Comparison of Compertz and Logistic Models in Estimating the Growth of Leptin-Deficient (ob/ob) Mice and Wild-Type Mice Fed an In-House Prepared High-Fat Diet

Corresponding Author: Beob Gyun Kim Department of Animal Science and Technology, Konkuk University, Seoul, Republic of Korea Email: bgkim@konkuk.ac.kr Abstract: This experiment was conducted to growth curve models for leptindeficient (ob/ob) and wild-type mice fed a meal-form High-Fat (HF) diet prepared in-house for over 120 weeks. Two hundred and twenty-four sets of Body Weight (BW) and age data were collected from ob/ob mice and 485 sets of data were collected from wild-type mice. All animals had free access to the HF diet (34.3% energy from fat) and water. Individual BW and survival rates of mice were measured. To develop growth curves of ob/ob mice and wild-type mice fed the HF diet, Gompertz and Logistic growth models were employed. The survival rates of ob/ob mice fed the HF diet were lower than those of wild-type mice. Models for estimating growth of ob/ob mice fed the HF diet were: BW, g = 61.36 × e -2.56×e^-0.13t in Gompertz model (Akaike’s Information Criterion, AIC = 1,694); and BW, g = 60.65 × (1 + 7.04 × e -0.18t ) -1 in Logistic model (AIC = 1,719). Growth models for wild-type mice fed the HF diet were: BW, g = 28.94 × e -1.75×e^-0.13t in Gompertz model (AIC = 3,171); and BW, g = 28.23 × (1 + 4.85 × e -0.25t ) -1


Introduction
Many obese models used in obesity or obesity-related disease research are gene-specific knockout or transgenic rodents and the obese model animals are often fed High-Fat (HF) diets (Woods et al., 2003;Buettner et al., 2007). Commercial pellet-type HF diets have been widely used to induce and maintain obesity in mouse models. In many nutrition experiments for testing dietary supplements, on the other hand, meal-form diets are preferred due to the convenience in preparing experimental diets (Niu et al., 2017;Elkahoui et al., 2018). However, meal-form HF diets for rodents are rarely available.
As elderly population increases, research on obese elderly men and women becomes more and more important (Vioque et al., 2007;Sardeli et al., 2018). Although young obese mice mostly less than 20 weeks of age are often used in nutrition research, older mice are more appropriate for mimicking adult or elderly human obesity and thus, have been recently used in elderly obesity research (Hunsche et al., 2016;Frasca et al., 2017). However, information on the survival rate and Body Weight (BW) changes of leptin-deficient (ob/ob) mice fed a HF diet for over 6 months is very limited. Therefore, the objectives of the present work were to test a novel meal-form HF diet fed to mice and to develop growth curve models for ob/ob and wild-type mice fed a meal-form HF diet for over 120 weeks.

Animals and Experimental Diet
Two hundred and twenty-four sets of BW and age data were collected from ob/ob mice and 485 sets of data were collected from wild-type mice. Body weight and age ranged from 6.3 to 77.7 g and from 3 to 127 weeks, respectively.
A HF diet (34.3% energy from fat) was prepared inhouse to contain 49% ground corn, 30% soybean meal (crude protein 48%), 20% fat powder and 1% vitaminmineral premix (Table 1). The fat powder is a proteincoated fat product that was mainly composed of 80% lard, whey and casein. All animals had free access to diet and water. Individual BW of mice was measured with approximately two-week intervals.

Growth Models and Statistical Analysis
For estimating the growth curves for ob/ob and wild-type mice, the NLMIXED procedure of SAS (SAS Inst. Inc., Cary, NC, USA) with nonlinear regression was used. The model equation used for the Gompertz model (Gompertz, 1825) is: The model equation used the Logistic model (Robertson, 1908) is: For both equations, 3 parameters in the equation were employed to estimate the growth model parameters in which Wt is the BW of mice at time t (week), A is the mature weight, b is the growth ratio, k is the maturing rate; and e is the natural logarithm. Based on the estimated 3 parameters, the age at point of inflection, gain at inflection and weight at inflection were calculated.
In the Gompertz model, equations for the age at point of inflection (t i ), body weight at inflection (Wt i ) and gain at inflection (δWt/δt) are: The inflection point, second derivative of function, means where the slope of the growth curve becomes maximum.

Results
The ob/ob mice gained more weight compared with the wild-type mice, and a hair loss was found on the neck area of ob/ob mice (Fig. 1). The survival rate of the ob/ob mice was lower than that of the wild-type mice (Fig. 2).
The average mature BW of ob/ob mice estimated from the Gompertz and Logistic models was approximately 61.0 g ( Table 2; Fig. 3 and 4). The average mature BW of wild-type mice estimated from the models was approximately 28.6 g. The age at point of inflection, BW at inflection and gain at inflection are presented in Table 3. The average BW at inflection of ob/ob and wild-type mice estimated from the models was approximately 26.5 and 12.4 g, respectively.

Discussion
It has been reported that the obesity of ob/ob mice can be visually identified after 4 weeks of age (Kennedy et al., 2010), and physical differences between the ob/ob and wild-type mice were also observed in the present study. Moreover, the wildtype mice had more sleek hairs and active movements than ob/ob mice in this study. This observation is supported by previous studies that reported critical role of leptin in hair cycle (Sumikawa et al., 2014;Tasaki et al., 2015;Sasaki et al., 2018). After depilation, elongation of telogen was observed in ob/ob mice compared with the wild-type mice (Tasaki et al., 2015). Moreover, the ob/ob mice and mice fed the high-fat diet suffered delayed wound closure (Seitz et al., 2010).
The lower survival rates of ob/ob mice compared with wild-type mice agreed with previous studies. Ren et al. (2010) reported reduced survival rates of the ob/ob mice compared with wild-type mice. In addition, the survival rate began to decline from 50 weeks of age ob/ob mice in the present study. This result agreed with Harrison and Archer (1987) reporting that the ob/ob mice showed high mortality after 52 weeks of age.    The greater mature BW of ob/ob mice than wildtype mice was observed in the present study. In both modes, the growth rate of the ob/ob mice was also greater compared with that of the wild-type mice. While the ob/ob mice reach the maximum BW of 60 to 70 g at 28 to 32 weeks, the lean-type mice reach the maximum BW of 30 to 40 g at 12 to 16 weeks (Hedrich, 2012). Greater energy and nutrient intake may have contributed to the rapid growth rate of ob/ob mice used in this study. Murphy et al. (2010) reported that the ob/ob mice gained more weight compared with lean mice, resulted from greater accumulative energy intake in the ob/ob mice. Moreover, a previous study reported that increases in weight gain and feed intake of the ob/ob mice fed HF diets containing corn oil or beef tallow were observed compared with the ob/ob mice fed a low-fat diet (Mercer and Trayhurn, 1987). In addition, the magnitude of increase was greater in the ob/ob mice than lean mice. Generally, the model with a lower value for Akaike's information criterion can be considered as a more appropriate model (Gbangboche et al., 2008). Based on the values for Akaike's information criterion, the Week Gompertz model was more appropriate for estimating BW of ob/ob and wild-type mice.
In the present study, the age at point of inflection, BW at inflection and gain at inflection were greater in the ob/ob mice compared with the wild-type mice. The greater BW and rapid growth rate of ob/ob mice compared with wild-type mice have also been reported in the literature (Harris et al., 1998;Sainsbury et al., 2002;Sun et al., 2006;Seitz et al., 2010).

Conclusion
In conclusion, the Gompertz models are more appropriate for estimating the growth of mice fed a highfat diet compared with Logistic models. The suggested growth curves and survival rate data for leptin-deficient mice fed a high-fat diet can be used for designing longterm experiments or planning to use elderly obesity models. Additionally, the high-fat diet formula for mealform feeds provided in the present work is useful in nutrition experiments employing leptin-deficient mice. Further research is warranted to develop growth models for other model animals for obesity or nutrition research.