Measuring Circadian Rhythms
Diurnal Variation as seen through the eyes of Intraocular Pressure
Harris A. Jaffee, Ph.D.
Department of Biostatistics, Johns Hopkins Bloomberg School of Public Health
Joint work with Myles J. Jaffe and Vince Galet, Merck & Co., Inc., Investigator Initiated Studies Program
Abstract: A new approach to diurnal variation is taken through the time development of
“circadian” IOP (Fig. 4). The diurnal range (Fig. 1) is deconstructed. Two variants, which
are more aware of fluctuation (Figs. 2, 3), are produced and examined.
Introduction
Caveats
• circadian rhythm is motivated by diurnal intraocular pressure and means the typical daily
pattern (usually manufactured). measuring alludes to a mathematical procedure sought to
quantify the diurnal variation in each instance of a given circadian rhythm. The traditional,
and conventional, definition is the “diurnal range-of-fluctuations”.
• Whatever clinical procedure (e.g. Goldman Applanation Tonometry, Tono-pen, implanted
sensor) yields the underlying data is assumed here, unspecified. A passage from observed
aperiodic clinical data to "circadian data”, or c-data, is performed via a statistical procedure
involving several “time-points” of the same individual, called diurnalization (Methods).
• Key to the variation of anything is its time derivative, in this setting as revealed by its c-data.
This is a “generalized velocity”, a viewpoint that offers a certain intuition (Appendix).
Clinical precedents
• Asrani-Zeimer et al (2000): The diurnal range-of-fluctuations (resp. diurnal mean) for an eye is
the range (mean) of the averages over at least 3 consecutive days of its IOP at like times. This
mitigates day-to-day variation, producing “typical” values. The early morning (averaged) value
will be replicated the next morning, here, with no effect on the range and yielding c-data. The
range was found to predict progression of open-angle glaucoma (better than the mean).
• Gumus et al (2006), on OAG with exfoliation syndrome (XS): IOP was highest in the morning,
showing a gradual decrease from 8am in the control group, but a second peak at 3pm in 53%
of the XS group. Both groups had about the same range-of-fluctuations.
Purpose
• Diurnal variation of intraocular pressure is felt to play a major role in the progression of openangle glaucoma, with the diurnal range (R), often inappropriately applied to a single (or even
partial) day of IOP data, taken as the virtual definition. But, R can only capture overall (daily)
fluctuation. For example, it seems helpless to indicate OAG with XS, even statistically.
• The proper meaning of DV remains obscure, and the goal here is a functional formulation. The
DV of diurnal IOP should reflect the mechanism of glaucomatous damage. Diurnal IOP will be
viewed through its c-data, produced by diurnalization, then explored as a periodic dynamical
process whose time development “carries” the fluctuation.
Mathematical background
• The "total increase" or positive variation as formulated by Camille Jordan in 1881:
Let pi = p(ti) be the observed (data) values of a function p at the time-points t0 < ... < tN, with
the difference pi+1 - pi denoted di. The positive variation V(p) is the sum of those di which are
positive. V(p) can be less than the range of p, even zero, but not under reasonable conditions.
Such p is circadian if p(t+24) = p(t) for all t. If so, and tN = t0+24, then pN = p0, and the N+1
pairs (ti, pi) amount to c-data for p. These N+1 points generate Jordan’s polygon, the graph of
a piecewise-linear function to which p is identified, in approximation, as usual. This graph is a
basic “V” if p is non-increasing from max(p) to min(p), then non-decreasing back to max(p). In
that case, the series for V(p) collapses to range(p). Otherwise, the graph is a generalized “W”
with some amount of “internal fluctuation”, by which V(p) exactly exceeds range(p). Thus,
“W Proposition” (Vince Galet): If p is circadian, V(p) = range(p) + (internal fluctuation)
• Geometric interpretation of the positive variation and its relation to the range:
A Fundamental Theorem of Diurnal Variation (adapted from Milnor and Brouwer)
Let p be circadian with data values min(p) = u0 < ... < uK = max(p). The values uj include the
singular values and, further, the extrema. For j<K, let Rj = uj+1 - uj, i.e. the length of the interval
Ij = [uj, uj+1], and write Oj for the interior of Ij. Every (regular value) z in any Oj occurs at least
twice, at which times p is either increasing or decreasing. Moreover, writing +(z) for the
number of times such a value z occurs while p is increasing, one has:
(Fundamental Theorem, continued)
1. The union of the Ij is disjoint away from the uj and equals [min(p), max(p)].
2. range(p) = max(p) - min(p) is the sum of the Rj.
3. +(z) is constant on each Oj so may be written mj independent of z.
4. The positive variation V(p) may be written Σ mjRj with each mj ≥1.
5. V(p) ≥ range(p) with equality if and only if all the mj are 1 (e.g. Figs. 1, 2B, 4, 5, 6D).
(For 3-4, see Appendix; for V>range, Figs. 2C, 3, 6L) The mj are multiplicities, and V is a range with multiplicity.
• Dynamical interpretation of the positive variation:
Form the differences si = ti+1-ti, and the ratios qi = di/si, which are the slopes of segments of the Jordan polygon, i.e.
average rates-of-change of p. Then V(p) may be written Σall i L(qi)  si, where L(q)=0 for q≤0, and L(q)=q otherwise.
Methods
• Unlike the diurnal mean, which benefits from linearity, the example of R already suggests that no measure of DV will
be assessable (say in aggregate) from an aggregate diurnal curve, but instead must be assessed case-by-case (and
then, if one likes, averaged).
• For a measure of DV to provide valid comparison (ranking) of cases, it should be based on 24-hour c-data produced
from observed data by diurnalization, the situation of multiple days of data having been described. For a partial day of
data, the replication step (Figs. 1-3) produces the only 24-hour pattern possible given the data. Otherwise, let (t0, p0)
and (tN, pN) be the first and last data-points with tN = t0+24-e, e fairly small, and pN = p0-E. To the extent that E is nonzero, these data cannot represent a persisting pattern. If P is the average of p0 and pN, a decent guess at the “implied
pattern” is made by replacing these two data-points with the virtual data points (t0-e/2, P) and (tN+e/2, P), especially if e
is zero (Fig. 4, Examples). If e is non-zero, a better candidate for P might be the average of the pseudo-values at t0-e/2
and tN+e/2 determined by linear interpolation/extrapolation, using the 2nd and 2nd-to-last data-points, respectively.
• Given c-data, compute 1: the time intervals s (in decimal hrs) between successive c-data points, 2: the differences d of
successive c-data values, and 3: the sequence of ratios q = d/s. Then the sum Σd>0 d (which is independent of the s)
expresses Jordan's positive variation and reflects all fluctuation in the c-data. It is also a range-with-multiplicity, in view
of the Fundamental Theorem, so might be called the m-range (M). Writing M in the form Σq>0 qs, perturbations are
suggested such as Σq>0 (eq - 1)  s which involve rate of fluctuation (yet coincide with M to first-order). This particular
one will be called the x-range (X). While the units of R and M are those of the data (here, mmHg), the units of X are
complex and involve time.
• In order to exercise the calculation of DV and the effect of a treatment on it, M and X were applied, along with R, to 24hour trial data (Konstas-Stewart) of two treatments for OAG, relative to Latanoprost monotherapy. IOP was observed
over 24 hours at 4-hour intervals starting at 8am, at baseline and after treatment. For each triple (study eye, treatment,
measure of DV), the DV’s of that eye's baseline and treatment IOP curves were computed from associated c-data. The
difference is the effect of that treatment on the DV of that eye's “inherent diurnal IOP”, i.e. under Latanoprost.
Examples
• Figure 1: With patients A and B, the proposed measure M coincides with R. In either measure, B has more DV than A.
• Figure 2: R(B) > R(C), but C exhibits an afternoon rise in IOP which is recognized by M, implying the reverse
relationship, M(C) > M(B). So in the measure M, C has more DV than B.
• Figure 3: Patients C and D have identical R and M. D has a sharper rise in IOP from 12n to 4pm, while C has a sharper
rise from 9pm to 7am. X(C) = (e2/4-1)4 + (e6/10-1)10 = 10.8, and X(D) = (e3/4-1)4 + (e5/10-1)10 = 11.0. In this measure,
D has slightly more DV.
• The Fundamental Theorem applied to patient C (Fig. 3). The extrema / singular- / data values are: 0 < 1 < 3 < 6; R0 = 1,
R1 = 2, R2 = 3; m0 = 1, m1 = 2, m2 = 1. V(p) = 2+6 = 8, alternatively, m0R0 + m1R1 + m2R2 = 11 + 22 + 13.
• Figure 4: Diurnalization and resulting c-data with a large correction: e=0, p0=26, pN=19, E=7, P = 22.5 (Methods)
• Figures 5, 6 (Methods, Results): A small, average reduction in DV of IOP under Bimatoprost; an average example of
diurnal IOP under Dorzolamide, with essentially no effect on DV
Results
• The three measures R, M, and X of DV rank cases of diurnal IOP slightly differently. M (resp. X) can also separate the
cases with like R (resp. M). For example, M and X can appreciate exfoliation syndrome, with an afternoon rise in IOP,
while R cannot.
• With respect to the measures (R, M, X) of DV applied to the Stewart trial data, relative to Latanoprost, Bimatoprost
reduced DV of IOP on average by (1.3, 1.2, 4.3) units, significantly, while Dorzolamide had no significant (average)
effect on DV of IOP.
Conclusions
• Two alternatives to the diurnal range were proposed. The m-range is a strong range, and the x-range a strong m-range.
• These alternatives may be better clinical “biomarkers” than the range and more useful in glaucoma research, depending on their correlation with
progression and visual-field loss, which are yet to be determined.
Acknowledgements: The author appreciates the advice and encouragement of Ran Zeimer, William Stewart, and David Guyton.
References
• Asrani-Zeimer et al, Large diurnal fluctuations in intraocular pressure are an independent risk factor in patients with glaucoma, J Glaucoma, 2000, Apr,
9(2), p. 134-142
• Brouwer L, Uber Abbildung von Mannigfaltigkeiten, Math Annalen, #71, 1912, p. 97-115
• Gumus et al, Diurnal variation of intraocular pressure and its correlation with retinal nerve fiber analysis in Turkish patients with exfoliation syndrome,
Graefe's Arch Clin Exp Ophth, 2006, #244, p. 170-176
• Jordan C, Sur la série de Fourier, Comptes Rendus (Math) Acad. Sci., Paris, #92, 1881, p. 228-230
• Konstas-Stewart et al, 24-hour intraocular pressures with brimonidine purite versus dorzolamide added to latanoprost in primary open-angle glaucoma
subjects, Ophthalmology, 2005, Apr;112(4), p. 603-608
• Milnor J, Topology from the Differentiable Viewpoint, University of Virginia Press, 1965
Appendix
• In the Fundamental Theorem, the numbers mj are well-defined and non-zero:
Let -(z) for z in some Oj be the number of times z occurs while p is decreasing, and set #(z) to the sum of +(z) and -(z). #(z) is non-zero by continuity (Intermediate Value Theorem)
and is also the number of intersections of the graph of p with the horizontal line at height z, so is locally constant. Following any occurrence of z while p is increasing (or decreasing),
there will eventually be another, similar occurrence by periodicity, hence in between an occurrence of the opposite kind, by continuity again. Thus, -(z) = +(z), and +(z) = #(z)/2, hence
+(z) is locally constant. The parity of #(z) is called the mod2 degree of p; the difference +(z) - -(z) is its Brouwer degree [Brouwer, Milnor, mathworld.wolfram.com/MapDegree.html].
Both of these constructs degenerate to zero in this primeval situation.
• Action:
Form S = Σq>0 L(q)s, where L is positive and strictly increasing for q>0, which is phrased more symmetrically as Σall q L(q)s with L also identically zero for q≤0. (M and X may be
expressed this way; R cannot.) S is an approximating Riemann sum for the definite integral ∫T L(p'(t))dt, where p'(t) is the time-derivative of the circadian process p, and T is any 24hour interval. The integral accumulates, over the day, the occurring values of instantaneous fluctuation, with a prescribed “weighting”. S is thus an instance of Feynman’s action
paradigm [Dirac, Feynman, Brown] with L a particularly simple form of Lagrangian; R is not.
Suggested reading:
• Brown L (editor), Feynman's Thesis: A New Approach to Quantum Theory, World Scientific, 2006
• Dirac P, The Lagrangian in Quantum Mechanics, Physik Zeits Sowjetunion, #3, 1933, p. 64• Feynman R, The Principle of Least Action in Quantum Mechanics, Princeton Thesis, 1942