Review of equations to predict methane emissions in dairy cows from milk fatty acid profiles and their application to commercial dairy farms

Greenhouse gas emission from the activities of all productive sectors is currently a topic of foremost importance. The major contributors in the livestock sector are ruminants, especially dairy cows. This study aimed to evaluate and compare 21 equations for predicting enteric methane emissions (EME) developed on the basis of milk traits and fatty acid profiles, which were selected from 46 retrieved through a literature review. We compiled a reference database of the detailed fatty acid profiles, determined by GC, of 992 lactating cows from 85 herds under 4 different dairy management systems. The cows were classified according to DIM, parity order, and dairy system. This database was the basis on which we estimated EME using the selected equations. The EME traits estimated were methane yield (20.63 ± 2.26 g/kg DMI, 7 equations), methane intensity (16.05 ± 2.76 g/kg of corrected milk, 4 equations), and daily methane production (385.4 ± 68.2 g/d, 10 equations). Methane production was also indirectly calculated by multiplying the daily corrected milk yield by the methane intensity (416.6 ± 134.7 g/d, 4 equations). We also tested for the effects of DIM, parity, and dairy system (as a correction factor) on the estimates. In general, we observed little consistency among the EME estimates obtained from the different equations, with exception of those obtained from meta-analyses of a range of data from different research centers. We found all the EME predictions to be highly affected by the sources of variation included in the statistical model: DIM significantly affected the results of 19 of the 21 equations, and parity order influenced the results of 13. Different patterns were observed for different equations with only some of them in accordance with expectations based on the cow’s physiology. Finally, the best predictions of daily methane production were obtained when a measure of milk yield was included in the equation or when the estimate was indirectly calculated from daily milk yield and methane intensity.


INTRODUCTION
Climate change is currently a very important and highly controversial topic.All productive sectors-whether agriculture, manufacturing, or services-are emitters of greenhouse gases, with livestock contributing significantly to emissions from agriculture (Knapp et al., 2014;IPCC, 2014;Moumen et al., 2016;Negussie et al., 2017;Berton et al., 2020).Dairy cattle, in particular, are among the greatest emitters of methane (CH 4 ), accounting for 30% of emissions within the agricultural sector (Tseten et al., 2022;FAO, 2023).Society demands that CH 4 emissions be reduced to mitigate environmental damage and increase feed efficiency (Vanlierde et al., 2016;Denninger et al., 2020).However, if this is to be achieved, methods and procedures to measure or predict CH 4 emission at the farm level are needed (Boadi et al., 2004;Zhao et al., 2020).
The gold standard method for directly measuring enteric methane emissions (EME) in vivo is the respiratory chamber (RC), but it is costly, time-consuming, and requires specialized equipment.Furthermore, the environmental conditions of cows in an RC are very different from those of commercial farms.Some other in vivo methods have been developed, such as the sulfur hexafluoride (SF 6 ) tracer, the GreenFeed system (GF), and the sniffer technique (Garnsworthy et al., 2019;Zhao et al., 2020), but they have certain economic, technical, and reproducibility concerns (Bovenhuis et al., 2018;Huhtanen and Hristov, 2018).The first method quantifies the CH 4 emitted in terms of its ratio with the tracer gas released at a known fixed rate.Some low-cost, indirect methods for predicting EME have also been developed and used on farms (Ellis et al., 2010;Bougouin et al., 2019).One of these consists of prediction equations based on various animal traits, such as feed or forage intake, BW, rumen parameters, or milk fatty acids (FA; Chilliard et al., 2009;Mohammed et al., 2011;Rico et al., 2016;Castro-Montoya et al., 2017;Engelke et al., 2018).These equations are often difficult to apply in the field as we cannot know each animal's exact intake as they are not weighed every day.Several studies have focused on the relationship between CH 4 emissions and the milk FA profiles of dairy cows.Methane mainly originates in the rumen as a result of the fermentation of feed components by rumen microbes.Methane formation in the rumen is related to the formation of specific FA that are transferred to the milk and that can be used as tracers for CH 4 production (Dijkstra et al., 2011).Equations were therefore developed using specific FA present in the milk (Chilliard et al., 2009;Dijkstra et al., 2011;Mohammed et al., 2011;van Gastelen et al., 2017).These studies have generally been single trials in which a group of cows fed one or more diets were tested for EME, often in a RC, and the traits were regressed on combinations of the FA in their milk analyzed by GC (Dijkstra et al., 2011;van Engelen et al., 2015;van Gastelen et al., 2017van Gastelen et al., , 2018a,b),b).The large majority of these trials were carried out in one or just a few experimental stations, on multiparous, mid-lactation Holstein cows.Moreover, the studies that also validated the prediction equations developed, compiled both the calibration and validation subsets from data, often randomly selected, from the same experimental dataset.Only 2 other studies were meta-analyses of some of those trials (van Lingen et al., 2014;Bougouin et al., 2019).
At the population level, however, EME is also affected by dairy system, individual farms within dairy system, and by animal characteristics, such as breed, parity, and lactation stage (Martínez-Marín et al., 2023).A few published equations have been applied and tested at the population level (Bittante et al., 2018) or used on FA profiles of dairy products other than milk (Bittante and Bergamaschi, 2019), but their ability to capture the effects of different sources of variation has not been greatly explored.
The present study aimed to evaluate and compare the EME estimates resulting from application of the literature equations to a database of milk FA profiles of a population of lactating cows from a range of farms, and of different parities and lactation stages; we also aimed to evaluate the ability of these equations to capture the effects of these factors (lactation stage, parity order, and farming system) on the predicted EME.

Experimental Design: Farming Systems, Herds, and Dairy Cows
For this study we retrieved from the scientific literature (through the Scopus key words "milk," "fatty acid," and "methane estimation") articles reporting equations for predicting the EME of dairy cows from the FA profiles of their milk.These equations were tested on a large database of FA profiles of milk samples obtained from a large population of cows from many commercial dairy herds operating different farming systems.
The FA database used for this study was compiled as part of the Cowability-Cowplus project.Details of the farming systems and the animals selected are reported in Bittante et al. (2018).Briefly, 85 herds were sampled in Trento province, northern Italy, of which 29 herds were classified as a "traditional" dairy farming system with tied cows and forages and concentrates administered separately.The other 56 herds were classified as a "modern" dairy farming system with freestall housing, and were subdivided into 3 different categories according to diet: the first category comprised modern farms not using TMR (30 herds, "no TMR"); the second category comprised modern farms using TMR with silages included (9 herds, "TMR with silage"); the last category comprised all of the modern farms using TMR with the addition of water but without silage (17 herds, "TMR with water").
Milk samples were collected once from 992 lactating cows, a maximum of 15 cows per herd, selected for good health and representing different parities and lactation stages.The samples were taken during the evening milking, refrigerated at +4°C, and transported to the laboratory within 20 h for analysis.A 50-mL subsample was taken from each sample for analysis of milk composition (fat, protein, lactose) in the Milk Quality Laboratory of the Breeders' Federation of Trento Province.Milk composition was determined using a Milkoscan FT6000 (Foss Electric A/S, Hillerød, Denmark).Another 50-mL subsample was used to determine the milk FA profiles by GC.The individual daily milk yield (dMY) of the cows of each farm were obtained from the national milk recording system database from daily milk measurements performed monthly.

Milk FA Analysis
The milk FA content was measured using the procedure proposed by Feng et al. (2004), as reported by Bittante et al. (2018) and more in detail by Pegolo et al. (2016).The fat was extracted from the milk and the FA methyl esters were obtained by an alkali-catalyzed transmethylation procedure.The FA profile was determined with the ThermoQuest GC instrument (Thermo Electron Corp., Waltham, MA) fitted with a flame-ionization detector and a high polar fused-silica capillary column (Chrompack CP-Sil88 Varian, Middelburg, the Netherlands).The individual FA were labeled according to the pure standard injection (52-Component FAME Mix, GLC-674; Nu-Chek Prep Inc., Elysian, MN).The values of individual FA were expressed as grams per 100 g of total FA content.

Identification and Selection of Methane Prediction Equations
The first literature search yielded 46 equations (Appendix Table A1), which were classified according to EME trait.Nine equations predicted methane yield (CH 4 /DMI, g/kg; i.e., grams of EME per kilogram of DMI); 7 equations predicted methane intensity (CH 4 /CM, g/kg; i.e., grams of EME per kilogram of fat-and protein-corrected milk [CM]); and 30 equations directly predicted methane production (dCH 4 , g/d; i.e., grams of EME per cow and day of lactation).We grouped the equations according to the independent factors that were used: the 9 equations predicting methane yield were based only on milk FA; 6 of the 7 equations predicting methane intensity were based only on milk FA, while one additionally included milk composition (fat, protein, and lactose); of the 30 equations directly predicting methane production, 14 were based only on the milk FA profile, other 14 on DMI and diet composition, and the other 2 on data from the rumen liquor (Appendix Table A1).These 2 last groups were not included in our comparisons due to the difficulty of obtaining dietary and rumen data for individual cows.
The remaining 21 equations were applied to all 992 cows in our database to obtain EME estimates for each of them.Additional estimates of (indirect) dCH 4 (g/d) were obtained by multiplying the estimates of the individual CH 4 /CM (g/kg), obtained using the above-described equations, by the corresponding individual daily corrected milk yields (dCMY, kg/d).To obtain the dCMY, a correction for milk fat and milk protein content was calculated and was then multiplied by the dMY (CVB, 2008): dCMY (kg/d) = [0.337+ 0.116 × milk fat (%) + 0.06 × milk protein (%)] × dMY.

Statistical Analysis
The statistical analysis was performed using the R software v.4.1.1 (R Core Team, 2021).A regression analysis was carried out between pairs of EME estimates obtained from the selected equations, and the coefficient of determination (R 2 ) of these regressions was used to evaluate the consistency of the equations (Appendix Table A2).We assumed consistency between the equations originated from different papers and dataset when the R 2 was greater than 0.50.
We carried out an ANOVA of the EME estimates obtained from each equation using the following linear mixed model: where y ijklm is the predicted EME; μ is the overall mean; dairy system i is the fixed effect of the ith dairy system (i = 1 to 4); DIM j is the fixed effect of the jth 30-d class of DIM (j = 11 classes; from 5 to 330 DIM); parity k is the fixed effect of the kth parity (k = 1 to 4 or more lactations); herd l (dairy system i ) is the random effect of the lth herd nested in the ith dairy system (l = 1 to 85); and e ijklm is the residual random error term.Herd and residuals were assumed to be normally distributed with a mean equal to zero, and variances of σ herd 2 and σ e 2 , respectively.Polynomial contrasts were run to evaluate the existence of linear, cubic, or quadratic patterns for DIM and parity classes.

Screening and Selection of the Literature Equations
All the equations for estimating EME that we retrieved from the literature are given in Appendix Table A1.The equations are ordered according to the trait of the estimated EME (methane yield, methane intensity, and daily methane production) and the sources of information.
Only 21 equations based on milk FA and milk traits were retained for further application.The major features of the studies from which these equations originated are given in Table 1, and the functional forms are reported in Table 2.In some cases, the research was based on a  Bougouin et al. (2019) were the only ones based on meta-analyses, the latter on a greater number of observations (825) than the other.With the sole exception of Engelke et al. (2018), who predicted the FA by mid-infrared spectroscopy (MIRs), the milk FA profile was determined by GC.In a later study, these authors also determined the FA profiles by GC (Engelke et al., 2019).In the majority of the studies examined, EME was measured in an RC.In addition to RC measurements, Bougouin et al. (2019) included EME measurements from SF 6 and GF in their meta-analysis, and Williams et al. (2014) included SF 6 .
All of the equations applied to the dataset were assigned an alphanumerical ID (a letter for the paper, a number for the number of equations developed in a given paper).For methane yield, all the equations were based only on milk FA; Bougouin et al. (2019) equation H2 for methane intensity included milk fat, protein, and lactose as independent variables in addition to FA.For direct prediction of daily methane production, Engelke et al. (2018) equation E2 included a correction for ECM yield.Similarly, Bougouin et al. (2019) included dMY in equa-tion H4, and dMY and milk fat content as independent factors in their equation H5.

Milk, FA Profiles, and EME Estimates
The constituents and FA compositions of the milk samples in our dataset are given in Table 3.The FA composition reflects the milk composition of a normal dairy population, with an average 4.19% fat (CV: 0.17), 3.64% protein (CV: 0.11), and 4.87% lactose (CV: 0.04; CV data not shown).The average SFA content was 72% (CV: 0.06), MUFA 25% (CV: 0.15), and PUFA 4% (CV: 0.22).Within the FA category, there was a wide variation in the CV of the various FA, averaging 0.45 for SFA, 0.35 for MUFA, and 0.46 for PUFA.
Various FA were included in the equations one or more times with either a positive or a negative sign.A positive sign indicates the factor potentially increases EME, and a negative sign potentially decreases it.Saturated FA were used 42 times in the equations (32 times with a positive sign, 10 with a negative sign); MUFA were used 51 times (8 times with a positive sign, 43 with a negative sign), and PUFA 22 times (7 times with a positive sign, 15 with a negative sign).The FA occurring most frequently in the equations with positive signs were C16:0 iso (5 times), C15:0 iso (4 times), and C18:0 (3 times).Those occurring most frequently with a negative sign were C18:1 c11+t15 (11 times), C18:1 t11 (10 times), C18:1 t10 (9 times), C18: 3n -3 (5 times), C4:0 (4 times), and C18:2 t11,c15 (4 times).
The EME estimates yielded by the different equations are reported in Table 4.The mean estimated values for methane yield ranged from 19 to 23 g/kg DMI, the CV  b The experiments come from different research groups.
2 Six experiments used the sulfur hexafluoride (SF 6 ) tracer technique and 3 used RC to measure the methane. 3The van Engelen et al. ( 2015) study is based on the database of Dijkstra et al. (2011).
5 Some studies used the SF 6 tracer technique, others used the GreedFeed method (GF), and some used RC.The letter refers to a given publication (author and year).The number identifies a single equation. 4 Methane yield = CH 4 /DMI (g/kg of DMI).

5
The equation was not used because in the present study C18:1 c13 was not detected. 6 The equation was not used because in the present study C7:0 and C16:1 c7 were not detected. 7 Methane intensity = CH 4 /CM (g/kg of fat-and protein-corrected milk). 8 The expression unit of the equation is grams of CH 4 /kilogram of milk.9 The equation was not used because in the present study C7:0, C18:1 c13, and C22: 6n -3 were not detected. 10 The equation was not used because in the present study C20:1 c11 was not detected. 11 The equation was not used because in the present study C18:1 c15 was not detected.
12 Daily methane production = dCH 4 (g/d). 13 The equation was not used because in the present study CLA t,t was not detected. 14 The equation was not used because in the present study C14:1 c11 and C18:2 t8,c13 were not detected. 15 The equation was not used because in the present study C16:1 c13, C16:1 t14, and C18:1 t12 were not detected. 16 The equations are expressed as liters per day, but they will be converted to grams per day with the following formula: (value/24) × 16. 17 The equation was not used because in the present study C20: 4n -3 was not detected.

Consistency Among the Equations
The degree of correlation between the estimates obtained with the different equations are given in Appendix Table A2.Consistency with respect to methane yield was considered relevant (R 2 > 0.50) only among the estimates yielded by equations A1 and H1 (R 2 : 0.88, Figure 1), both of which were retrieved from meta-analyses.No relevant consistency was found among the equations for methane intensity.There was moderate consistency among the estimates of direct daily methane production yielded between H3 versus C2, H3 versus C3, and E2 versus H5 (R 2 : 0.52, 0.62, and 0.55, Figure 2A, 2B and 2C, respectively).Indirect EME estimates, on the other hand, exhibited greater consistency with R 2 ranging from 0.59 to 0.68 (Figure 3), due to the fact that all of them used the same dCMY values.Finally, there was some consistency (R 2 from 0.51 to 0.63) between some direct and indirect EME estimates, as shown in Figure 4.

Effect of Parity and DIM on the Estimated Methane Emissions
In general, parity had little influence on the EME estimates obtained with the various equations (Figure 5), and only influenced methane yields (Figure 5A) estimated by equations B1, B2, and D1 (between P < 0.01 and P < 0.001).The first 2 equations suggested a linear increase in methane yield with increasing parities, whereas the third suggested a quadratic pattern.The estimates of methane intensity (Figure 5B) were also only weakly influenced by parity, although equations A2 and H2 suggested a positive, linear increase with parity order (P < 0.05).The estimates of direct daily methane production (Figure 5C) frequently suggested that parity order had a positive linear or quadratic effect (with P < 0.05 or P < 0.001).All the equations estimating indirect methane production (Figure 5D) suggested a potential increase in 8 All of the FA with this apex are FA that could not be separated at the moment of the integrations: C18:1 t6+t8 is the sum of C18:1 t6 and C18:1 t8; C18:1 c11+t15 is the sum of C18:1 c11 and C18:1 t15. 9 ΣMUFA = sum of monounsaturated fatty acids.

Table 3 (Continued).
Descriptive statistics for milk yield, corrected milk yield (MY), major milk constituents, and milk fatty acids (FA; g/100 g FA) of the database used in the current study (n = 992 individual records, 85 herds, 4 dairy systems) 1 methane production with increasing parity order with a quadratic relationship (P < 0.001).
The methane yield estimates were significantly influenced by stage of lactation (P < 0.001), except those yielded by equations F1 and G1 (Figure 6A).In general, methane yield increased with increasing DIM with a cubic pattern with a plateau in the middle.In contrast, the estimates obtained with equation D1 showed a linear decrease with increasing DIM.Methane intensity (Figure 6B) was positively influenced by DIM (P < 0.001) with a cubic or quadratic pattern with all the equations.Similarly, direct EME (Figure 6C) were significantly influenced by DIM with a linear, quadratic, or cubic pattern with all the equations.Only in the case of equation E2 did the estimates suggest that CH 4 production may decrease linearly with increasing DIM.Thus, almost all of the equations suggest a potential increase in CH 4 production 1 P1 = 1st percentile. 2 P99 = 99th percentile.
6 Daily methane production = dCH 4 (g/d). 7The equations are expressed as liters per day, but they will be converted to grams per day with the following formula: (value/24) × 16. 8 Calculated methane production per day from methane intensity × daily corrected milk yield (dCMY).Unit of measure: grams per day.
Figure 1.Linear regression of predicted methane yield (g/kg DMI) equations with R 2 greater than 0.50, excluding equations derived from the same study.References: Bougouin et al., 2019;van Lingen et al., 2014. up to a peak toward the fourth month of lactation, with a subsequent decrease.A similar pattern was observed with 2 of the 4 equations (A2, D2) yielding indirect methane production estimates (Figure 6D), while of the other 2 equations, G2 yielded a descending quadratic pattern, and equation H2 showed an opposite trend with a decrease in EME production when the other equations reported a peak in CH 4 production, and an increase in EME in the latter part of lactation.

Informative FA
In the literature equations, many milk FA were considered as independent variables, some of them with a positive sign and hence associated with the various forms of estimated EME.This suggests that some fermentation processes carried out by microbial populations in the rumen are associated with CH 4 formation (Lyons et al., 2018;Beauchemin et al., 2020;Pitta et al., 2022), which means that these FA can be used as tracers (Di-jkstra et al., 2011).As previously reported by Bittante et al. (2018) andCastro-Montoya et al. (2017), the milk FA generated by de novo synthesis (short-and mediumchain SFA) reflect the availability of acetyl-CoA, which is generated from acetate produced in the rumen.Acetate production is associated with that of hydrogen, which is used by methanogenic bacteria populations, such as the Archaea, to form CH 4 (Bauman et al., 2006).This explains the positive correlation found between de novo FA and CH 4 emissions (Chilliard et al., 2009;van Lingen et al., 2014).It is worth noting that other researchers found C4:0 and the sum of FA with 18 carbons to have no predictive ability in dairy cows fed forage-based diets (Williams et al., 2014).In some equations, furthermore, some branched-chain FA, mainly C16:0 iso and C15:0 iso, were found to be positively associated with EME.These branched-chain FA are produced in the rumen by the degradation activity of cellulolytic bacteria, which are major producers of EME.These branched-chain FA can be considered tracers of the presence of forages in the ration, and of the associated EME in the rumen (van  Bougouin et al., 2019;Engelke et al., 2018;Williams et al., 2014. Lingen et al., 2014;Bougouin et al., 2019;Palombo et al., 2020;Giannuzzi et al., 2022).
Other FA were included with a negative sign, those most frequently included being C18:1 t10, C18:1 t11, 18:1 c11+t15, C18: 3n -3, and C18:2 t11,c15.Many previous studies found a negative correlation between EME and long-chain UFA (Chilliard et al., 2009;Dijkstra et al., 2011;Vanrobays et al., 2016).In addition, long-chain UFA inhibit rumen fermentation of fiber, which is the major source of acetate and hydrogen production.The presence of UFA in the rumen also increases the rate of UFA hydrogenation, which makes less hydrogen available for CH 4 formation (Yang et al., 2022).Furthermore, a greater proportion of long-chain FA in the milk fat is related to a lower content of the short-to medium-chain de novo FA that are positively related to CH 4 formation (van Gastelen et al., 2017).This higher proportion of long-chain UFA in the milk fat is frequently found in the  first stage of lactation when the cow is in negative energy balance (NEB) and is forced to mobilize these FA from adipose fatty depots (Vanrobays et al., 2016).
Enteric methane emissions prediction based on milk FA is dependent on reliable detection of specific FA.These informative FA are frequently the minor ones produced in the rumen by microorganisms and are difficult to separate and quantify with current techniques.The analytical method used to analyze the milk FA profile is therefore a critical factor potentially limiting the range of applications of the equations.The gold standard analysis for milk FA profiling is GC, which is known for its high level of accuracy, even with regard to minor FA present in low concentrations (Maurice- Van Eijndhoven et al., 2013).However, it is too expensive and time-consuming for routine use on dairy farms.A novel technique that could also be used to determine milk FA profiles is MIRs (Engelke et al., 2018), a cheap, rapid, nondestructive technique based on multiparametric analysis, that is currently used in functional control systems for lactating cows for the genetic improvement of the dairy populations (Maurice- Van Eijndhoven et al., 2013;van Gastelen and Dijkstra, 2016;Manuelian et al., 2019;Samková et al., 2020).However, this method should be used with caution because it has low predictive ability for minor FA, particularly those in the region of UFA with 18 carbon chains (Maurice- Van Eijndhoven et al., 2013).Care should be also taken to develop and update the calibration settings (Samková et al., 2020).
Many minor FA in our database of milk samples were not identified and quantified, despite the use of GC techniques.In some cases, the FA were co-eluted, making it impossible to determine their precise proportions in the milk fat.In other cases, the lack of reference standard materials for pure FA made it impossible to identify the minor peaks.The absence of some minor FA meant that many equations had to be excluded.

Literature Equation for Estimating Methane Emissions
Successful implementation of strategies to reduce EME from ruminants must be supported by appropriate methods to measure or predict EME at the farm or the individual level.A promising tool for predicting EME could be based on the presence in the milk fat of specific FA reflecting the activity of the microbial population related to CH 4 formation in the rumen (Dijkstra et al., 2011;van Gastelen and Dijkstra, 2016).Several equations to predict EME on the basis of information to be gathered on the farm have been proposed in the scientific literature.The majority of these were developed from individual trials, and therefore reflect variations related to specific experimental conditions.We found little consistency among the estimates obtained from these equations when we compared them using FA profiles obtained from a large survey of several commercial herds operating different dairy farming systems, as is usually the case in practice.2019) (H1, H2, H3, H4, H5).For example, equations A1 and H1 gave estimates of methane yield that were strongly correlated (R 2 : 0.88), with a slope close to unity (0.993) and an intercept close to zero (−0.83 g/kg DMI).
Several equations produced estimates of methane yield.The estimates obtained with our database suggested that methane yield was on average 20.63 ± 2.26 g/ kg DMI, in very good agreement with the figure obtained by Marumo et al. (2023) from their meta-analysis (20.7 ± 2.64 g/kg DMI for lactating cows), and also similar to that obtained by the Jonker et al. ( 2020) meta-analysis (21.62 g/kg DMI).Daily methane production can be predicted as methane yield × DMI.However, information on DMI at the individual level is not generally available from commercial farms, although it is more frequently available in research centers.Unfortunately, few equations for estimating methane intensity can be found in the literature, even though the greenhouse gas emission per unit of product is the most effective way of expressing the environmental burden of a production chain, and also of carrying out genetic evaluation of the environmental impact of animals.In general, there was little consistency among the estimates of methane intensity yielded by the few equations available.Applying the different equations to our database, methane intensity averaged 16.05 ± 2.76 g/kg CM, which was in good agreement with the Marumo et al. (2023) figure for lactating cows (16.1 ± 2.83 g/kg MY) with an average MY comparable to that of the current work (25.6 kg/d).Methane intensity can be used to compute daily methane production when dMY is known.Unlike DMI, dMY is frequently known, even at the individual level on commercial farms, especially where the cows are under milk recording for genetic improvement purposes (Bittante and Cecchinato, 2020;Brito et al., 2021).
Many more equations for predicting daily methane production are available in the literature.When applied to our database of FA profiles these equations yielded an average daily methane production of 385.4 ± 68.The ratios CH 4 /DMI and CH 4 /CM are between the quantity of dCH 4 produced and, respectively, the quantity of DM consumed and the CM (which is related to the DM consumed).They can therefore be considered qualitative traits related to the type of rumen fermentation and consequently to the FA profile of the milk.In contrast, dCH 4 is a trait that is primarily dependent on the quantity of feed consumed (Hristov et al., 2018), which in turn depends on the cow's size and dMY and should therefore be considered a quantitative trait.The FA profile is a qualitative trait that reflects the type of fermentation in the rumen much more than the volume of rumen content and the flow of nutrients.This means that dCH 4 should be more accurately predicted by combining qualitative information (FA profile) with quantitative information, such as DMI (generally not available) or dMY.
The moderate degree of consistency observed among the results obtained from application of the different equations for predicting dCH 4 can therefore be explained by the fact that almost all of them are based only on FA profiles.Equations that do not consider quantitative information, such as dMY or DMI, may yield biased estimates.In fact, there was greater consistency among the estimates obtained from the equations developed by Bougouin et al. (2019) (H4 and H5) and Engelke et al.
We also took an alternative approach to indirectly calculating the cows' dCH 4 ; that is, we multiplied the qualitative trait methane intensity (CH 4 /CM) by the quantitative trait dCMY, as described in a previous study (Bittante and Cipolat-Gotet, 2018).
As expected, we found greater consistency among the EME estimates when dCH 4 was estimated as CH 4 /CM × dCMY, partly because all the estimates were multiplied by the cow's dCMY on the day of milk sampling.Thus, quantitative information inferred from the database was combined with qualitative information based on the FA profiles required by the different equations.In these cases, the estimates suggest that, across equations, daily methane production is in the order of 416.6 ± 134.7 g/d, a little higher than the average value obtained from the direct estimates previously examined.This would suggest that dCMY is more closely associated with CH 4 production than the milk FA profile.However, from a theoretical point of view equations based on the product of methane intensity × dCMY are biased as they do not take into account the CH 4 produced from the feed needed to satisfy the lactating cow's maintenance requirements.An improvement would be to use a factorial model, similar to that used to describe energy requirements, based on one factor that describes the impact of maintenance and another that describes the impact of the productivity level (NASEM, 2021).

Application of the Literature Equations to Data from Commercial Dairy Farms, and the Effects of Parity and Stage of Lactation
Statistical analysis of the effects of parity and stage of lactation on FA-predicted EME traits in a large test population from a range of commercial herds could be a means of assessing the reliability of the proposed equations.In fact, the LSM and their trend on predicted values can be compared with expectations of EME based on current knowledge of the physiology and metabolism of lactating cows.
As previously reported by Bittante et al. (2018), the effect of parity is mainly quantitative in nature, with dMY and DMI increasing with increasing parity order, particularly between the first and second parities.However, the diets of cows of different parities often have the same composition, so the rumen fermentation pattern is expected to also be similar.We expect, therefore, that CH 4 / DMI and CH 4 /CM will not be greatly affected by parity, whereas the expected pattern for dCH 4 is a quadratic increase from primiparous to secundiparous to multiparous cows.These expectations are confirmed by the majority of the equations tested.As can be seen from Figure 5, the cow's parity has little or no effect on the equations predicting CH 4 /DMI (Figure 5A) and CH 4 /CM (Figure 5B).In contrast, the majority of the equations predicting dCH 4 (Figure 5C) exhibited stronger linear and quadratic variations in the EME traits with increasing parity.In particular, the equations estimating dCH 4 as CH 4 /CM × dCMY (Figure 5D) all exhibited the increasing quadratic pattern expected for the variation in dCMY (Figure 5).Similar results emerged from our previous surveys at the population level (Pegolo et al., 2016).
Expectations for the variation in EME throughout lactation differ for the different EME traits.As previously mentioned, if the composition of the diet is roughly constant during lactation, CH 4 /DMI is expected to remain roughly constant or to present only small variations up to the end of lactation.Obviously, if the composition of the diet changes (especially the forage: concentrate ratio and the presence of oil supplements), CH 4 /DMI will be expected to change accordingly (Arndt et al., 2015;Drehmel et al., 2018;Beauchemin et al., 2022).The patterns in Figure 6A show there to be modest variations throughout lactation, except during the first month.The LSM are often much lower in fresh cows than in the following months of lactation, but sometimes they can be higher.It is possible that, as the experiments for developing the prediction equations were generally carried out on midor late-lactation cows, the resulting equations may not be applicable to cows at the beginning of lactation.Another possibility has to do with methodology.Polynomial regressions, like those used here, are very commonly used, but are not easy to interpret from a physiological point of view.More specific "lactation models" may be more useful in this regard.These were discussed in a parallel study on EME of Holstein and crossbred cows (Wilmink, 1987;Martínez-Marín et al., 2024), which found that the parameters of the lactation model for CH 4 /DMI were not significant, meaning that this trait was substantially constant throughout lactation.
If the diet is constant, dCH 4 is expected to exhibit the same pattern throughout lactation as the DMI of the lactating cow (i.e., to increase after calving to peak feed intake about a third of the way through lactation, and then to decrease slowly; Friggens et al., 1998;Harder et al., 2019).Six equations (C2, C3, E1, H3, H4, and H5) for predicting dCH 4 (Figure 6C) presented patterns very close to expectations; one of them (E2) yielded a linear decrease, whereas the others yielded curvilinear variations with modest variations in mid to late lactation.When dCH 4 is indirectly predicted as CH 4 /CM × dCMY (Figure 6D), we find, as expected, with the exception in particular of equation H2, a pattern similar to that anticipated for MY, but with a delayed peak.The pattern of DMI during lactation is not identical to that of MY because, as is well known, in the early stage the cow's appetite increases more slowly than its milk production resulting in NEB and depletion of body reserves (Stoop et al., 2009;Holtenius et al., 2018;Harder et al., 2019;Amalfitano et al., 2021).Negative energy balance is a condition that can increase the content of long-chain UFA and reduce the proportion of de novo FA in the milk fat.Cows are at zero energy balance (feed intake covers their requirements) after the lactation peak.In our study on EME of Holsteins and crossbreds using the Wilmink model, we estimated peaks at 37 d after calving for dMY, and at 53 d for dCH 4 (Martínez-Marín et al., 2024).After the feed intake peak, the energy balance tends to be positive (especially toward the end of lactation) to allow the cow to restore its body reserves and support pregnancy.This means that the expected pattern of CH 4 /CM during lactation is an increase from calving to the peak of feed intake, then stabilizing or increasing slowly afterward.The lower CH 4 /CM at the beginning of lactation reflects the quantity of milk produced through the mobilization of body reserves (no methane produced), whereas the slight increase after mid lactation reflects feed replacing the previously mobilized reserves and to support pregnancy.The Wilmink model used in the study on the EME of Holsteins and crossbred cows is better adapted to these expectations for CH 4 /CM (Martínez-Marín et al., 2024).In addition, the polynomial patterns displayed in Figure 6B are similar to expectations for 3 of the 4 equations compared, showing that the equations based on milk FA are generally able to capture the effect of NEB on CH 4 /CM at the beginning of lactation.The availability of reliable EME prediction equations is fundamental for establishing actions aimed at reducing the environmental impact of the dairy chain (Hristov et al., 2018;Beauchemin et al., 2022), an objective, however, that requires further research.

CONCLUSIONS
This research provides evidence that there is a poor consistency among the EME estimates yielded by the literature equations.However, a better consistency was found among equations developed from meta-analyses of data originated from experiments conducted under different productive conditions.The most reliable equations in terms of the ease of gathering information and responsiveness to variations in the cows' physiological conditions were those that included quantitative information (dMY) or were based on indirect estimation (methane intensity × dCMY).However, better predictions could be made by developing new equations for estimating EME, especially methane intensity, using a factorial approach that considers the contributions of maintenance, production level, and milk quality.

Figure 3 .
Figure 3. Linear regression of calculated daily methane production from methane intensity × daily corrected milk yield (methane intensity × dCMY, g/d), for equations with R 2 greater than 0.50.

Figure 5 .
Figure 5. Predicted methane yield (A), methane intensity (B), and daily methane production (C), and calculated daily methane production (methane intensity × dCMY; D) according to the 4 parity classes.*P < 0.05; **P < 0.01; ***P < 0.001.In the legend, near the ID of the equation, the significance of the linear (L), quadratic (Q), or cubic (C) patterns is reported.No trend line was reported when no L, Q, or C pattern was significant.CM = fat-and protein-corrected milk.
More promising are the equations developed from metaanalyses of more numerous observations obtained under different experimental conditions, such as those proposed by van Lingen et al. (2014) (A1, A2) and Bougouin et al. (

Figure 6 .
Figure 6.Predicted methane yield (A), intensity (B), daily production (C), and calculated daily methane production (methane intensity × dCMY; D) according to the 11 DIM classes (30-d lactation/class, 5-330 DIM).**P < 0.01; ***P < 0.001.In the legend, near the code of the equation, the significance of the linear (L), quadratic (Q) or cubic (C) patterns is reported.No trend line was reported when no L, Q, or C pattern was significant.CM = fat-and protein-corrected milk.
2 g/d, in good agreement with the Marumo et al. (2023) figure of 389.7 ± 102.7 g/d on lactating cows with similar MY.However, little consistency was found among the estimates of daily methane production yielded by the different equations.
Massaro et al.: COMPARISON OF EQUATIONS TO PREDICT METHANE Massaro et al.: COMPARISON OF EQUATIONS TO PREDICT METHANE Massaro et al.: COMPARISON OF EQUATIONS TO PREDICT METHANE

Table 1 .
Massaro et al.: COMPARISON OF EQUATIONS TO PREDICT METHANE Technical information on methane and milk fatty acid (FA) profile detection reported in the literature Reference Experiments, no.Diets, no.Observations Meta-analysis Milk FA detection Methane detection a The experiments come from the same research group.

Table 2 .
Equations to predict methane yield, methane intensity, and daily methane production 3Methane yield, 4

Table 3 .
Descriptive statistics for milk yield, corrected milk yield (MY), major milk constituents, and milk fatty acids (FA; g/100 g FA) of the database used in the current study (n = 992 individual records, 85 herds, 4 dairy systems) 1 Massaro et al.: COMPARISON OF EQUATIONS TO PREDICT METHANE 1 The table also reports the number of times in which the single trait had positive or negative sign in the different literature equations predicting methane emissions.c=cis;t=trans. 2t percentile.3=99thpercentile.4Number of literature equations that use the single FA or milk trait with a positive sign in the equations of Table2.5Number of literature equations that use the single FA or milk trait with a negative sign in the equations of Table2.6 ND = not detected. 7ΣSFA = sum of saturated fatty acids.

Table 4 .
Massaro et al.: COMPARISON OF EQUATIONS TO PREDICT METHANE Descriptive statistics for methane yield, intensity, and daily production estimates achieved from the literature equations used in the current study