Mathematics and Computers in Simulation 79 (2009) 2701–2712
Parameter estimation and sensitivity analysis of fat depositionmodels in beef steers using acslXtreme
Malcolm McPhee a,b,∗, Jim Oltjen c, James Fadel c, David Mayer a,d, Roberto Sainz c
a Cooperative Research Centre for Beef Genetic Technologies, Armidale, NSW 2351, Australiab NSW Department of Primary Industries Beef Industry Centre of Excellence, UNE, NSW 2351, Australia
c Department of Animal Science, University of California, Davis, CA 95616, USAd Department of Primary Industries & Fisheries, Yeerongpilly, Qld 4105, Australia
Received 3 July 2008; received in revised form 22 July 2008; accepted 23 August 2008Available online 10 September 2008
The Davis Growth Model, a dynamic steer growth model [9] that includes 4 fat deposition models [11] is currentlybeing used by the phenotypic prediction program of the Cooperative Research Centre (CRC) for Beef Genetic Tech-nologies to predict beef cattle fatness in the field [4]. Predicting beef cattle fatness will assist beef producers managecattle to meet stringent market specifications that are related to both weight and fatness for domestic and internationalmarkets.
∗ Corresponding author at: NSW Department of Primary Industries Beef Industry Centre of Excellence, UNE, NSW 2351, Australia.Tel.: +61 2 67701838; fax: +61 2 6770 1830.
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Fig. 1. Partitioning of net energy in the Davis Growth Model to total body fat and then the partitioning of total body fat to 4 fat depots.
The concepts of cellular hyperplasia and hypertrophy are integral components of the Davis Growth Model. The netsynthesis of total body fat is calculated from the net energy available after accounting for energy needs for maintenanceand protein synthesis. Total body fat (FAT) is then partitioned into 4 fat depots (intermuscular, intramuscular, subcu-taneous, and visceral) (Fig. 1). Three of the fat depots are then converted to carcass characteristics: intramuscular fat(IMF, kg) to IMF as a percentage (%), subcutaneous fat (kg) to 12/13th rib fat (mm) and subsequently to P8 fat (B.J.Walmsley, unpublished results), and visceral fat (kg) to kidney, pelvic, and heart fat (KPH, %) [8]. The 4th fat depot,intermuscular fat, is not converted to any carcass characteristic. Each of the 4 fat depots is derived by a first-orderdifferential equation. This paper describes the parameter estimation and sensitivity analysis of the DNA logistic growthequations and fat deposition first-order differential equations using acslXtreme (Hunstville, AL, USA, Xcellon). Datafrom Robelin [10] and Cianzio et al. [2] were used to parameterize the DNA equations and data from a meta-analysisstudy [7] were used to parameterize the fat deposition equations. The objectives of this study were: (1) describe thefat deposition models; (2) parameterize the DNA logistic growth equations and fat deposition first-order differentialequations and; (3) conduct a sensitivity analysis of the parameters.
2. Notation and units
A number of symbols and special nomenclature are used throughout this paper. Table 1 outlines the notation witha description, units, and value where appropriate.
3. Method
The DNA logistic growth equations and fat deposition equations have been parameterized using acslXtreme (a toolfor modelling and simulation of continuous dynamic systems and processes) and a sensitivity analysis was conductedusing the sensitivity analysis routine in acslXtreme. Data reported by Robelin [10] and Cianzio et al. [2] were used toparameterize the DNA logistic growth equations and data from a meta-analysis study [7] were used to parameterizethe fat deposition equations. The Ksyn and Kmaint parameters were initially parameterized against BW and FAT beforethe fat deposition parameters were estimated. It was assumed that fat was increasing in each of the depots thereforedata with fat deposition parameter coefficients less than zero were removed. An evaluation of each of the acslXtremeoptimization algorithms was conducted on 20 of the meta-analysis data sets. The carcass characteristics reported inthe meta-analysis study [7] were converted to kilograms of fat using the equations described by McPhee et al. [8].All BW values reported in this study are shrunk BW, i.e., the reported BW in the publications multiplied by 0.96.A summary of the inputs to initialize the Davis Growth Model simulations are shown in Tables 2 and 3 for non-implanted and implanted steers, respectively. A summary of the initial and final observed values to parameterize Ksynand Kmaint against BW and FAT in the first instance and subsequently the fat deposition parameter coefficients (kFATj)for intermuscular, intramuscular, subcutaneous, and visceral fat are shown in Tables 4 and 5 for non-implanted andimplanted steers, respectively.
3.1. Frame size
Frame size (1) was calculated based on the mean values of EBW reported in each of the publications in the meta-analysis study [7]. The industry scale of frame size is 1–9 corresponding to 550–950 kg MEBW, in steps of 50 kg
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Table 1Description of mnemonics, variables and coefficients used in this study.
Item Description Value
j Increment for each fat depotIntermuscular, 1Intramuscular, 2Subcutaneous, and 3Visceral 4
t (days) Time 1 to number of days on feedβ(t)j Proportion of total body fat gain in each fat depot j at time t in days –ADSMAXj (kg TG/g DNA) Maximum adipocyte size for each fat depot adipose j –BW (kg) Body weight –CS Condition score –DIAj (�m) Diameter of fat depot cells –DMI (kg/day) Dry matter intake –DNAj (g DNA) Deoxyribonucleic acid –DNAMAX1 (g DNA) Maximum DNA in intermuscular adipose 0.304DNAMAX2 (g DNA) Maximum DNA in intramuscular adipose 0.050DNAMAX3 (g DNA) Maximum DNA in subcutaneous adipose 0.175DNAMAX4 (g DNA) Maximum DNA in visceral adipose 0.211DOF Days on feed –EBW (kg) Empty body weight –FAT (kg) Total body fat –FATMAX (kg) Maximum total body fat –FMAXj (kg TG) Maximum amount of fat in each adipose j –FSIZj (kg TG/g DNA) Fat size in each adipose j –Fj (kg TG) Fat in each fat depot j –Inter (kg) Intermuscular fat –Intra (kg) Intramuscular fat –Ksyn (kg0.27) Protein synthesis coefficient –Kmaint (Mcal kg−0.75 day−1) Protein maintenance coefficient –kDNAj (1/g DNA) DNA parameter coefficient for each adipose j –kFATj (1/g DNA) Fat parameter coefficient for each fat depot j –MEBW (kg) Mature empty body weight –MEC (Mcal/kg DM) Metabolisable energy concentrate –Sub (kg) Subcutaneous fat –TG (kg) Triacylglygcerol –Vis (kg) Visceral fat –
respectively, [6]. Empty body weights of steers are adjusted to a stage of maturity based on the assumption of Fox andBlack [5] “that beef animals have equal body composition at similar stages of maturity”. When data were not availableto calculate frame size a scale was given based upon type of breed and the geographical location from which the steerswere sourced.
frame size =(
(MEBW, kg − 750)
50
)+ 5 (1)
Table 2Summary of inputs to the Davis Growth Model and initial observed values used to parameterize the fat deposition equations for non-implantedsteers (see Table 1 for a description of variables).
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Table 3Summary of inputs to the Davis Growth Model and initial observed values used to parameterize the fat deposition equations for implanted steers(see Table 1 for a description of variables).
MEBW (2) is calculated based on a ratio between reference (ref) values of EBW and MEBW.
MEBW, kg = MEBWref × EBW
EBWref(2)
3.2. DNA logistic growth equations
The DNA (DNAj) in each of the fat depots represents the number of cells, i.e., hyperplasia. Data from a studyconducted by Robelin [10] were used to parameterize the parameter coefficients kDNA1, kDNA3, and kDNA4 forintermuscular, subcutaneous, and visceral, respectively, in the DNA logistic growth equations (3) and data from a studyby Cianzio et al. [2] were used to parameterize the kDNA2 parameter coefficient for the intramuscular fat depot. TheDNA was estimated as the number of cells × 6.2 pg/cell [1]. The DNA data from Robelin [10] had a higher maximumDNA value than the Cianzio et al. [2] data and was considered suitable for developing the parameters. The Cianzioet al. [2] data for intramuscular fat were used because Robelin [10] did not determine DNA for the intramuscular fatdepot.
dDNAj
dt= kDNAj × DNAj(t) × (DNAMAXj − DNAj(t)) (3)
Table 5Summary of the initial and final observed values used to parameterize the fat deposition equations for implanted steers.
Item Initial observed values Final observed values
BW (kg) Fat (kg) Inter (kg) Intra (kg) Sub (kg) Vis (kg) BW (kg) Fat (kg) Inter (kg) Intra (kg) Sub (kg) Vis (kg)
The first-order differential Eq. (4) for each fat depot (Fj) is a proportion (5) of total body fat gain (kg/day). Theproportion (5) is a function of hyperplasia, DNAj, i.e., number of adipocytes; hypertrophy FSIZj, i.e., adipocyte size;and the maximum adipocyte size (ADSMAXj) of each fat depot adipose. The adipocyte size (FSIZj) is a function ofthe amount of fat (Fj) and DNA (DNAj) in the depot and ADSMAXj (7) is a function of the maximum amount of fat(FMAXj) and maximum amount of DNA (DNAMAXj) in each fat depot. The maximum amount of fat (FMAXj) wasdetermined from empirical equations reported by McPhee et al. [8] where the empirical equations are a function ofthe FAT. Total body fat maximum (FATMAX) was calculated as a proportion of MEBW (8) and the DNAMAXj for eachadipose is reported in Table 1. The maximum amount of DNA (DNAMAXj; Table 1) was determined by adding the S.D.to the maximum amount of DNA reported by Robelin [10] for intermuscular, subcutaneous, and visceral fat depotsand Cianzio et al. [2] for intramuscular fat.
dFj
dt= βj(t − 1) × dFAT
dt(4)
βj(t) = kFATj × DNAj(t) ×[
1 −(
FSIZj
ADSMAXj
)](5)
FSIZj = Fj(t)
DNAj(t)(6)
ADSMAXj = FMAXj
DNAMAXj
(7)
FATMAX = 0.44 × MEBW (8)
Fig. 2. Graph of the DNA logistic growth curves after the data of Robelin [10] and Cianzio et al. [2] had been used to parameterize the DNA logisticgrowth equations for each of the fat depots: (A) intermuscular DNA, (B) intramuscular DNA, (C) subcutaneous DNA, and (D) visceral DNA (g).
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3.4. Fat depot diameters
Fat depot diameters (9) were calculated for each fat depot. As stated in Section 3.2, the conversion from the numberof cells to grams of DNA was equal to the number of cells × 6.2 pg/cell [1], the 0.8 represents the conversion ofdissected fat to chemical fat as outlined in [8] and the density of fat was assumed to be 9.15 × 10−13 g/�m3.
DIAj = 2 × 3
√1000 × 3 × FSIZj × 6.2 × 10
0.8 × 4 × π × 9.15(9)
3.5. Parameterization
Each meta-analysis data set was parameterized for Ksyn and Kmaint to match BW and FAT using the Nelder-Meadoptimization algorithm in acslXtreme. Twenty meta-analysis data sets were then used to determine the most efficientand accurate algorithm to parameterize the 4 fat deposition parameters (kFATj). After determining the optimal algorithmthe fat deposition parameters were parameterized for each meta-analysis data set. The fat deposition model (5) wasconstrained to �βj = 1.
3.6. Statistical analysis
The intermuscular, intramuscular, subcutaneous, and visceral fat deposition parameter coefficients were analyzedusing the GLM procedure of SAS (SAS Inst., Inc., Cary, NC). The parameters were log transformed to normalize thedistributions. The effect of implant status and frame size was evaluated. Frame size was a continuous variable calculatedas described in Section 3.1. Eleven data sets had fat decreasing in some of their fat depots because animals in thesemeta-analysis studies were using their energy reserves. An assumption was made that data sets were only included inthe parameter estimation if fat was increasing in each of the fat depots. Therefore, 11 data sets were excluded fromthis analysis. The mean square error of prediction (MSEP) was used to evaluate the accuracy of predicting the finalobserved values for each of the fat depots.
3.7. Sensitivity analysis
3.7.1. acslXtreme sensitivity analysis of model responsesThe sensitivity analysis of model responses (Fj) with respect to model parameters (kDNAj and kFATj) was evalu-
ated. The acslXtreme sensitivity analysis calculates the partial derivatives of model responses with respect to modelparameters and the partial derivates are referred to as sensitivity coefficients. A forward difference method was usedand the parameters were normalized. A flat non-increasing slope indicates that the parameter is not sensitive.
Table 6Evaluation of acslXtreme optimization algorithms and mean square error of prediction for parameter estimates of fat depots.
Optimization algorithm n Objective functionevaluations
a Based on the difference between observed and simulated values at slaughter (i.e., final value).b Algorithm did not converge for any data set.c Algorithm did not converge for one data set.
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Fig. 3. Accuracy of simulating (lines) the observed values (symbols) using one meta-analysis data set after (a) Ksyn and Kmaint parameters had beenestimated to accurately predict body weight (�) and total body fat (�); and (b) the kFATj parameters had been estimated for each of the fat depots:intermuscular fat (�), intramuscular fat (�), subcutaneous fat (�), and visceral fat (�) (kg).
Fig. 4. Example of total body fat (kg) partitioned into 4 fat depots (kg) based on one meta-analysis data set.
Table 7A summary of the fat depot adipocyte diameters (�m).
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3.7.2. Sensitivity analysis of fat partition parameter effect on the prediction of fat in each fat depotThe sensitivity analysis of fat partition parameter effect of fat deposition for each of the fat depots for non-implanted
steers was evaluated where the inputs to the Davis Growth Model were set at the mean value of the meta-analysisdata (Table 2). Five incremental changes of ±10% to the mean fat partition parameters were made. A curve with anincreasing slope indicates that the parameter is sensitive. The y-axis for each fat depot was set at the maximum amountof fat for each fat depot based on the mean frame size (Table 2) where MEBW was calculated after re-arranging (1)subsequently FATMAX was calculated (8) and substituted into the regression equations as outlined in [8] to calculatemaximum amount of fat in each of the fat depots.
Fig. 5. Sensitivity analysis of model responses (Fj) with respect to model parameter coefficients [kDNAj (- -) and kFATj (—)] based on the meaninput values of non-implanted steers of the meta-analysis (Table 2) for: (a) intermuscular fat, (b) intramuscular fat, (c) subcutaneous fat, and (d)visceral fat.
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4. Results
4.1. Parameter estimation of DNA logistic growth equations
The DNA simulated logistic growth curve results are shown in Fig. 2. The DNA parameter coefficients (mean ± S.D.;1/g DNA) for each fat depot were 0.012 ± 0.0032, 0.19 ± 0.044, 0.048 ± 0.0047, and 0.019 ± 0.0045 for kDNA1,kDNA2, kDNA3, and kDNA4, respectively.
4.2. Evaluation of optimization algorithms for parameter estimation of fat deposition equations
Twenty meta-analysis data sets were used to evaluate the number of evaluations and processing time of 8 optimizationalgorithms (Nelder-Mead, Direction Set, Conjugate Gradient, Quasi-Newton, Simulated Annealing, Particle Swarm,Levenberg–Marquardt, and Generalized NL2SOL) [3]. The results of this evaluation are shown in Table 6 and indicatethat the Levenberg–Marquardt and Generalized NL2SOL algorithms accurately predict the slaughter values (i.e., finalvalue) for each fat depot (MSEP = 0 for all fat depots). The Generalized NL2SOL had the lowest number of evaluationsand the fastest processing time. Therefore, the Generalized NL2SOL algorithm was used to parameterize all of the fatdeposition parameters.
4.3. Parameter estimation of fat deposition equations
Initially, Ksyn and Kmaint were parameterized using the Nelder-Mead algorithm against BW and FAT so that theobserved values of BW and FAT were accurately simulated (Fig. 3(a)). The mean parameter estimates for Ksyn were
Fig. 6. Sensitivity analysis (±10% of the mean parameter value in 5 incremental steps) of fat partition parameter coefficient effect on the predictionof DNA for non-implanted steers with the y-axis scaled to the maximum amount of DNA for each depot: (a) intermuscular fat, (b) intramuscularfat, (c) subcutaneous fat, and (d) visceral fat (g) (� represents the mean parameter value).
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0.0470 ± 0.0008 and 0.0489 ± 0.0013 for non-implanted (n = 39) and implanted (n = 94) steers, respectively, and param-eter estimates of Kmaint were 0.0983 ± 0.0114 and 0.1083 ± 0.0156 for non-implanted (n = 39) and implanted (n = 94)steers, respectively. A 3.9% increase in Ksyn was detected between non-implanted and implanted steers.
The Ksyn and Kmaint parameters for each data set were used so that BW and FAT were accurately predicted. Thegeneralized NL2SOL algorithm as determined in Section 4.2 was used to parameterize the fat deposition coefficients(kFATj). All fat depots were optimized simultaneously and were accurately fitted against the observed values (Fig. 3(b)).The statistical analysis of the fat deposition coefficients indicated that implant status was not significantly different forany fat depot: P = 0.20, 0.31, 0.93, and 0.32 for intermuscular, intramuscular, subcutaneous, and visceral fat depots,respectively; and frame size was not significantly different for intermuscular (P = 0.17), intramuscular (P = 0.23), andvisceral fat (P = 0.32) depots but significant (P < 0.01) for subcutaneous fat. The mean parameter coefficients were0.81 ± 0.930, 0.53 ± 0.571, 0.18 ± 0.258 for kFAT1, kFAT2, and kFAT4, respectively; and a non-linear regression wasfitted to the subcutaneous coefficient kFAT3 with frame size as the independent variable (10). Fig. 4 illustrates thepartition of 152 kg of FAT into 4 fat depots and Table 7 summarises the fat depot adipocyte diameters. The results fromTable 7 indicate that the fat depot diameter ranges are within the published data [2,10] but they do not follow the order,i.e., the published data [2,10] in order from largest to smallest is Vis > Inter > Sub > Intra but Table 7 indicates that theorder is Inter > Intra > Sub > Vis.
Fig. 7. Sensitivity analysis (±10% of the mean parameter value in 5 incremental steps) of fat partition parameter coefficient effect on the predictionof fat for non-implanted steers with the y-axis scaled to the maximum amount of fat for each depot: (a) intermuscular fat, (b) intramuscular fat, (c)subcutaneous fat, and (d) visceral fat (kg) (� represents the mean parameter value and the subcutaneous coefficient was determined at the overallmean (n = 133) of non-implanted and implanted steers of frame size = 5.7).
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4.4. Sensitivity analysis
4.4.1. acslXtreme sensitivity analysis of model responsesThe results from the sensitivity analysis (Fig. 5) indicate that the kDNAj parameter coefficients are less sen-
sitive than the kFATj parameter coefficients to their response variables (Fj). The intermuscular and visceral DNAparameters (Fig. 5(a) and (d)) are the least sensitive. All fat deposition parameter coefficients (kFATj) were sen-sitive to there response variables. The kDNAj parameter coefficients in the first 50 days indicate that the DNAparameters are not sensitive to their response variables (Fj), however, an effect is clearly seen with DOF > 100 days(Fig. 5).
4.4.2. Sensitivity analysis of fat partition parameter effect on the prediction of fat in each fat depotThe effect of each partition parameter on the prediction of the respective fat depots (Figs. 6 and 7) was monotonic
increasing, indicating that the parameters are important determinants of model function. The subcutaneous coefficientkFAT3 was calculated using (10) with frame size set at the overall mean (n = 133) of non-implanted and implantedsteers of 5.7. Fig. 6 shows that the intramuscular fat DNA parameter coefficient was the least important and Fig. 7indicates that the intermuscular fat parameter coefficient was the least significant when each fat depot was scaled to alevel of maturity for each fat depot.
5. Conclusions
The results from this study indicate that the DNA (kDNAj) and fat deposition (kFATj) parameter coefficients aresensitive and important determinants of model function in the prediction of fat (Fj; kg) in each of the fat depots.This paper found that the generalized NL2SOL optimization algorithm, in acslXtreme, had the fastest processing timeand the minimum number of objective function evaluations when estimating the parameter coefficients of the first-order differential fat deposition equations with 2 observed values (initial and final fat) in each of the fat depots. Thegeneralized NL2SOL optimization algorithm also predicted fat at slaughter (i.e., final fat) for each of the fat depots(Fj; kg) with a MSEP = 0.
The statistical analysis of the fat deposition (kFATj) parameters indicate that they are not metabolically different fornon-implanted and implanted steers in all fat depots and not different for intermuscular, intramuscular, and visceral fatdepots for frame size at the level of aggregation used to simulate fat deposition in beef steers. The subcutaneous fatparameter coefficient did, however, indicate a metabolic difference for frame size therefore a non-linear relationshipwas developed for kFAT3 to adjust for differences in frame size.
Availability of data is a limiting factor because studies on the cellularity of bovine tissues are scarce. Nevertheless,the DNA parameter coefficient sensitivity analysis of model responses when DOF > 100 days and the DNA fat partitionof parameter coefficients indicated that the DNA parameter coefficients are important determinants of model function.The parameterization of the fat deposition first-order differential equations indicated that the parameter coefficientswere sensitive to model responses for all DOF and the fat depot partition of parameter coefficients indicated that thefat depot parameter coefficients are also important determinants of model function.
The DNA logistic growth curves (Fig. 2) were developed, as mentioned above, with a limited amount of data andtherefore may not accurately represent the shape of the relationship. This study highlights that further research inbovine adipose cellularity is required to develop and challenge DNA models. As additional data becomes availablethe DNA relationship may show that a secondary ‘wave’ of hyperplasia occurs as animals fatten. This secondary‘wave’ of hyperplasia may follow a more episodic pattern. Future research may also show that there are differences inhypertrophy between breeds.
A preliminary analysis of the field calculator that predicts P8 fat has shown some promising results (V.H., Oddy,unpublished results). The empirical equation in the field calculator was however, developed with the subcutaneousfat proportion of FAT fixed at 20%. This study and the parameterization of the DNA logistic growth equations andthe first-order differential fat deposition equations are dynamic not static! Therefore, the future development has thepotential to improve the prediction of fat in the Davis Growth Model. The simulated diameters (Table 7) indicate somediscrepancies with the limited amount of data that is available. Further development of the fat deposition models maycorrect this apparent anomaly.
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