Evolution of Deconvolution Analysis as a Hormone Pulse Detection Method

Michael L. Johnson; Johannes D. Veldhuis

Introduction

It is well known that the serum concentration of some hormones change by orders of magnitude multiple times within each day. Luteinizing hormone (LH), growth hormone (GH), prolactin (PRL), thyrotropin (TSH), and adrenocorticotropic hormone (ACTH) are examples of hormones that exhibit large, short duration fluctuations in concentration. The temporal variation in serum hormone concentration is believed to be a significant portion of the signaling pathway by which endocrine glands communicate with remote target organs (Desjardins, 1981; Urban et al., 1988a; Evans et al., 1992). It is thus important to be able to identify and quantify the pulsatile nature of time series of serum concentrations. Many algorithms have been developed for analysis of such data (Veldhuis and Johnson, 1986; Urban et al., 1988a,b; Evans et al., 1992).

In 1987 we proposed that the temporal shape of endocrine hormone pulses is a convolution integral of

(t)=C0+∫0tS(z)E(t−z)dz,

where C(t) is the concentration of serum hormone at any positive time, t ≥ 0; C0 is the concentration of hormone before any secretion at the first time point; S(z) is the amount of hormone secreted at time z per unit time and unit distribution volume; and E(t − z) is the amount of hormone elimination that has occurred in the time interval t − z (Veldhuis et al., 1987). This convolution process is shown in Fig. 1. The top curve in Fig. 1 depicts the rate of hormone secretion into the serum for a typical hormone pulse. The middle curve depicts a typical elimination pattern for an instantaneous bolus of hormone. This is the so-called instrument response function. The bottom curve in Fig. 1 depicts the convolution integral of the secretion and the elimination, that is, the hormone concentration as a function of time that results from the secretory event shown in the top curve of Fig. 1.

Our definition of the temporal shape of these hormone pulses has provided a significant and different prospective with which to identify and quantify endocrine pulses. With a deconvolution process it is now possible to evaluate the time course of the secretion of hormone into the serum. The temporal nature of the secretion can now be directly used to identify and characterize hormone pulses within the serum.

The deconvolution process is simply an algorithm to separate an observed concentration time series into its component parts, the secretion and the elimination as a function of time. However, the application of deconvolution algorithms to hormone time series data is not simple. Most deconvolution methods were developed for spectroscopic and/or engineering applications where data are abundant and the signal to noise ratio is high (i.e., low experimental uncertainties) (Jansson, 1984). The radioimmunoassays and other assays utilized to evaluate hormone concentrations in blood allow fewer observations and provide a substantially lower signal to noise ratio (i.e., significantly higher experimental uncertainties).

For the spectroscopic and/or engineering applications the convolution integral provides the observed signal as a combination of the actual signal and an instrument response function. Generally for the spectroscopic and/or engineering applications the instrument response function is easily characterized and defined, and thus the deconvolution process is comparatively simple. The analogous quantities for the hormone case are the observed concentrations, which are a convolution integral of the secretion (the actual signal) and the elimination (the bodily response function). The rate of hormone elimination is, of course, both hormone and patient specific and within a given patient can be a function of many variables such as diet and the presence of various drugs. Thus the deconvolution process for the hormone case is comparatively more difficult because both the secretion and elimination processes must be estimated simultaneously.

The purpose of this chapter is to outline and define the methodologies and algorithms that we have developed for the deconvolution analysis of hormone concentrations series. We also describe how the deconvolution process can be utilized as a pulse detection algorithm.

Deconvolution Methods

Since we introduced the concept of deconvolution analysis (Veldhuis et al., 1987) we have developed three different deconvolution algorithms for the evaluation of secretory rate as a function of time: DECONV, PULSE, and PULSE2. The primary difference among the algorithms is the assumed functional form of the secretory function S(t). Each of these strategies is unique with advantages and disadvantages. This section develops each of these methods in detail and explains the advantages and disadvantages of each.

All three methods (DECONV, PULSE, and PULSE2) are based on fitting Eq. (1) to the actual experimental data. The details of the various weighted nonlinear least-squares algorithms (damped Gauss–Newton, steepest descent, and Gauss–Newton) are presented elsewhere (Johnson and Frasier, 1985; Straume et al., 1991; Johnson, 1992; Johnson and Faunt, 1992) and are not repeated here. The algorithms for the evaluation of confidence intervals of estimated parameters (Johnson, 1983; Johnson and Frasier, 1985; Straume et al., 1991; Johnson and Faunt, 1992; Straume and Johnson, 1992a) and goodness-of-fit criteria (Straume et al., 1991; Straume and Johnson, 1992b) are also presented elsewhere.

Technically Eq. (1) is not a convolution integral because the limits of integration are written as 0 to t instead of the minus infinity to plus infinity that is characteristic of convolution integrals. To be rigorously correct Eq. (1) should be written as a convolution integral:

(t)=∫−∞∞S(z)H(t−z)E(t−z)dz=S(t)⊗[H(t)E(t)].

H(t − z) is a Heaviside step function:

(t−z)=(1,ift−z≥00,ift−z<0).

The purpose of the Heaviside step function is to eliminate the possibility of hormone being eliminated before it is secreted. In Eq. (1) this is accomplished by stopping the integration at an upper limit of t instead of infinity. The C0 in Eq. (1) corresponds to the integral in Eq. (2) integrated from minus infinity to zero:

0=∫−∞0S(z)[H(t−z)E(t−z)]dz.

Elimination Function: H(t − z)E(t − z)

The elimination function that we utilize for all the deconvolution methods is based on the classic one-compartment [Eq. (5)] or two-compartment [Eq. (6)] pharmacokinetic models for elimination:

(t−z)E(t−z)=(e−[ln2(t−z)]/HL,ift−z≥00,ift−z<0),

where HL is the one compartment elimination half-life and

(t−z)E(t−z)=((1−f2)e−[ln2(t−z)]/HL1+f2e−[ln2(t−z)]/HL2,ift−z≥00,ift−z<0),

where HL1 and HL2 are the two half-lives of the two compartment model and f2 is the fractional amplitude corresponding to HL2. For simplicity the Heaviside step function has been included in Eqs. (5) and (6).

DECONV

Our original deconvolution algorithm DECONV (Veldhuis et al., 1987) modeled the secretory event as a sum of a series of Gaussian shaped secretory events:

(z)=S0+∑l=1nelnHi−(1/2)[(z−PPi)/SD]2,

where In Hi, is the natural logarithm of the amplitude of the ith secretory event (in units of mass per unit time per unit distribution volume), PPi is the position of the ith secretory peak, SD is the standard deviation of the secretory events, and S0 corresponds to the basal secretion. The half-width at half-height of the secretory event is equal to 2.354 times the SD.

Once the mathematical forms of the secretion events, Eq....