Deconvolution of serum cortisol levels by using compressed sensing

Abstract

The pulsatile release of cortisol from the adrenal glands is controlled by a hierarchical system that involves corticotropin releasing hormone (CRH) from the hypothalamus, adrenocorticotropin hormone (ACTH) from the pituitary, and cortisol from the adrenal glands. Determining the number, timing, and amplitude of the cortisol secretory events and recovering the infusion and clearance rates from serial measurements of serum cortisol levels is a challenging problem. Despite many years of work on this problem, a complete satisfactory solution has been elusive. We formulate this question as a non-convex optimization problem, and solve it using a coordinate descent algorithm that has a principled combination of (i) compressed sensing for recovering the amplitude and timing of the secretory events, and (ii) generalized cross validation for choosing the regularization parameter. Using only the observed serum cortisol levels, we model cortisol secretion from the adrenal glands using a second-order linear differential equation with pulsatile inputs that represent cortisol pulses released in response to pulses of ACTH. Using our algorithm and the assumption that the number of pulses is between 15 to 22 pulses over 24 hours, we successfully deconvolve both simulated datasets and actual 24-hr serum cortisol datasets sampled every 10 minutes from 10 healthy women. Assuming a one-minute resolution for the secretory events, we obtain physiologically plausible timings and amplitudes of each cortisol secretory event with R (2) above 0.92. Identification of the amplitude and timing of pulsatile hormone release allows (i) quantifying of normal and abnormal secretion patterns towards the goal of understanding pathological neuroendocrine states, and (ii) potentially designing optimal approaches for treating hormonal disorders.

Figure 1. Estimated Deconvolution of the Experimental Twenty-Four-Hour Cortisol Levels in 10 Women.

Each panel shows the measured 24-hour cortisol time series (red stars), the estimated cortisol levels (black curve), the estimated pulse timing and amplitudes (blue vertical lines with dots) for one of the participants. The estimated model parameters are given in Table 2.

Figure 2. White Gaussian Structure in the Model Residuals of 10 Women.

In each panel, (i) the top sub-panel displays the autocorrelation function of the model residuals in one of the 10 participants; the graph shows that the model captures the dynamics and that residuals are white; (ii) the bottom sub-panel displays the quantile-quantile plot of the model residuals for that participant; the graph shows that the residuals are Gaussian.

Figure 3. Simulated Twenty-Four-Hour Cortisol Levels with Measurement Errors Corresponding to Datasets from 10 Women.

Each panel displays the simulated serum cortisol levels based on pulse patterns in Figure 1 and estimated model parameters and in Table 2 in one of the 10 participants, assuming a zero mean Gaussian measurement error with standard deviation in Table 3. In all simulations the initial conditions are , equals the initial cortisol level of the corresponding participant, and the cortisol levels are recorded every 10 minutes.

Figure 4. Estimated Deconvolution of Simulated Twenty-Four-Hour Cortisol Levels with Different Measurement Errors Corresponding to Datasets from 10 Women.

Each panel shows the simulated 24-hour cortisol time series (blue stars), the estimated cortisol levels (black curve), the simulated pulse timing and amplitudes (blue vertical lines with dots) and the estimated pulse timing and amplitudes (red vertical lines with empty circles) for one of the simulated datasets that each correspond to a participant. The estimated parameters are given in Table 3.