Zhihua Li

Xin Zhao

Curve Fitting Derivative Method and Its Application in Mouse Growth

Keywords: Mouse growth, Parameter estimation, Curve fitting, Derivative method

Exploring change rules of the mouse growth, providing genetic data for the prediction and production of genetic analysis in scientific research and basic data for the standards of laboratory animals.[Method] Mice were weighed at 10 time points, including 3,4, 5, 6, 7, 8, 9, 10, 11, 12 weeks of age, and the corresponding data were recorded. Both general and derivative methods of Gompertz curve parameter estimation were used to analyze the mice. Then, the parameter values and the fitting degree(𝑅𝑅 2 )were calculated.

[Result] The derivative method has the highest 𝑅𝑅 2 (0.993734), followed by the three-point method 𝑅𝑅 2 (0.990689). [Conclusion] The derivative method of Gompertz curve parameter estimation has the best effect on mouse fitting and can be used for the estimation and prediction of mouse growth.

The analysis of animal growth curve is one of the main methods to study the law of animal growth and development. It can describe the regularity and continuity of body weight as it increases with age. The essence of the growth curve analysis is to compress a series of age-weight data into several parameters in the growth mathematical model, effectively eliminate the influence of some experimental errors, and have very practical reference significance [START_REF] Cheng | Development and trend analysis of growth and development curve of Hu sheep [J][END_REF].

At present, the commonly used growth models are Logistic, Gompertz and Bertanlanffy fitting equations [START_REF] Fu | Interspecific mouse growth curve fitting[END_REF][START_REF] Yang | Analysis of three commonly used growth curves for poultry[END_REF][START_REF] Liu | BALB / c mouse growth curve analysis[END_REF]. Through the analysis of experimental animal growth curve, not only can dynamically understand the growth process of animals and predict the law of animal growth, but also can guide the management of breeders and improve the breeding effect. In the breeding of laboratory animals, we found that there were some differences in the quality of animals in different regions and different animal lines, and individual differences were observed in animal experiments. To this end, this article based on different gender animal body weight and age of the relationship between the experimental study and analysis [START_REF] Wang | Determination of growth curve, main organ parameters and physiological and biochemical indexes of GF grade KM mice [J][END_REF][START_REF] Xi | Determination of growth curve and weight of main organs of three SPF mice[END_REF].

By estimating the general method of Gompertz curve parameter estimation and the pre-statistical analysis of Gompertz curve parameter estimation derivative method, the parameter estimation and fitting degree 𝑅𝑅 2 and Gompertz curve parameter estimation derivative model are more suitable for the growth curve fitting, in this paper, Gompertz curve parameter estimation derivative model was used to analyze the mice.

2

Materials

Materials experimental animals were randomly divided into groups, free feeding and drinking water for feeding, Mice were weighed at 3, 4, 5, 6, 7, 8, 10, 9, 10, 11 and 12 weeks of age, and record the data as shown in Table 1 below.

𝑦𝑦 = 𝑎𝑎𝑒𝑒 -𝑏𝑏𝑒𝑒 -𝑐𝑐𝑐𝑐 (1)
Pairs of upper ends are obtained by logarithms:

𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 𝑎𝑎 𝑦𝑦 = 𝑙𝑙𝑙𝑙𝑙𝑙𝑒𝑒 -𝑐𝑐𝑐𝑐 (2)
Make:

𝑌𝑌 = 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 𝑎𝑎 𝑦𝑦 �A = lnb, B = -c
Then the above formula becomes:

𝑑𝑑𝑌𝑌 𝑑𝑑𝑑𝑑 = 1 𝑙𝑙𝑙𝑙 𝑎𝑎 𝑦𝑦 𝑎𝑎 𝑦𝑦 �- 𝑎𝑎 𝑦𝑦 2 � = - 𝑦𝑦 𝑦𝑦 𝑙𝑙𝑙𝑙 𝑎𝑎 𝑦𝑦 = - 1 𝑦𝑦(𝑙𝑙𝑙𝑙 𝑎𝑎 -𝑙𝑙𝑙𝑙 𝑦𝑦)
By the least squares method:

⎩ ⎪ ⎨ ⎪ ⎧ 𝐴𝐴 = 𝑌𝑌 -𝐵𝐵𝑋𝑋 𝐵𝐵 = ∑ 𝑋𝑋 𝑖𝑖 𝑌𝑌 𝑖𝑖 𝑛𝑛 𝑖𝑖=1 - 1 𝑙𝑙 (∑ 𝑋𝑋 𝑖𝑖 𝑛𝑛 𝑖𝑖=1 )(∑ 𝑌𝑌 𝑖𝑖 𝑛𝑛 𝑖𝑖=1 ) ∑ 𝑋𝑋 𝑖𝑖 2 𝑛𝑛 𝑖𝑖=1 - 1 𝑙𝑙 (∑ 𝑋𝑋 𝑖𝑖 𝑛𝑛 𝑖𝑖=1
) 2

(3)

Parameter estimation of a value

Three point method

First change the equation ( 1) to:

𝑦𝑦 𝑎𝑎 = 𝑒𝑒 -𝑏𝑏𝑒𝑒 -𝑐𝑐𝑐𝑐 (4)
On both sides of the above pairs of logarithm:

𝑙𝑙𝑙𝑙 𝑦𝑦 𝑎𝑎 = -𝑙𝑙𝑒𝑒 -𝑐𝑐𝑐𝑐 (5)
And then on both sides of the above pairs of logarithm:

𝑙𝑙𝑙𝑙 𝑙𝑙𝑙𝑙 𝑎𝑎 𝑦𝑦 = 𝑙𝑙𝑙𝑙 𝑙𝑙 -𝑐𝑐𝑐𝑐 (6)
Select the starting point, midpoint, and end point in the measured data, (𝑐𝑐 1 , 𝑦𝑦 1 ), (𝑐𝑐 2 , 𝑦𝑦 2 ),(𝑐𝑐 3 , 𝑦𝑦 3 ) and then the three points into the above formula:

𝑙𝑙𝑙𝑙 𝑙𝑙𝑙𝑙 𝑎𝑎 𝑦𝑦 1 = 𝑙𝑙𝑙𝑙 𝑙𝑙 -𝑐𝑐𝑐𝑐 1 𝑙𝑙𝑙𝑙 𝑙𝑙𝑙𝑙 𝑎𝑎 𝑦𝑦 2 = 𝑙𝑙𝑙𝑙 𝑙𝑙 -𝑐𝑐𝑐𝑐 2 𝑙𝑙𝑙𝑙 𝑙𝑙𝑙𝑙 𝑎𝑎 𝑦𝑦 3 = 𝑙𝑙𝑙𝑙 𝑙𝑙 -𝑐𝑐𝑐𝑐 3
Because2𝑐𝑐 2 = 𝑐𝑐 1 + 𝑐𝑐 3 , eliminate b and c get a dollar equation:

2 𝑙𝑙𝑙𝑙 𝑙𝑙𝑙𝑙 𝑎𝑎 𝑦𝑦 2 = 𝑙𝑙𝑙𝑙 𝑙𝑙𝑙𝑙 𝑎𝑎 𝑦𝑦 1 + 𝑙𝑙𝑙𝑙 𝑙𝑙𝑙𝑙 𝑎𝑎 𝑦𝑦 3 (7)
Solve the equation ( 7) to obtain the value of the three-point method to estimate the expression:

𝑙𝑙𝑙𝑙 𝑎𝑎 = 𝑙𝑙𝑙𝑙 𝑦𝑦 1 𝑙𝑙𝑙𝑙 𝑦𝑦 3 -(𝑙𝑙𝑙𝑙 𝑦𝑦 2 ) 2 𝑙𝑙𝑙𝑙 𝑦𝑦 1 + 𝑙𝑙𝑙𝑙 𝑦𝑦 3 -2 𝑙𝑙𝑙𝑙 𝑦𝑦 2 (8)

The derivative method of Gompertz curve parameter estimation

𝑦𝑦 = 𝑎𝑎𝑒𝑒 -𝑏𝑏𝑒𝑒 -𝑐𝑐𝑐𝑐 (9)
Gompertz is a commonly used growth curve, which occupies a certain position in the econometric economy. Where 𝑒𝑒 is the bottom of the natural logarithm, and 𝑎𝑎, 𝑙𝑙, and c are positive parameters. Its distribution is a S-shaped growth curve, the expression of the two ends of the natural logarithm:

𝑙𝑙𝑙𝑙 𝑦𝑦 = 𝑙𝑙𝑙𝑙 𝑎𝑎 -𝑙𝑙𝑒𝑒 -𝑐𝑐𝑐𝑐 (10)
On (10) at both ends of the derivative on t:

1 𝑦𝑦 𝑑𝑑𝑦𝑦 𝑑𝑑𝑐𝑐 = 𝑙𝑙𝑐𝑐𝑒𝑒 -𝑐𝑐𝑐𝑐 (11) (
𝑌𝑌 = 𝑑𝑑𝑒𝑒 -𝑐𝑐𝑐𝑐 (12)
(12) is an exponential function model with only two parameters of d and c. In order to estimate the parameters d and c of (12), we can take the natural logarithm at (12):

𝑙𝑙𝑙𝑙 𝑌𝑌 = 𝑙𝑙𝑙𝑙 𝑑𝑑 -𝑐𝑐𝑐𝑐 (13)
Let 𝑍𝑍 = 𝑙𝑙𝑙𝑙 𝑌𝑌 ,𝐴𝐴 = 𝑙𝑙𝑙𝑙 𝑑𝑑 ,𝐵𝐵 = -𝑐𝑐 (13) into a linear model:

𝑍𝑍 = 𝐴𝐴 + 𝐵𝐵𝑐𝑐 (14)
The parameters B and A in ( 14) can be estimated by the least squares method:

� 𝐴𝐴 ̂= 𝑍𝑍 -𝐵𝐵 � 𝑐𝑐 𝐵𝐵 = ∑ (𝑐𝑐 𝑖𝑖 -𝑐𝑐 ̅ ) 𝑛𝑛 𝑖𝑖=1 (𝑍𝑍 𝑖𝑖 -𝑍𝑍 ̅ ) ∑ (𝑐𝑐 𝑖𝑖 -𝑐𝑐 ̅ ) 2 𝑛𝑛 𝑖𝑖=1 (
′ = 𝑦𝑦 1 ′ (𝑐𝑐 𝑖𝑖 ) of 𝑐𝑐 𝑖𝑖 from 𝑦𝑦 𝑖𝑖 ′ = 𝑦𝑦 ′ (𝑐𝑐 𝑖𝑖 )(即𝑦𝑦 1𝑖𝑖 ′ ≈ 𝑦𝑦 1 ′ (𝑐𝑐 𝑖𝑖 ) = 𝑦𝑦 𝑖𝑖 ′
), we know that the Gompertz curve ( 9) is a growth curve, for which we can use the growth curve. But now the question is how to find 𝑦𝑦 𝑖𝑖 ′ = 𝑦𝑦 ′ (𝑐𝑐 𝑖𝑖 ), we use the curve fitting method:

𝑦𝑦 1 = 𝑦𝑦 1 (𝑐𝑐) = 𝛽𝛽 0 + 𝛽𝛽 1 * 𝑐𝑐 + 𝛽𝛽 2 * 𝑐𝑐 2 + 𝛽𝛽 3 * 𝑐𝑐 3 (16)
approximate instead of the Gompertz curve (9), so how to estimate (4) the four parameters 𝛽𝛽 0 , 𝛽𝛽 1 , 𝛽𝛽 2 , 𝛽𝛽 3 . This can be estimated by MATLAB based on the given data (𝑐𝑐 𝑖𝑖 , 𝑦𝑦 𝑖𝑖 ),(𝑖𝑖 = 1,2,. . . 𝑙𝑙). 16) type in:

Estimated by MATLAB 𝛽𝛽

0 � , 𝛽𝛽 1 � , 𝛽𝛽 2 � , 𝛽𝛽 3 � substitution (
𝑦𝑦 1 = 𝑦𝑦 1 (𝑐𝑐) = 𝛽𝛽 0 � + 𝛽𝛽 1 � * 𝑐𝑐 + 𝛽𝛽 2 � * 𝑐𝑐 2 + 𝛽𝛽 3 � * 𝑐𝑐 3 (17)
For (17) formula, the derivation of t is obtained:

𝑦𝑦 1 ′ = 𝑦𝑦 1 ′ (𝑐𝑐) = 𝛽𝛽 1 � + 2 * 𝛽𝛽 2 � * 𝑐𝑐 + 3 * 𝛽𝛽 3 � * 𝑐𝑐 2 (18)
Let 𝑦𝑦 1𝑖𝑖 ′ = 𝑦𝑦 1𝑖𝑖 ′ (𝑐𝑐 𝑖𝑖 ) = 𝛽𝛽 1 � + 2 * 𝛽𝛽 2 � * 𝑐𝑐 + 3 * 𝛽𝛽 3 � * 𝑐𝑐 2 , then 𝑦𝑦 1𝑖𝑖 ′ = 𝑦𝑦 1 ′ (𝑐𝑐 𝑖𝑖 ) ≈ 𝑦𝑦 𝑖𝑖 ′ = 𝑦𝑦 ′ (𝑐𝑐 𝑖𝑖 ),
the parameter b and c in the (10) formula are estimated by the least square method, and the parameter 𝑎𝑎 in the estimate (9) formula is obtained.

Fitting precision

In the Gompertz curve regression equation problem, it is generally used to use the correlation index as the fitting precision, that is, the test:

𝑅𝑅 2 = 1 - ∑ (𝑦𝑦 𝑖𝑖 -𝑦𝑦 � 𝑖𝑖 ) 𝑛𝑛 𝑖𝑖 2 ∑ (𝑦𝑦 𝑖𝑖 -𝑦𝑦 �) 𝑛𝑛 𝑖𝑖 2 (19)
the size of. The fitting accuracy of the curve is 𝑅𝑅 2 , that is, the closer the 𝑅𝑅 2 is to 1, the better the fitting accuracy of the curve.

Result

First, the parameter of 𝑎𝑎 is estimated as 𝑎𝑎 = 317.6852 according to the formula of the three-point method [START_REF] Xi | Determination of growth curve and weight of main organs of three SPF mice[END_REF], and then the parameters of b and c are estimated.

According to the general method of Gompertz curve parameter estimation, that is, by least squares solution B = -0.2598 , A = 1.3135 , so b = 3.7193 , c = 0.2598, Therefore Gompertz curve equation is:

𝑦𝑦 = 317.6852𝑒𝑒 -3.7193𝑒𝑒 -0.2598𝑐𝑐
The fitting accuracy 𝑅𝑅 1 2 , 𝑅𝑅 Use the formula instead of the derivative of the Gompertz curve, use its value to represent the derivative value of the Gompertz curve, can be obtained in Table 3 below: Through the MATLAB software drawing three-point method fitting curve 2:

1 Introduction*

 1 College of Information Technology, Jilin Agricultural University, Changchun City, Jilin Province

  11) where d = bc,𝑌𝑌 =

|

  𝑐𝑐→𝑐𝑐 𝑖𝑖 �. But because it only provides raw data (𝑐𝑐 𝑖𝑖 , 𝑦𝑦 𝑖𝑖 ), and thus cannot be obtained 𝑑𝑑𝑦𝑦 𝑑𝑑𝑐𝑐 | 𝑐𝑐→𝑐𝑐 𝑖𝑖 = 𝑦𝑦 ′ (𝑐𝑐 𝑖𝑖 ), and therefore cannot calculate the 𝑍𝑍 𝑖𝑖 value. If we are based on the known data(𝑐𝑐 𝑖𝑖 , 𝑦𝑦 𝑖𝑖 ),(𝑖𝑖 = 1,2,. . . 𝑙𝑙), find the approximate value of 𝑦𝑦 ′ (𝑐𝑐 𝑖𝑖 ), so that you can find the approximation of 𝑍𝑍 𝑖𝑖 (We still use 𝑍𝑍 𝑖𝑖 to represent the approximate values of 𝑍𝑍 𝑖𝑖 ) This can be done by the data (𝑐𝑐 𝑖𝑖 , 𝑍𝑍 𝑖𝑖 ),(𝑖𝑖 = 1,2,. . . 𝑙𝑙) substituting into (14) can estimate (15) where the parameters A and B. Thus 𝑐𝑐̂= -𝐵𝐵 � �𝑑𝑑 ̂= 𝑒𝑒 𝐴𝐴 � can be estimated, since 𝑑𝑑 ̂= 𝑙𝑙 � 𝑐𝑐, so the estimated value of b is 𝑙𝑙 � = In this way, the two parameter in the model (9) is estimated. b and c produce a curve 𝑦𝑦 1 = 𝑦𝑦 1 (𝑐𝑐) similar to the Gompertz curve to approximately replace the curve (9), then we derive the derivative 𝑦𝑦 1 = 𝑦𝑦 1 (𝑐𝑐) of the model (9) in 𝑐𝑐 𝑖𝑖 with the derivative 𝑦𝑦 1𝑖𝑖

Fig. 1 .

 1 Fig. 1. Three-point fitting curve

Fig. 2 .

 2 Fig. 2.fitting curve of derivative method

Table 1 . 3 to 12 Weeks old mice body mass data [7] .

 1

	Week age /t	3	4	5	6	7	8	9	10	11	12
	Body mass /y 44.3 82.9 122.8 151.5 172.5 210.4 227.3 242.5 255.3 262.9

3 Methods 3.1 A General Method of Gompertz Curve Parameter Estimation

Table 2 .

 2 [START_REF] Yang | Analysis of three commonly used growth curves for poultry[END_REF] To 12 Weeks old mice body weight and predicted body weight.

		1 2 = 1 -	479.662 51517.26	= 0.990689 is calculated by the
	formula (19)
	The Gompertz curve equation yields the following Table 2:
	Week age /t	3 4	5	6	7	8	9	10	11 12
	Body mass /y	44.3 82.9 122.8 151.5 172.5 210.4 227.3 242.5 255.3 262.9
	Predicted body mass /y 57.7 85.2 115.2 145.3 173.7 199.5 221.9 240.9 256.6 269.5

Table 3 .

 3 3 ~ 12 Weeks old mice body weight and predicted body weight.

	Week age /t	3 4	5	6	7	8	9	10 11 12
	Body mass /y	44.3 82.9 122.8 151.5 172.5 210.4 227.3 242.5 255.3 262.9
	Predicted body mass /y 41.2 75.0 112.9 149.5 181.0 206.4 225.8 240.1 250.4 257.7

Conclusion and discussion

From the fitting accuracy and image contrast, 𝑅𝑅 1 2 < 𝑅𝑅 2 2 , obviously, in mouse growth prediction, the derivative method of the Gompertz curve parameter is better than the general method of the Gompertz curve parameter, and is more suitable for statistical analysis of mice and can be used for mouse growth estimation and prediction.

The growth curve equation is used to summarize the information distributed to each sample point to a few parameters, the characteristics of the growth process can be described, the production process of the mice can also be predicted according to their curves, this has a lot of usefulness for making production plans, predicting selection reactions, and performing genetic analysis of growth processes. And for genetics, physiology nutrition and other disciplines to provide a new way and basis.

Yang Ning and other quail on the growth curve analysis, found that the parameters of the gender effect and its significant [START_REF] Yang | Analysis on Growth Curve of Quail in Different Strains[END_REF], in mice is also the case, pending further study and analysis.