Detecting Regulatory Mechanisms in Endocrine Time Series Measurements

The regulatory mechanisms underlying pulsatile secretion are complex, especially as it is partly controlled by other hormones and the combined action of multiple agents. Regulatory relations between hormones are not directly observable but may be deduced from time series measurements of plasma hormone concentrations. Variation in plasma hormone levels are the resultant of secretion and clearance from the circulation. A strategy is proposed to extract inhibition, activation, thresholds and circadian synchronicity from concentration data, using particular association methods. Time delayed associations between hormone concentrations and/or extracted secretion pulse profiles reveal the information on regulatory mechanisms. The above mentioned regulatory mechanisms are illustrated with simulated data. Additionally, data from a lean cohort of healthy control subjects is used to illustrate activation (ACTH and cortisol) and circadian synchronicity (ACTH and TSH) in real data. The simulation and the real data both consist of 145 equidistant samples per individual, matching a 24-hr time span with 10 minute intervals. The results of the simulation and the real data are in concordance.

Let x i be a concentration time series in which i represents the hormone. In the following sections we will use i = a or i = b to designate the two hormones. Let φ i (t) be the secretion amplitude at time t. The concentration x i (t) at a given time is determined by the previous concentration (x i (t − 1)), the exponential decay (β i ) and the secretion at that time (φ i (t)). A method for estimating β i and φ i (t) from measured series is described in Vis et al [1]. Hormone concentration time series are generated by: In the simulation studies, for all hormones, the decay parameter β i is set such that the half life is 5.5 time units; assuming that the time units are separated by 10 minutes this translates to a half life of 55 minutes. A shorter half life reduces the carry-over of events, when taken to the extreme, the concentration and secretion values will converge. A different half life value will affect AM 1, AM 3 and AM 4, but not AM 2.

S1.2 Calculation of metrics for real measured and simulated data
All presented association metrics are averaged cross-correlation profiles of 100 randomly generated datasets. The model mechanisms are simulated by independently generating 100 time series of two hormones, each at 145 time indices, the averaged result is presented in the figures. For the real data in Figures 5 and 10 the average profiles are based on data for 9 healthy subjects. The presented association measures are averaged over the 9 individual association measures.
The association metrics proposed are based on Pearson correlation which has a skewed sampling distribution. This skewing is lessened by taking the Fisher Z transform of the correlation which means taking the arc tangent hyperbolic [2,3]. The reported values are Fisher Z values that are transformed back to the original domain by taking the tangent hyperbolic. The results of the measured data are reported with the 95% confidence bounds of the mean. These values are calculated on the Fisher Z transformed data and then transformed back to the normal correlation space.

S1.3.1 Rapid activation
The activation mechanism dictates that a secretion pulse in one hormone will give a pulse in the second hormone, but with some delay. The following equations show how the time series data were generated: The η a (t) defines whether secretion occurs in the first hormone (a) at time t based on a Bernoulli distribution with success probability p, and Φ a (t) defines the amplitude of such a pulse. Each element of the pulse vector η a (t) is drawn from a binomial distribution with probability p. The realized secretion φ a (t) activates secretion of the other hormone (b) described by φ b (t + 1), where α is a positive scalar.
The delay of translating a pulse in a to the secretion of a pulse in b may vary. Equation 6 defines such variability resulting in Equation 7 in which ζ denotes the variable delay. As a result, with an equal probability, the pulse translation takes a delay of 1 or 2 time units. A tilde is added toφ b to distinguish it from φ b shown in Equation 5.

S1.3.2 Threshold activation
Concentration profiles of hormone a are generated using Equations 2 to 4 followed by Equation 1. A concentration threshold as a pulse triggering mechanism, is implemented such that when the concentration of a falls below a preset threshold (see Equation 8) this induces a single pulse in b.

S1.3.4 Activation and inhibition
For the sake of argument, the assumption is that a activates b very quickly (τ = 0), the responses in b being proportional (α) to those in a. Additionally, assume that the inhibition is very quick as well. For hormone a pulse and concentration profiles are generated using Equations 2 to 4 followed by Equation 1. Now Equation 2 and 3 are used to generate pulse indices and amplitudes for hormone b. Now Equation 12 describes the pulses induced in b, since the activation is very fast, the secretion of this hormone can already be seen at the same sampling time (that is, no delay).
The inhibition of the secretion of hormone a by hormone b can now be simulated using the following equations,φ a are the secretion values in the inhibited case: Equation 4 defines how the secretion vector is defined from the incidence vector and the amplitude vector. Equation 14 shows an example of inhibition that limits the amplitude by 90% and is active only when Equation 8 is 1 (true).
Inhibition results in lower than normal secretion amplitudes implying that the inhibited vector will be with lower values compared to the uninhibited amplitude vector. Equation 15 shows that behavior in symbolic form, stating that the average of the inhibited amplitudes is always lower than or equal to that of the uninhibited vector. In this simulation study the uninhibited vector is known and the previous statement can be corroborated, however, when the vector is estimated from real data only the inhibited vector is known.
However, the AM2 results for both sets of vectors are identical.
It is therefore concluded that in a system specified by Equations 13 and 11 it is impossible to assess whether ϑ < ∞ which means that inhibition cannot be identified in such an approach.

S1.3.5 Diurnal patterns
To illustrate the association patterns for two unrelated hormones, concentration time series are simulated using two unrelated and different event distributions over the day. The length of the vector of events is 145.
This creates two distinct, but unrelated, diurnal patterns.