Temperature curves for mice infected with F. tularensis A1a, A1b, A2 and type B (n = 14 mice per F. tularensis group) were generated and analyzed (Figure 1). Diagnostic plots created from the 56 individual temperature curves suggested that the time series is autoregressive with dependencies at lags of 1 time period (2 hours) and 12 time periods (24 hours). The lag at 2 hours indicates successive temperature measurements are correlated while the lag at 24 hours indicates temperature measurements taken at the same time each day are correlated. The method of Barry and Hartigan [10] identified two change points (a time point at which some fundamental characteristic such as the mean process or variance structure of the time series changes) in the temperature time series for all 56 F. tularensis infected mice. A representative temperature curve with the two change points, (CP1 and CP2), and intervening phases (P1, P2 and P3) is shown in Figure 2.

Comparison of mouse subcutaneous temperature in P1 indicated that the subcutaneous temperature of F. tularensis infected mice fell within the tolerance interval (35.3°C to 38.6°C) nearly 99% of the time. Temperature fluctuation during this phase ranged from 35°C to 39°C and was consistent with the temperature of uninfected animals. Mice were asymptomatic throughout this phase, therefore, P1 was termed the “normal phase”. The second phase (P2) of F. tularensis infection began with a rise in temperature that was sustained over the time course of this phase. In P2 the subcutaneous temperature of F. tularensis infected mice fell within the tolerance limits only 33% of the time. Mice showed slight clinical symptoms, primarily huddling within their enclosure, but remained responsive and continued to eat and drink throughout most of this phase. Due to the elevated temperature (>38°C) maintained by infected mice during this period, P2 was termed the “febrile phase”. The last temperature phase of F. tularensis infection consisted of a sudden drop from elevated temperatures with temperatures continuing to drop until death of the mouse. Thus, P3 was termed the ‘hypothermic phase”.

In addition to the three phases of F. tularensis infection demarcated by temperature CP1 and CP2, other similarities were noted among the temperature responses in mice infected with F. tularensis. Autocorrelation plots indicated strong evidence of a diurnal effect on temperature during the normal phase of F. tularensis infection. The temperature fluctuation during this phase (35°C to 39°C) correlated directly to mouse activity over a 24 h period. The daily temperature cycle was much less pronounced in the febrile phase, although this could be because the length of the febrile phase was generally much shorter (mean length of 38 to 42 h) than the normal phase (mean length ranging from 77 to 138 h) (see Table 1), but is more likely due to the decrease in mouse activity attributable to F. tularensis infection. The hypothermic phase averaged 10 to 15 h and the mice did not present daily cyclical behavior.

The temperature pattern of infection was similar for all mice, as all mice underwent P1 to P3 (Figure 1). The mean length of each phase differed based on the infection group (F. tularensis A1a, A1b, A2 or type B) (Table 1). It was also observed that mice infected with a single strain did not succumb to infection or reach a particular phase or change point at the same time (Figure 1, Table 1). Mice infected with different strains of the same group were only in the same phase for 33% (2 out of 6 days), 14.3% (1 out of 7 days), 12.5% (1 out of 8 days) and 30% (3 out of 10 days) of the time for A1a, A1b, A2 and type B, respectively (Table 2).

To determine whether temperature data serves as a correlate to death, the mean temperature during the normal phase, the mean temperature during the febrile phase, CP1, or CP2, for F. tularensis group (A1a, A1b, A2 and type B) were used in the linear mixed-effect model to predict time to death. The mean temperature during the normal and febrile phase were found to be uncorrelated with time to death, whereas CP1 and CP2 (Figure 3A) both correlated with the observed time to death. CP2 was pursued further as a time to death predictor as it occurred later in the progression of infection and therefore provided information for the infection through P2. The differences (in hours) between the observed and predicted times of death based on CP2 are shown in Figure 3B. The maximum error of prediction as compared to observed time to death is about 10 h with 71% of the predictions being within 5 h of the observed time to death. The square root of the mean square prediction error (RMSPE) for a leave-one-out validation is 3.89 h; for a leave-five-out validation, it is 3.92 h; and when half of the data set is left out (leave-28-out), the RMSPE is 4.09 h. Compared to the RMSPE for the full set of data (leave-zero-out) of 3.58 h, these suggest a strong predictive value of CP2 for time to death.

As determination of CP2 required complex statistical analyses, temperature curves where further analyzed to elucidate an easily identifiable point in the infection that could be readily applied as a surrogate for time to death. With only two exceptions (n = 56 mice) that were due to misreads by the probe, CP2 was found to be the first temperature measurement following the febrile phase to fall below the mean temperature of the normal phase (Figure 2). Therefore, this temperature change was termed “drop point”. The drop point differed slightly for each mouse given that it is directly dependent on the mean of the normal phase. Further investigation of the length and frequency of temperature monitoring necessary to determine drop point based solely on the mean temperature of the normal phase was performed. Calculating the 1^st percentile of the distribution of the length of P1 (mean length of time for P1 was 91.3, 76.5, 97.1 and 138.4 h in A1a, A1b, A2 and type B infected mice, respectively) indicated that the temperature of mice infected should be monitored for at least 56.6 h for A1a, 21.9 h for A1b, 22.3 h for A2, and 53.7 h for type B in order to accurately identify the drop point (Table 1). To determine the frequency of temperature monitoring necessary, the percent agreement between CP2 (based on joint likelihood analysis of time series plots) and drop point (the first temperature below the mean temperature of the normal phase) was computed for 5000 of temperature data for each of the 56 mice infected with F. tularensis. When temperature was sampled every 2 h, CP2 and drop point analyses identified the same time point 96% of the time. When temperature was sampled at 4, 6 and 12 h intervals, the agreements were 95.4%, 94.9%, and 92.1%, respectively. These results suggest that monitoring frequency should be no less than every 6 h to appropriately identify the drop point.

We previously showed that the parametric survival curves (modeled according to a Weibull distribution) for mice infected with F. tularensis A1a, A1b, A2 or type B are statistically different from one another in scale (i.e. mean and variability of time to death) and/or shape (i.e. symmetry of pattern of death) using observed time to death as the endpoint [9]. Therefore we investigated whether survival curves based on predicted time to death using CP2 or drop point yielded statistically similar results when compared with observed time to death. The differences observed among F. tularensis infection groups were the same as those obtained from observed time to death experiments when parametric survival curves were generated based on either CP2 (data not shown) or drop point (Figure 4A). Specifically, the scale parameter of survival curves differed significantly (p<0.008) between A1b infected mice and those of A1a, A2 and type B infected mice as well as between A1a and A2 infected mice. The survival curve for mice infected with A1a differed significantly (p<0.008) in shape from the survival curves of mice infected with A1b, A2 and type B. Although the prototypic F. tularensis virulent strain Schu S4 was not included in this study, it should be noted that this strain is classified as an A1a strain [11]–[12] Survival curves based on CP2 were statistically the same as the survival curves based on drop point (data not shown). These results indicate that predicted time to death based on CP2 or drop point can readily substitute as a non-biased surrogate endpoint for time to death experiments.

To further assess the utility of using drop point as an ethical endpoint for survival curve analysis, mice (n = 7) were infected with F. tularensis A1b strain MA00-2987, the same strain used in previous observed time to death experiments [9], and mouse temperature was monitored every two hours over the mean length of the normal phase for A1b (21.9 hours). The mean temperature of the normal phase was determined for each mouse and mice were euthanized once the drop point temperature was observed. A survival curve using drop point as the endpoint was generated. This was then compared to a F. tularensis MA00-2987 survival curve using drop point data calculated from a previous observed time to death experiment [9]. No significant differences (p = 0.89 for scale parameter; p = 0.95 for shape parameter; log-likelihood ratio p = 0.99) were observed in these two survival curves (Figure 4B), thus demonstrating the accuracy and robustness across experiments as well as the feasibility of using the drop point temperature as a surrogate endpoint for death.

All animal procedures were approved by the Division of Vector-Borne Diseases Institutional Animal Care and Use Committee (protocol number 08-012) and performed in accordance with the guidelines on the care and use of laboratory animals [20].

Specific-pathogen free 8–9 week old female inbred C57BL/6 mice (The Jackson Labortory, Bar harbor, ME) were used. Mice were implanted a week prior to F. tularensis infection with IPTT-300 transponders (BioMedic Data Systems Inc., Seaford, DE) programmed with mouse identifiers. Mice were anesthetized by inhalation with isoflurane and transponders were implanted below the dermis in the upper back region as per the manufacturer's instructions. Mouse subcutaneous temperature was monitored using the DAS-6007 probe (BioMedic Data Systems Inc.) once a day prior to infection and every 1 to 2 hours following infection until the mice expired. Control mice injected with saline were used to monitor normal mouse subcutaneous temperature throughout the study. Mice were given food and water ad libitum, and an exercise wheel was provided in every cage. Infection of mice with strains representing each of the four F. tularensis groups (A1a, A1b, A2 and type B) was accomplished by inoculation of 10–20 CFU intradermally (50 µl) via the tail dermis and colony forming units in each inoculum were verified as previously described [9]. For temperature model development, mice were allowed to expire. Mice (n = 7 in each group) were infected in 2 rounds. The first round consisted of infection with one strain representing each of the four F. tularensis groups (A1a, A1b, A2 and type B). The second round was a repeat of the first round using a different set of strains. A final animal infection was performed to test the temperature model. In this case mice were infected with F. tularensis strain MA00-2987 (n = 7) as described and temperature was monitored every two hours. The mean temperature of the normal phase was calculated once mice entered the febrile phase and mice were euthanized at their drop point. Two criteria needed to be satisfied to conclude a mouse had entered the febrile phase: first, the length of time passed must be greater than the 5^th percentile of the distribution of the length of the normal phase, and second, the temperature must remain above the cumulative mean temperature for 12 consecutive hours. All animal experiments were conducted in an ASBL3 facility.

F. tularensis strains (n = 8) used in this study originated from human tularemia cases [9]. Strains were grown from frozen stocks (−70°C) on cysteine heart agar supplemented with 9% sheep blood (CHAB) at 35°C for 48 h, followed by subculture onto CHAB for 24 h at 35°C. All F. tularensis strains were typed using pulsed-field gel electrophoresis (PFGE) as previously described [11], [21].

Time series plots of temperatures were generated using Spotfire S+ v8.1 (TIBCO Software, Inc.). Autocorrelation and partial autocorrelation plots were created for different contiguous subsets of the data in order to determine approximate structure of the moving average and autocorrelation components of the time series. To identify the number of change points (different segments of the time series), the method described by Barry and Hartigan [10] was implemented. To identify the location of the change points, the joint likelihood of the time series segments was constructed. Temperatures were assumed to be statistically independent among the segments. The mean process in all but the final segment was assumed constant (but different). The mean process in the final segment was modeled by a second degree polynomial function of time. The variances of the segments were allowed to differ by segment, but were assumed constant within each segment. The joint likelihood of the data was maximized over a grid of change point values.

For temperature comparisons between control and infected mice the marginal mean and variance of the distribution of temperatures in the control mice were estimated using a linear mixed-effects model accounting for mouse effects and AR1 correlation among temperatures within mice. Diagnostics, including quantile-quantile plots were used to verify the assumption of normality. Using the marginal mean and variance, a 95% tolerance interval for temperatures in the control mice was constructed. By construction, a temperature sequence consistent with that found in the control mice should fall within the tolerance limits 95% of the time. This tolerance interval was mapped onto the temperature sequences of the infected mice to determine if normal phase temperatures and febrile phase temperatures in infected mice were consistent with temperatures in control mice.

Using a linear mixed-effect model, time to death was modeled as a function of temperature data (time of change points, mean temperature during normal phase, and mean temperature during febrile phase). F. tularensis group (A1a, A1b, A2 and type B) was included as a random effect. Standard diagnostics, including residual analysis, were employed to ensure a valid model. The model was fit using data from all infections, and a cross-validation on the model was performed leaving out subsets of size one to 50% of the data and calculating the RMSPE (an estimate of the average number of hours by which the predicted times of death differed from the observed times of death).

Distribution plots of the lengths of the normal phase were made and Weibull densities were fit to these lengths for each F. tularensis group. The 1st, 5th, and 10th percentiles for each distribution were then estimated. Using the estimated models from the original data [9] we simulated 5000 new sets of temperature data for each mouse. Under sampling frequencies of 2, 4, 6, and 12 h and sampling to the 5^th percentile, the proportion of times among all simulations in which the sample mean temperature during the normal phase was higher than the sample minimum temperature during the febrile phase was computed.

As described previously [9], survival times were modeled according to a Weibull distribution (allowing both the scale and shape parameters to differ) using predicted time to death and drop point as endpoints. Standard diagnostics including residual plots and goodness of fit tests were used to validate the model fits. The survival curves generated here were compared to those created using observed time to death [9]. Differences in the parameter estimates (shape and scale) of the survival curves were considered statistically significant if p<0.008, with this level of significance determined using the Bonferroni adjustment for multiple comparisons.