## Abstract

Pulsatile hormonal secretion is a ubiquitous finding in endocrinology. However, typical protocols employed to generate data sets suitable for “pulsatility analysis” have required 60–300 samples, rendering such studies largely research methodologies, due primarily to considerable assay expense. One successful mathematical strategy in calibrating changes in pulsatility modalities is approximate entropy (ApEn), a quantification of sequential irregularity. Given the degree of differences between ApEn values in pathophysiological subjects vs. healthy controls reported in several recent studies, we queried to what extent coarser (less frequent) and shorter duration time sampling would still retain significant ApEn differences between clinically distinct cohorts. Accordingly, we reanalyzed data from two studies of 24-h profiles of healthy vs. tumoral hormone secretion:*1*) growth hormone comparisons of normal subjects vs. acromegalics, originally sampled every 5 min; and*2*) ACTH and cortisol comparisons of normal subjects vs. Cushing's disease patients, originally sampled every 10 min. By multiple statistical analyses, we consistently and highly significantly (*P* < 0.0001) established that serum concentration patterns in tumor patients are more irregular than those of controls, with high sensitivity and specificity, even at very coarse (e.g., 60 min) sampling regimens and over relatively short (2–4 h) time intervals. The consistency of these findings suggests a broadly based utility of such shorter and/or coarser sampling methodologies. Substantial reduction in sampling requirements holds the potential to move analysis of pulsatile hormone release from a primarily research tool to a clinically applicable protocol, in appropriate diagnostic and therapeutic contexts.

- irregularity
- approximate entropy
- tumor
- clinical utility

episodic (pulsatile) secretion of hormones is nearly universal in endocrinology, in both humans and diverse animal species (1, 2, 22, 24, 29). Furthermore, derangements in pulsatile hormone secretion are primary markers of endocrine pathophysiology and aging in numerous contexts, e.g., insidious endocrine tumors (4, 25, 27), insulin secretory patterns in diabetics (7), and normal reproductive aging (20). However, typical protocols employed to create a time series suitable for “pulsatility analysis” have utilized between 100–300 data points. This labor intensiveness currently renders pulsatility studies largely to research settings, rather than to more suitable general clinical contexts, due primarily to considerable expense (of numerous assays) and secondarily to protocol duration and blood loss.

In a broader context, it has been recognized that in many diverse time-series applications, the persistence of certain patterns, or shifts in an “aggregate amount of randomness,” provides the fundamental insight of subject status, reflecting essential biological information. Despite this recognition, formulas and algorithms to quantify an “extent of randomness” have not been developed and/or utilized in the desired settings, primarily because even within mathematics itself, such a quantification technology was lacking until very recently. Recently, a new mathematical approach and formula, approximate entropy (ApEn), have been introduced as a quantification of the regularity of sequential data, motivated both by application needs (9) and by fundamental questions within mathematics (16, 19). This approach calibrates an extent of sequential interrelationships, quantifying a continuum that ranges from totally ordered to completely random. In particular, ApEn is complementary to pulsatility modalities, and it has allowed the endocrinologist to monitor and quantify secretory patterns from a new viewpoint (17). Numerous applications, including those indicated in methods, have demonstrated the clinical utility of ApEn. Most pertinent to the present study (considered further in appendix ), ApEn can often effectively discriminate distinct cohorts based on partial system characterizations, without needing to fully model the underlying data (9).

Given the degree of significance and clarity of ApEn differences reported in the comparisons of pathophysiological subjects vs. healthy control subjects in several recent studies (3, 4, 25, 27), we queried whether, and to what extent, coarser (less frequent) and shorter duration (less extended) time sampling would still retain highly significant ApEn-based discrimination between clinically distinct cohorts. To gain insight into this perspective, we here reanalyze data from two studies of hormonal secretory data in control vs. tumoral states: *1*) growth hormone (GH) comparisons of normal subjects vs. patients with active acromegaly (4); and *2*) ACTH and cortisol comparisons of normal subjects vs. patients with Cushing's disease (27).

Importantly, substantial reduction in sampling requirements holds the potential to move hormone pulsatility analysis from a primarily research modality to a clinically applicable protocol, in appropriate contexts. Furthermore, such a perspective could have more general implications to numerous hormonal systems, both diagnostically and therapeutically.

## METHODS

*ACTH and cortisol data: normal subjects vs. subjects with Cushing's disease study*. Time series of serum ACTH and cortisol concentrations were obtained at 10-min intervals for 24 h (*n* = 144 points), from 0900–0900, in 17 patients with pituitary-dependent Cushing's disease (11 females, 6 males, overall mean age 38 yr) and 28 healthy matched controls (16 females, 12 males, overall mean age 42 yr) (27). Plasma ACTH concentrations were measured in duplicate by immunoradiometric assay, with reagents obtained from Nichols Laboratories (San Juan Capristrano, CA). The detection limit of this assay was 3.0 ng/l. The intra- and interassay precision varied from 2.8–7.5%. Plasma cortisol concentrations were measured by RIA (Sorin Biomedica, Milan, Italy). The detection limit of this assay was 25 nmol/l. The intra- and interassay precision varied from 2–4%. Detailed subject characterization and further assay description are given in Ref. 27.

*Growth hormone data: normal subjects vs. acromegalics study.* Time series of serum growth hormone (GH) concentrations were collected at 5-min intervals for 24 h (*n* = 288 points), from 0800–0800, in two groups (4). The control group comprised 20 healthy volunteers: 12 men (ages 22–28) and 8 women in the early follicular phase of the menstrual cycle (ages 23–25). Patients were 19 acromegalics (9 men, 10 women) with active disease. Serum GH concentrations were measured in duplicate by immunoradiometric assay (Nichols Laboratories). The sensitivity of the assay was 0.2 μg/l; samples with <0.2 μg/l GH were assigned a value of 0.2 μg/l for statistical analysis. A detailed description of the mean intra-assay coefficients of variation (CV) across a full range of serum GH concentrations has been previously reported (5). Detailed subject characterization and further assay description are given in Ref. 4.

*Quantification of irregularity*. To quantify the concept of irregularity, we utilized ApEn, defined in Ref.9, with further mathematical properties and representative biological applications given in Refs. 6, 11, 13, 16, 19, 21, and 23. ApEn is complementary to pulse detection algorithms widely employed to evaluate hormone secretion time series (26). ApEn evaluates both dominant and subordinant patterns in data; notably, it will detect changes in underlying episodic behavior not reflected in peak occurrences or amplitudes (17). Additionally, ApEn provides a direct barometer of feedback system change in many coupled systems (10, 17). Among numerous endocrine contexts, ApEn has shown vivid distinctions (*P* < 10^{−10}) between normal subjects and patients with endocrine tumors secreting GH (4), ACTH and cortisol (27), prolactin (3), and aldosterone (25). In addition, ApEn studies have demonstrated a pronounced and consistent gender difference in irregularity of GH secretion in both humans and rats (12) and a positive correlation between advancing age and greater irregularity of*1*) GH (28);*2*) luteinizing hormone and testosterone (18); and *3*) insulin secretion (8).

ApEn assigns a nonnegative number to a time series, with larger values corresponding to greater apparent process randomness or serial irregularity and smaller values corresponding to more instances of recognizable features or patterns in the data. Two input parameters, a run length *m* and a tolerance window*r*, must be specified to compute ApEn, formally denoted ApEn(*m*,*r*). The parameter *m* is dimensionless; it represents a window length of the number of contiguous time-series points that are compared with one another in forming the ApEn calculation. Briefly, ApEn measures the logarithmic likelihood that runs of patterns that are close (within*r*) for*m* contiguous observations remain close (within the same tolerance width *r*) on next incremental comparisons; the precise mathematical definition is given in appendix
. It is imperative to consider ApEn(*m*,*r*) as a family of parameters; comparisons are intended with fixed*m* and*r*.

When *m* = 1, as is employed herein, we interpret ApEn as a measure of the difference between the probability that runs of *length 1* will recur within tolerance*r* (i.e., that compared points have approximately equal values) and the probability that runs of*length 2* will recur to the same tolerance. A high degree of regularity in the data would imply that a given (matched) run of *length 1* would often continue with nearly the same second (next) value, producing a low value of ApEn.

For this study, we calculated ApEn(*m*,*r*) values for all data sets with *m* = 1 and *r* a fixed percentage of the SD of the individual subject time series, with the percentage a predetermined value that is a function of the time-series length*N* (number of data points), henceforth denoted as *r _{N}
*. For lengths

*N*≥ 60, apply

*r*= 20% SD; for 30 <

_{N}*N*< 60, apply

*r*= 35% SD; for 18 ≤

_{N}*N*≤ 30, apply

*r*= 50% SD; and for 10 ≤

_{N}*N*< 18, apply

*r*= 75% SD. Crucially, in each of the comparisons below between control and patient groups, we always utilize a common sampling procedure for both groups, hence a fixed data length

_{N}*N*throughout the specified comparison and thus a fixed ApEn(

*m*= 1,

*r*) statistic. It would be inappropriate to draw inferences based on ApEn values corresponding to data sets of differing lengths (and/or differing

_{N}*r*values).

_{N}Normalizing *r* to each time-series SD gives ApEn a translation and scale invariance to absolute serum concentration levels (11). Multiple previous studies that included both theoretical analysis (9, 14, 15) and clinical applications (4, 6, 11,13, 18, 21, 23, 25, 27, 28) have demonstrated that the choice of*m* = 1 and*r* = 20% SD as input parameters produce good statistical reproducibility for ApEn for time series of lengths *N* ≥ 60. As expanded on inappendix
, the aforementioned values of r_{N} provide similar replicability properties for the studies of lengths 10 ≤*N* < 60 reported inresults, balancing the statistical and probabilistic issues for these short data sets.

Further technical discussion of mathematical and statistical properties of ApEn, including robustness to noise and artifacts, mesh interplay, relative consistency of (*m*,*r*) pair choices, asymptotic normality under general assumptions, statistical bias, and error estimation for general processes can be found elsewhere (14, 15). To develop a more intuitive, physiological understanding of the ApEn definition, a multistep description of its typical algorithmic implementation, with figures, is developed in Ref.14.

*Statistical analysis*. All statistical comparisons in results for discrimination between two groups employed the two-sided*t*-test with unknown variance. Additionally, in many comparisons, we indicate the sensitivity (1-α) and specificity (1-β) of the diagnostic test and that ApEn values less than a specified value (denoted a cutpoint) are associated with healthy controls.

## RESULTS

*ACTH and cortisol data: normal subjects vs. patients with Cushing's disease.* For reanalysis at coarser frequency, data sets were evaluated by taking every 2nd point for 20-min sampling rate, every 3rd point for 30-min sampling rate, every 6th point for 60-min sampling rate, and every 12th point for 120-min sampling rate, from the baseline 10-min series. Statistical summaries (including the baseline analysis) are given in Table 1.

Although the ApEn values based on the distinct sampling intensities are somewhat different from one another, as anticipated, the relative normal vs. tumoral comparisons at each sampling frequency are similar to those of the baseline series. Specifically, normal secretory dynamics are consistently and highly significantly more regular or orderly (lower ApEn values) than are those for subjects with Cushing's disease, for both ACTH and cortisol and for the 10-min, 20-min, 30-min, and 60-min sampling rate comparisons. Only the very coarse 120-min sampling rate comparison (12 points/time series) failed to retain significant ApEn differences.

To elucidate several of the comparisons in Table 1, we consider Figs.1, 2, and3. Figure 1 shows representative serum cortisol concentrations (nmol/l) over 24 h, for a control subject and a Cushing's subject, both displayed for 10-min and 60-min sampling rates. Note from Fig. 1 that *1*) 60-min sampling loses quite a bit of the dynamics feature (insofar as certifiable pulse determination), compared with 10-min sampling, for each subject; and *2*) yet insofar as discrimination is concerned, the control subject's secretory dynamics at 60-min sampling still appear more regular than the corresponding cortisol series of the Cushing's patient and are so quantified by ApEn.

Analogous inferences follow from the data depicted in Figs. 2 and 3. These figures show individual subject ApEn values vs. mean serum ACTH (ng/l) and cortisol (nmol/l) concentrations at both 10-min (Fig. 2) and 60-min (Fig. 3) sampling frequencies. The separation realized by ApEn, in distinguishing controls from Cushing's subjects based on 10-min sampling, was nearly total for both ACTH and cortisol profiles, as noted previously (27). Namely, with a cutpoint of ApEn = 1.25 for ACTH, the specificity was 100% and the sensitivity 94%; with a cutpoint of ApEn = 1.2 for cortisol, the specificity was 96% and the sensitivity 94%. As one would anticipate, in the 60-min sampling comparisons, there is more (ApEn of controls vs. Cushing's subjects) cohort overlap, for both hormones. However, considerable (ApEn) separation is retained between the control and Cushing's subjects at 60-min sampling: with a cutpoint of ApEn = 0.85 for ACTH, the specificity was 68% and the sensitivity 94%; with a cutpoint of ApEn = 0.82 for cortisol, the specificity was 79% and the sensitivity 87%. Furthermore, as seen from Table 1, the controls vs. Cushing's subjects differed based on ApEn values for a 60-min sampling schedule at a significance level *P* < 0.0001 for both ACTH and cortisol.

To ascertain further statistical consistency of the coarser study findings, we performed additional analyses. We applied ApEn to each of the six constituent 60-min sampling interval time series derived from a parent 10-min time series for ACTH (the constituent series represent, respectively, starting points of either*t* = 10 min,*t* = 20 min,*t* = 30 min,*t* = 40 min,*t* = 50 min, or*t* = 60 min). We determined that these six constituents are indistinguishable as groups, with very similar mean ApEn values, significance level of ANOVA > 0.5, providing further confirmation of the 60-min results in this study. Similar indistinguishability of the three constituent series at 30-min sampling was also established for ACTH, with significance level of ANOVA > 0.7.

The three sets of shorter duration studies for each hormone were all based on 10-min sampling rate, with results indicated in Table 1. The 72-point analysis covered 9 AM-9 PM; and two distinct sets of 24-point (4 h) analyses were performed, with the 5 AM-9 AM epoch a period of known heightened ACTH-cortisol activity and the 11 AM-3 PM epoch generally a more quiescent period for this axis. We infer that the choice of time of day is consequential in assessing differences between control subjects and patients with Cushing's disease. Based on a 4-h interval from 5 AM-9 AM, for both ACTH and cortisol, Cushing's patients secrete more irregularly than controls,*P* < 0.005; whereas from 11 AM-3 PM, there is no significant difference (at*P* = 0.05) for either hormone. Figure4 shows representative serum cortisol concentrations for a normal subject and a Cushing's patient, at 10-min sampling, both from 5 AM-9 AM and from 11 AM-3 PM, to illustrate these findings. Figure 5 shows individual subject ApEn values vs. mean serum ACTH (ng/l) and cortisol (nmol/l) concentrations for the 5 AM-9 AM epoch.

*GH data: normal subjects vs. patients with active acromegaly.* Statistical summaries of the coarser frequency reanalysis of the GH data, originally sampled at 5-min intervals, are given in Table 2. The protocol to derive the coarser-sampled time series from baseline data was the same as that described for the ACTH-cortisol reanalysis. As is described in that section, whereas the ApEn values based on each distinct sampling rate are somewhat different from one another, GH secretory dynamics are consistently and significantly more regular for healthy subjects compared with tumor-bearing (here, active acromegalic) patients, for all the coarser regimens, including even the very coarse 120-min sampling rate. Furthermore, for each of the 5-min, 10-min, 20-min, and 30-min sampling reanalyses, ApEn separated acromegalic from normal GH secretion with both high (>85%) sensitivity and high (>95%) specificity.

Figure 6 shows representative serum GH concentrations (μg/l) over 24 h, for a male control subject and a male acromegalic patient, at both 5-min and 60-min sampling rates. Figure 7 shows individual subject ApEn values vs. mean serum GH (μg/l) concentrations, at both 10-min and 60-min sampling, for each of males and females. As for ACTH, we see for GH (Fig. 6) that much of the dynamical detail in the pulsatile pattern of hormone release is lost with 60-min sampling, compared with 5-min sampling, for each subject; e.g., the extent and magnitude of the peak activity near *t* = 1,000 min for the acromegalic subject, evident at 5-min sampling, are much less apparent for 60-min sampling. Yet insofar as discrimination is concerned, the control subject's secretory dynamics based on 60-min sampling still appear more regular than the corresponding GH series of the acromegalic patient, quantified by ApEn.

Analogous findings follow from the data depicted in Fig. 7. We considered males and females both separately and pooled, as a pronounced gender difference in GH secretory irregularity has previously been observed in healthy adults (4). The separation achieved by ApEn in distinguishing control subjects from acromegalic patients based on 10-min sampling was quite sharp both overall (100% specificity, 89% sensitivity) and separately for males (100% specificity, 89% sensitivity) and females (100% specificity, 90% sensitivity). There is complete (100% specificity and sensitivity) ApEn separation between the male control subjects and male acromegalic patients at 60-min sampling; whereas in counterpoint, for female subjects at 60-min sampling, no cutpoint yields simultaneously high sensitivity and specificity between the controls and acromegalics, an inference that is reinforced by the overlap (of control and acromegalic female 60-min ApEn values) seen in Fig. 7.

The foregoing observations in acromegaly are also confirmed by subgroup comparisons via *t*-test. Overall, as noted in Table 2, control subjects secrete GH more regularly (ApEn = 0.545 ± 0.273) than acromegalic patients (ApEn = 0.810 ± 0.116) at 60-min sampling, *P* < 0.0005. For male subjects considered alone, controls also secrete GH more regularly (ApEn = 0.388 ± 0.159) than acromegalics (ApEn = 0.835 ± 0.117) at 60-min sampling,*P* < 0.0001. For female subjects, there was no significant difference between controls (ApEn = 0.767 ± 0.203) and acromegalics (ApEn = 0.772 ± 0.131) at 60-min sampling, *P* > 0.95. Finally, the gender difference between healthy subjects persists at 60-min sampling, with male GH secretory dynamics more regular (ApEn = 0.388 ± 0.159) than that of females (ApEn = 0.767 ± 0.203),*P* < 0.001.

Multiple shorter duration studies were performed. First were five sets of analyses proximate to 12 PM-2 AM, a known interval of heightened GH secretory activity. These studies (results indicated in Table 2, “night period”) comprised three sets of 10-min resampling analyses: 11 PM-3 AM (24 points), 12 PM-3 AM (18 points), and 12 PM-2 AM (12 points); as well as two sets of 5-min analyses: 11 PM-3 AM (48 points) and 12 PM-2 AM (24 points). As indicated in Table 2, the control subjects' GH secretory dynamics are consistently and significantly more regular than are those for the sex-matched acromegalic patients, for each of the coarser sampling regimens, including the 10-min, 2-h (12 points) comparisons. Moreover, for each of these five sets of comparisons, ApEn separated acromegalic from control subjects with very high sensitivity and specificity, as indicated in Table 3. Figure8 shows representative serum GH concentrations for a control subject and an acromegalic volunteer, both from 11 PM-3 AM at 10-min sampling and from 12 PM-2 AM at 5-min sampling, to illustrate these findings. Figure9 shows individual subject ApEn values vs. mean serum GH concentrations for three sets of these reanalyses:*1*) 4-h duration: 11 PM-3 AM, at 10-min sampling; *2*) 2-h duration: 12 PM-2 AM, at 5-min sampling; and *3*) 2-h duration: 12 PM-2 AM, at 10-min sampling, which reinforces the statistical findings given in Tables 2 and 3.

Finally, we performed three sets of shorter duration studies at two distinct daytime periods, 9 AM-1 PM and 1 PM-5 PM. The results are indicated in Tables 2 and 3 in the corresponding format to those for the night period analyses. For each of the two daytime periods, chosen to fall within typical office hours of physicians, the three studies comprised a 5-min sampling procedure for 4 h, a 5-min sampling procedure for 2 h, and 10-min sampling for 2 h. As indicated in Tables2 and 3, the daytime results are qualitatively similar to those achieved during the night period analyses, insofar as the persistence of highly significant acromegalic vs. control subject distinctions on the basis of irregularity and by the degree of cohort separation realized as quantified by the reported sensitivities and specificities.

## DISCUSSION

Summarizing the statistical results, multiple analyses comparing either of two distinct classes of control vs. tumoral secretory states, involving three hormones, consistently and highly significantly establish that tumoral hormone concentration patterns are more irregular than those of matched control subjects, when evaluated with even very coarse sampling regimens and relatively short observational intervals. Thus we hypothesize that for a broad spectrum of autonomous endocrine tumors, coarse and/or short sampling protocols, which are insufficient to fully delineate (fine) pulse dynamics and circadian variation, may suffice to distinguish their resultant secretory dynamics as more irregular than that of healthy controls. More generally, our perspective is that if ApEn differences determined with frequent sampling are vivid between cohorts, irregularity analyses of the type performed herein with fewer samples may retain significant distinctions. Furthermore, the framework of the above statistical analyses (supported by the accompanying figures), in ensemble, establishes a general robustness of this perspective to the precise choice of coarsening or shortening protocol, up to the limits of statistically replicable boundaries.

There was considerable overlap between the distributions of individual subject's hormone concentrations for control and acromegalic mean GH levels (Fig. 7) and for control and Cushing's subject's mean cortisol and ACTH levels (Fig. 2). This observation reinforces the clinical utility of the distinct, complementary perspective taken by an assessment of irregularity of secretory patterns, beyond a mean serum hormone level, in determining clinical state.

Additionally, we have previously linked ApEn to mechanistic understanding via mathematical modeling studies (10, 17). There, we demonstrated that for diverse network models, greater regularity (lower ApEn) corresponds to greater component and subsystem autonomy, and conversely, greater irregularity corresponds to increased external influences and/or increased coupling strength. Based on this analysis, we propose that the greater hormonal secretory irregularity in the tumoral subjects implicates an autonomous external source or interacting factor atop the baseline relatively closed subnetwork seen in the control subjects. Moreover, theoretically, this association linking ApEn increase to network evolution is valid over a wide range of (*m*,*r*) choices in the ApEn implementation, including coarse*r* choices, as employed in this study.

The gender difference in the GH results is interesting statistically, because we infer that for males, 60-min sampling suffices as a screen to separate normal from acromegalic GH secretion, whereas for females, this protocol is too coarse. As well, this gender difference is interesting physiologically, and as discussed in Refs. 2 and 12, probably reflects a more complex hypothalamic control of pituitary GH secretion in the female, with greater somatostatin withdrawal in the female, with or without increased pituitary activation by GH-releasing hormone. The statistical finding also contains an important message, that hormone-specific knowledge is essential to determine how to best employ less intensive protocols. To reinforce this point, we did not even evaluate potential gender differences in ACTH and cortisol between control and Cushing's disease subjects in this study, because no such ApEn difference was seen for either hormone in the control or diseased state in the previous 10-min sampling study (27).

Also, it is reassuring, although somewhat expected, to note from Table1 that ACTH and cortisol, known to be causally correlated (as ACTH provokes cortisol secretion), show relatively similar ApEn values within each of the control and Cushing's disease groups, for all of the coarser and shorter sampling protocols.

*Toward clinical and experimental utility*. Whereas the statistical significance and extent of separation by ApEn of normal vs. tumoral hormone secretion decreases with coarser sampling time series, as expected, ApEn retains highly significant differences between the normal and the tumoral profiles: *P* < 0.0005, for both types of pituitary tumors and for each hormone under study, for protocols as coarse as 60-min sampling (24 total points required). Similarly, for the shorter interval studies, after appropriate time windows for each hormone were chosen, 24-point studies maintained highly significant (*P* < 0.005) normal-tumoral irregularity distinctions. Indeed, in the GH study, even 12- and 18-point shorter epoch analyses provided vivid cohort differences. Additionally, these reanalyses yielded simultaneously high sensitivity and specificity, as reported both inresults and in Table 3 and as inferred from Figs. 2, 3, 5, 7, and 9. The significance of these findings is that with no more than 24 points required to provide a good discriminator of tumoral from healthy subjects, the orientation of studying hormone pulsatility can potentially shift from an expensive research protocol to a justifiable clinical protocol in suitable contexts, based on total assay costs. Moreover, findings such as the 2-h GH reanalyses during typical office hours of physicians (both 11 AM-1 PM and 3 PM-5 PM) take on special importance, because such protocols could be routinely administered as an outpatient procedure.

Furthermore, the shorter and coarser ApEn analyses of this study generally provide efficient triage for very high and very low ApEn values; it is the middle range of ApEn values obtained in the coarser and shorter sampling protocols that is more difficult to interpret compared with more frequent sampling. This reinforces the utility of coarse-grained pulsatility analysis as a first-order clinical or experimental screen, from which further tests can be performed in instances in which abnormal ApEn values are measured.

Altogether, the findings in this study support a thematic shift, in which one often does not need to fully reconstruct pulses, or (mathematically) determine a complete model specification of secretory dynamics, to discriminate typical subjects from clinically distinct cohorts. As seen in results from comparisons of hormone dynamic patterns determined by 5- to 10-min vs. 60-min sampling (Figs. 1 and 6), one hardly achieves complete characterization of hormone pulsatility at the coarser sampling interval. Also, we do not require an assessment or estimate of the underlying true secretory dynamics (e.g., via deconvolution); the analyses in results are performed directly on the serum concentration levels.

Given a fixed number of points, we wish to inquire whether it is preferable to study *1*) shorter duration, more frequently sampled (e.g., 10-min sampling over 4 h); or*2*) longer duration, less frequently sampled (e.g., 60-min sampling over 24 h) segments, when the primary goal is a sharper level of discrimination. Clinically, in some contexts, we would likely prefer the former, for inpatient-outpatient considerations, hospital costs, and compliance, but the results here suggest that resolution of this issue should be based on the hormone and the context. For the Cushing's disease study, for 24-point analyses, the ACTH-cortisol ApEn values gave sharper control-tumor distinction for the coarser (60-min sampling) protocol, compared with the shorter duration, more frequently sampled protocol. Conversely, for the acromegalic study, for 12- or 24-point analysis of GH, the shorter duration analyses proved clearer than 60-min sampling over 24 h in distinguishing normal from tumoral secretion. Of course, these inferences have little to do with the statistical properties of ApEn but rather reflect the particular secretory characteristics of each hormone, in both physiological and pathophysiological states.

Similarly, the determination of optimal shorter duration time periods to realize clearest ApEn distinction between cohorts will require future studies. In the interim, the results of this study suggest that time periods of typically physiologically enhanced secretory activity expected for healthy controls will serve as good choices in many settings.

In conclusion, the primary inference here is that coarser and shorter sampling protocols, in conjunction with relevant hormone-specific implementation, can be employed to facilitate pulsatility analysis of hormonal secretion in suitably abbreviated clinical, as well as research, settings. Potential applications and further implications of this perspective are myriad and await further investigation.

## Acknowledgments

This work was supported in part by National Institute of Diabetes and Digestive and Kidney Diseases Grants R43-DK-54104 (to S. M. Pincus) and NIH R01-DK-32632 (to M. O. Thorner) and National Institute on Aging Grants R01-AG-01499 (to J. D. Veldhuis) and NIH R01 AG-10997 (to M. L. Hartman), the National Science Foundation Center for Biological Timing, and National Center for Research Resources Grant RR-00847 to the University of Virginia General Clinical Research Center.

## Appendix

*Mathematical definition of ApEn*. Given*N* data points*u*(1),*u*(2),...,*u*(*N*), two input parameters, *m* and*r*, must be fixed to compute ApEn [denoted precisely by ApEn(*m*,*r*,*N*)]. To define ApEn, first form vector sequences**x**(1) through**x**(*N*− *m* + 1) from the {*u*(*i*)}, defined by**x**(*i*) = [*u*(*i*),...,*u*(*i* +*m* − 1)]. These vectors represent *m* consecutive *u*values, commencing with the *i*th point. Define the distance*d*[**x**(*i*),**x**(*j*)] between vectors**x**(*i*) and**x**(*j*) as the maximum difference in their respective scalar components. Use the sequence **x**(1),**x**(2),...,**x**(*N*− *m* + 1) to construct, for each*i* ≤*N* −*m* + 1,
(*r*) = {no. of**x**(*j*) such that*d*[**x**(*i*),**x**(*j*)] ≤*r*}/(*N*− *m* + 1). The
(*r*) values measure within a tolerance *r*the regularity, or frequency, of patterns similar to a given pattern of window length *m*. Next, define Φ^{m}(*r*) as the average value of ln
(*r*), where ln is the natural logarithm. We define approximate entropy by ApEn(*m*,*r*,*N*) = Φ^{m}(*r*) − Φ^{m}
^{+ 1}(*r*).

## Appendix

*Choice of tolerance width r _{N} for ApEn input parameter.* It is useful to recall the technical intuition that motivates the ApEn development (9). A central observation is that one does not need to fully model a system to achieve effective statistical discrimination. Restated, one can often effectively distinguish two distinct system states by observing significant differences between respective (well-defined) partial characterizations. Subsequent modelling remains important, although the point here is that this task, which for physiological systems is oftentimes extremely difficult, can be separated from the application of effective discriminatory tools.

In the time series setting in which ApEn is cast here, we state this point more technically as follows: if joint probability measures (of contiguous observations) that describe the “dynamics” of each of two systems are different, then their marginal distributions on a fixed partition are often different. To distinguish two systems, we typically need orders of magnitude fewer points to accurately estimate these marginal probabilities than to perform accurate density estimation on the fully reconstructed measure that defines the process (9, 14). ApEn(*m*,*r*) then operates as a function of *m*+ 1 contiguous observations, at a level of resolution of tolerance width *r*, thus aggregating marginal probabilities with these specifications. The pulsatility differences between controls and tumor-bearing patients evaluated here are sufficiently pronounced that first-order differences are discernible at the level of statistics based on contiguous pairs of observations (thus ApEn with *m* = 1), even at a coarse level of resolution (i.e., a large choice for*r*).

As indicated in methods, for *N* > 60 points a number of both theoretical and clinical studies have established good statistical reproducibility for ApEn with parameter choices*m* = 1,*r* = 20% SD of the sequence (time series). However, for several of the coarser and shorter analyses, we needed to apply ApEn to yet a shorter length time series. Thus we further evaluated appropriate choices of*m* and*r* for application to the (different) very short length sequences. The trade-offs in (*m*,*r*) choices are statistical, as discussed in some detail in Refs. 14 and15. Larger *m* and smaller*r* describe sharper parameter (probabilistic) detail. However, larger*m* and smaller*r* also require time series of much longer lengths *N*, to ensure good statistical reproducibility [small SD of ApEn(*m*,*r*,*N*)] for general correlated processes.

For short data sets (*N* < 60 points), we retained *m* = 1 throughout, as this coarsest multivariate (contiguous measurement) probabilistic assessment is most compatible with both an irregularity perspective and good statistical properties. Smaller *N*required coarser (larger) r_{N}choices to ensure comparable statistical reproducibility of ApEn(1, r_{N}) with that for ApEn(1, 20% SD, *N*), where*N* > 60. The r_{N}% SD values indicated inmethods were chosen to satisfy the criterion SD of ApEn(*m* = 1, r_{N},*N*) ≤ 0.06 for length*N* sequences, for a wide range of parameter choices for three quite distinct autocorrelated processes: low-order autoregressive-moving average models; the composite oscillator-noise family of processes MIX(p) process defined in Ref. 9, for 0 ≤ p ≤ 1; and the logistics map, a representative deterministic dynamical system. The ApEn SD values were inferred from extensive Monte Carlo calculations. Methodologically, this procedure is virtually identical to the general protocol to establish the good reproducibility of ApEn(1, 20% SD,*N*) for*N* > 100 points, indicated in Refs.14 and 15, with the ApEn SD upper limit of 0.06 imposed here identical to that for the *N* > 100 setting.

## Footnotes

Address for reprint requests and other correspondence: S. M. Pincus, 990 Moose Hill Road, Guilford, CT 06437 (E-mail:stevepincus{at}alum.mit.edu).

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked “

*advertisement*” in accordance with 18 U.S.C. §1734 solely to indicate this fact.

- Copyright © 1999 the American Physiological Society