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ANIMAL GENETICS |
* American Simmental Association, Bozeman, MT 59715; and
and
Department of Animal Sciences, Colorado State University, Fort Collins 80523
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Abstract |
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 Top Abstract INTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONIMPLICATIONSLITERATURE CITED |
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Key Words: beef cattle • interaction • nonlinear equation • simulation • variability
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INTRODUCTION |
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TopAbstractINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONIMPLICATIONSLITERATURE CITED |
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A major rationale behind simulating at greater levels of aggregation is to streamline the process. Besides requiring stepped-up resources, refinement typically adds complexity, potentially reducing accuracy if the relationships at lower levels of aggregation are unknown. A common method of streamlining beef cattle simulation has been to disregard variability beyond that created by the model’s deterministic equations. Not surprisingly, this approach yields lower levels of simulated variability than that typically occurring in nature. To simulate more realistic levels of variability, stochastically generated variability must be supplemented.
Judging from the dearth of beef cattle simulation models with the capacity to generate realistic levels of variability for quantitative traits, it is reasonable to assume that researchers consider it unwarranted. Quite often, however, a researcher’s decision regarding adequate aggregation (e.g., the level of variability) is not a wholly informed one but rather is a decision frequently arrived at by default due to lower-level aggregation not being available for comparison.
The objective of this study was to determine if simulation with less than realistic levels of variability can yield unintended results when modeling cow-calf production.
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MATERIALS AND METHODS |
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TopAbstractINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONIMPLICATIONSLITERATURE CITED |
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Overview
Mathematical models used to simulate beef cattle production invariably include nonlinear functions to mimic underlying biological processes. It is possible for nonlinear equations to interact with the level of variability in an equation’s independent variables, leading to unintended outcomes if the level of variability is not accurately modeled. For example, maintenance is typically calculated as a function of weight to the ³/3 power. If we apply the function to individual animals weighing 900.00 and 300.0 kg, values of 164.3 and 72.1 are returned. If we disregard variability, a practice common in beef cattle simulation, and use the pair’s mean weight (600.0 kg), the function returns a value of 121.2 for each animal. Clearly, the sum of values is smaller in the scenario in which variability is accounted for (164.3 + 72.1 = 236.4) than in the scenario where an average animal is simulated (2 x 121.2 = 242.4). In a modeling situation, this difference has the potential to render faulty outcomes for any variable influenced by this function.
Examples could certainly be given for which the level of variation does not interact with nonlinear equations. This would occur in situations in which a function has a linear segment and all simulated values fall within that segment. For example, we may model the relationship between postpartum interval and body fat as a nonlinear function to a certain level of condition, after which nothing is to be gained with increased fatness. Under these circumstances, if all cows were above the threshold fatness, they would not be exposed to the nonlinear relationship. Even when some data points fall outside the linear section, the level of variation and the degree of nonlinearity, or both, may be insufficient to cause appreciable differences in outcome.
The level of simulated variability can also affect outcomes through culling and selection, or both. There are many conceivable circumstances in which culling and selection could interact with simulated variation. For example, in a situation of negative energy balance, a herd generated with realistic levels of variability in milk production is likely to see its mean production decline over time due to disproportionate numbers of heavy-milking cows failing to conceive. Besides impacting milk production, the reduction should have an effect on related traits (e.g., weaning weights, conception rates, and energy requirements). This decline and its cascade of related effects could not occur if there were little to no variability in milk production.
The interaction between the level of variability and culling or selection is not an issue, however, if conditions are such that they would not occur anyway (for example, if nutritional levels were sufficient so that heavier milking cows were not reproductively disadvantaged).
Given the possible simulation scenarios and the complexity of the array of deterministic equations in many simulation models, determining the impact of simulation with less than realistic levels of variability is not possible without actually testing the model. Therefore, we used the Colorado beef cattle production model (CBCPM), as described by Shafer et al. (2005)
, to determine if outcomes were affected by level of simulated variability.
By itself, CBCPM is described as a whole-herd, life-cycle simulation model that operates at an individual animal level. Through the integration of the simulation of production and utilization of rangeland (SPUR) model (Wight and Skiles, 1987; Hanson et al., 1992) via the interface FORAGE (Baker et al., 1992), CBCPM has the capacity to mimic rangeland production and grazing. To address the myriad of economic elements inextricably intertwined with beef cattle production, the firm level policy simulation model (FLIPSIM; Richardson and Nixon, 1986) was melded with CBCPM.
The Texas A&M model (Sanders and Cartwright, 1979a,b) forms the foundation of CBCPM. Further modifications and enhancements to the Texas A&M model undertaken by Notter et al. (1979a,b,c) and Bourdon and Brinks (1987a,b,c) were also incorporated. Though CBCPM’s precursors would be considered herd-wide, life-cycle biological simulations, none are individual animal models, which is a clear prerequisite for simulating realistic levels of variability.
Whereas a step in the right direction, an individual animal model alone does not ensure the generation of realistic variability. For instance, using extended time steps with an animal model will tend to diminish variation between animals because conditions remain static throughout the time step. Even with short time steps, it would be unusual for the level of variability expressed in a quantitative trait to be replicated via deterministic equations alone. In general, to simulate realistic levels of variability, stochastically generated variability must be supplemented.
Models described by Kahn and Spedding (1983) and Tess and Kolstad (2000a,b) are herd-wide, life-cycle biological models simulating at the individual animal level, though neither have the capacity to generate variability for quantitative traits beyond their deterministic equations. Conversely, though simulations such as the COWGAME, as referenced by Buchanan et al. (1988), can generate realistic levels of variability, they are devoid of the deterministic equations required to emulate the biology of beef production.
The CBCPM is uniquely qualified to test our hypothesis because it has the capacity to produce realistic variability for up to 20 traits. Phenotypes for these traits are derived in CBCPM via the following model:
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where PO is defined as an animal’s potential for the trait, and Es depicts environmental effects directly input to the simulation or generated through the model’s deterministic equations. The function is potentially nonlinear and represents the net effect of all equations that make up the biological simulation. The deterministic equations simulate quantifiable environmental effects such as age, environmental temperature, and forage quality. The equations are constructed so that in all cases they diminish an animal’s performance from the maximum achievable (i.e., the potential).
Potential is described as
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where the breeding value (BV) represents the sum of the trait mean, parental breeding value, and the animal’s breeding value deviate. Nonadditive value (NAV) is the sum of the mean nonadditive value for the cross and the animal’s nonadditive deviate. The environmental (E) deviate is calculated once for all traits. Temporary environment (TE) represents the temporary (regenerated as needed) environmental deviate essential for repeated traits in females (e.g., milk production). Environmental deviates simulate environmental variability not accounted for by the deterministic equations. All deviates are randomly generated and multinormally distributed with a mean of zero and (co)variance parameters provided through input. Baseline equations are representative of female performance. Therefore, potentials for males include a sex adjustment (SA) for traits in which sex influences performance.
It should be noted that, in the context of our model, BV and NAV variables differ somewhat from the true definition of breeding and nonadditive value. In CBCPM, BV and NAV refer to the maximum female performance achievable in a particular trait due to the additive and nonadditive portions of an animal’s genotype. This differs from the true definition in that BV and NAV are independent of the environment, population, or both in which the animal resides.
Parameterization
Due to CBCPM’s structure, altering variability in potentials was central to testing our hypothesis. Potentials for mature weight (POMWT), milk production (POMLK), gestation length (POGEST), maintenance requirement (POMAINT), and appetite (POAPP) were chosen as the traits to alter. With the exception of POG-EST, these potentials interact with nonlinear, deterministic equations embedded in the model and are linked to each other to varying degrees via those same equations.
Treatments with input variability (variability randomly generated according to input parameters) for POMWT, POMLK, POGEST, POMAINT, and POAPP, exclusively, were designated by MWTV, MLKV, GESTV, MAINTV, and APPV, respectively. Runs were also performed with input variability for all traits (ALLV), and for POMWT and POMLK simultaneously (MWTMLKV). Control (no input variability) was designated as NOV.
Means of 500 kg, 10 kg, 283 d, 1.0, and 1.0 were input for POMWT, POMLK, POGEST, POMAINT, and POAPP, respectively. Potential mature weight represents an animal’s maximum BW obtainable, excluding the conceptus, at 20% empty body fat (BCS 5.5; Herd and Sprott, 1986). The POMLK is the upper bound of a cow’s daily milk production. Gestation length is modeled without influence from deterministic equations, so POGEST is analogous to the actual (phenotypic) gestation length. Values for POMAINT and POAPP serve as multipliers of deterministically generated values for these traits. For example, a POMAINT of 1.05 would increase an animal’s maintenance requirements by 5% over the requirements generated by the model’s deterministic equations.
Input values for genetic and environmental variability are listed in Table 1. Values for POMWT and POMLK were adapted from Enns (1995), and estimates for POGEST were derived from data presented by Bourdon and Brinks (1982). Taylor and Young (1967, 1968) calculated a CV, independent of feed intake level, of 5.5 percent for maintenance requirements at a constant weight. This information, combined with an assumed heritability of 0.66, was used to calculate the parameters for POMAINT. Due to a lack of literature estimates, and for the sake of simplicity, the same values were used for POAPP. Input covariability between traits was not simulated. Although covariability was not provided through input, it is possible that the relationships between traits were created by the myriad of deterministic equations embedded in the model.
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To strike a balance between herd size, simulation length, and the number of replications required, we performed trial simulation runs. Also under consideration was the warm-up period (i.e., the time required for factors such as age distribution to reach a semblance of equilibrium). As a result of our trial runs, we chose to simulate a 500-cow herd over the course of 22 yr (with the first 2 warm-up years not reported to output). Each simulated scenario, including control, was replicated 3 times; because CBCPM generates puberty, estrus, conception, death, and sex outcomes stochastically, replicates were necessary even for runs without input variation in potentials.
The SE calculated via 1-way ANOVA are presented; however, we chose not to report the P-values because (1) not all traits are modeled with random variation, (2) simulated variation between replicates is not the same as would be measured among actual ranches, and (3) the homogeneity of variance assumption is directly violated in this study. Furthermore, we didn’t want the significance level to obfuscate the discussion of results; in our opinion, it was the movement of means rather than statistical significance that was germane to our hypotheses.
Physical Environment
The simulated environment was representative of the Pawnee National Grasslands in northeast Colorado. The area is considered semiarid, short-grass prairie with an average annual precipitation of 20 cm and typical stocking rates of 16.2 to 24.3 ha per cow-calf pair (Senft, 1983). To avoid the possibility of stocking pressure being a confounding factor, the cowherd had access to 20,245 ha of grassland (for practical purposes an unlimited area).
Input parameters for CBCPM’s plant component, the SPUR model (Wight and Skiles, 1987; Hanson et al., 1992) for climate (rainfall, temperature, solar radiation, and wind) as well as grass, wildlife, and insect species representative of that area were adapted from Hanson et al. (1992).
Culling, Replacement, and Mating
Cows were culled and sold at weaning if open, unsound, or 12 yr of age. Due to requirements of a concurrent study, no replacements were generated from within the herd. Instead, pregnant females were purchased when cows were culled (weaning) in a quantity necessary to keep the herd at its 500-cow target size. These females were generated with the same mean potentials as the original cowherd (500 kg, 10 kg, 283 d, 1.0, and 1.0 for POMWT, POMLK, POGEST, PO-MAINT, and POAPP, respectively). Replacements entered the simulation at 20% empty body fat. To avoid confounding between parity and service sire, cows and heifers were bred to medium-mature weight bulls as characterized by Enns (1995). Cows were exposed to bulls from June 1 through August 1. Castration and weaning dates were June 1 and November 1, respectively. All calves were removed from the simulation (i.e., sold) at weaning.
Nutrition
Although cattle had access to year-round grazing, the primary grazing months were May through October. During this time the herd could be supplemented with harvested forage if its average empty body fat was below 6% (BCS 2.5; Herd and Sprott, 1986), which would only occur during severe drought conditions. Supplementation typically occurred from November through April.
To test the premise that nutritional level affects interactions among variables, we ran contrasting simulations in which feed groups (animals fed together) were provided with high feed (HF) and low feed (LF) supplementation during these months. Although bred heifers were managed and fed separately from the rest of the cowherd, both groups were fed to the same target body fat within the simulation via a 63.5% TDN ration consisting of 80% hay and 20% corn (DM basis). Mean LF targets were 10, 12, and 14% for the periods of November through December, January, and February through April, respectively. Empty body fat targets for groups fed at the HF level were increased 8% relative to groups at the LF level.
Output
Rather than limiting output to the 5 treatment traits (those with input variation), we summarized and presented a wide array of characteristics. For the most part, variables chosen for output were those providing insight into the causal factors behind the outcomes, factors that can often be difficult to determine in a model of CBCPM’s size and complexity. Because economics are an important consideration in beef cattle production, we also included factors directly influencing profitability (for example, the number of replacement females bought, weight of weaned calves, and herd TDN intake). We also calculated the ratio of the weight of weaned calves to herd TDN intake. Although its use as a predictor of profitability is questionable, the ratio likely renders reasonable estimates of biological efficiency.
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RESULTS AND DISCUSSION |
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TopAbstractINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONIMPLICATIONSLITERATURE CITED |
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and 3 for HF and LF environments, respectively; outcome variables, control effects, and treatment effects are depicted in rows, right-hand columns, and remaining columns, respectively.
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Differences between control (no variation in potentials) and treatment (various combinations of potentials with variation) means were negligible under the HF environment (Table 2
), though substantial increases in the variability of those variables influenced by treatment occurred, and many of these variables are partial functions of nonlinear equations. The lack of treatment effects on means was presumably due to level of nutrition and body composition being beyond the nonlinear range for functions influenced by such things as intake and body fat. Except for treatment GESTV (variation in POGEST), disparities between control and treatment means were pronounced under the LF environment (Table 3). For brevity, the following discussion pertains only to the LF environment, excluding treatment GESTV, unless otherwise stated.
Effects of Simulated Variability on Trait Potentials
In these simulations, selection was limited almost entirely to the culling of open cows. As a result, differential culling (culling based on differences in trait potentials) occurred on any factors related directly or indirectly to fertility. Clearly, differential culling was only possible when potentials were allowed to vary (i.e., in all but the control scenario). The extent of differential culling is shown by differences between treatments and control in mean potentials of pregnant cows at weaning. In Table 3, mean potentials from all treatments with variability differ to some degree from control. Under treatments MAINTV and ALLV, POMAINT decreased by over a third of a standard deviation, whereas POAPP increased as much under treatments allowing it to vary. Mean POMWT decreased by 19.0, 17.0, and 13.0 kg under treatments MWTV, MWTMLKV, and ALLV, respectively. Reduction in POMLK over treatments MLKV, MWTMLKV, and ALLV was 1.32, 1.23, and 1.04 kg, respectively. As the number of potentials with variability increased from 1 (treatment MWTV or MLKV) to 2 (MWTMLKV) and finally to all (ALLV), culling pressure on POMWT and POMLK decreased. This was presumably due to culling on other variables, which eased pressure on milk and growth. For example, the efficiency gained by a reduction in POMAINT allowed more nutrients to be available for milk and growth, relaxing culling pressure on those traits.
Besides its direct effect on differential culling, simulating without individual animal variability prohibits modeling the change that would occur in replacements due to differential culling in parents. Replacement selection was not simulated in our study, but had we produced rather than purchased replacements, treatment effects on herd potentials would likely have been even more pronounced; their impact would have been cumulative over generations.
Effects of Simulated Variability on Performance Measures
Fertility.
Adding variability reduced pregnancy rates under all treatments. In some cases the reduction was dramatic. For example, varying all 5 potentials (treatment ALLV) caused pregnancy rates to drop from 0.94 to 0.80 (Table 3). Most of the reduction can be attributed to body condition at calving as measured by proportion of chemical fat in the empty body (PCFEB). Because groups of cattle were fed to meet targets for average body condition, mean PCFEB at calving did not vary across treatments; however, variability in PCFEB did. Pregnancy rates decreased as variability in PCFEB increased from the control level.
In CBCPM, the influence of body condition on pregnancy rate can be tracked mathematically through 2 intermediary variables: postpartum interval and CFFCON (correction factor for fertility on condition; measured in days), which increases postpartum interval in underweight cows. The operative equation is
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and applies only to animals with PCFEB under 0.20 at calving. Compared with control, the treatment with greatest variability in PCFEB (ALLV) added 9.3 d to CFFCON, 8.6 d to postpartum interval, and 18.5 d to the standard deviation of postpartum interval.
The biological explanation for the relationship between variability and pregnancy rate derives from the fact that, as body fat declines, fertility decreases at an increasing rate. Differences between animals at the lower ranges of body fat have a greater impact on fertility than differences at upper ranges. Increased variability in traits like growth and milk production causes increased variability in body condition, yielding both fatter and thinner animals, but because the thinner ones have more influence on fertility, the net result of increased variation is a reduction in pregnancy rates.
Production. Averages over lactation as well as peak levels of milk production were notably diminished in those treatments in which POMLK was allowed to vary. From control levels, peak production decreased 1.19, 1.05, and 1.08 kg/d, and average production over lactation was reduced by 0.71, 0.80, and 0.73 kg/d in treatments MLKV, MWTMLKV, and ALLV, respectively. The dampening of milk production in MLKV, MWTMLKV, and ALLV was associated with weaning weights that were 7.7, 9.6, and 8.2 kg lighter than controls, respectively.
Requirements and Intake.
We selected peak lactation as a point of reference to make inferences about requirements and intake because, unlike a specific date in time, it represents a clearly defined physiological state. As can be seen in Table 3, virtually all treatment effects pertaining to intake and requirements are different from control. Given that trait potentials were altered due to culling, this is not surprising. Moreover, in all instances, the difference is in the direction expected when considering the discrepancies in trait potentials. Specifically, maintenance requirements and consumed nutrients are reduced for treatments with reductions in POMWT, POMLK, and POMAINT. With an increase in POAPP (treatment APPV), however, maintenance requirements are held in check, whereas consumed nutrients are raised slightly.
Economics and Bioefficiency.
As can be seen from Table 3, large discrepancies in the number of cull cows and replacement heifers exist between treatments and control, the most glaring being treatment ALLV, under which the firm is required to purchase 55 more replacements annually compared with control. Even APPV, the treatment with the replacement rate most similar to control, requires 13.7 additional replacements.
Replacements required were consistently 9 to 10 animals more than the number of culls sold—the result of uniform death loss. Because death loss was steady, the dramatic differences in replacement rates can be fully attributed to the wide range in pregnancy rates.
Treatments in which POMWT, POMLK, or both varied were associated with lower total weight of weaned calves. This reduction was due to lighter weaning weights, as treatments did not appreciably affect the number of calves weaned. The reduction in weight was met with corresponding reductions in total herd TDN intake. Because of this, there was little difference between treatment and control for the ratio of weight of weaned calves to total TDN consumed.
Though ratios of weight of weaned calves to total TDN consumed were similar for treatments and control, they were arrived at via divergent routes. It is quite likely the different routes would yield disparity between treatments and control in profitability. For example, given the large discrepancy in replacement rates, if the cost of replacing females was relatively high we would expect treatment ALLV to be much less profitable than control despite their similar ratios. An additional caveat is that because pricing systems are typically nonlinear in nature, the unrealistic uniformity in weaning weights found in the control scenario may lead to inaccurate economic assumptions.
Differential Culling Vs. Interaction with Nonlinear Equations Independent of Culling
We did not attempt to separate treatment effects into those attributable to and those independent of culling. That would have required dubious simulation scenarios (e.g., not culling open cows), and we could see little utility in obtaining the breakdown. However, it is clear that outcomes were affected both by culling on potentials and by effects of variability independent of culling. The strongest evidence for the effect of culling is the movement of mean potentials, which occurred in directions that should improve pregnancy rates (decreased POMWT, POMLK, and POMAINT and increased POAPP). Nevertheless, pregnancy rates were reduced significantly by treatments—clear evidence of an effect of variation independent of culling.
Gestation Length: A Special Case
Unlike other traits, no deterministic equations directly affect gestation length in CBCPM; an animal’s POGEST is its phenotype for the trait. Therefore, any impact on outcomes caused by treatment (variability in POGEST) must result from differential culling (i.e., longer POGEST cows being culled for being open resulting in mean POGEST of pregnant cows being shorter). But because there were no appreciable differences between treatments and control in POGEST, we must conclude that longer gestation lengths weren’t discriminated against. An even poorer environment than our LF or a longer gestation genotype might have resulted in culling against longer POGEST.
Alternative Results
Relative to other traits, varying POMWT and POMLK had the most substantial impact on outcomes, varying POMAINT and POAPP affected results to a lesser degree, and varying POGEST appeared to have no impact. From these results, one could conclude that varying POMWT and POMLK is all that is necessary to avoid misleading outcomes. Although that may be true for the specific circumstances of our simulation, we can conceive of several, only slightly different scenarios that could lead to very different conclusions. A handful of examples follow. (1) If HF had been our only simulated environment, we could have concluded that using realistic levels of variability was not necessary for any traits. (2) By simulating longer gestation-length potentials or poorer environments, POGEST might have been affected by culling, possibly affecting other results. (3) Increasing the variability for POMAINT and POAPP would have likely heightened their influence. The increased variability could be justified biologically. In fact, Johnson et al. (2003) suggests approximately twice the variability we used in parameterizing POMAINT.
The inference from these examples is that only relatively minor changes have the potential to alter assessments about which traits need to be varied. Even within the confines of this study, these examples represent a small subsample of plausible configurations. Models like CBCPM are designed with the intention of producing valid results across a vast array of scenarios. If small changes in our limited study could yield such different conclusions about which traits to vary, a fair assumption may be that over the entire spectrum of possible scenarios in a model of CBCPM’s nature, most any trait will ultimately warrant variation.
It is worth mentioning that though our study focused on variance, it is quite conceivable that simulating without realistic levels of covariance between traits may also yield misleading results—certainly an area warranting future investigation.
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IMPLICATIONS |
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TopAbstractINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONIMPLICATIONSLITERATURE CITED |
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1 Corresponding author: wshafer{at}simmgene.com
Received for publication December 10, 2005. Accepted for publication September 29, 2006.
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LITERATURE CITED |
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TopAbstractINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONIMPLICATIONSLITERATURE CITED |
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