(4, s_0, 0.25)\;,\;\;\tau \geq 0\,.
\end{aligned}$$
(12)
where \(m_{0}\) and \(s_{0}\) are prior parameters, v = 2.5 for hyperparameters related to α as well as t0, and v = 1 otherwise. The three-parameter Student-t distributions used in Eqs. 9, 12 for non-negative τ are truncated distributions.
The correlation matrix LLT was assumed to be distributed following a LKJ distribution [28] with concentration parameter 2. This is a prior for correlation matrices where samples resemble the identity matrix more closely for concentration parameters closer to 1. Residual-error-model standard deviations ε and \(\sigma_{1}\) and \(\sigma_{2}\) were assumed to be positive and were given half-Cauchy prior distributions [29] with scale 2.5. Prior parameters were estimated from a meta-analysis of Ekstrand et al. [12] as described in the Appendix.
Analysis of the dexamethasone suppression test protocol
We simulated two different overnight DST protocols. Each consisted of a dexamethasone administration time and a sampling time on the following day. Cortisol concentration is analysed in the sampled blood plasma and the result of the DST is positive if concentration is above a prescribed threshold. The protocols analysed more closely are described in Dybdal et al. [1] (protocol A) and Frank et al. [11] (protocol B). Both protocols assume administration of 40 μg kg−1 of dexamethasone. The protocols differ in administration route. Protocol A assumes intramuscular (im) administration whereas protocol B assumes intravenous (iv) administration. Test starting times were at 9.00 a.m. (protocol B) and 5 p.m. (protocol A). Plasma sampling times for determination of cortisol concentration were after 19 h (protocol A) and 24 h (protocol B), respectively. In both protocols, the test is positive (indicating sick individuals) if measured cortisol concentration is above a threshold of 10 μg L−1.
The DST protocols were analysed in light of two different aspects. First, we performed a Monte Carlo study to visualise cortisol time courses for horses subjected to each protocol. A sample of 10,000 horses was simulated from the adjusted model. For this, residual variance parameters and hyper-parameters (N = 1000) were taken from the estimated posterior parameter distribution. Then, individual parameters (N = 10) were simulated from hyper-parameters and the distributions in Eqs. 9–11. Dexamethasone- and cortisol time courses were simulated under the two test protocols for the new subjects using Eqs. 1–4 as well as the measurement equation (Eq. 8). The investigated protocols assume administration of 40 μg kg−1 dexamethasone and the aim of this simulation was to determine whether this amount is necessary or if lower doses could be sufficient. Predicted cortisol concentration at sampling time was then used for further analysis.
These concentrations were then used in a second step to investigate both the sensitivity of the test, i.e., the probability that the test is positive for a sick subject, as well as the specificity of the test, i.e., the probability that the test is negative for a healthy subject [30]. The distributions of sensitivity and specificity were simulated through a combination of Monte Carlo and analytical steps. See the Appendix for the formulas used. In horses with PPID the mechanism for dexamethasone suppression of cortisol is challenged [6]. To quantify sensitivity, simulations from sick horses were needed. We hypothesized that dexamethasone has no suppression effect on sick individuals and therefore these horses were sampled at baseline. The studies reporting protocol A and B [1, 11] determined sensitivity and specificity experimentally and this analysis aimed to investigate if model predicted and experimentally determined values are aligned.
Numerical analysis and parameter estimation
The software Stan version 2.18.0 [31] was used for parameter inference through the interface CmdStan. Stan implements the NUTS sampler [32] that uses Hamiltonian Monte Carlo (HMC) [33] for estimation of the posterior parameter distribution and allows models with differential equations. PK and PD parameters were estimated in two stages. First, PK parameters in Eq. 1 were estimated. Each individual’s PK parameters were then summarised and fixed to the respective conditional mean. In a second stage, the PD parameters in Eqs. 2–4 were estimated. In each stage, four Markov chains were started at random initial parameters around the prior parameter means. Each chain was run for 250 iterations in warm-up and sampling, respectively. This led to a total of N = 1000 samples from the posterior.
The convergence of the HMC algorithm was checked in multiple ways. Numerical divergences during parameter estimation were observed and appropriate choices about parameter distributions were made and Stan settings were tuned to reduce and avoid divergences [34]. The Gelman–Rubin \(\hat{R}\) statistic [35] and trace plots were used to ensure proper mixing of the Markov chains. The effective sample size [25] was observed to be at least 10% of total samples size (N = 1000). The energy Bayesian fraction of missing information (E-BFMI) [36] was checked to ensure that the parameter space was properly and efficiently explored. No external validation data was available and therefore internal model checking was performed through posterior predictive checks (PPCs) [25]. These visualisations are similar to visual predictive checks (VPCs) [37]. However, PPCs include parameter uncertainty by simulating the response from the full estimated posterior distribution, whereas VPCs omit this. Estimated parameters were summarised by median and 95% credible intervals (CIs) [25]. Population predicted ranges were calculated as described in the Supplementary.