Figure 1
Principles of the ovariectomized rat model. The effects of ovariectomy (OVX) on circadian rhythms in the tail skin temperature of the rat. In the intact animal, the temperature baseline is lowered during the active nocturnal phase. In the ovariectomized animal, the temperature in this episode is elevated, which results in a reduction of the distance to the baseline of the quiescent phase. The graphics have been adapted from [3].
Figure 2
Telemetric device. Measurement device and transmitter used in the telemetry experiments (source: Bayer Pharma AG).
Figure 3
Different phases and treatment groups. Experimental protocol of the OVX experiments.
Figure 4
Example of model fit (E2 group). Examples of estimated model components belonging to an animal of the E2 group, separated by experimental phase (P1 - P3). (a) Tail skin temperature time measurements (yn); (b) circadian components ; (c) amplitude functions ; (d) trend curves ; (e) combination of estimated circadian components and trend curves , embedded in the time series. Daytime periods are shown in yellow; nighttime periods are shown in blue. The model parameter estimates are given in the Tables 1 and 2.
Figure 5
Example of model fit (Tibolone group). Examples of estimated model components belonging to an animal of the Tibolone group, separated by experimental phase (P1 to P3). (a) Tail skin temperature time measurements (yn); (b) circadian components ; (c) amplitude functions ; (d) trend curves ; (e) combination of estimated circadian components and trend curves , embedded in the time series. Daytime periods are shown in yellow; nighttime periods are shown in blue. The model parameter estimates are given in the Tables 1 and 2.
Table 1
Examples of estimated van der Pol parameters.
| Examples of estimated van der Pol parameters | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| E2 | Tibolone | |||||||||
| P1 | 0.5000 | 0.4969 | 0.0100 | 4.6416 | 4.6416 | 0.5000 | 0.3753 | 0.2452 | 3.6932 | 3.6931 |
| P2 | 2.7760 | 0.2000 | 0.4055 | 1.2618 | 1.2620 | 3.8264 | 0.2000 | -0.1275 | 1.4968 | 1.4971 |
| P3 | 0.5000 | 0.4552 | -0.1423 | 2.9695 | 2.9695 | 0.9150 | 0.2000 | -0.2483 | 3.8367 | 3.8363 |
Examples of van der Pol model parameters estimated for animals of the E2 group and the Tibolone group. The estimated circadian components and amplitude functions are illustrated in Figures 4 and 5. : initial amplitude, : flexibility parameter, : angular phase shift, : limit cycle amplitude, : last amplitude function value estimated for an experimental phase.
Table 2
Examples of estimated trend and variance parameters
| Examples of estimated trend and variance parameters | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| E2 | Tibolone | |||||||||
| P1 | 28.3782 | -0.0001 | 11.0133 | 3.9545 | 5.0698 | 29.7567 | -0.0001 | 8.9852 | 3.4665 | 7.7807 |
| P2 | 30.0259 | 0.0002 | 7.2154 | 4.4644 | 8.1160 | 28.7178 | 0.0005 | 12.5643 | 4.2927 | 8.5226 |
| P3 | 30.4638 | -0.0002 | 9.3479 | 3.9833 | 6.6188 | 31.8452 | -0.0005 | 17.5731 | 3.4213 | 7.9173 |
Examples of trend and variance parameters estimated for animals of the E2 group and the Tibolone group. The corresponding time series and the estimated trend functions are illustrated in Figures 4 and 5. : intercept, : slope, : variance of first daytime period (6 - 12 a.m.), : variance of second daytime period (12 a.m. - 6 p.m.), : variance of nighttime period (6 p.m. - 6 a.m.).
Table 3
Estimated van der Pol parameters
| Estimated van der Pol parameters | ||||||
|---|---|---|---|---|---|---|
| Group | Experimental phase | |||||
| E2 | P1 | 0.65 ± 0.32 | 0.52 ± 0.26 | 0.24 ± 0.19 | 3.97 ± 1.48 | 3.92 ± 1.40 |
| P2 | 2.48 ± 0.98 | 0.29 ± 0.25 | 0.40 ± 0.34 | 1.58 ± 0.55 | 1.58 ± 0.55 | |
| P3 | 1.43 ± 0.61 | 0.31 ± 0.26 | 0.16 ± 0.36 | 2.70 ± 1.00 | 2.70 ± 1.00 | |
| Tibolone | P1 | 0.57 ± 0.21 | 0.56 ± 0.34 | 0.25 ± 0.28 | 3.55 ± 0.75 | 3.52 ± 0.71 |
| P2 | 3.10 ± 1.37 | 0.36 ± 0.34 | 0.34 ± 0.37 | 1.61 ± 0.55 | 1.61 ± 0.55 | |
| P3 | 1.45 ± 0.67 | 0.20 ± 0.00 | 0.05 ± 0.21 | 3.29 ± 0.99 | 3.29 ± 0.99 | |
Mean values and standard deviations of the van der Pol model parameters estimated for the E2 and the Tibolone group (sample size: 10 animals per group). : initial amplitude, : flexibility parameter, : angular phase shift, : limit cycle amplitude, : last amplitude function value estimated for an experimental phase.
Table 4
Estimated trend and variance parameters
| Estimated estimated trend and variance parameters | ||||||
|---|---|---|---|---|---|---|
| Group | Experimental phase | |||||
| E2 | P1 | 29.53 ± 1.68 | (0.31 ± 1.99) · 10-4 | 9.87 ± 3.69 | 4.06 ± 1.28 | 10.39 ± 2.76 |
| P2 | 30.35 ± 1.12 | (1.37 ± 1.39) · 10-4 | 9.56 ± 3.25 | 4.91 ± 3.55 | 11.17 ± 2.40 | |
| P3 | 30.71 ± 0.74 | (-1.46 ± 0.62) · 10-4 | 10.22 ± 3.71 | 3.55 ± 0.90 | 9.75 ± 2.07 | |
| Tibolone | P1 | 30.02 ± 1.01 | (1.14 ± 2.85) · 10-4 | 9.83 ± 2.78 | 5.25 ± 1.79 | 15.07 ± 5.24 |
| P2 | 30.23 ± 1.86 | (1.53 ± 2.58) · 10-4 | 10.78 ± 2.51 | 5.85 ± 1.33 | 11.97 ± 2.74 | |
| P3 | 30.86 ± 1.28 | (-3.15 ± 1.15) · 10-4 | 14.77 ± 3.50 | 4.57 ± 1.34 | 11.28 ± 2.68 | |
Mean values and standard deviations of the trend and variance parameters estimated for the E2 and the Tibolone group (sample size: 10 animals per group). : intercept, : slope, : variance of first daytime period (6-12 a.m.), : variance of second daytime period (12 a.m.- 6 p.m.), : variance of nighttime period (6 p.m. - 6 a.m.).
Figure 6
Estimated limit cycle amplitudes per phase and treatment group. Limit cycle amplitudes occurring at the end of each experimental phase. (a) E2 group, (b) Tibolone group.
Figure 7
Changes in estimated amplitudes between vehicle and treatment phase. Pairwise differences (P2 - P3) of limit cycle amplitudes estimated in both treatment groups.
Table 5
Mean differences between vehicle and treatment phase
| Mean differences between vehicle and treatment phase | ||
|---|---|---|
| Treatment group | CI | |
| E2 | -1.1265 | (- ∞, - 0.8247) |
| Tibolone | -1.6807 | (- ∞, - 1.3224) |
Mean estimates and one-sided 95%-bootstrap confidence intervals (CI) of the pairwise differences (P2 - P3) in limit cycle amplitudes γ after treatment with E2 and Tibolone in P3.
Table 6
Differences between the amounts of amplitude reconstruction in the E2 and the Tibolone group
| Differences between the amounts of amplitude reconstruction in the E2 and the Tibolone group | |||
|---|---|---|---|
| Estimator | CI | p 0.2 | |
| θ(E2) - θ (TI) | -0.2294 | (- ∞, 0.0078) | 0.0031 |
Median estimates and one-sided 95% - bootstrap confidence intervals (CI) of the differences between the limit cycle amplitudes ratios obtained in both treatment groups. The p-value (p0.2) was obtained by applying the hypothesis pair in equation (7) to test whether this difference was below the threshold 0.2.