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175
Buletin USAMV-CN, 63/2005 (175-181)
ISSN 1454-2382
THE ESTIMATION OF THE GROWTH CURVE AT DOG
Ardelean A. I
1
., E. Suteu
2
1
Directia Sanitara Veterinara si pentru Siguranta Alimentelor Cluj, Laborator de Histopatologie,
P-Ńa Mărăşti, nr. 1, Cluj-Napoca, jud. Cluj, cod poştal: 400609 adrian.a@mail.dntcj.ro,
2
Universitatea de Stiinte Agricole si Medicina Veterinara Cluj Napoca, Disciplina de Boli Parazitare,
Calea Mănăştur, nr.3-5, Cluj-Napoca, jud. Cluj, cod poştal: 400372
Key words: growth curve, dog, curve fit, maximum weight
Abstract: The present studies have the purpose the anticipation of growth of the dogs and the
estimation of the associated critical period. For the estimation of growth curve was used the results obtained after
the measurement done at a group of 83 dogs, and was applied three mathematic models. The curves fitted using
the linear, the Gompertz modified and the logarithmic modified, has lead at the determination of the individual
specific values. These models permit the achievement of some supplementary parameters: the maximum weight,
the time of maximum growth and the maximum speed of growth. The substantiate of results have permit the
anticipation of the high intensity growth period (the time of maximum growth) placed between 2 and 5 months
of age, individual variable in this interval. At the same time, the application of the mathematic models permit the
estimation of the development potential (the maximum weight ) and the specific period of growths (the
maximum speed of growth) that can be used to adjust the nutrition and the life stile.
INTRODUCTION
.
In the course of growth the body development is dinamic, so it is very important to
anticipate the critical period like that to assure the proper management. The specific weight
development of the dog measured at the different age was compared with a standard
development curve. These was the purpose to detect the possible deviation from the normal
development. The growth impairment can be one frome the early signs who can appear in
many disease with subclinical evolution like nutritional diseases, hormonal imbalance, and
chronic infectious diseases, and the estimation of these are useful. From the specific growth
curve it can be possible to anticipate the maximum weight of dog at the adult age, who can be
used in nutrition and life stile management.
MATERIAL AND METHOD.
In the present study was measured the weight in the growth stage for dogs, values who
was used in the determination of growth curve through three models. Was fallowed the
growth of 83 dogs from 20 breeds with different size. The surveillance was making at up to 35
months of age period. From the weight values obtained, 13 representative dogs selected for
presentation, showed in table 1. For estimation were used three published models: linear,
Gompertz, and logarithmic modified.
176
Chow-chow Schnautzer
Irish Setter Boxer Pit-bull
Doberman1
Age
month Masa
(kg) Age
month Weight
(kg) Age
month Weight
(kg) Age
month Weight
(kg) Age
month Weight
(kg) Age
month Weight
(kg)
3.83 13
4.68 14
5.21 17
7.57 20
12.95 24
3.63 14
4.32 18
13.44 35
25.34 36
4.03 16
4.88 19
8.29 23
12.68 25
24.29 30
4.16 10
5.6 15
8.91 22
12.68 27
4.45 13
5.77 18
6.85 20
12.16 23
3.27 13
4.59 19
5.57 24
7.34 29
Doberman2 Rottweiler2 Great dane German
shepard1 German shepard2
Age
month Weight
(kg) Age
month Weight
(kg) Age
month Weight
(kg) Age
month
Age
month Weight
(kg)
3.93 15
6 21
14.85 30
29.93 35
3.31 10
4.29 17
6.03 27
6.49 28
12.03 39
3.14 10
4.13 17
5.37 23
8.26 28
13.11 35
3.14 10
4.13 17
5.37 23
8.26 28
13.11 35
3.34 13
4.85 20
5.93 22
13.01 30
Table 1. The results obtained after the weight measurement and the age when was done.
177
RESULTS AND DISCUSION
The constant values who participate in the functions compute using from mathematic
considerations
1
n
+
experimental data, n is the number of parameters that used in the function
what describe the mathematic model. Just the values of the weight measured at the age older
than three months (t=3 months) are useful, all the models of calculation are valid if the age of
the dog are greater than 3 months (t>3 months). The constants obtained from fitting the
functions used in the three models, are presented in the table 2. The fitting was made by using
software for curve fitting and graphic analysis.
The linear model is a logarithmic one, which describe the rise of weight M of the dog
in function of the time of age t, express in months and measured from the date of birth.
1 2
ln
M L t L
= +
(1)
This is a most simple model what need a less veterinarian consult, which offer values
approaching the real measurement for low time of age t. The linear model did not permit the
obtaining of the maximum weight of the adult, and can estimate the growth of the dog just
until the age of ten months.
The Gompertz model describes the rise of weight in time in the form of function:
3
(1 )
1 2
G t
e
M G eG
−
−
=
(e= 2,7182)
From the Gomertz model, we have changed the formula:
( 3)
3
2
(1 )
1
G t
G e
M G e
− −
−
=
(2)
The point of inflexion of the curve describe the maximum growth moment. This is the
t* moment, which in a second derivation of the functions (2) canceled itself.
*2
3
ln
3
G
tG
= +
(3)
The G
1
constant have a concrete meaning, it is the weight at 3 months of age (M
3
) of
the dog [t=3 months substituted by expression (2)]. The weight of the adult was obtained for
t
→∞
:
2
m a x 1
G
M G e
=
(4)
0 5 10 15 20 25
0
10
20
30
40
50
60
70
Dog:
Gompertz: G1=15,868, G2=1,192,
G3=0,355
Liniar: L1=28,478, L2=-16,712
masa (kg)
timp (luni)
0
10
20
30
40
50
60
0 5 10 15 20
Dog:
Logaritmic: P1=3,106, P2=1,906, P3=0,847
timp (luni)
masa (kg)
178
0 5 10 15 20 25
5
10
15
20
25
30
35
40
45
50
Ciobanesc german 1:
Gompertz: G1=10,467, G2=1,201, G3=0,397
Liniar: L1=16,841, L2=-7,471
masa (kg)
timp (luni)
0
5
10
15
20
25
30
35
40
0 5 10 15 20 25 30 35
Ciobanesc german 1
Logaritmic: P1=-2,048, P2=5,580, P3=25.874
timp (luni)
masa (kg)
0
5
10
15
20
25
30
35
0 5 10 15 20 25 30 35
Boxer
Logaritmic: P1=1,595, P2=3,724, P3=11,562
timp (luni)
masa (kg)
0 5 10 15 20 25
0
5
10
15
20
25
30
35
40
Boxer:
Gompertz: G1=6,889, G2=1,463, G3=0,275
Liniar: L1=15,210, L2=-11,447
masa (kg)
timp (luni)
0 5 10 15 20 25
0
10
20
30
40
50
60
Rottweiler 2
Gompertz: G1=8,479, G2=1,553, G3=0,438
Liniar: L1=22,554, L2=-15,531
masa (kg)
timp (luni)
0
5
10
15
20
25
30
35
40
45
0 5 10 15 20 25 30
Rottweiler 2
Logaritmic: P1=0,230, P2=3,100, P3=13,333
timp (luni)
masa (kg)
0 10 20 30 40
0
10
20
30
40
Setter
Gompertz: G1=15,418, G2=0,689, G3=0,144
Liniar: L1=5,993, L2=9,992
masa (kg)
timp (luni)
0
5
10
15
20
25
30
35
0 5 10 15 20 25 30 35 40
Setter:
Logaritmic: P1=3,919, P2=6,838, P3=11,100
timp (luni)
masa (kg)
Fig. 1. The graphics results after the compute at the : Great Dane (Dog), Rottweiler2 and Irish Setter.
( masa= weight; timp= time; luni=month)
179
Fig. 2. The graphics results after the compute at the : German shepard 1(Ciobanesc german1) Boxer and
Doberman1. (masa= weight; timp= time; luni=month)
The G
2
constant can be express like:
max
2
3
ln M
G
M
= (5)
and is named the growth speed [2].
The d`Aquilar model (logarithmic) express in logarithmic form the age t related with
the weight:
1 2 max
ln ln( )
t P M P M M M
= − − +
(6)
The M
max
is the maximum weight and the M is polynomials with n degree [2]. The rise
of weight according with this model was not presented. We replaced the polynomials with P
3
constant, in this formula (the logarithmic modified):
1 2 max 3
ln ln( )
t P M P M M P
= − − +
(7)
For the purpose to obtain the growth curve, we had limited the number of the constants
using the maximum weight resulted from the Gompertz modified model, weight which are
like the values for each breed what mentioned in literature. The constants resulted from fitting
are presented in the table 2.
The curve resulted from fitting the linear model (dotted curve) and Gompertz modified
(continuous curve) are showed together with data resulted from measurement, in the left
figure from each group corresponding with a particular breed. Data resulted from weight
measurement along with the curve obtained from fitting the logarithmic modified model (7),
are represented in the right figure of each set. All fitting was made from a least four
measurement. The logarithmic modified model permits the theoretical estimation until the
values reach the maximum limit of weight. Also for this model, was finding out that the
theoretical curve describe correct the real values for the measurement for a time of life shorter
than the moment who limited the growth. Parameters who participate in these models, resulted
from fits are presented in the table 2.
0 5 10 15 20 25
0
10
20
30
40
50
60
Doberman 1:
Gompertz: G1=11,494, P2=1,142, P3=0,386
Liniar: L1=20,104, L2=-11,014
masa (kg)
timp (luni)
0
5
10
15
20
25
30
35
40
0 2 4 6 8 10 12 14 16 18 20
Doberman 1:
Logaritmic: P1=1,000, P2=2,698, P3=9,194
timp (luni)
masa (kg)
180
Table 2. The contant obtained after fitted and application of formula
Formula Gompertz modificat Liniar Logaritmic modificat
Breed G1=M
3
G2 G3 L1 L2 P1 P2 P3
Chow-chow 10,166 0,888 0,332 9,642 -0,097 0.858 3.407 10,25
Schnautzer 10,661 1,21 0,42 11,867 0,285
Irish Setter 15,418 0,689 0,144 5,993 9,992 3,919 6,838 11,100
Boxer 6,889 1,463 0,275 15,21 -11,447 -0,833 8,232 32,561
Pit-bull 5,914 1,361 0,599 9,255 0,758 -7,771 6,739 41,164
Doberman1 11,494 1,142 0,386 20,104 -11,014 1 2,698 9,194
Doberman2 13,87 0,927 0,162 9,76 2,661
Rottweiler2 8,479 1,553 0,438 22,55 -15,53 0,23 3,1 13,333
Great dane 15,868 1,192 0,355 28,478 -16,712 3,106 1,906 0,847
German
shepard1 10,467 1,201 0,397 16,841 -7,471 -2,048 5,58 25,874
German
shepard2 11,747 0,953 0,395 12,015 -0,169
After applying the mathematic models and the fitting was obtained the fallow
parameters: the weight at 3 months of age, the maximum time of growth, the speed of growth,
and the maximum weight, with the values presented in the table 3.
Table 3. The parameters who results after the application of the compute formula
Breed The weight at 3
months of age
The
maximum
time of
growth
(months)
The maximum
speed of
growth
The maximum
weight (kg)
Chow-chow 10,166 2,64 0,888 24,7
Schnautzer 10,661 1,21 35,75
Irish Setter 15,418 3,45 0,689 37,54
Boxer 6,889 4,38 1,463 29,75
Pit-bull 5,914 3,51 1,361 23,06
Doberman1 11,494 3,34 1,142 36,01
Doberman2 13,87 2,54 0,927 35,04
Rottweiler2 8,479 4,00 1,553 40,07
Great dane 15,868 3,49 1,192 52,3
German shepard1 10,467 3,46 1,201 34,8
German shepard2 11,747 2,88 12,015 30,46
After the balance of the results compute with the measured values, can be find out that
the applying the models used in this trial can be achieve, those having a high level of
estimation. The parameters resulted after the using the compute formula, can describe the
development of the body. Thus, can be assess the maximum development period of the dog
after the calculation of the maximum time of growth. The speed of growth G2 correlated with
181
the growth graphic showed that a high speed is attended with a fast flattening of the growth
curve. The estimation of the maximum weight and of the speed of growth, are useful for the
management and individualization of the nutrition and physical training. Thus, in the case of
the giant breeds, the flatten of the growth curve become latter (are dogs with low speed of
growth and high maximum weight), so the physical training must be graduate. This must be
done in such way that to not produce any overloading of the bones and the joint before the
flattening period, for the purpose to prevent the development of the osteo- and arthropaties.
The applying of these models permits the calculation of the requirement of the
nutrients for each period of the growth. Through the confront of the maximum time of growth
with the age who are more often saw pediatrics diseases, can be ascertain that the infectious
diseases appear more frequent in this period, with more severe evolution [1]. Can be define
like a risk period the age between 2 and 5 months. Thus, the disease management takes to the
dog can be individualized. If the time of maximum growth is high, the dog will need
supplementary prophylactic actions, an optimum diet, and supplementary accurate assistance.
The applying of modified formulas has proved a high estimative potential. The results
obtained are in concordance with the literature data.
CONCLUSION
The studied and modified mathematic models have certain applicability bin veterinary
pediatry. The results obtained are superior if they are used togheter.
It can be define like a critical period for dogs the age between 2 and 5 months, or
greater in the caseof dog with high time of maximum growth.
The nutrition must be individualized and optimized in the case of dogs with high time
of maximum growth and low speed of growth, could be estimate the age of finishing the body
development.
In the case of dogs with high speed of growth and low maximum time of growth, the
individualisation of the nutrition permit the avoid of overfeeding.
The estimation of the growth curve can be use in the modeling of the life style,
nutrition and disease management.
BIBLIOGRAPHY
1. Ardelean A.I. , E. Şuteu, V. Cozma (2005) Epidemiology of digestive helminthosis from urban area
of Cluj-Napoca, Romania, Buletin USAMV-CN, 62 /2005
2. Lucile Martin (1996) Approche pratique de la modelisation de la corbe de croissance du chiot, Rev.
Med.Vet172 (9-10), 465-475
