Analyzing pulsatile hormone dynamics using the hormoneBayes framework

The hormoneBayes framework allows inference of key physiological parameters describing pulsatile hormone dynamics. The framework utilizes stochastic mathematical models describing circulating hormone levels and state-of-the-art Bayesian machinery to calibrate these models to data of hormone profiles and infer model parameters. Fig 1 presents a parsimonious model (a simple model with great explanatory power) describing circulating LH levels. The model assumes that there are two modes of LH secretion, namely pulsatile and basal. The former corresponds to an on/off signal that randomly switches between two states (corresponding to a high and a low activity) while the latter corresponds to a continuous noisy signal. Furthermore, the model incorporates LH clearance as a linear first order process leading to an exponential decay of LH levels following a pulse [3]. We note that more detailed clearance models, such as the bi-exponential model [3], could be easily integrated. By considering the processes of LH release and clearance, the model predicts LH circulating dynamics in terms of five key parameters that can be recovered from data: 1. LH clearance rate; 2. maximum LH release rate; 3. strength of the pulsatile signal relative to the basal signal; 4. mean time in the on state; 5. mean time in the off state. Moreover, the model incorporates measurement error as an additional parameter that is determined based on the assay coefficient of variation (CV).

HormoneBayes relies on the Bayesian paradigm to extract information from the observed data. Using the Bayes theorem, the method revises our prior beliefs regarding model parameters by transforming the parameters’ prior probability density distributions into posterior distributions. Parameter prior distributions enable the user to input context-specific information into the analysis, hence enhancing the flexibility of the method to handle different datasets. For example, when dealing with data from post-menopausal women the user could choose to adjust the parameter priors to acknowledge a higher LH secretion rate and/or more sustained basal secretion. Fig 1 illustrates how the Bayesian machinery allows us to calibrate the model to the data and extract information regarding model parameter and latent hypothalamic signals with an estimate of certainty. As we explain in the sections to follow, this information can be used to identify pulses; summarise the data (e.g., providing mean and standard deviation estimates); and perform statistical tests.

Identifying LH pulses

Using data to infer the latent hypothalamic signal provides a transparent way to identify pulses based on their likelihood under the model. As explained above, the model assumes LH pulses are partly driven by an on/off hypothalamic signal. This latent variable (i.e., inferred variable) takes two values indicating the ‘on’ (1) and ‘off’ (0) state of the hypothalamic pulse generator, and therefore the expected posterior estimate (inferred from LH profiling data) can be interpreted as the probability that at any given time the hypothalamic pulse generator is ‘on’. This quantitative measure for accessing the likelihood of a pulse can significantly ease the job a clinician trying to decide whether an upstroke in the LH profile represents a pulse or not. Fig 2 illustrates a representative example of an LH trace with two obvious pulses occurring at times 100min and 300min. These are indeed identified by inspecting the hypothalamic signal profile which peaks at around those times. At time 200min a less pronounced bump in LH could be indicative of an LH pulse, however the inferred pulsatile hypothalamic signal remains well below 0.5, indicating that under the current model there is higher probability that the bump is a measurement artefact rather than a pulse.

To validate our pulse identification method, we used a database of LH profiles obtained from healthy pre-menopausal women and compared the identified pulses with those previously obtained by the deconvolution method [19], which uses a maximum likelihood approach to fit a series of pulses to the data [3]. Here, we identify a pulse when the posterior probability that the hypothalamic signal is ‘on’ exceeds a threshold value. We use 0.5 as the threshold value, which signifies there is higher probability that the hypothalamic pulse generator is on (rather than off). This value yields the highest agreement between hormoneBayes and the deconvolution method (see Fig D in S1 Text). We find that hormoneBayes agrees with the deconvolution method in 77 out of 87 identified pulses (89%). Moreover, 13 pulses identified by hormoneBayes were not identified by the deconvolution method. Overall, this suggests a good agreement between the two methods, with hormoneBayes having the added advantage of providing a measure for the likelihood of each pulse that clinicians and researchers can use to inform their clinical decision making.

Variation of model parameter within and across groups

To test the applicability of hormoneBayes in different contexts we compile a database of LH profiles from four groups with diverse LH dynamics (men, healthy pre-menopausal women with regular menstrual cycles, post-menopausal women, women with HA, and women with PCOS). As illustrated in Fig 3, the model successfully captures the differences in LH dynamics in all four groups. Moreover, the model allows us to summarize LH data through model parameters and assess the variability across and within groups. We find that two model parameters explain most of the variability between groups, namely the maximum secretion rate and pulsatility strength (Fig 3C). The first parameter describes how much LH can be secreted over time, whilst the second parameter quantifies the strength of the pulsatile signal relative to the basal signal. Based on these two parameters there is a strong distinction between women with PCOS and women with HA, who have lower LH secretion rates and/or diminished LH pulsatility strength (i.e., pulsatile signal is weak relative to the basal signal). Furthermore, post-menopausal women display higher secretion rates compared to pre-menopausal women but also reduced pulsatility strength (i.e., pulses are less pronounced when higher level of LH are established post menopause). Interestingly, healthy men and women illustrate a much lower parameter variability as compared to HA and post-menopausal women, which could be indicative of various degrees of severity of HA/PCOS and tighter LH regulation in healthy individuals of reproductive age.

Ethics statement

Data included in this manuscript were obtained from five different clinical research studies, involving healthy men [25], healthy pre-menopausal women [19], post-menopausal women [26], women with PCOS and women with HA [8]. Ethical approval for these studies was granted by: the Hammersmith and Queen Charlotte’s and Chelsea Hospitals Research Ethics Committee (registration number 05/Q0406/142) [8,19]; the UK National Research Ethics Committee-Central London (Research Ethics Committee number 14/LO/1098) [25]; and the West London Regional Ethics Committee (15/LO/1481) [26]. Written informed consent was obtained from all subjects. All studies were conducted according to Good Clinical Practice Guidelines.

Data collection

Participants attended a clinical research facility for 8 hour study visits hat included baseline (vehicle treatment) LH measurements according to the relevant trial protocol as previously described [8,19,25,26]. A cannula was inserted into a peripheral vein under aseptic conditions (time at least -30 minutes), through which all subsequent blood samples were taken every 10 minutes from time 0 until 480 minutes. All participants were ambulatory and could eat and drink freely during the study visit. All blood samples were left to clot for at least 30 minutes prior to centrifugation at 503 rcf for 10 minutes, after which the serum supernatant was extracted and immediately frozen at -20°C prior to subsequent analysis using an automated chemiluminescent immunoassay method (Abbott Diagnostics, Maidenhead, UK) in batches after study completion. Reference ranges were as follows: LH 4–14 IU/L; respective intra-assay and inter-assay coefficients of variation were 4.1% and 2.7%; analytical sensitivity was 0.5 IU/L.

Stochastic model of LH

We used a discrete-time, stochastic model to describe pulsatile LH dynamics. The model comprises of three dynamical variables, P_t, B_t, and LH_t, that describe the pulsatile and basal hypothalamic signals and the LH concertation in circulation, respectively.

The pulsatile hypothalamic signal P_t can take two values: H_t = 0 corresponding to the ON state; and H_t = 1 corresponding to the OFF state. The stochastic dynamics of H_t are governed by the following probability matrix

i.e., parameters τ_ON and τ_OFF govern the probabilities that the value of H will either flip or remain the same over the time interval (s, s+δt).

The evolution of the basal hypothalamic signal, B_t, is described using a discrete time autoregressive model obeying the following equation where ε_t is a normally distributed random variable with zero mean and unit variance. Note that both B_t and H_t are bounded in the interval [0,1].

The two hypothalamic signals drive LH secretion, and along with LH clearance dictate the circulating LH levels, LH_t. The equation describing the time evolution of LH_t, is where d denotes the clearance rate, k denotes the maximum secretion rate, and parameter f (termed pulsatility strength) describes the relative strength of the two hypothalamic signals. Finally, the model assumes measurement error in the form: where η_t is a normally distributed random variable with zero mean and std. deviation equal to the CV of the assay. Throughout our analysis we have used δt = 1 min, hence, assuming the system dynamics do not change significantly over shorter times.

Bayesian inference

The hormoneBayes framework uses Bayesian inference to obtain model parameters Θ = (τ_ON, τ_OFF, k, d, f) and latent variable (H_t, B_t) dynamics from LH profiling data D. In particular, hormoneBayes solves the inference problem by sampling from the target posterior distribution: where P(Θ) is the prior parameter distribution; is the likelihood of the data given the parameters; and is the marginal likelihood or model evidence.

Sampling from the full posterior distribution is performed using a Gibbs sampler, which is an iterative Monte Carlo Markov Chain (MCMC) scheme. The algorithm is initialised with parameter values drawn from the prior distribution, i.e., Θ⁰~P(Θ) and each subsequent iteration, i = 1,…,M involves two steps: (1) sampling latent variables (H_t, B_t)ⁱ given the data, D, and the current parameter values Θⁱ⁻¹ and (2) sampling new parameter values Θⁱ given D and the latent variables (H_t, B_t)ⁱ. The first step is performed using Sequential Monte Carlo (SMC) with ancestral sampling [27]. The second step is further broken down into two parts, first parameters (τ_ON, τ_OFF) are sampled using an adaptive Metropolis-Hastings sampler and then parameters (k, d, f) are sampled using the simplified version of the manifold Metropolis adjusted Langevin algorithm (sMMALA) presented in [28].

For the analysis of all datasets in this study we considered the following independent prior distributions: and , based on the sampling rate (10min) and duration (480min) used in the LH profiling studies; , set as a broad uninformative prior; , based on LH half-life data; and log(f)~U(0,1), to accommodate analysis of different LH profiles with high or low pulsatility. Evaluation of the algorithm on synthetic dataset can be found in Fig A-C in S1 Text.

HormoneBayes also allows the user to access the effect of pharmacological interventions on LH pulsatility, by fitting in tandem two LH profiling datasets: corresponding to periods before and after the intervention. In this case a composite model is used to allow inference of parameters Θ_c = (τ_ON, τ_OFF, k, d, f), corresponding to the baseline period (before the intervention), and parameters Θ_p = (τ_ON,_i, τ_OFF,_i, k_i, f_i) corresponding to the period after the intervention. Here we assume the intervention does not affect the clearance rate d, hence this parameter does not appear in Θ_p. In mathematical terms the target posterior is now given by and sampling from the posterior is performed as described above. An example of this analysis is Fig E of the S1 Text.

An open access C++ implementation of HormoneBayes accompanied with a graphical interface implemented in Python and a user manual can be found at https://git.exeter.ac.uk/mv286/hormonebayes.