Variability in rat weight gain during development

Abstract

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Spanish

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The rat is one of the most employed animal models in biomedicine. Traditionally, weight gain has been utilized to gauge development and compare across species. Numerous studies have conducted longitudinal analyses of rat development, with emphasis on weight gain analysis. Given the high variability in these patterns, experimental data from a single laboratory may not be reliable for generalized estimation. This study aimed to analyze the effect of different factors on the pattern of weight gain during rat development. A literature survey was conducted to compile a database comprising nearly 300 data points of age and weight from 15 longitudinal studies. The database comprised both pre- and postnatal data. Utilizing the Gompertz equation, the data was analyzed to formulate a comprehensive model describing rat development. Differences in growth patterns became increasingly evident at later developmental stages, when significant differences in the maximum asymptote between sexes and strains were reached.

Keywords

Development weight rat Gompertz mixed model

Introduction

The generation of animal models in biomedicine involves multiple challenges. The similarities and differences between laboratory animals and humans are complex, often making it difficult to obtain and translate information between the species. When attempting to model a pathology that occurs during development, the problem becomes even more complex.¹ Therefore, it is crucial to gain a comprehensive and generalized understanding of the development patterns in laboratory animals.

The rat is one of the most utilized animal models in biomedicine. Numerous studies have conducted longitudinal analyses of rat development, with emphasis on weight gain analysis. Among these studies, the majority focused on postnatal weight gain, while only a few delved into intrauterine weight gain. No studies were found where both prenatal and postnatal periods were analyzed. The complexity of studying this phenomenon increases when considering variations between sexes, strains, diets, or even across different research laboratories.

In individual studies, it has been observed that diets high in calories generally result in an increase in adult weight.^2,3 Differences between males and females have also been examined.⁴ Sex differences in intake may be divergent in a process that involves the male rat’s tendency to eat larger meals.^5,6

A study from 1975 conducted on Wistar rats aimed to discern patterns of differences in weight gain between the sexes. The author proposed that female rats reach their peak rate of weight gain before males, and that this peak is smaller than the one observed in males.⁷ A subsequent study examined these differences in more detail, revealing the contribution of distinct body parts to sexual dimorphism.⁴ While muscle mass makes the highest contribution, other organs also exhibit an increase in weight.

The Sprague Dawley (SD) is an outbred strain generated by Robert S. Dawley during the 1920s through crossbreeding Wistar rats with their wild counterparts. While there is limited research comparing SD and Wistar rats, it has been reported that SD rats typically exhibit higher adult body weights compared to Wistar rats.^8,9

These analyses are valid; however, the issue lies in the limited translatability of the data across various laboratories. To gain a broader understanding of this phenomenon, it is essential to analyze data from diverse sources.

The Gompertz model, introduced by Benjamin Gompertz in 1825,¹⁰ is an asymmetric sigmoidal function¹¹ that allows the modeling of population sizes, growth of organisms, disease progression, and drug dose–response relationships, among other uses. The basic equation has been modified elsewhere to facilitate a more intuitive understanding of the parameters.^12,13

In this study, data was compiled from 15 scientific papers: 11 that comprise postnatal development,^14
–24 and four that cover prenatal and perinatal periods.^25
–28 The resulting database enabled the analysis of weight gain variability across different studies. It comprises information extracted from public datasets, tables, and graphs covering age and weight for the two most used rat strains, Wistar and SD. The dataset covers the entire rat development from fetal stages to adulthood and includes information about sex differences. Furthermore, the Gompertz equation was employed to model the data and a mixed model was used for group comparisons.

In a recent study conducted by our group,²⁹ the comparative development between humans and rats was analyzed. This investigation relied on specific homologous milestones to gauge how the developmental patterns of both species correlate. A crucial aspect of this analysis involved comparing quantitative continuous data. In this context, dealing with the variability in rat weight growth presented a challenge, and standardizing it became an essential initial step. The aim of this study was to unravel the patterns of body weight variations throughout the development in rats. This is a fundamental inquiry with broad implications across different scientific disciplines, such as nutrition, toxicology, and pathophysiology. A more generalized understanding of these patterns would allow comparisons of growth among different laboratories and provide a generalized tool to assess differences between strains and sexes. Understanding the generalized pattern of weight gain during development in both humans and rats is necessary to conduct comparative developmental analyses between the species.

Methods

Data

A survey was conducted on the developmental weight changes of rats using published literature. Data was gathered from multiple sources, including the National Center for Biotechnology Information/PubMed, Google Scholar, ResearchGate, and Elsevier/ScienceDirect, without imposing any restrictions on publication years. These databases were chosen for their comprehensive coverage of academic literature pertinent to our study's focus. The criteria for study selection aimed to include diverse perspectives by encompassing research conducted across different laboratories, time periods, and even multiple countries, with a particular emphasis on articles sourced from reputable journals. The search terms employed for this work included key phrases such as “rat weight” and “development.” All data presented in this study was sourced entirely from original empirical works. While some literature reviews were consulted to locate certain research papers, any information not traceable to its original source was excluded. This approach should be mandatory for this kind of study, yet it is not consistently adhered to.

The extraction of data from graphs is common practice, and specialized tools have been developed for this purpose.³⁰ In this study it was decided to use a free online tool, http://www.graphreader.com/. The data obtained from tables in pdf. format was extracted manually or using Microsoft PowerToys.

All data was reformatted to the structure explained in the following section.

Database structure

Each entry in the database corresponds to the average weight reported for each measurement within the respective studies. The weight of each study was not adjusted based on either the sample size or the number of measurements. The quantity of data points extracted per study matched the number of measurements taken. Ultimately, each study contributed a set of Gompertz parameters that were equally considered.

No studies were found that analyzed both intrauterine and postnatal weight gain. To compare these two periods, two groups were created. Intrauterine data remained aligned with the dates commonly referenced in studies, usually embryonic days, labeled in this work as DAF (days after fertilization). For postnatal studies, an additional 22 days were added to account for intrauterine development and to convert the age to DAF, allowing for a comparison between the groups.

Only data from control rats was employed. The database encompasses measurements from 15 distinct studies. Some studies conducted measurements for both male and female subjects. Altogether, the database is composed of 297 rows, with columns organized as follows:

Species: a placeholder for future comparative analysis. In all cases in this study it corresponds to “Rat”;

Parameter: a placeholder for future comparative analysis. In all cases in this study it corresponds to “weight”;

DAF: indicates the age as days after fertilization. When the data was reported as postnatal days, 22 was added to account for intrauterine development;

Value: indicates body weight in grams;

Sex: indicates the sex of the rat;

DOI: indicates the Digital Object Identifier of the source study;

Strain: indicates the specific rat strain;

Reference: indicates the first author's name and publication year.

Statistics and code

The Gompertz equation involves an initial stage of exponential growth followed by a deceleration after the increase rate has peaked (Figure 1),

Weight (DAF) = A e ˄ (- e ˄ (- K (DAF - T)))

First derivative (velocity of weight increase):

v (t) = A K e ˄ (- e ˄ (- K (DAF - T)) - K (DAF - T))

Figure 1.

(a) Pattern of increase in weight throughout development for all the studies included in this analysis, (b) a magnification of the perinatal period, and (c) scheme explaining the Gompertz model used to analyze the data.

Second derivative (acceleration in weight increase):

a (t) = (K e ˄ (- K (DAF - T)) - K) \cdot A K e ˄ (- e ˄ (- K (DAF - T)) - K (DAF - T))

For more information about these functions^31,32 and the methodological considerations in the discussion. Parameter K is proportional to the maximum weight gain rate, parameter T indicates the time at K in days and parameter A is maximum asymptote compatible with the adult weight in grams.

In a preliminary analysis, it was concluded that a sample size of 13 was deemed acceptable based on the desired level of statistical power of 80%. This determination allowed for a balance between practical constraints and the need to achieve a statistically reliable outcome. The code was developed using Python 3.9 within the Anaconda distribution. The libraries utilized included Seaborn, Matplotlib, Patsy, and Statsmodels. The Gompertz equation was fitted using the SciPy library through least minimum squares.

Although the overall postnatal fit was consistent with pre- and perinatal datasets, a constraint was introduced to ensure the fit closely aligned with zero around the 13th day of intrauterine development. This constraint was selected taking into consideration the findings of Norman and Bruce,²⁸ indicating a fetal weight of 0.018 g on day 13. Preliminary analyses for this study suggested that the rat’s weight was unlikely to deviate significantly from this value. The constraint prevents values from exceeding a range of 0 to 0.036 g at the mentioned date. With this constraint, the model adjustment is compelled to pass through the selected range at DAF 13. This could result in the model adjusting consistently with the expected value from a theoretical perspective or, conversely, the parameters might deviate significantly from observed values, thereby becoming outliers and not being considered in the analysis. In any case, the constraint was necessary to ensure the theoretical coherence of the model.

RMSE (root mean square error) quantifies the average difference between predicted values by a model and the actual observed values, providing insight into how well the model fits the data: lower RMSE values indicate a better fit.

RMSE = \sqrt{\frac{{Σ (yreal - ypred)}^{2}}{n}}

A mixed model was used to compare the Gompertz equation parameters through different groups (sex and strain). The normality of the residuals in the mixed model was assessed using the Jarque–Bera test, which showed non-significant results, indicating their approximation to a normal distribution. Homoscedasticity of residuals was examined with the Breusch–Pagan test, revealing no significant deviations, thus supporting consistent variability across independent variable levels.

The code and databases are accessible to the public through the following link: https://github.com/Vazquez-Borsetti/rat-weight-gain-through-development, as well as in the permanent repository, https://doi.org/10.6084/m9.figshare.24943395.

Results

As development progresses, the pattern of weight increase diverges from a similar starting point to show substantial augmentation of variability

The data collected in this study clearly demonstrated an increase in variability as development progressed among different laboratories, strains, and sexes (Figure 1). Prenatal development exhibited lower levels of dispersion. Even when considering only postnatal weight gain studies, it became evident that the patterns appeared to diverge from a common starting point. Consistent with previous findings from other laboratories, there was a tendency for female rats to reach a lower final weight than males, although the distinction between the two was not as clearly pronounced as when analyzing data from a single laboratory.

Utilizing the Gompertz equation to model weight gain in rat development

Based on what was observed, it did not seem unreasonable to assume a similar pattern of weight gain during perinatal and intrauterine development, even among the group of postnatal studies. Using the Gompertz equation as a mathematical model, the pattern of weight gain was extrapolated from postnatal development backward into prenatal development. The weight of the embryo/fetus remains negligible until days 10 to 13 if assessed against the studied ranges. The Gompertz equation demonstrated a strong fit to the data (Figures 2 and 3 and Table 1) for each of the studies within the postnatal study group. In the figures, data points from the prenatal and perinatal study group provide a visual guide for how closely the extrapolation describes early progression in weight gain. The figures revealed that despite the variability in postnatal measurements, the fitted Gompertz equation closely described both prenatal and perinatal weights. This descriptive capability was enhanced by the constraint. However, in most cases, an acceptable approximation could still be achieved even without it. It is worth mentioning that an attempt was made to fit the same data with a logistic equation without achieving satisfactory results. The parameters obtained for each postnatal study were compared after grouping them by sex and strain. The boxplots in Figure 4 show the distribution of the parameters obtained. A couple of outliers were observed for parameter T. Those values were removed for the subsequent analyses. It can also be appreciated from Figure 4 that the major differences between sex and strains were from variations in parameter A, whereas the other parameters remained similar among the groups. A mixed model was used to assess whether there were significant differences among these parameters.

Figure 2.

Least squares fit of the Gompertz equation for each study and sex in Wistar rats.

Figure 3.

Least squares fit of the Gompertz equation for each study and sex in Sprague Dawley rats. The last subplot shows the scatter plot of the perinatal group of studies.

Table 1.

Parameters A, K, and T of the Gompertz equation fit for each study and sex.

Reference	Strain	Sex	A	K	T	RMSE
Pahl²⁵	ND	Female	185.377	0.048	51.989	3.740
Pahl²⁵	ND	Male	420.475	0.034	68.414	3.062
Alemáan et al.²⁴	SD	Female	363.502	0.022	85.336	7.048
Moon et al.²²	SD	Female	322.000	0.044	58.330	4.626
Nohynek et al.²¹	SD	Female	426.014	0.024	81.261	18.448
Alemáan et al.²⁴	SD	Male	566.559	0.016	110.487	6.612
Brower et al.²³	SD	Male	522.557	0.034	69.162	6.944
Moon et al.²²	SD	Male	647.145	0.038	68.402	5.481
Nohynek²¹	SD	Male	701.500	0.031	76.540	19.274
Abramova et al.¹⁷	Wistar	Female	253.111	0.033	64.615	3.566
Hernández-Álvarez et al.²⁰	Wistar	Female	575.208	0.010	151.181	17.947
Munhoz et al.¹⁹	Wistar	Female	261.860	0.044	54.131	7.770
Nivoit et al.¹⁴	Wistar	Female	301.623	0.040	58.334	9.055
Abramova et al.¹⁷	Wistar	Male	394.873	0.024	81.792	3.268
Coêlho et al.¹⁶	Wistar	Male	474.541	0.043	63.434	4.553
Iossa et al.¹⁵	Wistar	Male	530.333	0.037	66.461	9.678
Nivoit et al.¹⁴	Wistar	Male	478.342	0.036	65.102	10.217
Average	Wistar		398.776	0.036	74.627
		Male	458.614	0.037	68.049
		Female	338.939	0.035	81.205
	SD		504.406	0.032	80.571
		Male	606.56	0.031	82.550
		Female	368.191	0.033	77.932
Total average			410.046	0.038	67.451

Note: SD: Sprague Dawley rat; ND: non-declared; RMSE: root mean square error.

The bolded rows indicate the outliers.

Figure 4.

Boxplot of the parameters obtained after fitting the Gompertz equation to each study. A couple of outliers can be observed for parameter T: those studies were removed for the subsequent analyses.

Mixed model analysis of the parameters

To analyze these observations, three mixed linear models were performed, one for each parameter of the Gompertz equation. Two categories, strain and sex, were used as predictors, and were converted to dummy variables. The parameters from studies with Wistar rats were encoded as 1, while SD studies were encoded as 0. A similar strategy was used for sex, where male was represented as 1, and female as 0. The variability between different research was included as a random effect because several works included measurements for both sexes.

Given the similarity of prenatal data and the fact that the model was constrained to resemble those results, the four prenatal studies were not used for the mixed model analysis, only the 11 postnatal studies were included. Among these, two were excluded for displaying outliers, yet the total was 13 because several studies contained data for both sexes, resulting in balanced data with a minimum of three values for each sex within each strain.

Parameter A, the maximum asymptote in grams, showed greater and clearer differences (n = 13, intercept = 381.129, SE = 33.440, p < 0.001; coef.Wistar =−130.013, SE = 39.157, p = 0.001; coef.Male =221.170, SE = 33.679, p < 0.001). The factor strain indicated that adult SD rats weighed more than Wistar rats, yielding significant differences. The factor sex also yielded significant differences where adult male rats weighed more than adult female rats. These results confirmed the differences between strains and sexes, but most important, they provided generalized quantification of the difference.

The other two parameters did not yield significant differences. Parameter K showed no significant differences among the groups (n = 13, intercept = 0.035, SE = 0.004, p < 0.001; coef.Wistar = 0.008, SE = 0.004, p = 0.088; coef.male = –0.003, SE = 0.003, p = 0.214); and parameter T, the time until inflection point in DAF, showed no significant differences among the groups (n = 13, intercept = 73.538, SE = 5.586, p < 0.001; coef.Wistar = –13.223, SE = 7.049, p = 0.061; coef.male = 4.615, SE = 4.352, p = 0.289).

Gompertz functions were used to showcase the average parameters across all analyzed studies, as well as the averages for each strain and the averages for each sex within each strain (Figure 5). The size differences between strains appeared to be primarily driven by variations among the males, whereas the development patterns of females from both strains were more similar.

Figure 5.

Gompertz functions were fitted (a) with parameters corresponding to the overall average for all studies analyzed and (b) the average for each strain.

Consistent with that already observed, especially regarding the divergent pattern of the data, it was observed that the parameters corresponding to earlier periods were similar among groups, whereas the parameters that affected the model at the later stages exhibited greater differences. To analyze this phenomenon more thoroughly, the first and second derivatives of the Gompertz equation were examined (Figure 6). As before, three mixed linear models were also conducted revealing significant differences in maximum velocity (coef. = 0.070, SE = 0.026, p = 0.007), maximum acceleration (coef. = 0.070, SE = 0.026, p = 0.007), and maximum deceleration (coef. = –0.036, SE = 0.014, p = 0.007) with sex as a factor. Males exhibited higher values than females. No significant differences were observed with strain as a factor in these cases. From these results, it can be concluded that at an early age, differences have already manifested in higher growth acceleration. However, this effect does not lead to perceptible changes until its accumulation causes noticeable differences between sexes and strains.

Figure 6.

Gompertz functions with parameters corresponding to (a) the average for each sex within each strain, (b) the first derivative of the function indicating the velocity in growth as a function of time, and (c) the second derivative indicating the acceleration as a function of time. Significant differences in maximum velocity, maximum acceleration, and maximum deceleration were observed with sex as a factor, where males exhibited higher values than females (p < 0.05; see supplementary material).

Discussion

This study aimed to uncover and describe the patterns and variations in weight gain throughout development in rats. A divergent trend was observed across various studies, strains, and sexes. The influence of late development on this variability was found to be significant. To study this pattern, the data was fitted to the logistic and Gompertz equations. The first adjusted poorly whereas the latter showed an excellent fit in most cases. Gompertz equation is a mathematical formulation historically utilized to describe diverse biological phenomena. It was also used to study postnatal rat development.^4,33
–36 In this work, it has been utilized to assess rat development, spanning both intrauterine and postnatal phases. In most cases, the generated curve did not substantially deviate from the perinatal data group.

The variability in weight development and its generalization and standardization are widely discussed topics in the case of humans, given that their deviations have significant medical implications. The recognition of pathological growth relies on the existence of reliable standards. This has led to the establishment of normal charts for key biometric variables, including weight^37,38 (see also research by Ohadike et al.³⁹ and Ioannou et al.⁴⁰).

Regarding animals, numerous studies have been conducted on commercially valuable domestic animals, both in the prenatal stage⁴¹ and the postnatal stage.⁴² However, there hasn't been as much focus on laboratory animals.

In a recent study with a research strategy similar to this work, the relationship between feed intake and liveweight in domestic animals was analyzed. The authors conducted bibliographic surveys, modeled the data with the Gompertz equation, and used a mixed model to compare data. The objective differed slightly, focusing solely on the postnatal period with one or two data points per animal.⁴³

Methodological considerations

In many studies, the age of the animals in DAF is not always precisely known. In most cases, the studies report postnatal days. The full estrous cycle of the rat occurs over 4 or 5 days. Thus, the DAF transformation may include an error of up to a couple of days in the best scenario. In certain studies, such as the one conducted by Munhoz,¹⁸ rats were sourced from an animal facility. In the study, there appears to be a leftward shift in the curve. It is plausible that a decrease in the precision of age reports occurs when animals are acquired from a commercial breeder. Inaccuracies in age reporting by the technicians may have contributed to these discrepancies. Possible sources of such inaccuracies include mistakes made by the technicians, selection of animals based on weight rather than age, or provision of animals within an age range, meaning not every animal in a group will be of the exact same age, among other factors.

The data were acquired from databases, tables, and, in some instances, extracted from graphs. With a few exceptions (i.e., points identified as outliers), internal consistency among the data was observed, supporting the employment of these strategies.

Data integration in biological sciences involves combining, unifying, and analyzing data from diverse sources and formats within the field of biology. This is a standard practice in omics fields such as genomics, transcriptomics, and proteomics, as well as in meta-analyses, given the extensive amount of information.⁴⁴ Another common approach in systematic reviews and meta-analyses is the extraction of data directly from graphs.

As stated in the methods, a preliminary analysis indicated that a sample size of 13 was considered adequate, given the desired statistical power level of 80%. Moreover, the sample was balanced across sexes and strains, and the fact that internal differences were observed among these categories suggests that an acceptable degree of precision has been achieved. Additionally, there is no apparent reason for our sample to be biased. While precision could be further enhanced with a larger sample size, there is no reason to believe that the obtained results are not representative of the population.

Both the database and the code are publicly available on GitHub and figureshare. This facilitates and enhances transparency and reproducibility. Moreover, the database is accessible for other groups to conduct their own analyses or to expand this one.

Final considerations

The obtained results provide insights into a general pattern of rat development beyond the outcomes of a single laboratory. This can be important in nutrition, toxicological or pathophysiological studies to know if the starting point of the controls is below or above the average, which can influence the margin of change and the reproducibility of the experiment. Another possible benefit of knowing these patterns is to compare studies that involve weight gain but that have been carried out in different strains and/or sexes. Animal facilities could conduct their own non-linear fitting to the Gompertz equation (using either the public code designed for this purpose or more user-friendly alternatives available in commercial software) to assess if the obtained parameters deviate from the expected values or if there's poor fit to the equation, which could indicate issues in development, feeding, or age estimation.

The standardization of laboratory rat growth is a tool that carries implications for animal care, enabling consistent health monitoring, early issue detection, and precise experimental design not only in adults but throughout the entire lifespan of the animal.

But above all, this work provides a generalized model that is useful for conducting comparative studies across different species.

Supplemental Material

sj-zip-1-lan-10.1177_00236772241246370 - Supplemental material for Variability in rat weight gain during development

Supplemental material, sj-zip-1-lan-10.1177_00236772241246370 for Variability in rat weight gain during development by Pablo Vázquez-Borsetti in Laboratory Animals

Footnotes

Data availability

Data described in the manuscript, code book, and analytic code will be made publicly and freely available without restriction at https://github.com/Vazquez-Borsetti/rat-weight-gain-through-development and in the permanent repository, .

Declaration of conflicting interest

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Ethical statement

Our study did not require an ethical board approval because it did not contain human or animal trials.

Funding

The author received no financial support for the research, authorship, and/or publication of this article.

During the preparation of this work, the author used OpenAI’s GPT 3.5 in order to improve the language and readability. After using this tool/service, the author reviewed and edited the content as needed and takes full responsibility for the content of the publication.

ORCID iD

Pablo Vazquez-Borsetti

Supplemental material

Supplemental material for this article is available online.

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