Application of a Statistical Dynamic Model Investigating the Short-Term Cellular Kinetics Induced by Riddelliine, a Hepatic Endothelial Carcinogen

Marjo V. Smith*,1, Abraham Nyska{dagger} and Chris Portier{dagger}

* Constella Health Sciences, 2605 Meridian Parkway, Durham, North Carolina 27713 and {dagger} National Institute of Environmental Health Sciences, National Institutes of Health, Research Triangle Park, North Carolina 27713

Received December 15, 2004; accepted April 20, 2004


    ABSTRACT
 TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 REFERENCES
 
In recent studies, riddelliine, a pyrrolizidine alkaloid, was found to increase rates of replication and apoptosis and induce hemangiosarcoma in the liver of rats and mice. To analyze DNA replication and apoptosis data taken from the same animals, we have developed a predictive mathematical model for describing BrdU labeling and apoptotic processes. The model allows the incorporation of simple diurnal patterns in cellular kinetics and is applied to data on hepatocytes and endothelial cells taken from riddelliine exposed rats. Predictions from the model were used with multivariable nonlinear regression techniques to estimate replication and apoptotic rate constants for both cell types and all treatment groups. Hypothesis tests were used with the predicted rates to separate the competing effects of riddelliine on replication and apoptosis of hepatocytes and endothelial cells as well as compare replication rates between cell types. That estimated replication rates were found to be significantly higher for endothelial cells supports the supposition of induction of hemangiosarcoma by riddelliine in the liver.

Key Words: riddelliine; mathematical model; cellular kinetics; hemangiosarcoma.


    INTRODUCTION
 TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 REFERENCES
 
Riddelliine is a pyrrolizidine alkaloid existing in plants of the genera Sencio, Crotalaria, and Amsinckia found in the rangeland of the western United States. Early studies established that riddelliine induced unscheduled DNA synthesis in rat hepatocytes in vitro and in vivo (Mirsalis, 1987Go; Mirsalis et al., 1983Go, 1993Go). A later study showed evidence for a genotoxic mechanism for liver tumor induction in rats by riddelliine (Yang et al., 2001Go). The National Toxicology Program bioassay of riddelliine (NTP TR-508) found that this chemical induced a high incidence of hemangiosarcomas in rats and mice (Chan et al., 2003Go). Hemangiosarcomas develop from endothelial cells that line the blood vessels; replication of endothelial cells has been linked to release of vascular endothelial growth factor (VEGF; Maragoudakis et al., 2000Go).

Nyska et al. (2002)Go, while investigating the mechanism of riddelliine induction of hemangiosarcomas, exposed male F334/N rats to three dosages of riddelliine (0, 1, and 2.5 mg/kg) using two different protocols, a one-week and a six-week exposure. Apoptotic cells were measured by the number of TUNEL-positive cells for both hepatocytes and endothelial cells, while DNA replication was measured by BrdU administered in the drinking water three days before sacrifice. In both cell types, only the highest dose group with the longer exposure showed significant changes in apoptotic numbers, found to be lower than those of the control group. Significant differences from the control groups in BrdU labeling index were found for the highest dose level groups in both protocols and for both cell types: a significant increase for the shorter exposure (both cell types) and for the longer exposure, a significant decrease for hepatocytes and increase for endothelial cells.

This article provides an extended analysis of these data using a dynamic model describing predictively the replication and apoptosis for both hepatocytes and endothelial cells during the period of active BrdU uptake in both protocols. The present analysis extends that of an earlier article in that it can incorporate directly and simultaneously in the same analysis information about individual liver size and average cell density for each treatment group as well as BrdU and apoptosis data. Furthermore, multivariate regression techniques can be used with the model predictions to take into account the correlation between hepatocyte and endothelial cell counts taken from the same animal. Since the residuals from the model did not differ significantly from the normal distribution, parametric (i.e., F distribution based) tests could be used to test the significant treatment effects of riddelliine.

In addition DNA replication rates and apoptotic rates could be compared between cell types. In so doing, the application of this dynamic model may help to explain hemangiosarcoma development in terms of the relative sizes of the kinetic parameters of the two cell types investigated. The presented technique is more generally applicable to any experiment combining labeling data with apoptotic counts.


    MATERIALS AND METHODS
 TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 REFERENCES
 
Data. The experimental data used in this article were previously described in detail (Nyska et al., 2002Go). Briefly, six groups of seven male F334/N rats were divided into two different experimental protocols with three treatment levels each. The two protocols were started at the same time, when the rats were about six weeks old. Each protocol incorporated a control group, a low-dose group (1 mg/kg body weight), and a high-dose group (2.5 mg/kg body weight). For the eight-dose protocol, the rats were dosed daily for eight days and then sacrificed. For the 30-dose protocol, the rats were dosed daily Monday through Friday (but not on the weekends) for six weeks before sacrifice. In both protocols, BrdU was administered in the drinking water at a concentration of 80 mg/100 ml during the last three days before sacrifice.

Slides were prepared from liver sections (about 6 µm thick) from each rat. Each slide contained approximately 112 fields for microscopic examination. For three rats from each treatment group, five fields were randomly chosen, and the cell counts averaged to estimate the average number of hepatocytes and endothelial cells per field for each treatment group. It is assumed that the same volume (Vliv) of liver is examined for each field, so that results are comparable between fields. If a constant density of the liver ({delta}liv) is also assumed, then each field examined Vliv {delta}liv gm of liver. The total number of hepatocytes or endothelial cells can then be approximated for each rat by the following:

Where CTh and CTe are the total number of hepatocytes and endothelial cells respectively, CAhepa and CAendo are the average number of cells per field as described above, and LW is the liver weight. In further computations, the common factor, 1/Vliv {delta}liv gm, will drop out, so that the actual density of the liver is neither used nor estimated. Table 1 shows the means of the estimated numbers of hepatocytes and hepatic endothelial cells for each rat up to the factor 1/Vliv {delta}liv.


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TABLE 1 Summary of Means of Estimated Cell Numbers

 
For each rat one liver section (about 112 fields) was perused, and the apoptotic hepatocytes and endothelial cells were identified using the TUNEL assay and counted. For each cell type, the total number of apoptotic cells was calculated by the following:

for hepatocytes, and for endothelial cells,

where CTh, CTe, CAhepa, and CAendo are as above, and Ahepa and Aendo are the numbers of apoptotic cells counted for a section.

DNA replication was measured in hepatocytes by counting both labeled and unlabeled cells within five randomly selected fields for each rat. At least 1500 hepatocytes were scored for each animal. The total number of unlabeled hepatocytes is calculated as

For endothelial cells only the labeled cells could be counted. Since the same five fields were used to evaluate proliferation for both endothelial cells and hepatocytes, the total number of endothelial cells present in the five fields could be calculated by multiplying the total number of hepatocytes in the five fields by the ratio of average cell counts per field of each cell type found above:

Thus, for each rat, C5fields(endo) is the number of endothelial cells found in the five random fields, CAendo and CAhepa are as above, and Chepa is the number of all hepatocytes counted on the five random fields. The total number of unlabeled hepatic endothelial cells is calculated as

In summary, the data can be used to calculate six quantities for each rat: total cell count, unlabeled cell count, and apoptotic cell count for each of the two cell types. The means and SEs of these calculated cell counts up to the factor 1/Vliv {delta}liv are given in Table 1.

Because cell counts per slide vary a great deal between hepatocytes and the less common endothelial cells, scientists often compare cell counts indices. Thus, Table 2 shows the mean BrdU labeling indices (percent of labeled cells of total counted cells) and the mean apoptotic indices (percent of apoptotic cells of total counted cells).


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TABLE 2 Means of Labeling and Apoptotic Indices for Control Groups

 
The model. A differential equation system is set up to describe the transition of cells from the unlabeled to the labeled state and from both states to apoptosis. The differential equations are solvable analytically for each cell type. A link between the two cell types was not explicitly modeled.

A complicating feature of the experiment was the use of young rats, which grew appreciably through the course of the experiment. To avoid having to model the growth process or possible effects of BrdU on the growth rate (Ton et al., 1997Go) in addition to the effects of riddelliine exposure, only the period of active BrdU uptake was modeled. By beginning the modeling period at the point of active BrdU uptake, all cells may be assumed to be unlabeled at t = 0. As cells cycle, they incorporate BrdU into newly formed DNA during the synthesis phase and are counted as labeled cells. A cell with replicated DNA is assumed to divide into two labeled daughter cells. During this same period, cells may also enter apoptosis. The remnants are assumed to remain visible for a fixed period of time before being absorbed by surrounding cells. This period of time will be referred to as the apoptotic duration of a cell. If L refers to labeled cells, U to unlabeled cells, and A to apoptotic cells, then Equations 1 Go3 give the basic model. The model is applied separately with potentially different parameter values for each treatment/protocol/cell type.

(1)

(2)

(3)

The solution to the last equation in the model denotes the visible and identifiable apoptotic cells or bodies. Apoptotic bodies produced earlier than d days (the apoptotic duration) before the end of the experiment are no longer identifiable. Thus, the last equation is integrated over the interval, [T – d, T]. Bursch et al. (1990)Go estimate the average duration of the visible stages of apoptosis to be about 3 h for hepatocytes in female Wistar rats, but the apoptotic duration for endothelial cells has not been experimentally determined. The model was initially applied to the data assuming a common apoptotic duration for both cell types of 3 h.

Rats are strongly nocturnal, neither eating nor drinking much during periods with light (0600–1800 h for this experiment.), as long as food and water are supplied ad libitum as in this experiment. Thus, although BrdU was supplied in the drinking water in the morning three days before sacrifice, the first 12 h were not modeled. If the active uptake of BrdU starts at t = 0, then the time of death, T, is 2.583 days, 0800 h of the third day.

Evidence suggests that the animals' feeding and drinking habits may in turn affect the cell kinetics. During the day the animals eat little food, triggering apoptosis that peaks towards the end of the day. Nightly feeding triggers DNA replication albeit with a 5–8 h delay, resulting in peak DNA replication in the early morning (Dragan et al., 2004Go; Grasl-Kraupp et al., 1994Go; Schulte-Hermann, 1976Go, 2004Go). Diurnal rhythms become visible only when treatment increases the kinetic rates sufficiently (Grasl-Kraupp et al., 1994Go; Schulte-Hermann, 2004Go). Although the present data set does not include observations at enough time points to detect diurnal rhythm, the data do show a significant increase in the number labeled endothelial cells in riddelliine treated groups (Nyska et al., 2002Go; Tables 1 and 2].

For this reason the data were analyzed a second time assuming a simplified diurnal schedule for both apoptosis and DNA replication as shown in Figure 1. As the light period in the cages lasted from 0600–1800 h, the diurnal schedule assumes DNA replication takes place primarily from 0400–1000 h, and apoptosis takes place only during the day, with a low rate during the morning hours (0600–1200 h) and at a high rate during the afternoon (1200–1800 h).



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FIG. 1. The simplified diurnal pattern under which the model was fit. The double headed arrows in the lower right show the apoptotic durations of hepatocytes (the short gray arrow on top, denoting 3 h) and the endothelial cells (longer black arrow underneath, denoting 17 h).

 
The larger apoptotic index for endothelial cells as compared to hepatocytes (see Table 2) could be explained without requiring a higher apoptotic rate for unexposed cells if the apoptotic bodies from endothelial cells were absorbed more slowly, i.e., if the apoptotic duration were longer for endothelial cells. Examination of Figure 1 shows that the 3-h apoptotic duration of hepatocytes just before sacrifice occurs during a period of low apoptosis. If apoptotic duration for endothelial cells can be assumed to last as long as 17 h, then apoptotic bodies remaining from the previous period of high apoptosis (1500–1800 h) would still be visible at sacrifice. The combination of the diurnal pattern with the 17 h apoptotic duration for endothelial cells allows the apoptotic rates to be the same for hepatocytes and endothelial cells. Finally, for comparison, the data was fit again to the model, with the assumption of constant rates but with the same 17-h apoptotic duration for endothelial cells.

In the Appendix, Equations 1Go3 are solved for general time-dependent rate parameters, and adapted to the assumptions of constant rate parameters or the diurnal schedule shown in Figure 1.

Equations 4GoGo7 show the predictive equations for the constant rates (see Appendix).

(4)

(5)

(6)

(7)

U and A denote, respectively, the imputed number of remaining unlabeled cells, and the imputed number of apoptotic cells of each cell type as indicated by the subscript (h = hepatocytes; e = endothelial cells). C denotes the total number of nonapoptotic cells, labeled and unlabeled. The parameters dh and de denote the period of identifiability of apoptotic bodies for hepatocytes and endothelial cells, respectively. The four equations require four parameters values from the fit ({lambda}h, {alpha}h, {lambda}e, and {alpha}e) for each animal. Note that the common factor, 1/(Vliv {delta}liv), cancels out in each equation. Ch(T) and Ce(T) function as covariates in the right hand side of the equations, generating individual predictions for each rat.

To apply the model to the simplified diurnal schedule, the rate parameters are considered to be step functions with different values over subintervals delineated in Figure 1. Since an apoptotic duration of 3 h continues to be assumed for hepatocytes, the hepatocytes apoptotic observations are used to estimate the apoptotic rate for periods of low apoptosis, while the endothelial apoptotic observations are used to estimate the apoptotic rate for periods of high apoptosis. The replication rates are allowed to be different between cell types. Equations 8GoGo11 show the predictive equations for the diurnal schedule.

(8)

(9)

(10)

(11)
In these equations, {lambda}h or {lambda}e refers to the average replication rate over the modeled interval. The four parameter values required for Equations 8GoGo11 are ({lambda}h, {alpha}lo, {lambda}e, {alpha}hi).

Statistical analysis. Equations 4GoGo7 or 8GoGo11 constitute a multivariable statistical model, predicting all four measurements made on each animal. The model may be rewritten for the ith animal in a particular protocol as follows

where the four-tuple on the left represents the four measurements taken on the ith rat, namely, the numbers of hepatocytes, apoptotic hepatocytes, endothelial cells, and apoptotic endothelial cells. The arrows indicate the quantity underneath is a vector, containing several elements: four for , , and ; 12 for . The predicting expression, , represents the right hand sides of Equations 4 GoGo7 or 8GoGo11 and is nonlinear in the parameters. Note that since T represents the same length of time interval (2.583 days) for all animals, it is not explicitly listed as a variable. The parameter vector, , consists of four parameters ({lambda}h, {alpha}h, {lambda}e, {alpha}e) or ({lambda}h, {alpha}lo, {lambda}e, {alpha}hi) for each dose group or 12 parameters in all. The two protocols were analyzed separately. The method of estimation used in this paper is taken from Gallant's book on statistical methods, especially Chapter 5 (Gallant, 1987Go). Matlab was used throughout the analysis (The Math Works, Inc., Natick, MA).

The residual vector as it stands is not independent or homoscedastic. Thus for each experimental protocol, the parameters were estimated in two stages. First, the observations were averaged over each treatment group and used with Equations 4GoGo7 or 8GoGo11 to find the method-of-moment estimates for all parameters. These rate estimates were used with Equations 4GoGo7 or 8GoGo11 to make predictions for each animal, and the residuals of the three treatment groups within each protocol were combined and used to estimate the variance covariance matrix for that protocol. These variance covariance matrices were used as weights to find the weighted least squares parameters for each set of 12 parameters. The expression to be optimized can be written as

The residuals from the weighted least squares estimates for each protocol were tested for normality using the Kolmogorov-Smirnov test (Lindgren, 1968Go). In the case that the residuals did not deviate significantly from normality, multiple partial F-tests were used to test the significance of the difference between the treatment rate parameters and the appropriate control parameters. For example, to test whether the low dose replication parameter for hepatocytes of the eight-dose protocol was significantly different from that of the corresponding control group, the same parameter would be used in both positions in . In this case, the optimized would have only 11 distinct values, not 12. The residual sums of squares using 11 and 12 parameters, respectively, would be compared using an F-statistic. (See Kleinbaum et al., 1988, p. 132 for linear models, and Gallant, 1987Go, p. 56, for the extension to nonlinear models.) Note we did not test for trend, but only compared the treatment groups separately to the controls. The large number of individual tests done requires care with the significance level. In the final summary of the treatment comparisons, those significant at p = 0.025 and p = 0.005 are shown.

The rate parameter estimates could be compared to each other, even between hepatocytes and endothelial cells, because the estimated rates were multiplied by the number of cells to predict the number of replicated or apoptotic cells. In other words, the rates were normed with respect to the total number of cells of each type. The significance of the difference between any two parameters can thus be statistically tested, e.g., parameters between dose levels in the same experiment, replication rates between hepatocytes and endothelial cells, or even between replication rates and apoptotic rates.


    RESULTS
 TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 REFERENCES
 
In all, the model was fit under three different scenarios. With the assumption of constant kinetic rates (i.e., no diurnal rhythm in cell kinetics) the model was fit with an endothelial apoptotic duration of 3 h (the same as for hepatocytes) and again with an endothelial apoptotic duration of 17 h. The model was fit a third time using a simple diurnal pattern as shown in Figure 1. The estimated variance covariance matrices for the two protocols were found to be the same for all three scenarios. Similarly, the residual from the optimizations under the three scenarios was found to be the same.

Figure 2 shows the values predicted by the model against the observations. As expected given the equivalence of fit under the three scenarios, the graphs of predicted vs. observed values do not change under the different scenarios. Therefore only one set is shown. Of the three plots, the worst fit is for apoptotic predictions (Fig. 2A) with r2 = 0.61. The only covariates used in the model are the imputed final numbers of endothelial cells and hepatocytes at sacrifice. The variability of the number of apoptotic cells between slides probably contributes to this poor fit. Figure 2B shows the predicted values for the number of unlabeled cells. The r2 = 0.9992, but this is largely an artifact of there being far fewer labeled than unlabeled cells. Thus the observed number of unlabeled cells is strongly correlated with the total number of cells, which is also used to compute the predicted number of unlabeled cells. For comparison, Figure 2C shows the predicted values of the number of labeled cells, although these were not directly used in the analysis. Here r2 = 0.86, so the correlation is still higher than for the apoptotic cell predictions.



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FIG. 2. The predicted values for apoptotic cells (A), unlabeled cells (B), and labeled cells (C) up to a factor of 1/Vliv {delta}liv gm of liver against the values imputed from observations. The r2 values are 0.61 for the predicted apoptotic cells (A), 0.9992 for the predicted unlabeled cells (B), and 0.86 for the predicted labeled cells. Each of the four groups of cells had different estimated variances.

 
Tables 3GoGo through 6 show the parameter values and their SEs. For the two scenarios with constant rate parameters (no diurnal rhythm) the different assumed values for the endothelial apoptotic duration did not affect the estimates of the replication rate. Under the assumed diurnal rhythm, DNA replication occurred only from 4:00 a.m. to 10:00 a.m. As shown in the Appendix, the average DNA replication rate over the entire modeled interval is the same as if a constant rate had been assumed, so that the average replication rates for the diurnal scenario are also the same as those in Table 3. To find the DNA replication rate over just the (4:00 a.m. to 10:00 a.m.) interval under the diurnal scenario, multiply the values in Table 3 by 62/16.


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TABLE 3 Predicted Replication Rates

 

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TABLE 4 Predicted Apoptotic Rates

 

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TABLE 5 Predicted Apoptotic Rates

 

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TABLE 6 Predicted Apoptotic Rates

 
The final residuals from the weighted least squares fit of the full model did not deviate significantly from the normal distribution for either experimental protocol (Kolmogorov-Smirnov test, p > 0.10), supporting the notion that hypotheses can be tested using the F-test. In Tables 3GoGo through 6 the p-values are shown for testing the significance of the difference between each treated rate parameter and its corresponding control value. Single asterisks show significance at the 0.025 level, while double asterisks show significance at the 0.005 level.

Tables 4Go through 6 show the estimated apoptotic rates under the three scenarios. As expected, the longer assumed apoptotic duration for endothelial cells predicts a lower estimated apoptotic rate. Even with an apoptotic duration of 17 h for endothelial cells, however, the estimated apoptotic rates for endothelial cells are still larger than those for hepatocytes, significantly so for the control group in the 30 dose protocol (p < 0.001) Under the assumption of the diurnal pattern of Figure 1, high and low apoptotic rates can not be estimated separately for hepatocytes and endothelial cells. In this case only two of the last 3 h before sacrifice have positive apoptosis, so the low apoptotic rate is estimated to be about 50% larger than the estimated hepatic apoptotic rate. The high apoptotic rates were significantly higher than the low rates for both control groups (p = 2.76e-2 for the 8-dose protocol and p < 0.001 for the 30-dose protocol) predicting a strong diurnal pattern.

Although the estimated apoptotic rates are different for endothelial cells under the three different scenarios, the hypothesis tests between treated groups and the appropriate control groups show the same results. The analysis does not show significant change due to riddelliine in the apoptotic rates. The increase in apoptotic rates in endothelial cells at the end of the eight-dose protocol is suggestive, but not statistically significant. Both hepatocytes and endothelial cells showed significant riddelliine effects in their estimated replication rates. The highest dose group of the eight-dose protocol showed a significant increase in replication rates for both cell types, most strongly for the endothelial cells. Hepatocytes of the 30-dose protocol showed a significant decline in predicted replication rates at both the low and the high dose levels. The endothelial cells on the other hand showed significantly increased replication rates at both the low and high dose levels.

The results from the present analysis do differ somewhat from those found in the original paper. While the original analysis found a significant decrease in the number of apoptotic cells at the highest dose level after the 30-dose protocol, we did not find that decrease to be significant. On the other hand, in addition to finding significant differences in replication rates at the highest dose level for both protocols and cells types, we also found significant differences in replication rates at the low dose for the 30-dose protocol for both cell types.

Table 7 confirms these results by additionally testing the difference between replication rates of hepatocytes and endothelial cells directly. Note the F-test just tests whether or not the difference between two rate parameters is significant, so that the p-values in Table 7 correspond to two-sided tests. All treatment groups show the endothelial replication rate to be significantly larger than the hepatocyte rate.


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TABLE 7 Comparing Replication Rates for Different Cell Types

 
In summary, a model-based analysis of cell kinetics data confirmed and strengthened results of an earlier analysis indicating a strong treatment effect of riddelliine on replication rates of hepatocytes and endothelial cells (Nyska et al., 2002Go). In addition, replication rates of hepatocytes and endothelial cells were compared directly, showing the endothelial cells to have a significantly higher predicted replication rate in all treatment groups. Although the unknown endothelial apoptotic duration is used by the model, the results of the model-based testing were shown to be completely robust to different assumed values.


    DISCUSSION
 TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 REFERENCES
 
These data have demonstrated a model-based analysis of replication and apoptosis rates in rats exposed to riddelliine. This analysis, like any other assessment, was dependent on the accuracy of the model and the assumptions that accompanied it. In our model, a simple first-order kinetic model was assumed to describe cellular replication, label uptake, and apoptosis of cells. Real growth of the animal was not explicitly modeled, although the effects of five additional weeks of growth were included in the different parameter values found in the eight-dose and the 30-dose protocols.

The different scenarios (constant kinetic rates vs. the assumption of a diurnal pattern) did not lead to different results or even different qualities of fit. Upon reflection these results are not surprising: each treatment group had two observed quantities (apoptotic and unlabeled cells) and two estimated parameters ({lambda} and {alpha}) for each cell type. The model closely predicted mean values for each of the observed quantities with the same scatter around the means for each scenario. Even so, fitting the three scenarios fit gave interesting results, all reflecting the much higher observed apoptotic index for endothelial cells as compared to hepatocytes in the presence of roughly similar replication rates. The examination of the apoptotic rates in the control groups of the two protocols suggest endothelial cells may have a much longer apoptotic duration than hepatocytes, since the significantly higher apoptotic rates under the assumption of a 3 h apoptotic duration are difficult to explain. The assumption of a diurnal rhythm led to significant differences between the high and low apoptotic rates, which are not usually found in untreated animals (Grasl-Kraupp et al., 1994Go). Testing for significant differences between kinetic rates of treated groups and the appropriate control groups was shown to be very robust, even though no information was available on the apoptotic duration for endothelial cells.

A mild increase in karyomegaly accompanying riddelliine exposure was observed in endothelial cells but not measured quantitatively, so that incorporation in any quantitative analysis is difficult. Mitotic activity was observed to diminish for hepatocytes exposed to riddelliine. Possible increases in ploidy without quantitative data would be a problem in any statistical analysis and was not addressed in the original article (Nyska et al., 2002Go). The resulting decrease in the number of cells per slide was accounted for by using the observed density of cells for each treatment group in the estimation of the total number of cells used in the estimation of rates (see Data section). However, the model as presented assumes that a nucleus picking up BrdU label through DNA synthesis divides into two daughter nuclei. Ploidy change can easily be incorporated by changing Equations 1 Go3 to following

(12)

(13)

(14)
where {lambda} specifically denotes the DNA replication rate, while the quantity {gamma} – 1 denotes the probability of cell division: if {gamma} = 1, cells may still replicate DNA and acquire label, but no division takes place; if {gamma} = 2, all cells that replicate also divide.

In order to evaluate {gamma}, quantitative information of the number of cells showing increased ploidy is needed and was not available for the present experiment. There are two reasons for thinking the effect on the analysis presented in this article may be slight, however. First, only the last three days of both experiments were modeled, so that the effect of the mistakenly assuming two daughter cells did not accumulate over the entire course of the experiment. Second, as shown in the Appendix, the mistaken assumption of two daughter cells results in a conservative estimate of the replication rate, {lambda}. Since the principal finding is that the endothelial replication rate increased with riddelliine exposure (and therefore, increased incidence of karyomegaly), those results are not affected.

The proposed mechanism for the induction of liver hemangiosarcoma (Nyska et al., 2002Go) suggests that the active metabolite of riddelliine interacts with endothelial DNA, causing damage to endothelial cells of the liver that includes karyomegaly, cytomegaly, and apoptosis. The enlarged endothelial cells obstruct the blood vessels causing local hypoxia. Hepatic hypoxia was earlier shown to induce VEGF production by hepatocytes (Rosmorduc et al., 1999Go). Increases in VEGF then induce increases in endothelial cell replication. The increased replication enhances the probability that DNA damage, either spontaneous or drug-induced, will escape repair and become fixed as mutations that eventually lead to hemangiosarcomas.

The results presented in Tables 3GoGoGo through 7 support the above stated mechanism for inducing hemangiosarcoma (Nyska et al., 2002Go). At the end of the eight-dose protocol, predicted endothelial and hepatocyte replication rates were significantly increased at the high-dose level. That the hypoxia also triggered replication in the endothelial cells was suggested by Nyska et al. (2002)Go. At the termination of the 30-dose protocol, the predicted hepatocyte replication rates demonstrated a dose-related antimitogenic effect associated with riddelliine (Wilson et al., 2000Go), decreasing at every dose level. On the other hand the endothelial cells showed a dose-related increase in predicted replication rates. Table 7 shows the replication rates predicted for endothelial cells to be significantly higher than hepatocytes for all treatment groups, though similar in control groups for both protocols. The "advantage" of the endothelial cells over the hepatocytes regarding replicative parameters suggests the eventual carcinogenic outcome, i.e., hemangiosarcoma development.


    APPENDIX
 TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 REFERENCES
 
Computation of Equations 4GoGo7 and 8GoGo11
The model shown in Equations 1Go3 in the text can be solved more easily by adding the equations for unlabeled and labeled cells to form a differential equation for all non-apoptotic cells, C(t):

(A1)
The following system of equations is equivalent to Equation 1, but easier to solve:

(A2)

(A3)

(A4)

Note that in this formulation, the equations in U and C can be solved independently, for general time-dependent rate parameters. We get

and

By design, U(0) = C(0), so that we may divide and find the ratio

or

(A5)

In the case that kinetic parameters are assumed constant, the right hand side of equation [A5] becomes 2{lambda}cT, which can be readily solved for {lambda}. In general is the time average of the time-dependent replication rate, denoted , so that the right hand side of A5 becomes . If the simplified diurnal schedule outlined in Figure 1 is used, , where {lambda} is the replication rate in units per cell per day from 4:00 a.m. to 10:00 a.m. (see Fig. 1). The average replication rate over the modeled interval (0, T = 2.583 days), , is the same as the constant replication rate, {lambda}c. The prediction equation in general for the number of unlabeled cells is

(A6)

To find the prediction equation for the number of apoptotic cells, a formulation of C(t) is needed where t is in the interval [0, T = 2.583]. To find this function in general, we integrate the differential equation for dC/dt, over the interval [t,T], where t is in the interval [0, T]. Solving the resulting equation for C(t), we get

Then the general predictive equation for the number of apoptotic cells is

(A7)
In the case that the kinetic parameters may be assumed to be constant, this equation simplifies to

(A8)
which is used in Equations 4GoGo7 in the text.

Applying Equation A7 to the simplified diurnal schedule (see Fig. 1) requires dividing the apoptotic durations into subintervals with constant kinetic rates for the endothelial cells. The assumed period of apoptotic duration for endothelial cells (17 h) is [1.875, 2.583], thus the subintervals would be: [1.875, 2.0] with no replication and a high rate of apoptosis; [2.0, 2.417] with neither replication nor apoptosis; [2.417, 2.5] with replication but no apoptosis; and [2.5, 2.583] with replication and a low rate of apoptosis. The resulting equation is then,

(A9)
where {lambda} is the time-specific replication rate from 4:00 a.m. to 10:00 a.m. This value is related to the average values (found in Table 3) by multiplying the tabled values by 62/16, since 16/62 is the portion of the modeled period with DNA replication under the diurnal pattern in Figure 1. Equation A9 is used for Equation 11 in the text. Equation 9 in the text, predicting the number of apoptotic hepatocytes is found by using only the latest subinterval, [2.5, 2.583].

The Effect of Unmodeled Ploidy Changes
The ploidy version of the model shown in Equations 12Go14 in the text can again be solved more easily by adding the equations for unlabeled and labeled cells to form a differential equation for all non-apoptotic cells, C(t). The equivalent system of equations is:



where {lambda} specifically denotes the DNA replication rate, while the quantity {gamma} – 1 denotes the probability of cell division. Then



Let the estimated replication rate (found by assuming two daughter cells are formed every time DNA is replicated) be and the "true" replication rate (found by assuming an average of {gamma} daughter cells are formed every time DNA is replicated) by {lambda}true. Then

Since {gamma} < 2, .

So the estimate of the replication rate tends to be somewhat underestimated, with the underestimation increasing with dose. In the article we found the estimated replication rate increasing with dose for endothelial cells in spite of this bias, indicating these results are conservative.


    NOTES
 

1 To whom correspondence should be addressed. Fax: (919) 544-7507. E-mail: msmith{at}constellagroup.com.


    REFERENCES
 TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 APPENDIX
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