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ANIMAL GROWTH, PHYSIOLOGY, AND REPRODUCTION |
* Department of Animal and Poultry Sciences, Virginia Polytechnic Institute and State University, Blacksburg 24061-0306 and
and
Equine Studies Group, Waltham Centre for Pet Nutrition, Melton Mowbray, United Kingdom
Abstract
The objective of this study was to establish a procedure for differentiating a baseline curve from a systematic deviation in weight-age data, and hence to develop a physiological growth model for the Thoroughbred. A total of 2,698 records for 175 foals was obtained during a period of 8 yr (1994 to 2001). Weight-age data were fit with a sigmoid growth equation, W = A(1 + be-kt)M, where W is BW at age t, A is the asymptotic value of W, b is a scaling parameter that defines the degree of maturity at t = 0, k is a rate constant, and M defines the point of inflection in the sigmoid curve in relation to age. Short-term systematic deviations in the weight-age data were identified by a goodness-of-fit procedure and illustrated in three-dimensional contour plots of the sigmoid equation parameters as they changed upon removal of selected subsets of the data. Based on features of the contour plots, a negative deviation between 210 and 420 d of age was set aside, with the remaining data establishing the baseline data set. The sigmoid growth equation was fit to the baseline data set using a nonlinear mixed model with repeated measures, and indicated a mature weight of 542 ± 6.2 kg reached at 7 yr. The systematic deviation identified in this weight-age data set is present in other published Thoroughbred growth data and is likely to result in erroneous parameter estimates if not set aside before fitting sigmoid growth equations to the thus-modified weight-age data set. The techniques developed in this study enable identification of short-term systematic deviations in weight-age data and define a realistic baseline growth curve. Differentiation of these two components enables the development of a physiological model of growth that distinguishes between baseline growth and environmental influences, represented respectively, by the baseline curve and the systematic deviation.
Key Words: Growth • Physiological Model • Sigmoid Function • Thoroughbred
Introduction
"A description of growth must necessarily precede an investigation of growth processes. . ." (Medawar, 1950
). Growth has been described by numerous empirical equations and physiological models. Empirical equations evolve as the best mathematical representation of data sets, whereas physiological models are representations of biologic processes and the model parameters may have a biologic interpretation. Physiological models are more likely to be robust in prediction of other data sets. The latter are therefore useful for describing growth with future objectives of investigating genetic and environmental influences on growth.
The pattern of animal growth can be considered as analogous to that of autocatalytic enzyme reactions (Lopez et al., 2000; Reed and Holland, 1919; Thompson, 1942). The best-fitting physiological model is often a flexible sigmoid curve with parameter values that represent biological processes. For example, parameters in the Hill equation modeled biochemical processes (Hill, 1913). One difficulty in describing growth is separating the long-term sigmoid pattern from short-term deviations due to genetics or environment. Although the flexibility of the generalized sigmoid model aids in finding an overall best fit to growth data, the parameter estimates are strongly affected by short-term deviations in growth. The objective of this study was to establish a technique for separating weight-age data into two components, a baseline curve and a systematic deviation, that together can be regarded as a physiological model of growth.
Materials and Methods
The growth of 175 foals born at the Middleburg Agricultural Research and Extension Center (Middleburg, VA) was monitored from birth to approximately 16 mo. Two types of supplements were fed over the 8 yr of this study. One was abundant in sugar and starch, the other in fat and fiber (Hoffman and Kronfeld, 1999). Supplements were designed to be isoenergetic, with mineral and vitamin (DSM Nutritional Products, Inc., Parsippany, NJ) contents balanced to complement the pastures in central and north-central Virginia and meet or exceed current recommendations (Griewe-Crandell et al., 1995; Hoffman and Kronfeld, 1999; NRC, 1989). Feeding management was described previously (Staniar et al., 2001). In order to maintain body condition, the supplement:forage ratio of DE during months of good pasture growth was approximately 1:2, whereas during dry or cold months it was increased to about 1:1. Mares were maintained on the supplements from May until time of weaning, at which point the foals were continued on the supplements until July or August of the following year. The protocol used in this study was approved by the institutional animal care and use committee.
All horses on pasture had ad libitum access to water. Horses remained on mixed grass and ladino clover pasture at all times, except when housed in stalls for medical treatment. Shelter was provided for each group by three-sided run-in sheds (5.5 x 18.3 m). Mares and foals were on previously described anthelmintic, vaccination, and hoof-trimming schedules (Staniar et al., 2001).
Annually, broodmares were paired by weight, foaling date, and sire and were then randomly assigned to two dietary groups. Mares were bred over a 10-wk period from April 15 to June 22, and 95% of the foals were born in April and May. Colts were gelded at 1 yr of age in the first and second year and at 3 to 4 wk of age in subsequent years. Foals were weaned gradually, at approximately 6 mo of age, by the removal of two mares from each group every 4 d.
Measurements
A total of 2,698 BW records was obtained during a period of 8 yr (i.e., 1994 to 2001) and represent the primary data set (PDS; Figure 1A, B). Foals were first measured 24 h or 1 mo following birth, depending on the year of the study. Subsequent measurements were taken at approximately 28-d intervals for the following 16 to 19 mo. Body weights were measured using a portable electronic walk-on scale with a precision of ± 2% (Digi-Star, LLC, Fort Atkinson, WI). The scale was calibrated annually.
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). The best-fit equation was used for subsequent analysis.
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 | [1] |
Body weight (W, kg) is described as a function of age (t, d). The biological interpretations of the four parameters have been discussed previously (Fitzhugh, 1976; Notter et al., 1990; Richards, 1959). Briefly, A is the asymptotic limit for BW as t approaches infinity; b is a scaling parameter that defines the degree of maturity at t = 0 (1 ± b)M; k is a rate constant that establishes the spread of the sigmoid curve along the time axis and serves both as a measure of growth rate and of rate of change in growth rate; M defines the point of inflection in the sigmoid curve in relation to age; for 0 < M < 1, the point of inflection is undefined. Growth rate (ADG, kg/d) was calculated with the first derivative of Eq. [1]
, and proportion of mature size at time t(ut) was calculated as
 | [2] |
The sigmoid growth equation was fit to the weight-age data using the nonlinear mixed procedure of SAS (SAS Inst., Inc., Cary, NC). Because best-fit values of k were close to zero, the equation was modified to facilitate computation and convergence (Eq. [3]), where k' = ln(k).
 | [3] |
Both fixed Eq. [3] and mixed Eq. [4] were fit to the data.
 | [4] |
The mixed equation had random effects associated with foal added to both A and M. Random effects were added to these two parameters after an iterative procedure of all equations with single and double random effects indicated that Akaike’s information criteria (AIC) and the residual standard deviation (RSD) were lowest for this equation. The addition of the two random effects accounts for within-foal variation and different patterns of growth between foals. An improved description of the variance structure also enhances the precision of the other parameter estimates in the equation. Predicted values for the random effect of each foal, variance estimates for each random effect, residual variance, and the estimated covariance between the random effects were obtained using the NLMIXED procedure of SAS. Approximate standard errors of the equation parameters, variance estimates, and covariance estimates are based on the second derivative matrix of the likelihood function. The fixed and mixed equations were evaluated using AIC, RSD, and R2 calculated as squared correlations between the predicted and actual observations. Similar analysis has been applied in characterizing swine growth (Schinckel and Craig, 2002).
Separation of the weight-age data into two components, a baseline curve and a systematic deviation, was accomplished using two techniques. First, the ADG-age data were fit using four sequential linear regressions in a segmented model. The start and end of each line were selected to minimize the combined sum of residuals of all likely combinations of the four regressions fit to the data from 30 to 541 d of age. Second, 1,485 weight-age data sets were created, each with a defined portion of the PDS removed. The removed portion was defined in terms of d of age, such that the start of the removed portion equaled 0 to 540 d by 10-d increments and end of the removed portion equaled start + 10 d by 10-d increments. The Richards equation (Eq. [3]) was individually fit to each of the 1,485 data sets. The fixed equation was used because the computational difficulty of running the mixed equation was cumbersome. Estimates for parameter A were depicted in contour plots with start of the removed portion on the x-axis, end of the removed portion on the y-axis, and changes in A represented by contour lines (Figure 2A). Parameter estimates from the 359 data sets that did not result in convergence were not plotted. The positive slope of the lines fit to ADG and prominent features of the contour plot were evidence of the presence and location of a systematic deviation in the PDS. These features were used to accurately identify and set aside the data points that make up the systematic deviation, creating the baseline data set (BDS).
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fit to the BDS (Table 2). This adjusted data set is illustrated in Figure 1C. The adjusted data set, still containing 75% of the PDS, was subjected to the same iterative procedure as described above (Figure 2B) and characteristics of the contour plots compared.
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The Richards equation best fit the PDS by all criteria and was used for all subsequent analysis (Table 1). Body weight was 57 ± 0.8 kg at birth and increased to 434 ± 4.3 kg at 504 d of age (Figure 1A). Predicted BW at birth and 504 d of age using Eq. [4] fit to the BDS were 58 and 437 kg, respectively. The four linear regressions fit to the ADG-age data illustrated that the ADG decreased moderately through 208 d of age, at which point the negative slope increased until 291 d of age, and then the slope changed to a positive and ADG reached a peak at 381 d of age, after which ADG decreased to 541 d of age. The actual and predicted growth pattern and the residuals from the best-fit equation in this study were similar to those in other equine growth studies (Figure 3A, B). This is further illustrated in the agreement between percent maturities calculated using the best-fit equation to the BDS and published values (Table 3). There was no effect of sex on BW (P = 0.97) or ADG (P = 0.42).
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only) all had similar characteristics. Figure 2A shows the plot of the A parameter as it changes upon the removal of different subsets of the PDS. Estimates of A ranged from 362 to 6,495 kg with a mean of 842 ± 16 kg. The estimate of A where no data had been removed was 824 ± 86 kg. The highest estimates of A as well as a region characterized by a lack of convergence were centralized around an approximate start-to-end region of 100 to 300 d. There was a similar region with starts below 100 d and ends above 450 d. There are two regions characterized by local minimum estimates of A. The first is centralized around a minimum A (362 ± 4.8 kg; start and end, 320 and 550 d, respectively) and the second around another minimum A (531 ± 12 kg; start and end, 210 and 420 d, respectively). The second nadir was selected as the data set with the least influence from the systematic deviation, and established as the BDS.
The parameter estimates for fixed and mixed Richards equations fit to the primary and baseline data sets are summarized in Table 2. The fixed effects equation fit to the PDS indicates a mature weight of 824 ± 86 kg reached at 25 yr. The mixed effects equation fit to the PDS indicates a mature weight of 752 ± 22 kg reached at 20 yr. The approximate SE for A, b, k, and M were reduced by 76, 67, 72, and 56%, respectively, indicating an improvement in precision seen in the residuals (Figure 4A, B). The residuals from the mixed equation also illustrate three possible systematic deviations from the best fit sigmoid curve, with maximum deviations at approximately 200, 325, and 500 d of age. The improved fit of the mixed equation is further demonstrated by reductions in the AIC and RSD values and the increased R2.
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). The residuals from the mixed equation fit to the BDS illustrate only one systematic deviation, with a maximum deviation at approximately 325 d of age. The improved fit of the mixed equation is further clarified by the reduction in the AIC and RSD values and the increased R2. Discussion
The results characterize a baseline curve and systematic deviation in the weight-age data for this group of young Thoroughbreds. A common characteristic of Thoroughbred growth curves is a slowing of growth and subsequent period of compensatory growth that may reflect environmental influences. Fitting a sigmoid equation to data that include this systematic deviation results in an inaccurate physiological model of growth. This is demonstrated in the parameter estimates of the Richards equation fit to the PDS. This study developed a technique for accurately separating the long-term sigmoid pattern of growth from the short-term systematic deviations by means of contour plots. Separation of baseline and systematic deviations in a physiological model of growth should enable more accurate evaluation of independent variables, such as mo of birth and supplements, in subsequent studies.
Body weights for age in this study are similar to those in one report (Thompson, 1995) but larger than those in others (Hintz et al., 1979; Jelan et al., 1996; Pagan et al., 1996). The higher rates of early growth of foals in this study are characteristic of foals born later in the year (Goater et al., 1983); 95% of the foals in this study were born in April and May, whereas over 50% in one study were born in January, February, and March (Pagan et al., 1996). If weights are compared in relation to the day of year, the foals in this study are initially lighter, but rapid growth in the first 6 mo results in their catching up in weight to foals born 1 or 2 mo earlier.
The parameter estimates from the initial fitting of the Richards equation to the PDS did not fit with general conceptions and previously published characteristics of the Thoroughbred growth curve. An anthology of the literature before 1975 lists a range of mature weights from 505 to 544 kg (Willoughby, 1975). A recent study listed mature weight of 472 Thoroughbred mares 70 d postpartum as 570 kg and mature weight of 25 Thoroughbred stallions as 580 kg (Pagan et al., 1996). The lack of fit of the growth model using all the data to conventional Thoroughbred growth is illustrated in the high estimated mature weight of 752 ± 22 kg, a low k value, and low percent maturities. We hypothesized that a systematic deviation in the weight-age data was resulting in mathematically correct but physiologically misleading parameter estimates when fitting the sigmoid equation. A goodness-of-fit procedure that identified the best period corresponding to the systematic deviation enabled development of a more correct and useful baseline growth curve.
The systematic deviation in the weight-age data is characterized by a decrease and subsequent increase in the rate of growth that occurs between 210 and 420 d of age. This deviation is well defined in the growth of horses in this study. We suggest that the size of the deviation is due to a greater environmental influence on the growth rate of foals kept in pasture than in stalls. It may be suggested that this deviation is due to a nonenvironmental influence, such as puberty (Nogueira et al., 1997). The lack of a sex effect between geldings and fillies on growth in this study diminishes the likelihood that puberty plays a large role in the deviation. When the baseline growth curve developed in this study is compared to growth data from other studies, it is clear that the deviation is a consistent characteristic of Thoroughbred growth because there is a maximum deviation from the physiological baseline growth model between 325 and 375 d of age. Further evidence that this deviation is due mainly to environment is that the peaks in the residuals of previous studies occur in foals that are approximately 25 d older than foals in this study, suggesting that if the data were on a day-of-year basis the peaks of residuals would be closer together due to the later parturition dates of foals in this study.
The first derivative of a sigmoid growth curve with 0 < M < 1 approximates exponential decay and illustrates the growth rates of horses in this study. Four sequential linear regressions fitted to this derivative curve would be expected to exhibit negative slopes, with each line being less steep than the previous line. For the ADG-age data, the departure of fitted regression lines from this pattern indicates a deviant slowing of growth from 200 to 300 d of age, followed by a period of compensatory growth peaking at approximately 380 d.
The negative deviation is further identified by setting aside 1,485 separate portions of the PDS and fitting the sigmoid equation to each remaining portion. The sigmoid equation was able to converge for 1,126 of these subsets; thus, there were 1,126 individual estimates for each of the four main parameters. Contour plots of these parameter estimates illustrated how the parameter estimates changed when subsets of the data were removed from the analysis. In this way, the contour plots become a visual tool for identifying deviations that are affecting parameter estimates. To further illustrate this concept, we hypothesized that a data set without a deviation would result in a relatively flat contour plot; in effect, removal of portions of the data set would have little effect on parameter estimates. The contour plot of the adjusted data set confirmed this hypothesis.
Three prominent features in the contour plot of the PDS are evidence of the systematic deviation in the weight-age data. The most easily understood feature is the minimum centralized around a start of 320 d and an end of 550 d. The start of 320 d is near the lowest point in a negative deviation in the weight-age data. Because this subset includes only data up to 320 d of age, it follows that this curve would have the lowest estimate of A. There is a second minimum, centralized around a start of 210 d and an end of 420 d. This feature is due to the removal of data surrounding the approximate midpoint of the deviation, 320 d, defined clearly by the first feature. The data set that resulted in the minimum in the second feature (A = 531.1 kg) was selected as the BDS. The third feature is the prominence, with approximate borders defined by starts of 50 to 200 d and ends of 220 to 360 d. Inside these borders, there was a general lack of convergence. This feature results from the removal of portions of the deviation in the weight-age data, leaving a data set whose characteristics cannot be fit with a sigmoid growth equation. Each feature is evidence of a systematic negative deviation in the weight-age data, centralized around 320 d of age, with an approximate beginning of 210 d of age and end of 420 d of age.
Fitting the sigmoid equation to the BDS yielded parameter estimates that were in agreement with published values. The estimated mature weight was 542 ± 6.2 kg, and in better agreement with both the estimates of Willoughby (1975) and Pagan (1996). The higher growth rate parameter was more in line with animals that reach mature size between 5 and 7 yr. The higher growth rate is also evident in the ut values and is likely due to the April and May foaling dates. The lack of a point of inflection is similar to the pattern of growth seen in a wide range of horses (Walton and Hammond, 1938). The residuals illustrate only one systematic deviation in the data vs. at least three seen fitting the equation to the PDS. The M parameter (0.89 ± 0.02) is significantly different from 1 (P < 0.0001), illustrating the necessity of using a sigmoid equation that does not assume a value for M.
The variation in the BW of foals in this study increases with time. A mixed nonlinear equation was used to better account for within- and between-animal differences in the growth curves. The A parameter was made random due to the likelihood that mature weight was expected to vary considerably between animals. The M parameter was made random due to this parameter’s influence on the shape of the growth curves. The improved precision of the mixed nonlinear equation’s fit to the data is clear in the plot of the residuals as well as the RSD. The decrease in the AIC statistic and increased R2 indicate the improved accuracy of the mixed equation.
The objective of this study was to develop an accurate physiological baseline growth model from which systematic deviations could be differentiated. To achieve this objective, systematic deviations in the weight-age data were carefully characterized and removed using a novel method. The described deviation was removed, eliminating its influence on the baseline curve. The improved accuracy and precision of this model provide better resolution of the long-term sigmoid growth pattern and short-term environmental influences on growth.
Implications
This study applied a goodness-of-fit procedure to differentiate between a systematic deviation in weight-age data and a baseline curve, and thereby enabled development of a physiological model of growth in Thoroughbreds. This novel procedure could be applied to other growth scenarios. A systematic deviation was identified from 210 to 420 d, which was attributed mainly to changes in the environment. The baseline curve predicted mature weights in line with the literature, of approximately 542 ± 6.2 kg. Improved characterization of growth should lead to advances in determining nutrient and energy requirements, as well as the development and testing of novel feeding management practices that promote desired growth rates.
Footnotes
1 Supported by Paul Mellon (deceased), the John Lee Pratt Graduate Program in Animal Nutrition, and the Waltham Centre for Pet Nutrition. The technical assistance of the staff at the Middleburg Agricultural Research and Extension Center is gratefully acknowledged. 
2 Correspondence: Middleburg Agricultural Research and Extension Center, 5527 Sullivans Mill Rd., Middleburg, VA 20117 (phone: 540-687-3521; fax: 540-687-5362; e-mail: wstaniar{at}vt.edu).
Received for publication September 5, 2003. Accepted for publication November 28, 2003.
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| HOME | HELP | FEEDBACK | SUBSCRIPTIONS | ARCHIVE | SEARCH | TABLE OF CONTENTS |
represents data points from the primary data set that make up the baseline data set and • represents data points from the primary data set, deemed part of a systematic deviation from the sigmoid growth pattern, that were excluded from the baseline data set. In panel C, 
represents predicted data points generated using the mixed sigmoid growth model that were substituted for data points excluded from the baseline data set.
), Pagan et al. (1996)
), and Jelan et al. (1996)
). Residuals of those weights from the baseline growth curve (Panel B).