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The human body exhibits a variety of biological rhythms. There are patterns that correspond, among others, to the daily wake/sleep cycle, a yearly seasonal cycle and, in women, the menstrual cycle. Sine/cosine functions are often used to model biological patterns for continuous data, but this model is not appropriate for analysis of biological rhythms in failure time data.

We adapt the cosinor method to the proportional hazards model and present a method to provide an estimate and confidence interval of the time when the minimum hazard is achieved. We then apply this model to data taken from a clinical trial of adjuvant of pre-menopausal breast cancer patients.

The application of this technique to the breast cancer data revealed that the optimal day for pre-resection incisional or excisional biopsy of 28-day cycle (i. e. the day associated with the lowest recurrence rate) is day 8 with 95% confidence interval of 4–12 days. We found that older age, fewer positive nodes, smaller tumor size, and experimental treatment were predictive of longer relapse-free survival.

In this paper we have described a method for modeling failure time data with an underlying biological rhythm. The advantage of adapting a cosinor model to proportional hazards model is its ability to model right censored data. We have presented a method to provide an estimate and confidence interval of the day in the menstrual cycle where the minimum hazard is achieved. This method is not limited to breast cancer data, and may be applied to any biological rhythms linked to right censored data.

The human body exhibits a variety of biological rhythms. There are patterns that
correspond, among others, to the daily wake/sleep cycle, a yearly seasonal cycle and, in
women, the menstrual cycle. The clinical relevance of circadian rhythm has been
demonstrated in multi-center randomized trials [

Various mathematical models have been used to assess the suitability of periodic
functions associated with biological rhythms. The most common approach is that of "cosinor
rhythmometry", in which a linear least squares regression is used to fit a sinusoidal
curve to time-series data [

Failure may be broadly defined as the occurrence of a pre-specified event. Events of this nature include time of death, disease occurrence or recurrence and remission. One important aspect of FTD is that the anticipated event may not occur for each individual under study. This situation is referred to as censoring and the study subject for which no failure time is available is referred to as censored. Censored data analysis requires special methods to compensate for the information lost by not knowing the time of failure of all individuals. The literature is short of methodologies that deal with circadian or biological rhythms in failure time data.

This article models the biological rhythms in censored data. It presents a method to estimate the time that achieves the minimum hazard along with its associated confidence interval. The model is then used to predict the optimal day in the menstrual cycle for breast cancer surgery (i.e. day associated with the lowest recurrence rate) in pre-menopausal women using data from the National Cancer Institute of Canada's Clinical Trial Group MA.5 study.

The most common approach to the analysis of biological rhythm is that of "cosinor
rhythmometry" (see Nelson et al. [

_{i}_{i }_{i }
(1)

where t_{i }represents the time of measurements for the i^{th
}individual, M the mean level (termed mesor) of the cosine curve, A is the
amplitude of the function, ω is the angular frequency (period) of the curve, and
φ is the acrophase (horizontal shift) of the curve. It is assumed that the errors,
ε_{i}, are independent and normally distributed with means zero and a
common residual variance σ^{2}. It is also possible to use more than one
cosine function with different values of ω (whether or not in harmonic relation)
or a combined linear-nonlinear rhythmometry [

The Cox regression model (proportional hazard model) [^{T }= (X_{1}, ...,
X_{p}) denote

where λ_{0}(t) is a baseline hazard corresponding to X^{T }= (0,
..., 0), β^{T }= (β_{1}, ...,
β_{p}) is a vector of regression coefficients, and
β^{T}X is an inner product. The important inference questions in this
setting are about the conditional distribution of failure, given the covariates. In
order to examine the effect of biological rhythm upon survival, one needs to adapt
cosinor rhythmometry to the proportional hazard model.

Let us assume that we have _{i}, an indicator variable, δ_{i},
with a value of 1 if ξ_{i }is uncensored or a value of 0
if ξ_{i }is censored, and X_{i
}= [_{1i
}_{2i}]^{T},
so that the observed data are (ξ_{i},
δ_{i}, _{i}). Here
_{1i }=
cos_{i}_{2i
}= sin_{i }and ω
= 2π/τ. Hence _{1 }= A cos_{2 }= - A sin

The proportional hazards model (2) for the

_{i}(_{0}(_{1}_{1i
}+
_{2}_{2i})
(3)

This equation is almost the same as the cosinor equations with the same meaning. We
should notice that in this model the effects are multiplicative instead of additive. The
mesor parameter M is taken up in λ_{0}(t), and it is difficult to draw the
cosine curve on top of the data as in the continuous case.

The β-coefficients in the proportional hazards model, which are the unknown
parameters in the model, can be estimated using the _{1 }< ...
<_{L }denote the _{0 }so the covariates
associated with the _{(1)}, ...,
X_{(L) }. The log partial likelihood is given by

where ℜ_{i }is the set of cases at risk at time
_{i}. The efficient score for β,
Z(β) = ∂/∂β log L(β), is

Maximum partial likelihood estimates β are found by solving the

Let _{1 }and β_{2
}respectively. Then, the parameter estimates

An approximate (1 - α) 100% confidence interval is _{α/2 }is the (1 - α/2) 100% cut off point of the standard
normal distribution.

The estimation of the amplitude can be obtained by

An approximate (1 - α) 100% confidence interval is

Some may determine the optimal time by trying different partitions to the data where the variation looks cyclical. The simplest cyclical pattern is the sine wave with its associated parameters of amplitude, mean level (mesor), angle frequency, and phase angle (acrophase). This section will establish and construct an estimate and confidence interval of the day where the minimum hazard is achieved. The objective is to locate the optimal time for intervention, which is the time where the curve is at a minimum.

Since we know that cos π = -1, the optimum time must be when

The variance of

The asymptotic 95% confidence intervals will be based on the standard errors using an
assumption of normality

Bootstrapping also can be used to estimate the variability of the estimated function,
and to provide information on whether certain features of the estimated function are
true features of the data or just random noise [

While seasonality affects us all, the menstrual cycle directly affects about 52% of the
world's inhabitants. Each of the members of this small global majority spends about half
of her life participating regularly and continuously in this powerful biological rhythm.
Many diverse disease activities have been demonstrated to be affected by this cycle.
Cancer is one of these [

The timing of surgical intervention for breast cancer may have an influence on the
outcome of these interventions [

The MA.5 study was a multi-center clinical trial conducted by the National Cancer
Institute of Canada Clinical Trial Group (NCIC CTG) [

All analyses were conducted with SAS Version 9.1 and S-Plus for Windows Version 6.0.
All tests were two-sided, and a level of α = 0.05 was used to determine a
significant result. Product-limit survival curves were calculated by the method of
Kaplan-Meier. The Cox proportional hazards model [

The smoothed plot of the proportion of patients who relapsed (Figure

Proportion of recurrence according to day of the menstrual cycle at time of tumor excision.

Multivariate analyses using the Cox Proportional Hazards model identified age, positive nodes, pathologic stage, and experimental treatment as the most significant factors related to disease free survival. The time of surgery within the menstrual cycle was a significant independent predictor of disease-free survival.

Assuming that the length of the cycle is 28 days for all women, and using back-
transformation, we obtained the acrophase

Using this optimal interval, the disease recurred in 55 patients (30%) in the group were
LMP was 0–3 and 13–40 days, whereas 10 patients (13%) developed disease in
mid-cycle (4–12 days) group. Figure

Relapse-Free Survival by Timing of Surgery: Proposed Definition.

Since different comparisons had been made in the past based on retrospective analyses, it was of interest to compare results from approaches used in the past with the approach proposed in this study:

The risk for recurrence differed between the two phases: 30% of patients developed
recurrence after surgery in the luteal group compared with 20% in the follicular group.
Figure

Relapse-Free Survival by Timing of Surgery: Senie's Definition.

In this kind of menstrual interval, tumor recurred in 29% of Perimenstrual patients and
in 20% of mid-cycle patients. Figure

Relapse-Free Survival by Timing of Surgery: Hrushesky's Definition.

In this paper we have described a method for modeling failure time data with an underlying biological rhythm. The advantage of adapting a cosinor model to proportional hazard model is its ability to model right censored data. We have presented a method to provide an estimate and confidence interval of the day where the minimum hazard is achieved.

The application of this technique to breast cancer data revealed that the optimal days for pre-resection incisional or excisional biopsy of 28-day cycle (i. e. the days associated with the lowest recurrence rate) are days 4–12. This represents the putative follicular phase for women with 28 to 36 day cycle duration and the luteal phase for those with the usual cycle length between 21 and 28 days. This is in agreement with the contention that disease recurrence and metastasis are more frequent and appear more rapidly in women who have had their initial breast cancer resection during days 0–6 and 21–36 of the menstrual cycle.

Most of studies designed to assess the efficacy of breast surgery in relation to the timing of the intervention during the menstrual cycle were retrospective. This article presents a prospective investigation of menstrual cycle operative timing. The proposed analytical technique is not limited to breast cancer data and may be applied to any biological rhythms linked to right censored data.

We did not identify any situation that might be perceived as a conflict of interest.

The authors contributed equally to this work.