A model to describe growth patterns of the mammary gland during pregnancy and lactation

Feed restriction of lactating cows triggers acute downregulation of mammary mTOR signaling and chronic reduction of mammary epithelial mass

While there is generally no consensus about how nutrients determine milk synthesis in the mammary gland, it is likely that the mechanistic target of rapamycin complex 1 (mTORC1) plays a role as a key integrator of nutritional and mitogenic signals that can influence a multitude of catabolic and anabolic pathways. The objectives of this study were to evaluate acute changes (<24 h) in translational signaling, in addition to chronic changes (14 d) in mammary gland structure and composition, in response to a severe feed restriction. Fourteen lactating Holstein dairy cows were assigned to either ad libitum feeding (n = 7), or a restricted feeding program (n = 7). Feed-restricted cows had feed removed after the evening milking on d 0. Mammary biopsies and blood samples were collected 16 h after feed removal, after which cows in the restricted group were fed 60% of their previously observed ad lib intake for the remainder of the study. On d 14, animals were sacrificed and mammary glands dissected. In response to feed removal, an acute increase in plasma nonesterified fatty acid concentration was observed, concurrent to a decrease in milk yield. In mammary tissue, we observed downregulation of the mTORC1-S6K1 signaling cascade, in addition to reductions in mRNA expression of markers of protein synthesis, endoplasmic reticulum biogenesis, and cell turnover (i.e., transcripts associated with apoptosis or cell proliferation). During the 14 d of restricted feeding, animals underwent homeorhetic adaptation to 40% lower nutrient intake, achieving a new setpoint of 14% reduced milk yield with 18% and 29% smaller mammary secretory tissue dry matter and crude protein masses, respectively. On d 14, no treatment differences were observed in markers of protein synthesis or mammary cell turnover evaluated using gene transcripts and immunohistochemical staining. These findings implicate mTORC1-S6K1 in the early phase of the adaptation of the mammary gland's capacity for milk synthesis in response to changes in nutrient supply. Additionally, changes in rates of mammary cell turnover may be transient in nature, returning to basal levels following brief alterations that have sustained effects.

Modeling daily yields of milk, fat, protein, and lactose of New Zealand dairy goats undergoing standard and extended lactations

This study aimed to assess the milk production data for New Zealand dairy goats in either a standard lactation (SL; ≤305 d in milk [DIM]) or extended lactation (EL; >305 and ≤670 DIM) using a random regression (RR) with third- and fifth-order Legendre polynomials, respectively. Persistency of EL was defined as (B/A) × 100, where A was the accumulated yield from d 1 to 305, and B was the accumulated yield from d 366 to 670. On average, goats in SL produced 1,183 kg of milk, 37 kg of fat, 37 kg of protein, and 54 kg of lactose. The average production of milk, fat, protein, and lactose in EL were 2,473 kg, 78 kg, 79 kg, and 112 kg, respectively. The average persistences for milk, fat, protein, and lactose yields during EL were 98%, 98%, 102%, and 96%, respectively. The relative prediction errors were close to 10% and the concordance correlation coefficients >0.92, indicating that the RR model with Legendre polynomials is adequate for modeling lactation curves for both SL and EL. Total yields and persistency were analyzed with a mixed model that included the fixed effects (year, month of kidding, parity, and proportion of Saanen) as covariates and the random effects of animal and residual errors. Effects of year, month of kidding, and parity were significant on the total yields of milk, fat, protein, and lactose for both SL and EL. The total milk yield of first-parity goats with SL was 946 kg and the total milk yield of second-parity goats with SL was 1,284 kg, making a total of 2,230 kg over 2 years. The total milk yield of a first-parity goat with EL was 2,140 kg. Thus, on average, a goat with SL for the first and second parity produced 90 kg more milk than a first-parity goat subjected to EL. However, a second-parity goat subjected to EL produced 43 kg more milk (2,639 kg) than a goat with SL following the second and third parity (1,284 kg + 1,312 kg). These data, along with the various other benefits of EL (e.g., fewer offspring born and reduced risk of mastitis, lameness, and metabolic problems in early lactation), indicate that EL as a management strategy holds the potential to improve dairy goat longevity and lifetime efficiency without compromising milk production.

Fitting mathematical functions to extended lactation curves and forecasting late-lactation milk yields of dairy cows

A 305-d lactation followed by a 60-d dry period has traditionally been considered economically optimal, yet dairy cows in modern intensive dairy systems are frequently dried off while still producing significant quantities of milk. Managing cows for an extended lactation has reported production, welfare, and economic benefits, but not all cows are suitable for an extended lactation. Implementation of an extended lactation strategy on-farm could benefit from use of a decision support system, based on a mathematical lactation model, that can identify suitable cows during early lactation that have a high likelihood of producing above a target milk yield (MY) at 305 d in milk (DIM). Therefore, our objectives were (1) to compare the suitability of 3 commonly used lactation models for modeling extended lactations (Dijkstra, Wood, and Wilmink) in primiparous and multiparous cows under a variety of lactation lengths, and (2) to determine the amount of early-lactation daily MY data needed to accurately forecast MY at d 305 by using the most suitable model and determine whether this is sufficient for identifying cows suitable for an extended lactation before the end of a typical voluntary waiting period (50-90 d). Daily MY data from 467 individual Holstein-Friesian lactations (DIM >305 d; 379 ± 65-d lactation length [mean ± SD]) were fitted by the 3 lactation models using a nonlinear regression procedure. The parameter estimates of these models, lactation characteristics (peak yield, time to peak yield, and persistency), and goodness-of-fit were compared between parity and different lactation lengths. The models had similar performance, and differences between parity groups were consistent with previous literature. Then, data from only the first i DIM for each individual lactation, where i was incremented by 30 d from 30 to 150 DIM and by 50 d from 150 to 300 DIM, were fitted by each model to forecast MY at d 305. The Dijkstra model was selected for further analysis, as it had superior goodness-of-fit statistics for i= 30 and 60. The data set was fit twice by the Dijkstra model, with parameter bounds either unconstrained or constrained. The quality of predictions of MY at d 305 improved with increasing data availability for both models and assisting the model fitting procedure with more biologically relevant constraints on parameters improved the predictions, but neither was reliable enough for practical use on-farm due to the high uncertainty of forecasted predictions. Using 90 d of data, the constrained model correctly classified 66% of lactations as being above or below a target MY at d 305 of 25 kg/d, with a probability threshold of 0.95. The proportion of correct classifications became smaller at lower targets of MY at d 305 and became greater when using more lactation days. Overall, further work is required to develop a model that can forecast late-lactation MY with sufficient accuracy for practical use. We envisage that a hybridized machine learning and mechanistic model that incorporates additional historical and genetic information with early-lactation MY could produce meaningful lactation curve forecasts.

Effect of hydrogenated fat-embedded calcium gluconate on lactation performance in dairy cows

Hydrogenated fat-embedded calcium gluconate (HFCG), a prebiotic mixture designed to target the hindgut, has improved milk and component yields when supplemented in mid-lactation cows, likely due to improved hindgut health. The objective of this study was to evaluate production responses to HFCG when fed to lactating dairy cattle over a full lactation. Seventy-four Holstein cows (21 primiparous, 53 multiparous) were used in a randomized complete block design comparing supplementation with either HFCG (approximately 16 g/d of supplement delivering approximately 6.4 g of active ingredient) or a negative control from approximately 21 days prior to calving until the end of lactation. In multiparous cattle supplemented with HFCG, average daily milk protein yield (P = 0.037) was increased during the first 8 weeks of lactation, while average daily yields of milk fat, fat- and energy-corrected milk tended (P ≤ 0.075) to increase over the same period of time. Increased yields were likely supported by the concurrent increase in dry matter intake (P = 0.036). Future work is needed to characterize the mode of action of this product within both the hindgut lumen and host, as well as investigate the potential differential responses between primiparous and multiparous animals over the course of lactation.

Modelling Extended Lactations in Polish Holstein-Friesian Cows

Simple Summary

Mathematical models of lactation curves are functions that describe milk production on each day of lactation. These models are able to predict milk yields as well as provide valuable information applicable in breeding and management decisions. The aim of the present study was to examine different shapes of lactation curves for milk traits (i.e., milk, fat, protein and lactose yields and urea content in milk) modelled by the Wilmink function and by linear or squared functions between 306 and 400 days in milk (DIM). The results suggested that the course of extended lactations could be modelled by a nonlinear model, for example, the Wilmink function, for up to 305 DIM, and the linear or squared function could be more appropriate afterwards.

Abstract

The objectives of this study were (1) to examine different shapes of lactation curves for milk, fat, protein and lactose yields and urea content in milk fitted by the Wilmink function using the test-day (TD) records and (2) to find the function that best describes test-day records beyond 305 days in milk (DIM) for Polish Holstein–Friesian cows. The data were 6,955,768 TD records from the 702,830 first six lactations of 284,193 Polish Holstein–Friesian cows. Cows calved in 19,102 herds between 2001 and 2018. The following functions were fitted to TD data from each lactation: (1) Wilmink model fitted to the whole lactation, (2) Wilmink model fitted to TD records from 5 to 305 DIM and linear function fitted to TD records from 306 to 400 DIM, (3) Wilmink model fitted to TD records from 5 to 305 DIM and squared function fitted to TD records from 306 to 400 DIM. The present study showed that urea content in milk was modelled slightly worse than other milk traits. The results suggested that the course of lactation could be successfully modelled by a nonlinear model, for example, the Wilmink function, for up to 305 DIM, and by the linear or squared function afterwards.

Associations between feed efficiency and aspects of lactation curves in primiparous Holstein dairy cattle

Genetic selection for improved feed efficiency in dairy cattle has received renewed attention over the last decade to address the needs of a growing global population. As milk yield is a critical component of feed efficiency metrics in dairy animals, our objective was to evaluate the associations between feed efficiency in primiparous Holstein cattle and parameters of a mathematical model describing individual lactation curves. The Dijkstra lactation curve model was fit to individual lactation records from 34 Holstein heifers with previously estimated measures of feed efficiency. We found that the optimal fit of the Dijkstra model was achieved using daily milk yield records up to 21 d in milk to capture the rise to peak milk yield and using monthly dairy herd improvement records for the remainder of lactation to accurately characterize lactation persistency. In the period of lactation before peak milk yield, improved feed efficiency was associated with a faster increase in daily milk yield over a shorter period of time at the expense of increased mobilization of body reserves; this serves to reinforce the concept that dairy cattle are primarily capital breeders versus income breeders. Feed efficiency in the period following peak lactation, as measured by gross feed efficiency, return over feed costs, and net energy efficiency of lactation, was positively associated with higher peak milk yield. The findings in early lactation suggest that estimates of feed efficiency could be improved by evaluating feed efficiency relative to conception, rather than parturition and lactation, to better account for the energy stored and released from body reserves in capital breeding.

Milk losses and dynamics during perturbations in dairy cows differ with parity and lactation stage

Milk yield dynamics during perturbations reflect how cows respond to challenges. This study investigated the characteristics of 62,406 perturbations from 16,604 lactation curves of dairy cows milked with an automated milking system at 50 Belgian, Dutch, and English farms. The unperturbed lactation curve representing the theoretical milk yield dynamics was estimated with an iterative procedure fitting a model on the daily milk yield data that was not part of a perturbation. Perturbations were defined as periods of at least 5 d of negative residuals having at least 1 day that the total daily milk production was below 80% of the estimated unperturbed lactation curve. Every perturbation was characterized and split in a development and a recovery phase. Based hereon, we calculated both the characteristics of the perturbation as a whole, and the duration, slopes, and milk losses in the phases separately. A 2-way ANOVA followed by a pairwise comparison of group means was carried out to detect differences between these characteristics in different lactation stages (early, mid-early, mid-late, and late) and parities (first, second, and third or higher). On average, 3.8 ± 1.9 (mean ± standard deviation) perturbations were detected per lactation in the first 305 d after calving, corresponding to an estimated 92.1 ± 135.8 kg of milk loss. Only 1% of the lactations had no perturbations. On average, 2.3 kg of milk was lost per day in the development phase, while the recovery phase corresponded to an average increase in milk production of 1.5 kg/d, and these phases lasted an average of 10.1 and 11.6 d, respectively. Perturbation characteristics were significantly different across parity and lactation stage groups, and early and mid-early perturbations in higher parities were found to be more severe with faster development rates, slower recovery rates, and higher milk losses. The method to characterize perturbations can be used for precision phenotyping purposes that look into the response of cows to challenges or that monitor applications (e.g., to evaluate the development and recovery of diseases and how these are affected by preventive actions or treatments).

Genome-wide association study for mammary structure in Canadian Angus cows

Functional and enduring mammary structure is pivotal for producer profitability, and animal health and welfare in beef production. Genetic evaluations for teat and udder score in Canadian Angus cattle have previously been developed. The aim of this study was to identify genomic regions associated with teat and udder structure in Canadian Angus cows thereby enhancing knowledge of the biological architecture of these traits. Thus, we performed a weighted single-step genome wide association study (WssGWAS) to identify candidate genes for teat and udder score in 1,582 Canadian Angus cows typed with the GeneSeek^® Genomic Profiler Bovine 130K SNP array. Genomically enhanced estimated breeding values (GEBVs) were converted to SNP marker effects using unequal variances for markers to calculate weights for each SNP over three iterations. At the genome wide level, we detected windows of 20 consecutive SNPs that explained more than 0.5% of the variance observed in these traits. A total of 35 and 28 windows were identified for teat and udder score, respectively, with two SNP windows in common for both traits. Using Ensembl, the SNP windows were used to search for candidate genes and quantitative trait loci (QTL). A total of 94 and 71 characterized genes were identified in the regions for teat and udder score, respectively. Of these, 7 genes were common for both traits. Gene network and enrichment analysis, using Ingenuity Pathway Analysis (IPA), signified key pathways unique to each trait. Genes of interest were associated with immune response and wound healing, adipose tissue development and morphology, and epithelial and vascular development and morphology. Genetic architecture from this GWAS confirms that teat and udder score are distinct, polygenic traits involving varying and complex biological pathways, and that genetic selection for improved teat and udder score is possible.

Clustering and Characterization of the Lactation Curves of Dairy Cows Using K-Medoids Clustering Algorithm

Simple Summary

A lactation curve (LC) provides valuable insights in planning appropriate management strategies related to health, nutrition, and breeding in dairy cows. A clustering based approach on LC patterns analysis is presented. The k-medoids algorithm is adopted for the clustering. This approach generates several clusters which have similar milking characteristics of total milk yield, peak milk yield, and days in milk at peak yield. The LCs of some groups represent characteristics of atypical milking patterns which are not considered much in previous approaches, whereas LCs of the other groups show the typical LC patterns similar to the results of previous methods. This approach could be used as a tool to manage an abnormal herd of cows.

Abstract

The aim of the study was to group the lactation curve (LC) of Holstein cows in several clusters based on their milking characteristics and to investigate physiological differences among the clusters. Milking data of 330 lactations which have a milk yield per day during entire lactation period were used. The data were obtained by refinement from 1332 lactations from 724 cows collected from commercial farms. Based on the similarity measures, clustering was performed using the k-medoids algorithm; the number of clusters was determined to be six, following the elbow method. Significant differences on parity, peak milk yield, DIM at peak milk yield, and average and total milk yield (p < 0.01) were observed among the clusters. Four clusters, which include 82% of data, show typical LC patterns. The other two clusters represent atypical patterns. Comparing to the LCs generated from the previous models, Wood, Wilmink and Dijsktra, it is observed that the prediction errors in the atypical patterns of the two clusters are much larger than those of the other four cases of typical patterns. The presented model can be used as a tool to refine characterization on the typical LC patterns, excluding atypical patterns as exceptional cases.

The use of dijkstra mechanistic model for genetic analysis of the lactation curve characteristics and their relationships with age at first calving and somatic cell score of Iranian dairy cows

The objective of this study was to estimate lactation curve parameters with Dijkstra mechanistic model and to evaluate genetic and phenotypic relationships between the parameters and the average somatic cell count in primiparous cows. The finding indicated that heritability estimates for partial milk yield (PMY1, PMY2 and PMY3), total 305-day milk yield (TMY305), decay parameter (λ2), age at first calving (AFC) and peak yield (PY) were moderate while the heritability of persistency (PS%), average somatic cell score (AVGSCS), time to peak yield (TP), initial milk production (λ0), specific rate of cell proliferation at parturition (λ1), and specific rate of cell death (λ3) were quite low. Genetic correlations between both AFC and PS% traits with average somatic cell scores was negative (-0.047 and -0.060) but low positive genetic correlation were between partial milk yields (PMY1 and PMY3) while negative genetic correlation (-0.06) was obtained between TMY305 and AVGSCS. Differences between TMY305 of cows with less than 100000 cells mL-1 and cows with >1,500,000 cells mL-1 was approximately 708 Kg and is equivalent to 8% loss of milk yield/cow during lactation period and also loss of persistency (11.1 %( was shown for the extreme classes of SCC in this study.

Review: Synergy between mechanistic modelling and data-driven models for modern animal production systems in the era of big data

Mechanistic models (MMs) have served as causal pathway analysis and 'decision-support' tools within animal production systems for decades. Such models quantitatively define how a biological system works based on causal relationships and use that cumulative biological knowledge to generate predictions and recommendations (in practice) and generate/evaluate hypotheses (in research). Their limitations revolve around obtaining sufficiently accurate inputs, user training and accuracy/precision of predictions on-farm. The new wave in digitalization technologies may negate some of these challenges. New data-driven (DD) modelling methods such as machine learning (ML) and deep learning (DL) examine patterns in data to produce accurate predictions (forecasting, classification of animals, etc.). The deluge of sensor data and new self-learning modelling techniques may address some of the limitations of traditional MM approaches - access to input data (e.g. sensors) and on-farm calibration. However, most of these new methods lack transparency in the reasoning behind predictions, in contrast to MM that have historically been used to translate knowledge into wisdom. The objective of this paper is to propose means to hybridize these two seemingly divergent methodologies to advance the models we use in animal production systems and support movement towards truly knowledge-based precision agriculture. In order to identify potential niches for models in animal production of the future, a cross-species (dairy, swine and poultry) examination of the current state of the art in MM and new DD methodologies (ML, DL analytics) is undertaken. We hypothesize that there are several ways via which synergy may be achieved to advance both our predictive capabilities and system understanding, being: (1) building and utilizing data streams (e.g. intake, rumination behaviour, rumen sensors, activity sensors, environmental sensors, cameras and near IR) to apply MM in real-time and/or with new resolution and capabilities; (2) hybridization of MM and DD approaches where, for example, a ML framework is augmented by MM-generated parameters or predicted outcomes and (3) hybridization of the MM and DD approaches, where biological bounds are placed on parameters within a MM framework, and the DD system parameterizes the MM for individual animals, farms or other such clusters of data. As animal systems modellers, we should expand our toolbox to explore new DD approaches and big data to find opportunities to increase understanding of biological systems, find new patterns in data and move the field towards intelligent, knowledge-based precision agriculture systems.

Use of Mechanistic Nutrition Models to Identify Sustainable Food Animal Production

To feed people in the coming decades, an increase in sustainable animal food production is required. The efficiency of the global food production system is dependent on the knowledge and improvement of its submodels, such as food animal production. Scientists use statistical models to interpret their data, but models are also used to understand systems and to integrate their components. However, empirical models cannot explain systems. Mechanistic models yield insight into the mechanism and provide guidance regarding the exploration of the system. This review offers an overview of models, from simple empirical to more mechanistic models. We demonstrate their applications to amino acid transport, mass balance, whole-tissue metabolism, digestion and absorption, growth curves, lactation, and nutrient excretion. These mechanistic models need to be integrated into a full model using big data from sensors, which represents a new challenge. Soon, training in quantitative and computer science skills will be required to develop, test, and maintain advanced food system models.

A redefinition of the modeled responses of mammary glands to once-daily milking

Milking cows once daily is a management tool that has been implemented to improve physical and financial results of seasonal pasture-based dairy farms. The Molly cow model integrates physiology and metabolism of dairy cattle; however, milk production during short-term changes in milking frequency (e.g., 1× milking) is not well represented. The model includes a representation of variable rates of cell quiescence and death. However, the rate constants governing cell death and the return of quiescent to active cells are not affected by milking frequency. An empirical assessment of the problem was conducted, and it was hypothesized that changing the current representation of the rate of cell death in response to short-term 1× milking would more accurately represent active and quiescent cells and improve predictions of milk production. An extra senescent cell flux was added to account for cell loss during periods of 1× milking. Additional changes included a gradual decline in the rate of 1× stimulated senescence during 1× milking, and a structural change in cell cycling between active and quiescent cells during and after short-term 1× milking. Data used for parameter estimation were obtained from 5 studies where 1× milking or different feeding strategies were tested. Parameter estimates of cell loss indicated that 1× milking would affect a small proportion of quiescent cells to cause extra cell death. This added cell senescence was influenced by the length of 1× milking such that cell senescence peaked on d 1 of 1× milking and decayed from that point. The new structure in the model includes a variable rate of cell death in response to 1× milking and a gradual rate of return of quiescent cells back to the active pool in response to switching to 2× milking after short-term 1× milking. Root mean square errors, mean bias, and slope bias declined by at least 50% for predictions of energy-corrected milk yield and fat percent. The model showed quantitative agreement with production data from short-term 1× milking. The accuracy of predictions was improved and the error was reduced by implementing modifications in the model in response to changes in milking frequency.

Using quantile regression for fitting lactation curve in dairy cows

The main objective of this study was to compare the performance of different 'nonlinear quantile regression' models evaluated at the τth quantile (0·25, 0·50, and 0·75) of milk production traits and somatic cell score (SCS) in Iranian Holstein dairy cows. Data were collected by the Animal Breeding Center of Iran from 1991 to 2011, comprising 101 051 monthly milk production traits and SCS records of 13 977 cows in 183 herds. Incomplete gamma (Wood), exponential (Wilmink), Dijkstra and polynomial (Ali &amp; Schaeffer) functions were implemented in the quantile regression. Residual mean square, Akaike information criterion and log-likelihood from different models and quantiles indicated that in the same quantile, the best models were Wilmink for milk yield, Dijkstra for fat percentage and Ali &amp; Schaeffer for protein percentage. Over all models the best model fit occurred at quantile 0·50 for milk yield, fat and protein percentage, whereas, for SCS the 0·25th quantile was best. The best model to describe SCS was Dijkstra at quantiles 0·25 and 0·50, and Ali &amp; Schaeffer at quantile 0·75. Wood function had the worst performance amongst all traits. Quantile regression is specifically appropriate for SCS which has a mixed multimodal distribution.

Application of non-linear mathematical models to describe effect of twinning on the lactation curve features in Holstein cows

The objective of this study was to evaluate effect of twinning on the lactation curve characteristics for milk yield (MY), fat (FP) and protein (PP) percentages, fat to protein ratio (FPR) and somatic cell score (SCS) in Holstein cows. The data set consisted of 5,917,677 test day production of 643,625 first lactation cows from 3146 herds in Iran. Calvings were classified into single or twin births. Six non-linear models (Brody, Wood, Sikka, Nelder, Dijkstra and Rook) were fitted to monthly productive records of single or twin calvers. The Rook model provided the best fit of the lactation curve for MY and FP in single and twin calvers and single calvers for SCS and twin calvers for FPR. The Dijkstra model provided the best fit of the lactation curve for PP in single and twin calvers and single calvers for FP. Also, the Wood model provided the best fit of lactation curve for SCS in twin calvers. Twin calvers had greater predicted 200-day and 305-day cumulative milk yield than single calvers. Time to the peak milk yield was observed later for twin calvers (92 days in milk vs. 80 days in milk) with greater peak milk yield (32.16 kg vs. 31.70 kg) compared with single calvers. The results indicated the efficiency of different models for modelling productive performance of single or twin calvers. The application of the association between production and twinning in dairy cows is crucial because present strategies used in the dairy industry are intended to maximize production performance.

Modeling homeorhetic trajectories of milk component yields, body composition and dry-matter intake in dairy cows: Influence of parity, milk production potential and breed

The control of nutrient partitioning is complex and affected by many factors, among them physiological state and production potential. Therefore, the current model aims to provide for dairy cows a dynamic framework to predict a consistent set of reference performance patterns (milk component yields, body composition change, dry-matter intake) sensitive to physiological status across a range of milk production potentials (within and between breeds). Flows and partition of net energy toward maintenance, growth, gestation, body reserves and milk components are described in the model. The structure of the model is characterized by two sub-models, a regulating sub-model of homeorhetic control which sets dynamic partitioning rules along the lactation, and an operating sub-model that translates this into animal performance. The regulating sub-model describes lactation as the result of three driving forces: (1) use of previously acquired resources through mobilization, (2) acquisition of new resources with a priority of partition towards milk and (3) subsequent use of resources towards body reserves gain. The dynamics of these three driving forces were adjusted separately for fat (milk and body), protein (milk and body) and lactose (milk). Milk yield is predicted from lactose and protein yields with an empirical equation developed from literature data. The model predicts desired dry-matter intake as an outcome of net energy requirements for a given dietary net energy content. The parameters controlling milk component yields and body composition changes were calibrated using two data sets in which the diet was the same for all animals. Weekly data from Holstein dairy cows was used to calibrate the model within-breed across milk production potentials. A second data set was used to evaluate the model and to calibrate it for breed differences (Holstein, Danish Red and Jersey) on the mobilization/reconstitution of body composition and on the yield of individual milk components. These calibrations showed that the model framework was able to adequately simulate milk yield, milk component yields, body composition changes and dry-matter intake throughout lactation for primiparous and multiparous cows differing in their production level.

A new standard model for milk yield in dairy cows based on udder physiology at the milking-session level

Milk production in dairy cow udders is a complex and dynamic physiological process that has resisted explanatory modelling thus far. The current standard model, Wood’s model, is empirical in nature, represents yield in daily terms, and was published in 1967. Here, we have developed a dynamic and integrated explanatory model that describes milk yield at the scale of the milking session. Our approach allowed us to formally represent and mathematically relate biological features of known relevance while accounting for stochasticity and conditional elements in the form of explicit hypotheses, which could then be tested and validated using real-life data. Using an explanatory mathematical and biological model to explore a physiological process and pinpoint potential problems (i.e., “problem finding”), it is possible to filter out unimportant variables that can be ignored, retaining only those essential to generating the most realistic model possible. Such modelling efforts are multidisciplinary by necessity. It is also helpful downstream because model results can be compared with observed data, via parameter estimation using maximum likelihood and statistical testing using model residuals. The process in its entirety yields a coherent, robust, and thus repeatable, model.

Modelling lactation curve for milk fat to protein ratio in Iranian buffaloes (Bubalus bubalis) using non-linear mixed models

The aim of this study was to compare seven non-linear mathematical models (Brody, Wood, Dhanoa, Sikka, Nelder, Rook and Dijkstra) to examine their efficiency in describing the lactation curves for milk fat to protein ratio (FPR) in Iranian buffaloes. Data were 43 818 test-day records for FPR from the first three lactations of Iranian buffaloes which were collected on 523 dairy herds in the period from 1996 to 2012 by the Animal Breeding Center of Iran. Each model was fitted to monthly FPR records of buffaloes using the non-linear mixed model procedure (PROC NLMIXED) in SAS and the parameters were estimated. The models were tested for goodness of fit using Akaike's information criterion (AIC), Bayesian information criterion (BIC) and log maximum likelihood (-2 Log L). The Nelder and Sikka mixed models provided the best fit of lactation curve for FPR in the first and second lactations of Iranian buffaloes, respectively. However, Wood, Dhanoa and Sikka mixed models provided the best fit of lactation curve for FPR in the third parity buffaloes. Evaluation of first, second and third lactation features showed that all models, except for Dijkstra model in the third lactation, under-predicted test time at which daily FPR was minimum. On the other hand, minimum FPR was over-predicted by all equations. Evaluation of the different models used in this study indicated that non-linear mixed models were sufficient for fitting test-day FPR records of Iranian buffaloes.

Effect of parity on changes in udder traits, milk yield and composition of West African dwarf sheep during lactation

Twelve West African Dwarf sheep, comprising four ewes in each of the Parities 1, 2 and 3, were used to determine the effect of parity on udder traits during lactation, milk yield and composition. Udder length, udder width, udder circumference, udder volume, teat length, teat width, teat circumference, distance between the teats and teat height from the ground of sheep were measured in centimetres (cm) weekly for 12 weeks of lactation, commencing from Day 4 postpartum. Parity highly influenced (P < 0.05) udder traits, except udder volume (P > 0.05). The udder length, width and circumference were higher (P < 0.05) in Parity-3 ewes than Parity-1 and -2 ewes. A similar trend was shown for teat length, width and circumference. Parity had no effect (P > 0.05) on udder volume, but significantly (P > 0.05) affected distance between the teats and teat height from the ground. With the exception of udder volume, all traits peaked by the 3rd week of lactation, and gradually declined thereafter. Milk yield and milk weight peaked by the 3rd week postpartum and these were higher for ewes in Parity 3. Milk yield and milk weight declined faster in ewes in Parities 1 and 2 than they did in Parity-3 ewes (P < 0.05) as lactation length increased. In regard to milk composition, moisture, protein and fat, but not total solid, solid-not-fat and lactose, were greater (P < 0.05) in Parity-3 ewes both at peak milk yield (3rd week of lactation) and end of lactation. From these findings, we conclude that parity type plays a significant role in influencing udder traits, milk yield and milk quality in ewes.

Appropriate mathematical models for describing the complete lactation of dairy sheep

Despite milk being an important product from sheep, there are very few reports of milk production from the complete lactation of dairy sheep. The Improved Awassi in Israel is kept under an intensive system of management with lambs being weaned soon after birth. Records from one such flock were analysed to investigate the suitability of various mathematical functions for describing milk yield from the complete lactations of dairy sheep. This included a consideration of whether the functions could cope with short lactations, a characteristic of dairy sheep, and a limited number of test-day records per lactation.

Four non-linear mathematical functions were investigated (Wood, Morant, Grossman and Pollott), two of which could also be fitted in a linear and a linear weighted form (Wood and Morant). These functions were fitted to weekly data from a ‘typical Awassi lactation curve’, represented by least squares means of daily milk yield from each week of a 40-week lactation derived from an analysis of 25605 test day records. Characteristics of the lactation were calculated from the functions, such as total milk yield, day and level of peak yield and persistency. These functions were also fitted to 1416 individual lactation records of up to 10 test-day records per lactation. The value of the functions was investigated using the residual mean square (RMS) of the fitted curve as an indicator of how well each function described the lactation. Forms of these functions with a reduced number of parameters were also investigated.

The non-linear functions always fitted the data with a lower RMS than their linear equivalent and the weighted form of the linear functions always had a lower RMS than the unweighted form. Of the linear functions, Morant fitted better than Wood. Of the non-linear functions Grossman, Morant and Pollott (additive and multiplicative) fitted the data equally as well but better than Wood. The various functions predicted characteristics of the lactation curve differently; the Wood functions tended to overestimate yield in early lactation and the Morant functions underestimated peak yield.

No function was better suited to short lactations than another. However the three-parameter function of Morant, Pollott multiplicative and Pollott additive were considered to be the most suitable for describing the complete lactation of dairy sheep.

Fitting mathematical models to lactation curves from Holstein cows in the southwestern region of the state of Parana, Brazil

The objective of this study was to evaluate four mathematical models with regards to their fit to lactation curves of Holstein cows from herds raised in the southwestern region of the state of Parana, Brazil. Initially, 42,281 milk production records from 2005 to 2011 were obtained from "Associação Paranaense de Criadores de Bovinos da Raça Holandesa (APCBRH)". Data lacking dates of drying and total milk production at 305 days of lactation were excluded, resulting in a remaining 15,142 records corresponding to 2,441 Holstein cows. Data were sorted according to the parity order (ranging from one to six), and within each parity order the animals were divided into quartiles (Q25%, Q50%, Q75% and Q100%) corresponding to 305-day lactation yield. Within each parity order, for each quartile, four mathematical models were adjusted, two of which were predominantly empirical (Brody and Wood) whereas the other two presented more mechanistic characteristics (models Dijkstra and Pollott). The quality of fit was evaluated by the corrected Akaike information criterion. The Wood model showed the best fit in almost all evaluated situations and, therefore, may be considered as the most suitable model to describe, at least empirically, the lactation curves of Holstein cows raised in Southwestern Parana.

Comparison of non-linear models to describe the lactation curves for milk yield and composition in buffaloes (Bubalus bubalis)

In order to describe the lactation curves of milk yield (MY) and composition in buffaloes, seven non-linear mathematical equations (Wood, Dhanoa, Sikka, Nelder, Brody, Dijkstra and Rook) were used. Data were 116,117 test-day records for MY, fat (FP) and protein (PP) percentages of milk from the first three lactations of buffaloes which were collected from 893 herds in the period from 1992 to 2012 by the Animal Breeding Center of Iran. Each model was fitted to monthly production records of dairy buffaloes using the NLIN and MODEL procedures in SAS and the parameters were estimated. The models were tested for goodness of fit using adjusted coefficient of determination (Radj(2)), root means square error (RMSE), Durbin-Watson statistic and Akaike's information criterion (AIC). The Dijkstra model provided the best fit of MY and PP of milk for the first three parities of buffaloes due to the lower values of RMSE and AIC than other models. For the first-parity buffaloes, Sikka and Brody models provided the best fit of FP, but for the second- and third-parity buffaloes, Sikka model and Brody equation provided the best fit of lactation curve for FP, respectively. The results of this study showed that the Wood and Dhanoa equations were able to estimate the time to the peak MY more accurately than the other equations. In addition, Nelder and Dijkstra equations were able to estimate the peak time at second and third parities more accurately than other equations, respectively. Brody function provided more accurate predictions of peak MY over the first three parities of buffaloes. There was generally a positive relationship between 305-day MY and persistency measures and also between peak yield and 305-day MY, calculated by different models, within each lactation in the current study. Overall, evaluation of the different equations used in the current study indicated the potential of the non-linear models for fitting monthly productive records of buffaloes.

On the analysis of Canadian Holstein dairy cow lactation curves using standard growth functions

Six classical growth functions (monomolecular, Schumacher, Gompertz, logistic, Richards, and Morgan) were fitted to individual and average (by parity) cumulative milk production curves of Canadian Holstein dairy cows. The data analyzed consisted of approximately 91,000 daily milk yield records corresponding to 122 first, 99 second, and 92 third parity individual lactation curves. The functions were fitted using nonlinear regression procedures, and their performance was assessed using goodness-of-fit statistics (coefficient of determination, residual mean squares, Akaike information criterion, and the correlation and concordance coefficients between observed and adjusted milk yields at several days in milk). Overall, all the growth functions evaluated showed an acceptable fit to the cumulative milk production curves, with the Richards equation ranking first (smallest Akaike information criterion) followed by the Morgan equation. Differences among the functions in their goodness-of-fit were enlarged when fitted to average curves by parity, where the sigmoidal functions with a variable point of inflection (Richards and Morgan) outperformed the other 4 equations. All the functions provided satisfactory predictions of milk yield (calculated from the first derivative of the functions) at different lactation stages, from early to late lactation. The Richards and Morgan equations provided the most accurate estimates of peak yield and total milk production per 305-d lactation, whereas the least accurate estimates were obtained with the logistic equation. In conclusion, classical growth functions (especially sigmoidal functions with a variable point of inflection) proved to be feasible alternatives to fit cumulative milk production curves of dairy cows, resulting in suitable statistical performance and accurate estimates of lactation traits.

A review of the genetic and non-genetic factors affecting extended lactation in pasture-based dairy systems

Milk production per cow has significantly increased over the last 50 years due to the strong genetic selection for increased milk production; associated with this increased production has been a decline in reproductive performance. As a result, superior-yielding cows that have failed to get into calf in a traditional 12-month calving system may be carried over and milked continuously for another 6 months instead of being culled. Studies indicate that cows are able to achieve lactations greater than 305 days and up to 670 days under pasture-based systems, with and without the use of supplementary feeds. Extended lactations of 16 months are most common and economically viable in Australian dairy systems. These findings indicate a potential role for extended lactation in countries such as Australia, where pasture-based dairy systems in which Holstein-Friesian dairy cows predominate. However, variation between cows in their milk yield profiles and the ability of cows to reach a planned dry-off date over an extended lactation occurs depending on the cow’s genetic strain, nutrition and environmental interactions, with certain strains of cow being better suited to extended lactation than others. The focus of this review is to examine the scope for genetic improvement as well as important considerations (non-genetic factors) when selecting suitable animals for extended lactation in pasture-based dairy systems, with an emphasis on Australian systems. These considerations include the impacts of cow strain, nutrition, milk production, and biological and economical costs associated with extended lactation. Methods for modelling extended lactation and estimating genetic parameters of lactation persistency, milk yield and component traits under extended lactation will be addressed and future directions for further research suggested.

Whole-body retention of α-linolenic acid and its apparent conversion to other n-3 PUFA in growing pigs are reduced with the duration of feeding α-linolenic acid

In the present study, fifteen growing pigs were used to determine the whole-body oxidation, retention efficiency (RE) and apparent conversion (AC) of α-linolenic acid (18 : 3n-3) to n-3 highly unsaturated fatty acids (HUFA), including EPA (20 : 5n-3) and DHA (22 : 6n-3). The pigs were fed a diet containing 10 % flaxseed for 30 d. Whole-body fatty acid composition was determined at initial (27·7 (se 1·9) kg), intermediate (day 15; 39·2 (se 1·4) kg) and final (45·7 (se 2·2) kg) body weight. On day 12, four pigs were fed 10 mg/kg of uniformly labelled 13C-18 : 3n-3 (single-bolus dose) to determine the oxidation of 18 : 3n-3. Expired $$CO_{2} $$ samples were collected for 24 h thereafter. The whole-body content of n-3 PUFA increased linearly (P< 0·0001) with time; however, the content of 22 : 6n-3 exhibited a quadratic response (P< 0·01) with a peak occurring at 15 h. As a proportion of intake, the RE of 18 : 3n-3 tended to reduce with time (P= 0·098). The AC of ingested 18 : 3n-3 to the sum of n-3 HUFA was reduced with time (P< 0·05; 12·2 v. 7·53 % for days 0–15 and days 15–30, respectively). The AC of 18 : 3n-3 to 20 : 5n-3 or 22 : 6n-3 was lower than that to 20 : 3n-3, both for days 0–15 (P< 0·05; 1·14 or 1·07 v. 7·06 %) and for days 15–30 (P< 0·05; 1·51 or 0·33 v. 4·29 %). The direct oxidation of 18 : 3n-3 was 7·91 (se 0·98) % and was similar to the calculated disappearance of 18 : 3n-3 between days 0 and 30 (8·81 (se 5·24) %). The oxidation of 18 : 3n-3 was much lower than that reported in other species. The AC of 18 : 3n-3 to n-3 HUFA was reduced over time and that to 20 : 3n-3 in the present study was much higher than that reported in other species and should be explored further.

Effects of seasonal change and parity on raw milk composition and related indices in Chinese Holstein cows in northern China

This study was to investigate the effects of seasonal change and parity on milk composition and related indices, and to analyze the relationships among milk indices in Chinese Holstein cows from an intensive dairy farm in northern China. The 6,520 sets of complete Dairy Herd Improvement data were obtained and grouped by natural month and parity. The data included daily milk yield (DMY), milk solids percentage (MSP), milk fat percentage (MFP), milk protein percentage (MPP), milk lactose percentage (MLP), somatic cell count (SCC), somatic cell score (SCS), milk production loss (MPL), and fat-to-protein ratio (FPR). Data analysis showed that the above 9 indices were affected by both seasonal change and parity. However, the interaction between parity and seasonal change showed effects on MLP, SCS, MPL, and DMY, but no effects on MFP, MPP, MSP, and FPR. Duncan's multiple comparison on seasonal change showed that DMY (23.58 kg/d), MSP (12.35%), MPP (3.02%), and MFP (3.81%) were the lowest in June, but SCC (288.7 × 10(3)/mL) and MPL (0.69 kg/d) were the lowest in January; FPR (1.32) was the highest in February. Meanwhile, Duncan's multiple comparison on parities showed that MSP, MPP, and MLP were reduced rapidly in the fourth lactation, but SCC and MPL increased with increasing parities. The canonical correlation analysis for indices showed that SCS had high positive correlation with MPL (0.8360). Therefore, a few models were developed to quantify the effects of seasonal change and parity on raw milk composition using the Wood model. The changing patterns of milk composition and related indices in different months and parities could provide scientific evidence for improving feeding management and nutritional supplementation of Chinese Holstein cows.

Interpreting experimental data on egg production--applications of dynamic differential equations

This contribution focuses on applying mathematical models based on systems of ordinary first-order differential equations to synthesize and interpret data from egg production experiments. Models based on linear systems of differential equations are contrasted with those based on nonlinear systems. Regression equations arising from analytical solutions to linear compartmental schemes are considered as candidate functions for describing egg production curves, together with aspects of parameter estimation. Extant candidate functions are reviewed, a role for growth functions such as the Gompertz equation suggested, and a function based on a simple new model outlined. Structurally, the new model comprises a single pool with an inflow and an outflow. Compartmental simulation models based on nonlinear systems of differential equations, and thus requiring numerical solution, are next discussed, and aspects of parameter estimation considered. This type of model is illustrated in relation to development and evaluation of a dynamic model of calcium and phosphorus flows in layers. The model consists of 8 state variables representing calcium and phosphorus pools in the crop, stomachs, plasma, and bone. The flow equations are described by Michaelis-Menten or mass action forms. Experiments that measure Ca and P uptake in layers fed different calcium concentrations during shell-forming days are used to evaluate the model. In addition to providing a useful management tool, such a simulation model also provides a means to evaluate feeding strategies aimed at reducing excretion of potential pollutants in poultry manure to the environment.

Comparison of mathematical models applied to f1 dairy sheep lactations in organic farm and environmental factors affecting lactation curve parameter

THE OBJECTIVE OF THIS STUDY WAS TO COMPARE THE GOODNESS OF FIT OF FOUR LACTATION CURVE MODELS: Wood's Gamma model (WD), Wilmink (WL), and Pollott's multiplicative two (POL2) and three parameters (POL3) and to determine the environmental factors affecting the complete lactation curve of F1 dairy sheep under organic management. A total of 5,382 weekly milk yields records from 150 ewes, under organic management were used. Residual mean square (RMS), determination coefficients (R(2)), and correlation (r) analysis were used as an indicator of goodness of fit for each model. WL model best fitted the lactation curves as indicated by the lower RMS values (0.019), followed by WD (0.023), POL2 (0.025) and POL3 (0.029). The four models provided total milk yield (TMY) estimations that were highly correlated (0.93 to 0.97) with observed TMY (89.9 kg). The four models under estimated peak yield (PY), whereas POL2 and POL3 gave nearer peak time lactation estimations. Ewes lambing in autumn had higher TMY and showed a typical curve shape. Higher TMY were recorded in second and third lambing. Season of lambing, number of lambing and type of lambing had a great influenced over TMY shaping the complete lactation curve of F1 dairy sheep. In general terms WL model showed the best fit to the F1 dairy sheep lactation curve under organic management.

Quantifying inter-group variability in lactation curve shape and magnitude with the MilkBot(®) lactation model

Genetic selection programs have driven development of most lactation models, to estimate the magnitude of animals’ productive capacity from sampled milk production data. There has been less attention to management and research applications, where it may also be important to quantify the shape of lactation curves, and predict future daily milk production for incomplete lactations since residuals between predicted and actual daily production can be used to quantify the response to an intervention. A model may decrease the confounding effects of lactation stage, parity, breed, and possibly other factors depending on how the model is constructed and used, thus increasing the power of statistical analyses. Models with a mechanistic derivation may allow direct inference about biology from fitted production data. The MilkBot^® lactation model is derived from abstract suppositions about growth of udder capacity. This permits inference about shape of the lactation curve directly from parameter values, but not direct conclusions about physiology. Individual parameters relate to the overall scale of the lactation, the ramp, or rate of growth around parturition, decay describing the senescence of productive capacity (inversely related to persistence), and the relatively insignificant time offset between calving and the physiological start of milk secretion. A proprietary algorithm was used to fit monthly test data from two parity groups in 21 randomly selected herds, and results displayed in box-and-whisker charts and Z-test tables. Fitted curves are constrained by the MilkBot^® equation to a single peak that blends into an exponential decline in late lactation. This is seen as an abstraction of productive capacity, with actual daily production higher or lower due to random error plus short-term environmental effects. The four MilkBot^® parameters, and metrics calculated directly from them including fitting error, peak milk and cumulative production, can be used to describe and compare individual lactations or groups of lactations. There is considerable intra-herd and inter-herd variability in scale, ramp, decay, RMSE, peak milk, and cumulative production, suggesting that management and environment have significant influence on both shape and magnitude of normal lactation curves.

A teleonomic model describing performance (body, milk and intake) during growth and over repeated reproductive cycles throughout the lifespan of dairy cattle. 1. Trajectories of life function priorities and genetic scaling

The prediction of the control of nutrient partitioning, particularly energy, is a major issue in modelling dairy cattle performance. The proportions of energy channelled to physiological functions (growth, maintenance, gestation and lactation) change as the animal ages and reproduces, and according to its genotype and nutritional environment. This is the first of two papers describing a teleonomic model of individual performance during growth and over repeated reproductive cycles throughout the lifespan of dairy cattle. The conceptual framework is based on the coupling of a regulating sub-model providing teleonomic drives to govern the work of an operating sub-model scaled with genetic parameters. The regulating sub-model describes the dynamic partitioning of a mammal female's priority between life functions targeted to growth (G), ageing (A), balance of body reserves (R) and nutrient supply of the unborn (U), newborn (N) and suckling (S) calf. The so-called GARUNS dynamic pattern defines a trajectory of relative priorities, goal directed towards the survival of the individual for the continuation of the specie. The operating sub-model describes changes in body weight (BW) and composition, foetal growth, milk yield and composition and food intake in dairy cows throughout their lifespan, that is, during growth, over successive reproductive cycles and through ageing. This dynamic pattern of performance defines a reference trajectory of a cow under normal husbandry conditions and feed regimen. Genetic parameters are incorporated in the model to scale individual performance and simulate differences within and between breeds. The model was calibrated for dairy cows with literature data. The model was evaluated by comparison with simulations of previously published empirical equations of BW, body condition score, milk yield and composition and feed intake. This evaluation showed that the model adequately simulates these production variables throughout the lifespan, and across a range of dairy cattle genotypes.

Analysis of lactation shapes in extended lactations

In order to describe the temporal evolution of milk yield (MY) and composition in extended lactations, 21 658 lactations of Italian Holstein cows were analyzed. Six empirical mathematical models currently used to fit 305 standard lactations (Wood, Wilmink, Legendre, Ali and Schaeffer, quadratic and cubic splines) and one function developed specifically for extended lactations (a modification of the Dijkstra model) were tested to identify a suitable function for describing patterns until 1000 days in milk (DIM). Comparison was performed on individual patterns and on average curves grouped according to parity (primiparous and multiparous) and lactation length (standard ≤305 days, and extended from 600 to 1000 days). For average patterns, polynomial models showed better fitting performances when compared with the three or four parameters models. However, LEG and spline regression, showed poor prediction ability at the extremes of the lactation trajectory. The Ali and Schaeffer polynomial and Dijkstra function were effective in modelling average curves for MY and protein percentage, whereas a reduced fitting ability was observed for fat percentage and somatic cell score. When individual patterns were fitted, polynomial models outperformed nonlinear functions. No detectable differences were observed between standard and extended patterns in the initial phase of lactation, with similar values of peak production and time at peak. A considerable difference in persistency was observed between 200 and 305 DIM. Such a difference resulted in an estimated difference between standard and extended cycle of about 7 and 9 kg/day for daily yield at 305 DIM and of 463 and 677 kg of cumulated milk production at 305 DIM for the first- and second-parity groups, respectively. For first and later lactation animals, peak yield estimates were nearly 31 and 38 kg, respectively, and occurred at around 65 and 40 days. The asymptotic level of production was around 9 kg for multiparous cows, whereas the estimate was negative for first parity.