Evaluation of different mathematical models on fitting the in vitro gas production parameters in beef cattle

Evaluation of different mathematical models on fitting the in vitro gas production parameters in beef cattle | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article Evaluation of different mathematical models on fitting the in vitro gas production parameters in beef cattle Jinze Wang, Zhiyu Zhou, Yang Cheng, Hao Wu, Xin Yi, Ziqi Deng, and 9 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4413816/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 24 Feb, 2025 Read the published version in Scientific Reports → Version 1 posted 11 You are reading this latest preprint version Abstract In vitro rumen gas production experiment was conducted with 57 kinds of feedstuff, which were categorized into energy feed, protein feed, and roughage, collected within China. Eight mathematical models were employed to describe the kinetics of in vitro rumen gas production. The results found that for energy feeds, protein feeds, and roughages, respectively, MM or Logistic-Exponential with lag (LEL), MM, and Mitscherlich (MIT) exhibited the highest or shown no significant difference compared to the highest coefficient of determination (R 2 ) ( P < 0.05) for all categories of feed. Furthermore, regression estimation of intercept and slope for regression estimates of intercept and slope for Observed versus Predicted of aforementioned models shown no significant difference from 0 and 1, respectively ( P < 0.05), except LEL for energy feed. Mean absolute error (MAE), root mean squared error of prediction (RMSEP), mean squared error of prediction (MSEP) of those models were relatively lower, with minimal systematic bias and regression bias. Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) rankings were higher compared with other models. Given these results, in studies where feedstuff categories are not distinguished or multiple feedstuffs categories are included, the MM model proves to be a good choice. MM or LEL was considered to better fit energy foodstuffs. The MM model was the optimal choice for fitting protein foodstuffs. MIT provided the best accuracy and moderate precision when fitting roughages. Biological sciences/Biological techniques/Biological models/Gastrointestinal models Biological sciences/Zoology in vitro gas production rumen fermentation mathematical model model evaluation Figures Figure 1 Figure 2 Figure 3 Figure 4 Introduction In vitro gas production method is commonly used for evaluating ruminant feed and additives 1,2 , microbial fermentation process 3,4 and the effect of anti-nutrient factors on microbial activity 5,6 . For better understanding of the rumen fermentation, the kinetic parameters of degradation are crucial, as they not only describe digestion but also characterize the intrinsic properties of substrates. Mathematical models can effectively parameterize the degradation or fermentation profiles of each category of feedstuff 7 . Exponential models were among the earliest mathematical models applied to the processes of digestion. In 1979, Ørskov and McDonald 8 used exponential model to describe the temporal changes of substrates in the nylon bags during the in situ digestion experiment. Similarly, Krishnamoorthy 9 employed a modified version of the model most commonly used to describe fiber digestion 10 , which is now the most widely used exponential model. The exponential model assumed that the digestion rate is proportional to the substrate concentration and independent of microbial biomass. Depending on whether the lag time is considered, exponential models include either 2 (EXP0) or 3 (EXPL) parameters. However, the exponential model suffered from inflexibility issues and failed to adequately describe the gas production process. Schofield 11 first applied the Gompertz (GOM) and Logistic (LOG) models, which originated from microbial growth kinetics, to calculate in vitro rumen gas production parameters. Both models assumed that the rate of digestion is directly proportional to the current microbial population and substrate level, but they differ in the mathematical effect of substrate limitation on microbial growth rate. The LOG model assumes a linear relationship, while the GOM model assumes a logarithmic relationship 11 . However, due to the presence of a positive intercept and a fixed inflection point occurring at half of the asymptotic gas production, the LOG model may not be suitable in certain situations 12 . Wang et al. 13 introduced a shape parameter into the model, constructing the Logistic-Exponential (LE) model. This model is more flexible and variable, with a non-fixed inflection point, and eliminates the positive intercept on the y-axis. France et al. 14 decomposed the fractional rate of degradation µ into a function related to the square root of time, establishing the Mitscherlich (MIT) model. According to France 12 , the relationship between µ and the square root of time may be associated with the diffusion of microbial enzymes. Groot et al. 7 proposed a modified Michaelis–Menten (MM) model based on the Michaelis–Menten equation in enzyme kinetics. MM model makes more flexible assumptions about fractional degradation rate, which can be continuously decreased over time or increase initially before decreasing 12 . Similar to the MIT model, the increase in the degradation rate reflects the increased accessibility of substrates due to particle hydration, microbial attachment, and an increase in microbial population, while the decrease in degradation rate may reflect the imposition of chemical and structural constraints 12 . Additionally, the gas production process is closely associated with the chemical composition and the physical structure of the substrate. Getachew et al. 15 investigated the relationship between in vitro rumen gas production and chemical nutrients in 12 kinds of feedstuff comprising 38 samples. They found a negative correlation between crude protein (CP) and gas production ( P < 0.001), while non-fiber carbohydrate (NFC) levels were positively correlated with gas production ( P < 0.01). It is generally believed that this is because the fermentative capacity of protein is approximately 0.3 times of carbohydrates 16 . Moreover, gas production may decrease due to the reaction between ammonia and carbon dioxide 17 . Different types of carbohydrates produce varying volumes of gas depending on their chemical properties and interactions with other feed components 2 . Compared to the slowly degradable fraction, rapidly degradable carbohydrates produce more propionate, which is a process with lower gas potential compared to acetate 18 . In ruminant nutrition, feedstuff can be simply categorized into energy feed, protein feed, and roughage. Therefore, distinct models should be used based on the nutrient characteristics of each feedstuff. Though previous studies have investigated the comparison of the goodness of fit of different models, they may suffer from issues such as small sample sizes and undivided feed categories 1,11,13,19–24 . To address these issues, this study aims to improve the reliability and accuracy of the research by increasing the sample size and categorizing the data according to feedstuff types. Through these methods, the accuracy, precision, and complexity of different models can be compared more effectively, thereby providing more valuable insights for future research. Results Feedstuffs and in vitro rumen gas production profile The 57 feedstuff materials were categorized into energy feeds, protein feeds, and roughages based on their common use in ruminant production. Descriptive statistics of the nutritional components for these three categories of feedstuff were presented in Table 1 , and gas production profiles were depicted in Fig. 1 . Energy feeds primarily consisted of grains and brans, characterized by a high starch content (43.31 ± 5.03%) and low levels of structural carbohydrates. Protein feeds exhibited high crude protein (CP) content (33.87 ± 2.4%) and lower carbohydrate proportions. Additionally, due to the inclusion of various by-product from oil industry, protein feeds had higher levels of crude fat (EE) (6.61 ± 0.77%). Roughages mainly comprised hays and straws, containing significant amounts of structural carbohydrates (67.86 ± 2.59% NDF, 44.78 ± 2.51% ADF, 10.21 ± 1.89% ADL). Fitting of the gas production profiles Figure 2 illustrated the fitting of different models for the three categories of feedstuff. There were significant differences in the shape of the fitting curves for the three categories. Gas production from energy feeds was notably higher than that from the other two categories, and both energy and protein feeds exhibited higher gas production rates than roughages during the same period, reaching the plateau phase earlier. The gas production curves for roughages were the most gradual, reaching the plateau phase later, with relatively lower gas production. Regarding different models, GOM and LOG models shown sigmoidal curves, with similar shapes, and almost completely overlap in energy feed. The LOG model reached the plateau phase earlier compared to other models within the same category of feedstuff, while the GOM model only exhibited similar characteristics in energy and protein feed, with lower gas production after reaching the plateau phase compared to other models. Other models fitted non-sigmoidal curves for the three categories of feedstuff, with the MM model showing the highest final gas production compared to other models within the same category. The mean values of Vf, t 0.5 , and µ 0.5 predicted by the 8 models for different categories of feedstuff were presented in Fig. 3 . There were significant differences in gas production parameters among different categories of feedstuff. The Vf for energy feed (61.98–71.68 ml/200mg) was much higher than that for protein feed (37.63–45.77 ml/200mg) and roughage (35.67–46.86 ml/200mg). Across all categories, MM yielded the highest predicted Vf values, and LOG produced the lowest. Predicted t 0.5 values did not show significant differences among different model for any category. However, GOM and LOG consistently predicted significantly higher µ 0.5 values compared to other models in all categories. Model comparison The average R 2 for each category of feedstuff fitted by different models were depicted in Fig. 4 , with all models achieving R 2 values above 0.94. Across different groups, a similar ranking of model performance was observed. Within each group, MM consistently exhibited the highest R 2 values, followed by MIT, while LOG consistently showed significantly lower R 2 compared to other models. In the energy feeds and roughages groups, MIT, LEL, and LE0 did not exhibit significant differences compared to MM, whereas in the protein feed group, MM significantly outperformed all other models. And among all feedstuffs without categorization, the MM model exhibits significantly higher R 2 values compared to other models. The results of regressing observed versus predicted gas production were illustrated in Fig. 5 and summarized in Table 2 . Among all 8 models, EXPL,MIT and MM exhibited non-significant slopes and intercepts across the three categories of feedstuff. In the energy feeds group, significant differences were observed in the slopes compared to 1 for EXP0, GOM, LOG, LE0, and LEL, with significant non-zero intercepts for all models except LEL. In the protein feeds and roughages groups, LE0 and LEL also demonstrated good fitting performance. However, EXP0, GOM, and LOG exhibited significantly non-unity slopes, while EXP0 and LOG had intercepts significantly different from 0. In the uncategorized feedstuffs group, only EXPL and MM exhibited non-significant slopes and intercepts. Error analysis of the 8 models fitting different categories of feedstuff was presented in Table 3 . Across the four groups, the MAE, RMSEP, and MSEP of different models were consistently similar, with MM or MIT demonstrating the smallest values, followed by LE0 and LEL. Focusing on the MSEP decomposition parts of these models, it was observed that MIT exhibits no systematic bias in four groups and only shows a regression bias of 0.01% when fitting roughages. Apart from MIT, in energy feed, LEL and LE0 also demonstrated low system bias, with LEL exhibiting lower regression bias as well. In protein feed, MM shows minimal systematic bias and regression bias other than MIT. And, all models exhibited system bias and regression bias in roughage. In the group comprising all feedstuffs, LE0 also displayed no bias. The AIC and BIC values were calculated for each model fit. And the models were ranked based on these criteria for each sample, as shown in Table 4 . A higher rank indicated better model performance. MM consistently ranked highest in terms of AIC (BIC) across all four groups. Among energy feeds, MM (MM) exhibited the highest performance, followed by LE0, LEL, and EXPL (LE0, LEL and EXPL) according to AIC (BIC) criteria. In the case of protein feed, MM (MM) outperformed other models, with EXP0, EXPL, and LE0 (EXP0, EXPL and LE0) following in AIC (BIC) ranking. Regarding roughage, MM (MM) ranked highest, followed by EXPL, MIT, and LE0 (LE0, EXPL and EXP0). For the entire group of feedstuffs, MM (MM) was superior, with LE0, EXPL, and LEL trailing behind (EXPL, EXP0 and LE0). Discussion The 68 samples selected for this experiment comprised 57 kinds of feedstuff, categorized into three groups commonly used for ruminants: energy feeds, protein feeds, and roughages. The nutritional and gas production data exhibit a wide range and substantial quantity, indicating their representativeness. The chemical composition and the physical structure of feedstuff are closely related to gas production curves. It is generally believed that the main sources of gas in in vitro rumen gas production experiments are carbohydrate degradation and the reaction of volatile fatty acids (VFAs) with substrate buffer solution 25,26 .CP fermentation generated less gas, while EE hardly produces any gas 2 . Structural carbohydrates, being more resistant to degradation, often have a significant impact on both gas volume and gas production rate 27 . This also explains why the final gas production and Vf of most protein and roughage feeds were lower than those of energy feeds. Additionally, roughages typically exhibited flatter gas production curves, longer t 0.5 , and smaller µ 0.5 values. Huhtanen et al. 28 , through a comparison of residuals over time, observed that EXP0 and MM tended to overestimate asymptotic gas volume. This trend appeared to be present in both energy feeds and roughages in this experiment. Huhtanen explained the overestimation of asymptotic gas volume by the EXP0 model as its inability to describe lag phenomena 28 . However, this did not explain why EXP0 does not exhibit higher Vf in protein feed, and why there was no significant difference between EXP0 and EXPL in all feedstuff categories. Considering that Huhtanen's samples were roughages, the overestimation of Vf by EXP0 might be due to its exponential mathematical structure being unsuitable for energy and roughage feedstuffs. Additionally, the decreasing fractional degradation rate of MM in the later stages of fermentation might not be reasonable, which led to the overestimation of asymptotic gas volume 21,28 . Schofield et al. 11 found in their model derivation that the maximum rate (i.e., the curve inflection point) of LOG and GOM models occurs at V = Vf/2 and V = Vf/e. This might explain why µ 0.5 was significantly higher than GOM, while GOM was significantly higher than other models in all feedstuff categories. This was consistent with previous research results 13,28 . Considering that there was no significant difference in t 0.5 between models within each feedstuff category, different models may not exhibit differently in the fractional rate of gas production when fitting the same category of feedstuff. The models extended to multi-pool analysis, with the fundamental assumption that the incubated feedstuff was not homogeneous 7,29 . Cone et al. 30 classified gas production into three stages: the initial stage involves the neutral detergent soluble fraction (NDS), the subsequent stage entails NDF fermentation, and the final stage relates to microbial population limitation. Some researchers believe that compared to single-pool analysis, this nonlinear multi-pool analysis increased the number of parameters in the model, potentially compromising its robustness and posing numerous challenges to fitting 1,19,31 . Therefore, this study only selected single-pool models for analysis and comparison. When comparing mathematical models, no single indicator was enough to fully evaluate their performance. Each indicator highlighted a specific aspect of the comparison 32,33 . Therefore, a combination of various statistical analyses must be employed to assess the adequacy of mathematical models properly, along with a rational analysis of the original conceptualization and development purposes of the mathematical models. In addition to comparing among the three categories of feedstuffs, this study also included a group containing all feedstuffs to investigate whether it is necessary to differentiate from feedstuff categories when comparing models fitting performance. The results revealed that it exhibited performance distinct from the three groups in terms of Observed versus Predicted (OvP) and error analysis, particularly in relation to changes in the distribution and mean of predicted values. Therefore, using different models for different categories of feedstuff may be necessary, as it would lead to more accurate and realistic results. The coefficient of determination (R 2 ) serves as a reliable measure of precision, with higher R 2 values indicating greater precision 32 . The MM model exhibited the highest R 2 across all four groups which surpassed all other models significantly in the protein feeds and the uncategorized group. This finding aligned with the conclusions drawn by most previous studies 1,20,23,31 . After deriving the formulas, France et al. 12 noted that the MM model's assumption regarding gas production rate was the most flexible. Compared to carbohydrates, the energy obtained by rumen microbes from protein fermentation was lower, leading to slower microbial growth 16,34 . Moreover, protein itself yields less gas, and the NH 3 produced during fermentation consumes CO 2 2 . These variations resulted in irregular and challenging-to-fit gas production curves, which the highly flexible MM model can effectively accommodate. A study found that the GOM model outperforms MM when roughages were used as a substrate 13 . Considering MM's characteristic of decreasing residuals over time, this could be attributed to differences in fermentation time between the study (72h) and the experiments referenced (48h). Additionally, the MIT, LE0, and LEL models also exhibited high R 2 values when fitting energy and roughage feedstuffs, consistent with a few prior studies 13,22,23,35,36 . These findings further underscore the advantage of employing flexible and adaptable curve shapes to enhance model accuracy. Generally, accuracy is considered the most important criterion as it measures the model's ability to predict true values 32 . The regression estimates of intercept and slope for OvP values serve as reliable indicators of accuracy, with intercepts and slopes closer to 0 and 1 indicating higher levels of precision 32 . Apart from the flexible and adaptable MIT and MM models, the introduction of lag time in EXPL relative to EXP0 also contributed to higher accuracy. Lag time can be attributed to hydration, removal of digestive inhibitors, or microbial substrate attachment 7,12 . Although lag time may not fully describe the phenomenon of hysteresis, it provides a simple quantitative measure of delayed effects. Allen and Mertens 37 proposed a two-step model describing the digestion process: the first step involves the transformation of substrate from an unavailable to an available form (i.e., the lag pool concept), and the second step represents substrate digestion. While theoretically more accurate than discrete lag models, lag pool models did not fit the data better 28 , and were not widely used due to their complexity. GOM and LOG inevitably introduce an initial gas volume (V0) during the assumption stage 11 , whereas V0 is clearly zero at the start of the experiment. According to Schofield et al. 11 , the error in V0 (i.e., intercept on the y-axis) is most severe for GOM and LOG when the digestion rate is low and lag time is short, which evidently leads to poorer accuracy when fitting energy feed. However, the intercept decreased when fitting protein and roughage feedstuffs, which validated the findings of Schofield et al. 11 . The slopes of LE0 and LEL were significantly different from 1 when fitting energy feed. LE0 and LEL might underestimate the gas production in the later stages of energy feed fermentation. This was consistent with the significantly lower Vf for LE0 and LEL when fitting energy feeds compared to half of the models. RMSEP and MAE are two standard indicators for evaluating the accuracy of models. Under different error distributions, each has its own advantages and disadvantages 38 . This study presented both indicators to provide a comprehensive assessment. MSEP is perhaps the most common and reliable estimate for evaluating the accuracy of model predictions, and further decomposition of MSEP allows for the analysis of model adequacy by identifying the sources of variation 32 . If there is a significant system bias, the model's predicted mean may be larger or smaller than the observed mean, and if there is significant regression bias, it indicates potential inadequacies in the structural equation of the model in describing the analyzed data 39 . In this study, the results of linear fitting for Observed versus Predicted (OvP) and error analysis were inconsistent in some aspects. This inconsistency might stem from the different principles underlying their analyses, with the former focusing on the distribution of predicted values and the latter on the mean of predicted values. This experiment found that fitting different fermentation substrates does not affect the level of errors but may result in different sources of errors. It is generally believed that the fractional rate of gas production µ of substrates during fermentation is not constant. An initial increase in µ reflects particle hydration, microbial attachment, and an increase in microbial numbers, while a decrease in µ may indicate limitations in chemical nutrients and structure within the substrate 12 . Models like LE0, LEL, MIT, and MM have the ability to allow µ to vary freely over time, leading to smaller errors when fitting the three types of feedstuff 7,13,14 . Camila et al. 1 found that the MM model had the lowest RMSEP when fitting corn silage using various models. Dhanoa et al. 19 , in comparing the performance of different models after fitting roughages and silage feedstuffs, found that the residual mean squares of MM and MIT were lower than those of other models. Wang et al. 13 further validated the models and found that the RMSEP of LE0 and LEL was lower than that of most models. MIT exhibited near-zero SB and RB in all three categories of feedstuff, while the performance of LE0, LEL, and MM varied depending on the categories of feedstuff being fitted. This may be attributed to the different assumptions underlying LE0 and LEL (microbial growth 13 ), MIT (degradation by surface erosion and complete invasion 14 ), and MM (formation of microbial film 7 ). For energy and roughage feedstuffs, despite its advantages, the MM model exhibited systematic bias and regression bias. Considering the previous discussion, it seems that the MM model may have overestimated the gas production values 28 . This is an important factor to consider when using the MM model for energy and roughage feedstuffs, as it could impact the accuracy of the results. In addition to accuracy and precision, complexity directly impacts the performance of the model on both training and unseen data, affecting its interpretability and fitting cost. AIC and BIC serve as indicators that balance complexity and precision. For small datasets, the performance of AIC is similar to or even better than BIC, while BIC becomes the preferred criterion for selecting the correct model, for datasets of moderate size 40 . In this study, both AIC and BIC were chosen to evaluate the complexity and precision of the models, because each sample comprised approximately 40 to 100 data points, falling the range between small and moderate sample sizes 41 . Among the models selected in this study, EXP0 has 2 parameters, while LEL and MIT have 4 parameters, with the rest having 3 parameters. Table 4 revealed that the model with 2 parameters, EXP0, did not exhibit significant advantages, ranking 3.51 (3.10) in AIC (BIC) only for the protein feed where other models struggled to fit. On the other hand, the 3-parameter MM model ensured model precision with a slight increase in complexity. This finding aligned with previous research results 13,23 . MIT outperformed other models in roughages, indicating that its theoretical assumptions regarding microbial enzyme diffusion and surface erosion are better suited for difficult-to-degrade roughage 14 . The MM model appears to demonstrate high precision, moderate to high accuracy, and low complexity across all three categories of feed. Therefore, the MM model is a good choice for studies that do not differentiate between feed categories or include multiple categories of feed. However, considering the MM model’s overestimation of asymptotic gas production, it is advisable to choose a model based on its specific performance for different feeds. If only accuracy is considered, models such as LEL, MIT and MM for energy feed, MIT and MM for protein and roughage feedstuff have performed well. However, it is unrealistic to rely solely on an accurate but imprecise model for predictions. Conclusions In summary, due to the involvement of different fermentation processes, the chemical composition and the physical structure of feed, could influence the gas production process, resulting in gas production curves with different characteristics. EXP0 and MM tend to overestimate Vf for energy and roughage feeds. In studies where feedstuff categories are not distinguished or multiple feedstuffs categories are included, the MM model proves to be a good choice. When fitting various categories of feedstuff, the following conclusions were drawn:(1) For energy feed, MM showed relatively highest R 2 among the 8 models ( P < 0.05). The slope and intercept in OvP did not significantly differ from 1 and 0 ( P < 0.05). MAE, RMSEP, and MSE were lowest (1/8), but there was a presence of systematic bias and regression bias. The AIC (BIC) ranking was 2.29 (2.29), exceeding all other models. LEL performs worse on certain indicators compared with MM, but exhibits no bias, which should be taken into consideration. (2) For protein feed, MM's R 2 was significantly higher than other models ( P < 0.05). The regression estimates of intercept and slope in OvP had no significant difference from 0 and 1, respectively ( P < 0.05). MAE, RMSEP, and MSE were the lowest (1/8) with almost no systematic bias and regression bias. The AIC (BIC) ranking was 1.62 (1.62). (3) For roughage, MIT showed relatively higher R 2 (2/8) with no significant difference from the highest value ( P < 0.05). The intercept and slope in OvP had no significant difference from 0 and 1, respectively ( P < 0.05). MAE, RMSEP, and MSE were low (1/8) with no systematic bias and regression bias. The AIC ranks third only to MM and EXPL. Due to limitations in sample quantity and diversity, the categorization of feeds in this study may not be sufficiently specified or comprehensive. In the future, collecting more extensive and diverse gas production data will enable a better exploration of the suitability of models for different feed categories. Materials and Methods Animals and diets All animal procedures used in this study were approved by the Animal Care and Use Committee of China Agricultural University (Approval No.AW81404202-1-9), and the experimental procedures used in this study were in accordance with the university’s guidelines for animal research. The rumen fluid used in this experiment was collected from three Angus steers of similar weight (400.00 ± 50.00 kg, mean ± SD) with permanent rumen fistulas. The experimental animals had ad libitum access to water and were fed twice daily at 8:00 and 17:00. The diets were formulated based on the body weight and nutritional requirements to meet 1.3 times maintenance needs of beef cattle according to NASEM 2016 42 (Table 5 ). Feed samples preparation and nutrient analysis A total of 57 kinds of feedstuffs, which were commonly used for ruminants, were collected within China and categorized into three groups, with some feedstuffs comprising two varieties, resulting in a total of 68 samples. 57 feedstuffs were categorized into three groups commonly used for ruminants. Among them, there were 20 types of energy feed, totaling 24 samples, including: barley (2), beet pulp, buckwheat, corn, corn husk, dried foxnut, flaked corn, highland barley, millet, millet bran, oat, oat bran, rice, rice bran (2), rice bran meal, rye grain, sorghum (2), soybean hulls, wheat, and wheat bran (2). For protein feed, there were 18 types totaling 22 samples including: bean curd residue (2), brewer's grains, cottonseed meal, DDGS (2), distiller's grain, flax seed meal (2), linseed cake, mulberry leaf, palm meal, paper mulberry leaf, peanut meal, rapeseed cake, rapeseed meal, sesame cake, soy sauce residue, soybean meal, sunflower meal (2), and vinegar residue. For roughage feedstuffs, there were 19 kinds totaling 22 samples including: alfalfa, apple pomace, Chinese wildrye, corncob, dried corn stalk (2), highland barley stalk, oat grass (2), oat wheat straw, peanut seedling, peanut shell, pear pomace, ryegrass, sorghum bran, sorghum stalk, soybean straw, straw, sugarcane bagasse, sweet potato seedling, and wheat straw (2). The samples were dried in a 65°C air oven for 48 hours, followed by grinding through a 0.5-millimeter sieve, for chemical analysis and in vitro rumen gas production. Dry matter (DM), ash, and starch were determined following the AOAC 43 . Nitrogen content was determined using the Dumas combustion method with nitrogen analyzer (Rapid III, Elementar, Germany), and crude protein (CP) content was calculated as nitrogen content times 6.25. Ether extract (EE) was determined by an automatic fat analyzer (XT15, ANKOM Technology, USA).Neutral detergent fiber (NDF), acid detergent fiber (ADF), and acid detergent lignin (ADL) were determined following the method of Van Soest et al. 44 with automatic fiber analyzer (A2000i, ANKOM Technology, USA). In vitro rumen gas production In vitro rumen gas production was conducted following the method outlined by Menke et al. 45 . Accurately weighing 0.200g (DM) air-dried feedstuff sample, and placing at the bottom of a glass syringe. In the morning of the experiment day, prior to feeding, the contents and rumen fluid were uniformly collected from the steer. After filtering through four layers of cheesecloth, the mixed rumen fluid were placed in a preheated thermos bottle at 39°C. The rumen fluid was mixed with CO 2 -saturated rumen buffer solution at a ratio of 1:2 to prepare rumen inoculum. Then, 30 mL of the inoculum was added to the glass syringe using an automatic pipette and placed in a 39°C water baths shaker with 60 revolutions per minute. Models and curve-fitting Table 6 presented 8 commonly used single-pool models describing in vitro rumen gas production kinetics. The exponential model included two types: EXP0 (without lag phase) and EXPL (with lag phase), which were among the most used models in in vitro rumen gas production. The Gompertz (GOM) and Logistic (LOG) models, initially proposed in microbiology, were first applied to in vitro rumen gas production kinetics by Schofield et al 11 . Wang et al. 13 introduced the shape parameter d to the Logistic (LOG) model, developing a more flexible Logistic-Exponential (LE) model. France et al. 14 incorporated the concept of degradation by surface erosion and by complete invasion into mathematical formulas, resulting in the Mitscherlich (MIT) model. The Michaelis-Menten (MM) model originated in enzymatic kinetics, and Groot et al. 7 were the first to apply it to describe in vitro rumen gas production kinetics. Table 6 provided the time (t 0.5 ) and fractional rate of gas production (µ 0.5 ) for each model at V = Vf/2. Here, t 0.5 was calculated as the t when V = Vf/2, and µ 0.5 was the value obtained by substituting t 0.5 into dV/dt/V. Statistical analysis In R version 4.3.2 (R Core Team, Vienna, Austria), each sample was fitted using the nonlinear least squares method via the NLS function. The stats package was used to compute the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) for the fits. Gas production attributes (Vf, t0.5, µ0.5) for each category feedstuff were compared across different models using repeated measures one-way ANOVA with Greenhouse-Geisser correction. Bar charts were plotted using the ggplot2 package. The determination coefficient (R2) for each feedstuff category was calculated separately, and comparisons were made using repeated measures one-way ANOVA with Greenhouse-Geisser correction. R2 was calculated as: $${\text{R}}^{\text{2}}\text{=1-}\frac{\text{∑}{\text{(}{\text{y}}_{\text{i}}\text{-}\widehat{{\text{y}}_{\text{i}}}\text{)}}^{\text{2}}}{\sum \text{}{\text{(}{\text{y}}_{\text{i}}\text{-}\stackrel{\text{-}}{\text{y}}\text{)}}^{\text{2}}}$$ Here, \({\text{y}}_{\text{i}}\) represents the actual value of the 𝑖 -th observed data, \(\widehat{{\text{y}}_{\text{i}}}\) represents the predicted value of the model for the𝑖 -th observed data, \(\stackrel{\text{-}}{\text{y}}\) denotes the mean value of the observed data. Using GraphPad Prism 9.5.0 (GraphPad Software, San Diego, CA, USA), linear regression was performed on the observed gas production values (x-axis) versus the predicted values (y-axis) for each feedstuff category. The intercepts and slopes of each model were evaluated, and fitted equations were plotted. Statistical analysis was conducted to test the significance of the regression parameters, examining the hypotheses "slope = 1" and "intercept = 0". Error analysis refers to the systematic analysis and interpretation of the differences between model predictions and actual observed values. The following methods were used to assess the errors of the model: Mean Absolute Error (MAE), Root Mean Squared Error of Prediction (RMSEP), and Mean Squared Error of Prediction (MSEP). MSEP is further decomposed into system bias (SB), regression bias (RB), and random error (RE) 32,46 .The calculation formulas were as follows: $$\text{SB=}{\left(\frac{\text{∑}{\text{y}}_{\text{i}}}{\text{n}}\text{-}\frac{\text{∑}\widehat{{\text{y}}_{\text{i}}}}{\text{n}}\right)}^{\text{2}}$$ $$\text{RB=}{\text{(}{\text{S}}_{\widehat{\text{y}}}\text{-r×}{\text{S}}_{\text{y}}\text{)}}^{\text{2}}$$ $$\text{RE=(1-}{\text{r}}^{\text{2}}\text{)×}{\text{S}}_{\text{y}}$$ $$\text{r=}\frac{\sum \frac{（{\text{y}}_{\text{i}}\text{-}\frac{\text{∑}{\text{y}}_{\text{i}}}{\text{n}}）\text{×(}\widehat{{\text{y}}_{\text{i}}}\text{-}\frac{\text{∑}\widehat{{\text{y}}_{\text{i}}}}{\text{n}}\text{)}}{{\text{S}}_{\text{y}}\text{×}{\text{S}}_{\widehat{\text{y}}}}}{\text{n}}$$ Where n represents the number of data points, \({\text{S}}_{\text{y}}\) denotes the variance of the observed data, and \({\text{S}}_{\widehat{\text{y}}}\) represents the variance of the model-predicted data. Declarations Acknowledgments We are grateful to the staff of Beef Cattle Research Center for their help with sampling and laboratory analyses. Study Funding This work was supported by the National Key R&D Program of China (2022YFA1304204) and the Government Purchase Service (16200158, 16190050). Author contributions statement JW designed the study, analyzed the data, and wrote the manuscript. ZZ and YC performed the experiments and sample analyses.BY, SL, LR, NL, XZ and WL conducted the trial,. XY, ZD and SY assisted with the manuscript preparation. ZZ, HW and QM revised the manuscript and provided experimental guidance. All authors read and approved the final manuscript. Additional information The authors declare no real or perceived conflicts of interest. Ethical approval All animal procedures used in this study were approved by the Animal Care and Use Committee of China Agricultural University (Approval No.AW81404202-1-9). 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Ciência Rural 51 (2021). Tedeschi, L. O., Assessment of the adequacy of mathematical models. AGR SYST 89 225 (2006). GREEN, I. R. A. & STEPHENSON, D., Criteria for comparison of single event models. Hydrological Sciences Journal 31 395 (1986). Russell, J. B., Sniffen, C. J. & Van Soest, P. J., Effect of Carbohydrate Limitation on Degradation and Utilization of Casein by Mixed Rumen Bacteria. J DAIRY SCI 66 763 (1983). Mello, R., Magalhães, A. L. R., Breda, F. C. & Regazzi, A. J., Modelos para ajuste da produção de gases em silagens de girassol e milho. Pesquisa Agropecuária Brasileira 43 (2008). Azevedo, M. et al. , MODELOS MATEMÁTICOS PARA ESTIMATIVA DA CINÉTICA DE FERMENTAÇÃO RUMINAL DO PSEUDOFRUTO DO CAJUEIRO ATRAVÉS DA TÉCNICA IN VITRO SEMI-AUTOMÁTICA DE PRODUÇÃO DE GASES / MATHEMATICAL MODELS TO ESTIMATE THE CASHEW TREE FALSE FRUIT RUMINAL FERMENTATION KINETIC THROUGH THE SEMI-AUTOMATIC IN VITRO GAS PRODUCTION TECHNIQUE. Brazilian Journal of Development 6 73534 (2020). Allen, M. S. & Mertens, D. R., Evaluating Constraints on Fiber Digestion by Rumen Microbes. The Journal of Nutrition 118 261 (1988). Hodson, T. O., Root-mean-square error (RMSE) or mean absolute error (MAE): when to use them or not. GEOSCI MODEL DEV 15 5481 (2022). Traxler, M. J. et al. , Predicting forage indigestible NDF from lignin concentration. J ANIM SCI 76 1469 (1998). Emiliano, P. C., Vivanco, M. J. F. & de Menezes, F. S., Information criteria: How do they behave in different models? COMPUT STAT DATA AN 69 141 (2014). Kuha, J., AIC and BIC. SOCIOL METHOD RES 33 188 (2004). National Academies Of Sciences, E. A. M., Nutrient requirements of beef cattle , 8th revised ed ed. (The National Academies Press, Washington, DC, 2016). AOAC, A. O. O. A., Official methods of analysis of AOAC international, 18th ed. Arlington, VA: Association of Official Analytical Chemists (2006). Van Soest, P. V., Robertson, J. B. & Lewis, B. A., Methods for dietary fiber, neutral detergent fiber, and nonstarch polysaccharides in relation to animal nutrition. J DAIRY SCI 3583 (1991). H, M. K. et al. , The estimation of the digestibility and metabolizable energy content of ruminant feed stuffs from the gas production when they are incubated with rumen liquor in vitro. Journal of Agricultural Science (Cambridge) 93 217 (1979). Kohn, R. A., Kalscheur, K. F. & Hanigan, M., Evaluation of Models for Balancing the Protein Requirements of Dairy Cows. J DAIRY SCI 81 3402 (1998). Tables Tables 1 to 6 are available in the Supplementary Files section. Additional Declarations No competing interests reported. Supplementary Files Tables.docx TableS1.xlsx Table S1. Chemical composition of the feedstuffs (Dry matter basis, %). 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14:18:12","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4413816/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4413816/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1038/s41598-025-90189-8","type":"published","date":"2025-02-24T15:56:55+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":57084600,"identity":"08f9f4b5-fdf7-443a-98a7-f767018f1242","added_by":"auto","created_at":"2024-05-24 11:27:33","extension":"jpg","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":4197774,"visible":true,"origin":"","legend":"\u003cp\u003eThe \u003cem\u003ein vitro \u003c/em\u003egas production profiles. Fermentation of energy and protein feeds for 60 hours, and roughages for 72 hours. (a)24 energy feed samples; (b) 22 protein feed samples; (c) 22 roughage samples.\u003c/p\u003e","description":"","filename":"floatimage1.jpg","url":"https://assets-eu.researchsquare.com/files/rs-4413816/v1/a0acb21b6ce3fb47927c09dc.jpg"},{"id":57084598,"identity":"43806876-57c9-4476-9da6-c189601c4d77","added_by":"auto","created_at":"2024-05-24 11:27:33","extension":"jpg","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":921933,"visible":true,"origin":"","legend":"\u003cp\u003eFitting profiles of 8 models for 3 categories of feedstuff. The curves of energy and protein feeds for 60 hours, and roughages for 72 hours. (a)8 curves for energy feeds; (b) 8 curves for protein feeds; (c) 8 curves for roughages. EXP0: exponential without lag phase; EXPL: exponential with lag phase; GOM: Gompertz; LOG: logistic; LE0: logistic-exponential without lag phase; LEL: logistic-exponential with lag phase MIT: Mitscherlich; MM: Michaelis‒Menten.\u003c/p\u003e","description":"","filename":"floatimage2.jpg","url":"https://assets-eu.researchsquare.com/files/rs-4413816/v1/3cc2a3376ca33d0864541a98.jpg"},{"id":57084613,"identity":"122e7d93-317c-48ca-b6bf-831b908e10fe","added_by":"auto","created_at":"2024-05-24 11:27:36","extension":"jpg","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":14915522,"visible":true,"origin":"","legend":"\u003cp\u003eComparison of gas production parameters for different models. Vf, t\u003csub\u003e0.5\u003c/sub\u003e, μ\u003csub\u003e0.5\u003c/sub\u003e are the average of each category of feedstuff. Means with different letters differ (\u003cem\u003eP\u003c/em\u003e\u0026lt; 0.05). EXP0: exponential without lag phase; EXPL: exponential with lag phase; GOM: Gompertz; LOG: logistic; LE0: logistic-exponential without lag phase; LEL: logistic-exponential with lag phase; MIT: Mitscherlich; MM: Michaelis‒Menten; Vf: final asymptotic gas production; t\u003csub\u003e0.5\u003c/sub\u003e: the time at Vf/2; μ\u003csub\u003e0.5\u003c/sub\u003e: fractional rate of gas production at Vf/2.\u003c/p\u003e","description":"","filename":"floatimage3.jpg","url":"https://assets-eu.researchsquare.com/files/rs-4413816/v1/5e564596ebcb2eb3f0f5406b.jpg"},{"id":57084599,"identity":"cfc99fae-5fd9-4d62-9694-1787bc5a7535","added_by":"auto","created_at":"2024-05-24 11:27:33","extension":"jpg","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":2362974,"visible":true,"origin":"","legend":"\u003cp\u003eThe R2 of fitted curves of 8 models for 3 categories of feedstuff. R2 are the average of each category of feedstuff. (a) energy feeds; (b) protein feeds; (c) roughages; (d) all feedstuffs. Means with different letters differ (\u003cem\u003eP\u003c/em\u003e\u0026lt; 0.05).EXP0: exponential without lag phase; EXPL: exponential with lag phase; GOM: Gompertz; LOG: logistic; LE0: logistic-exponential without lag phase; LEL: logistic-exponential with lag phase; MIT: Mitscherlich; MM: Michaelis‒Menten.\u003c/p\u003e","description":"","filename":"floatimage4.jpg","url":"https://assets-eu.researchsquare.com/files/rs-4413816/v1/ed02e4781d167054552d20cd.jpg"},{"id":77622335,"identity":"37c769c9-1a22-418c-a854-3d49866d9e8e","added_by":"auto","created_at":"2025-03-03 16:04:47","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":23157406,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4413816/v1/54fc47ab-45b5-4f17-92ee-e1407d6827eb.pdf"},{"id":57084614,"identity":"11f432d1-fed7-4db0-9214-40779f46cbd0","added_by":"auto","created_at":"2024-05-24 11:27:37","extension":"docx","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":47401,"visible":true,"origin":"","legend":"","description":"","filename":"Tables.docx","url":"https://assets-eu.researchsquare.com/files/rs-4413816/v1/5fcfca1e46a676e958319c61.docx"},{"id":57084601,"identity":"30b92960-6d65-47ba-8d4b-8cd1a0fb0efe","added_by":"auto","created_at":"2024-05-24 11:27:34","extension":"xlsx","order_by":2,"title":"","display":"","copyAsset":false,"role":"supplement","size":14298,"visible":true,"origin":"","legend":"\u003cp\u003eTable S1. Chemical composition of the feedstuffs (Dry matter basis, %).\u003c/p\u003e","description":"","filename":"TableS1.xlsx","url":"https://assets-eu.researchsquare.com/files/rs-4413816/v1/a2d97906e5f907c0efeec181.xlsx"}],"financialInterests":"No competing interests reported.","formattedTitle":"Evaluation of different mathematical models on fitting the in vitro gas production parameters in beef cattle","fulltext":[{"header":"Introduction","content":"\u003cp\u003e \u003cem\u003eIn vitro\u003c/em\u003e gas production method is commonly used for evaluating ruminant feed and additives\u003csup\u003e1,2\u003c/sup\u003e, microbial fermentation process\u003csup\u003e3,4\u003c/sup\u003e and the effect of anti-nutrient factors on microbial activity\u003csup\u003e5,6\u003c/sup\u003e. For better understanding of the rumen fermentation, the kinetic parameters of degradation are crucial, as they not only describe digestion but also characterize the intrinsic properties of substrates.\u003c/p\u003e \u003cp\u003eMathematical models can effectively parameterize the degradation or fermentation profiles of each category of feedstuff\u003csup\u003e7\u003c/sup\u003e. Exponential models were among the earliest mathematical models applied to the processes of digestion. In 1979, \u0026Oslash;rskov and McDonald \u003csup\u003e8\u003c/sup\u003e used exponential model to describe the temporal changes of substrates in the nylon bags during the \u003cem\u003ein situ\u003c/em\u003e digestion experiment. Similarly, Krishnamoorthy\u003csup\u003e9\u003c/sup\u003e employed a modified version of the model most commonly used to describe fiber digestion\u003csup\u003e10\u003c/sup\u003e, which is now the most widely used exponential model. The exponential model assumed that the digestion rate is proportional to the substrate concentration and independent of microbial biomass. Depending on whether the lag time is considered, exponential models include either 2 (EXP0) or 3 (EXPL) parameters. However, the exponential model suffered from inflexibility issues and failed to adequately describe the gas production process. Schofield\u003csup\u003e11\u003c/sup\u003e first applied the Gompertz (GOM) and Logistic (LOG) models, which originated from microbial growth kinetics, to calculate \u003cem\u003ein vitro\u003c/em\u003e rumen gas production parameters. Both models assumed that the rate of digestion is directly proportional to the current microbial population and substrate level, but they differ in the mathematical effect of substrate limitation on microbial growth rate. The LOG model assumes a linear relationship, while the GOM model assumes a logarithmic relationship\u003csup\u003e11\u003c/sup\u003e. However, due to the presence of a positive intercept and a fixed inflection point occurring at half of the asymptotic gas production, the LOG model may not be suitable in certain situations\u003csup\u003e12\u003c/sup\u003e. Wang et al.\u003csup\u003e13\u003c/sup\u003e introduced a shape parameter into the model, constructing the Logistic-Exponential (LE) model. This model is more flexible and variable, with a non-fixed inflection point, and eliminates the positive intercept on the y-axis. France et al.\u003csup\u003e14\u003c/sup\u003edecomposed the fractional rate of degradation \u0026micro; into a function related to the square root of time, establishing the Mitscherlich (MIT) model. According to France\u003csup\u003e12\u003c/sup\u003e, the relationship between \u0026micro; and the square root of time may be associated with the diffusion of microbial enzymes. Groot et al.\u003csup\u003e7\u003c/sup\u003e proposed a modified Michaelis\u0026ndash;Menten (MM) model based on the Michaelis\u0026ndash;Menten equation in enzyme kinetics. MM model makes more flexible assumptions about fractional degradation rate, which can be continuously decreased over time or increase initially before decreasing\u003csup\u003e12\u003c/sup\u003e. Similar to the MIT model, the increase in the degradation rate reflects the increased accessibility of substrates due to particle hydration, microbial attachment, and an increase in microbial population, while the decrease in degradation rate may reflect the imposition of chemical and structural constraints\u003csup\u003e12\u003c/sup\u003e.\u003c/p\u003e \u003cp\u003eAdditionally, the gas production process is closely associated with the chemical composition and the physical structure of the substrate. Getachew et al.\u003csup\u003e15\u003c/sup\u003e investigated the relationship between \u003cem\u003ein vitro\u003c/em\u003e rumen gas production and chemical nutrients in 12 kinds of feedstuff comprising 38 samples. They found a negative correlation between crude protein (CP) and gas production (\u003cem\u003eP\u003c/em\u003e\u0026thinsp;\u0026lt;\u0026thinsp;0.001), while non-fiber carbohydrate (NFC) levels were positively correlated with gas production (\u003cem\u003eP\u003c/em\u003e\u0026thinsp;\u0026lt;\u0026thinsp;0.01). It is generally believed that this is because the fermentative capacity of protein is approximately 0.3 times of carbohydrates\u003csup\u003e16\u003c/sup\u003e. Moreover, gas production may decrease due to the reaction between ammonia and carbon dioxide\u003csup\u003e17\u003c/sup\u003e. Different types of carbohydrates produce varying volumes of gas depending on their chemical properties and interactions with other feed components\u003csup\u003e2\u003c/sup\u003e. Compared to the slowly degradable fraction, rapidly degradable carbohydrates produce more propionate, which is a process with lower gas potential compared to acetate\u003csup\u003e18\u003c/sup\u003e. In ruminant nutrition, feedstuff can be simply categorized into energy feed, protein feed, and roughage. Therefore, distinct models should be used based on the nutrient characteristics of each feedstuff.\u003c/p\u003e \u003cp\u003eThough previous studies have investigated the comparison of the goodness of fit of different models, they may suffer from issues such as small sample sizes and undivided feed categories\u003csup\u003e1,11,13,19\u0026ndash;24\u003c/sup\u003e. To address these issues, this study aims to improve the reliability and accuracy of the research by increasing the sample size and categorizing the data according to feedstuff types. Through these methods, the accuracy, precision, and complexity of different models can be compared more effectively, thereby providing more valuable insights for future research.\u003c/p\u003e"},{"header":"Results","content":"\u003cp\u003e \u003cb\u003eFeedstuffs and\u003c/b\u003e \u003cb\u003ein vitro\u003c/b\u003e \u003cb\u003erumen gas production profile\u003c/b\u003e\u003c/p\u003e \u003cp\u003eThe 57 feedstuff materials were categorized into energy feeds, protein feeds, and roughages based on their common use in ruminant production. Descriptive statistics of the nutritional components for these three categories of feedstuff were presented in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e, and gas production profiles were depicted in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e. Energy feeds primarily consisted of grains and brans, characterized by a high starch content (43.31\u0026thinsp;\u0026plusmn;\u0026thinsp;5.03%) and low levels of structural carbohydrates. Protein feeds exhibited high crude protein (CP) content (33.87\u0026thinsp;\u0026plusmn;\u0026thinsp;2.4%) and lower carbohydrate proportions. Additionally, due to the inclusion of various by-product from oil industry, protein feeds had higher levels of crude fat (EE) (6.61\u0026thinsp;\u0026plusmn;\u0026thinsp;0.77%). Roughages mainly comprised hays and straws, containing significant amounts of structural carbohydrates (67.86\u0026thinsp;\u0026plusmn;\u0026thinsp;2.59% NDF, 44.78\u0026thinsp;\u0026plusmn;\u0026thinsp;2.51% ADF, 10.21\u0026thinsp;\u0026plusmn;\u0026thinsp;1.89% ADL).\u003c/p\u003e \u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003eFitting of the gas production profiles\u003c/h2\u003e \u003cp\u003eFigure\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e illustrated the fitting of different models for the three categories of feedstuff. There were significant differences in the shape of the fitting curves for the three categories. Gas production from energy feeds was notably higher than that from the other two categories, and both energy and protein feeds exhibited higher gas production rates than roughages during the same period, reaching the plateau phase earlier. The gas production curves for roughages were the most gradual, reaching the plateau phase later, with relatively lower gas production. Regarding different models, GOM and LOG models shown sigmoidal curves, with similar shapes, and almost completely overlap in energy feed. The LOG model reached the plateau phase earlier compared to other models within the same category of feedstuff, while the GOM model only exhibited similar characteristics in energy and protein feed, with lower gas production after reaching the plateau phase compared to other models. Other models fitted non-sigmoidal curves for the three categories of feedstuff, with the MM model showing the highest final gas production compared to other models within the same category.\u003c/p\u003e \u003cp\u003eThe mean values of Vf, t\u003csub\u003e0.5\u003c/sub\u003e, and \u0026micro;\u003csub\u003e0.5\u003c/sub\u003e predicted by the 8 models for different categories of feedstuff were presented in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e. There were significant differences in gas production parameters among different categories of feedstuff. The Vf for energy feed (61.98\u0026ndash;71.68 ml/200mg) was much higher than that for protein feed (37.63\u0026ndash;45.77 ml/200mg) and roughage (35.67\u0026ndash;46.86 ml/200mg). Across all categories, MM yielded the highest predicted Vf values, and LOG produced the lowest. Predicted t\u003csub\u003e0.5\u003c/sub\u003e values did not show significant differences among different model for any category. However, GOM and LOG consistently predicted significantly higher \u0026micro;\u003csub\u003e0.5\u003c/sub\u003e values compared to other models in all categories.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003eModel comparison\u003c/h2\u003e \u003cp\u003eThe average R\u003csup\u003e2\u003c/sup\u003e for each category of feedstuff fitted by different models were depicted in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e, with all models achieving R\u003csup\u003e2\u003c/sup\u003e values above 0.94. Across different groups, a similar ranking of model performance was observed. Within each group, MM consistently exhibited the highest R\u003csup\u003e2\u003c/sup\u003e values, followed by MIT, while LOG consistently showed significantly lower R\u003csup\u003e2\u003c/sup\u003e compared to other models. In the energy feeds and roughages groups, MIT, LEL, and LE0 did not exhibit significant differences compared to MM, whereas in the protein feed group, MM significantly outperformed all other models. And among all feedstuffs without categorization, the MM model exhibits significantly higher R\u003csup\u003e2\u003c/sup\u003e values compared to other models.\u003c/p\u003e \u003cp\u003eThe results of regressing observed versus predicted gas production were illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e and summarized in Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e. Among all 8 models, EXPL,MIT and MM exhibited non-significant slopes and intercepts across the three categories of feedstuff. In the energy feeds group, significant differences were observed in the slopes compared to 1 for EXP0, GOM, LOG, LE0, and LEL, with significant non-zero intercepts for all models except LEL. In the protein feeds and roughages groups, LE0 and LEL also demonstrated good fitting performance. However, EXP0, GOM, and LOG exhibited significantly non-unity slopes, while EXP0 and LOG had intercepts significantly different from 0. In the uncategorized feedstuffs group, only EXPL and MM exhibited non-significant slopes and intercepts.\u003c/p\u003e \u003cp\u003eError analysis of the 8 models fitting different categories of feedstuff was presented in Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e. Across the four groups, the MAE, RMSEP, and MSEP of different models were consistently similar, with MM or MIT demonstrating the smallest values, followed by LE0 and LEL. Focusing on the MSEP decomposition parts of these models, it was observed that MIT exhibits no systematic bias in four groups and only shows a regression bias of 0.01% when fitting roughages. Apart from MIT, in energy feed, LEL and LE0 also demonstrated low system bias, with LEL exhibiting lower regression bias as well. In protein feed, MM shows minimal systematic bias and regression bias other than MIT. And, all models exhibited system bias and regression bias in roughage. In the group comprising all feedstuffs, LE0 also displayed no bias.\u003c/p\u003e \u003cp\u003eThe AIC and BIC values were calculated for each model fit. And the models were ranked based on these criteria for each sample, as shown in Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e. A higher rank indicated better model performance. MM consistently ranked highest in terms of AIC (BIC) across all four groups. Among energy feeds, MM (MM) exhibited the highest performance, followed by LE0, LEL, and EXPL (LE0, LEL and EXPL) according to AIC (BIC) criteria. In the case of protein feed, MM (MM) outperformed other models, with EXP0, EXPL, and LE0 (EXP0, EXPL and LE0) following in AIC (BIC) ranking. Regarding roughage, MM (MM) ranked highest, followed by EXPL, MIT, and LE0 (LE0, EXPL and EXP0). For the entire group of feedstuffs, MM (MM) was superior, with LE0, EXPL, and LEL trailing behind (EXPL, EXP0 and LE0).\u003c/p\u003e \u003c/div\u003e"},{"header":"Discussion","content":"\u003cp\u003eThe 68 samples selected for this experiment comprised 57 kinds of feedstuff, categorized into three groups commonly used for ruminants: energy feeds, protein feeds, and roughages. The nutritional and gas production data exhibit a wide range and substantial quantity, indicating their representativeness.\u003c/p\u003e \u003cp\u003eThe chemical composition and the physical structure of feedstuff are closely related to gas production curves. It is generally believed that the main sources of gas in \u003cem\u003ein vitro\u003c/em\u003e rumen gas production experiments are carbohydrate degradation and the reaction of volatile fatty acids (VFAs) with substrate buffer solution\u003csup\u003e25,26\u003c/sup\u003e.CP fermentation generated less gas, while EE hardly produces any gas\u003csup\u003e2\u003c/sup\u003e. Structural carbohydrates, being more resistant to degradation, often have a significant impact on both gas volume and gas production rate\u003csup\u003e27\u003c/sup\u003e. This also explains why the final gas production and Vf of most protein and roughage feeds were lower than those of energy feeds. Additionally, roughages typically exhibited flatter gas production curves, longer t\u003csub\u003e0.5\u003c/sub\u003e, and smaller \u0026micro;\u003csub\u003e0.5\u003c/sub\u003e values.\u003c/p\u003e \u003cp\u003eHuhtanen et al.\u003csup\u003e28\u003c/sup\u003e, through a comparison of residuals over time, observed that EXP0 and MM tended to overestimate asymptotic gas volume. This trend appeared to be present in both energy feeds and roughages in this experiment. Huhtanen explained the overestimation of asymptotic gas volume by the EXP0 model as its inability to describe lag phenomena\u003csup\u003e28\u003c/sup\u003e. However, this did not explain why EXP0 does not exhibit higher Vf in protein feed, and why there was no significant difference between EXP0 and EXPL in all feedstuff categories. Considering that Huhtanen's samples were roughages, the overestimation of Vf by EXP0 might be due to its exponential mathematical structure being unsuitable for energy and roughage feedstuffs. Additionally, the decreasing fractional degradation rate of MM in the later stages of fermentation might not be reasonable, which led to the overestimation of asymptotic gas volume\u003csup\u003e21,28\u003c/sup\u003e. Schofield et al.\u003csup\u003e11\u003c/sup\u003e found in their model derivation that the maximum rate (i.e., the curve inflection point) of LOG and GOM models occurs at V\u0026thinsp;=\u0026thinsp;Vf/2 and V\u0026thinsp;=\u0026thinsp;Vf/e. This might explain why \u0026micro;\u003csub\u003e0.5\u003c/sub\u003ewas significantly higher than GOM, while GOM was significantly higher than other models in all feedstuff categories. This was consistent with previous research results\u003csup\u003e13,28\u003c/sup\u003e. Considering that there was no significant difference in t\u003csub\u003e0.5\u003c/sub\u003e between models within each feedstuff category, different models may not exhibit differently in the fractional rate of gas production when fitting the same category of feedstuff.\u003c/p\u003e \u003cp\u003eThe models extended to multi-pool analysis, with the fundamental assumption that the incubated feedstuff was not homogeneous\u003csup\u003e7,29\u003c/sup\u003e. Cone et al.\u003csup\u003e30\u003c/sup\u003e classified gas production into three stages: the initial stage involves the neutral detergent soluble fraction (NDS), the subsequent stage entails NDF fermentation, and the final stage relates to microbial population limitation. Some researchers believe that compared to single-pool analysis, this nonlinear multi-pool analysis increased the number of parameters in the model, potentially compromising its robustness and posing numerous challenges to fitting \u003csup\u003e1,19,31\u003c/sup\u003e. Therefore, this study only selected single-pool models for analysis and comparison.\u003c/p\u003e \u003cp\u003eWhen comparing mathematical models, no single indicator was enough to fully evaluate their performance. Each indicator highlighted a specific aspect of the comparison\u003csup\u003e32,33\u003c/sup\u003e. Therefore, a combination of various statistical analyses must be employed to assess the adequacy of mathematical models properly, along with a rational analysis of the original conceptualization and development purposes of the mathematical models.\u003c/p\u003e \u003cp\u003eIn addition to comparing among the three categories of feedstuffs, this study also included a group containing all feedstuffs to investigate whether it is necessary to differentiate from feedstuff categories when comparing models fitting performance. The results revealed that it exhibited performance distinct from the three groups in terms of Observed versus Predicted (OvP) and error analysis, particularly in relation to changes in the distribution and mean of predicted values. Therefore, using different models for different categories of feedstuff may be necessary, as it would lead to more accurate and realistic results.\u003c/p\u003e \u003cp\u003eThe coefficient of determination (R\u003csup\u003e2\u003c/sup\u003e) serves as a reliable measure of precision, with higher R\u003csup\u003e2\u003c/sup\u003e values indicating greater precision\u003csup\u003e32\u003c/sup\u003e. The MM model exhibited the highest R\u003csup\u003e2\u003c/sup\u003eacross all four groups which surpassed all other models significantly in the protein feeds and the uncategorized group. This finding aligned with the conclusions drawn by most previous studies\u003csup\u003e1,20,23,31\u003c/sup\u003e. After deriving the formulas, France et al.\u003csup\u003e12\u003c/sup\u003e noted that the MM model's assumption regarding gas production rate was the most flexible. Compared to carbohydrates, the energy obtained by rumen microbes from protein fermentation was lower, leading to slower microbial growth\u003csup\u003e16,34\u003c/sup\u003e. Moreover, protein itself yields less gas, and the NH\u003csub\u003e3\u003c/sub\u003e produced during fermentation consumes CO\u003csub\u003e2\u003c/sub\u003e\u003csup\u003e2\u003c/sup\u003e. These variations resulted in irregular and challenging-to-fit gas production curves, which the highly flexible MM model can effectively accommodate. A study found that the GOM model outperforms MM when roughages were used as a substrate\u003csup\u003e13\u003c/sup\u003e. Considering MM's characteristic of decreasing residuals over time, this could be attributed to differences in fermentation time between the study (72h) and the experiments referenced (48h). Additionally, the MIT, LE0, and LEL models also exhibited high R\u003csup\u003e2\u003c/sup\u003e values when fitting energy and roughage feedstuffs, consistent with a few prior studies\u003csup\u003e13,22,23,35,36\u003c/sup\u003e. These findings further underscore the advantage of employing flexible and adaptable curve shapes to enhance model accuracy.\u003c/p\u003e \u003cp\u003eGenerally, accuracy is considered the most important criterion as it measures the model's ability to predict true values\u003csup\u003e32\u003c/sup\u003e. The regression estimates of intercept and slope for OvP values serve as reliable indicators of accuracy, with intercepts and slopes closer to 0 and 1 indicating higher levels of precision\u003csup\u003e32\u003c/sup\u003e. Apart from the flexible and adaptable MIT and MM models, the introduction of lag time in EXPL relative to EXP0 also contributed to higher accuracy. Lag time can be attributed to hydration, removal of digestive inhibitors, or microbial substrate attachment\u003csup\u003e7,12\u003c/sup\u003e. Although lag time may not fully describe the phenomenon of hysteresis, it provides a simple quantitative measure of delayed effects. Allen and Mertens\u003csup\u003e37\u003c/sup\u003e proposed a two-step model describing the digestion process: the first step involves the transformation of substrate from an unavailable to an available form (i.e., the lag pool concept), and the second step represents substrate digestion. While theoretically more accurate than discrete lag models, lag pool models did not fit the data better\u003csup\u003e28\u003c/sup\u003e, and were not widely used due to their complexity. GOM and LOG inevitably introduce an initial gas volume (V0) during the assumption stage\u003csup\u003e11\u003c/sup\u003e, whereas V0 is clearly zero at the start of the experiment. According to Schofield et al.\u003csup\u003e11\u003c/sup\u003e, the error in V0 (i.e., intercept on the y-axis) is most severe for GOM and LOG when the digestion rate is low and lag time is short, which evidently leads to poorer accuracy when fitting energy feed. However, the intercept decreased when fitting protein and roughage feedstuffs, which validated the findings of Schofield et al.\u003csup\u003e11\u003c/sup\u003e. The slopes of LE0 and LEL were significantly different from 1 when fitting energy feed. LE0 and LEL might underestimate the gas production in the later stages of energy feed fermentation. This was consistent with the significantly lower Vf for LE0 and LEL when fitting energy feeds compared to half of the models.\u003c/p\u003e \u003cp\u003eRMSEP and MAE are two standard indicators for evaluating the accuracy of models. Under different error distributions, each has its own advantages and disadvantages\u003csup\u003e38\u003c/sup\u003e. This study presented both indicators to provide a comprehensive assessment. MSEP is perhaps the most common and reliable estimate for evaluating the accuracy of model predictions, and further decomposition of MSEP allows for the analysis of model adequacy by identifying the sources of variation\u003csup\u003e32\u003c/sup\u003e. If there is a significant system bias, the model's predicted mean may be larger or smaller than the observed mean, and if there is significant regression bias, it indicates potential inadequacies in the structural equation of the model in describing the analyzed data\u003csup\u003e39\u003c/sup\u003e. In this study, the results of linear fitting for Observed versus Predicted (OvP) and error analysis were inconsistent in some aspects. This inconsistency might stem from the different principles underlying their analyses, with the former focusing on the distribution of predicted values and the latter on the mean of predicted values. This experiment found that fitting different fermentation substrates does not affect the level of errors but may result in different sources of errors. It is generally believed that the fractional rate of gas production \u0026micro; of substrates during fermentation is not constant. An initial increase in \u0026micro; reflects particle hydration, microbial attachment, and an increase in microbial numbers, while a decrease in \u0026micro; may indicate limitations in chemical nutrients and structure within the substrate\u003csup\u003e12\u003c/sup\u003e. Models like LE0, LEL, MIT, and MM have the ability to allow \u0026micro; to vary freely over time, leading to smaller errors when fitting the three types of feedstuff\u003csup\u003e7,13,14\u003c/sup\u003e. Camila et al. \u003csup\u003e1\u003c/sup\u003e found that the MM model had the lowest RMSEP when fitting corn silage using various models. Dhanoa et al. \u003csup\u003e19\u003c/sup\u003e, in comparing the performance of different models after fitting roughages and silage feedstuffs, found that the residual mean squares of MM and MIT were lower than those of other models. Wang et al. \u003csup\u003e13\u003c/sup\u003e further validated the models and found that the RMSEP of LE0 and LEL was lower than that of most models. MIT exhibited near-zero SB and RB in all three categories of feedstuff, while the performance of LE0, LEL, and MM varied depending on the categories of feedstuff being fitted. This may be attributed to the different assumptions underlying LE0 and LEL (microbial growth\u003csup\u003e13\u003c/sup\u003e), MIT (degradation by surface erosion and complete invasion\u003csup\u003e14\u003c/sup\u003e), and MM (formation of microbial film\u003csup\u003e7\u003c/sup\u003e). For energy and roughage feedstuffs, despite its advantages, the MM model exhibited systematic bias and regression bias. Considering the previous discussion, it seems that the MM model may have overestimated the gas production values\u003csup\u003e28\u003c/sup\u003e. This is an important factor to consider when using the MM model for energy and roughage feedstuffs, as it could impact the accuracy of the results.\u003c/p\u003e \u003cp\u003eIn addition to accuracy and precision, complexity directly impacts the performance of the model on both training and unseen data, affecting its interpretability and fitting cost. AIC and BIC serve as indicators that balance complexity and precision. For small datasets, the performance of AIC is similar to or even better than BIC, while BIC becomes the preferred criterion for selecting the correct model, for datasets of moderate size \u003csup\u003e40\u003c/sup\u003e. In this study, both AIC and BIC were chosen to evaluate the complexity and precision of the models, because each sample comprised approximately 40 to 100 data points, falling the range between small and moderate sample sizes\u003csup\u003e41\u003c/sup\u003e. Among the models selected in this study, EXP0 has 2 parameters, while LEL and MIT have 4 parameters, with the rest having 3 parameters. Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e revealed that the model with 2 parameters, EXP0, did not exhibit significant advantages, ranking 3.51 (3.10) in AIC (BIC) only for the protein feed where other models struggled to fit. On the other hand, the 3-parameter MM model ensured model precision with a slight increase in complexity. This finding aligned with previous research results\u003csup\u003e13,23\u003c/sup\u003e. MIT outperformed other models in roughages, indicating that its theoretical assumptions regarding microbial enzyme diffusion and surface erosion are better suited for difficult-to-degrade roughage\u003csup\u003e14\u003c/sup\u003e.\u003c/p\u003e \u003cp\u003eThe MM model appears to demonstrate high precision, moderate to high accuracy, and low complexity across all three categories of feed. Therefore, the MM model is a good choice for studies that do not differentiate between feed categories or include multiple categories of feed. However, considering the MM model\u0026rsquo;s overestimation of asymptotic gas production, it is advisable to choose a model based on its specific performance for different feeds. If only accuracy is considered, models such as LEL, MIT and MM for energy feed, MIT and MM for protein and roughage feedstuff have performed well. However, it is unrealistic to rely solely on an accurate but imprecise model for predictions.\u003c/p\u003e"},{"header":"Conclusions","content":"\u003cp\u003eIn summary, due to the involvement of different fermentation processes, the chemical composition and the physical structure of feed, could influence the gas production process, resulting in gas production curves with different characteristics. EXP0 and MM tend to overestimate Vf for energy and roughage feeds. In studies where feedstuff categories are not distinguished or multiple feedstuffs categories are included, the MM model proves to be a good choice. When fitting various categories of feedstuff, the following conclusions were drawn:(1) For energy feed, MM showed relatively highest R\u003csup\u003e2\u003c/sup\u003e among the 8 models (\u003cem\u003eP\u003c/em\u003e\u0026thinsp;\u0026lt;\u0026thinsp;0.05). The slope and intercept in OvP did not significantly differ from 1 and 0 (\u003cem\u003eP\u003c/em\u003e\u0026thinsp;\u0026lt;\u0026thinsp;0.05). MAE, RMSEP, and MSE were lowest (1/8), but there was a presence of systematic bias and regression bias. The AIC (BIC) ranking was 2.29 (2.29), exceeding all other models. LEL performs worse on certain indicators compared with MM, but exhibits no bias, which should be taken into consideration. (2) For protein feed, MM's R\u003csup\u003e2\u003c/sup\u003e was significantly higher than other models (\u003cem\u003eP\u003c/em\u003e\u0026thinsp;\u0026lt;\u0026thinsp;0.05). The regression estimates of intercept and slope in OvP had no significant difference from 0 and 1, respectively (\u003cem\u003eP\u003c/em\u003e\u0026thinsp;\u0026lt;\u0026thinsp;0.05). MAE, RMSEP, and MSE were the lowest (1/8) with almost no systematic bias and regression bias. The AIC (BIC) ranking was 1.62 (1.62). (3) For roughage, MIT showed relatively higher R\u003csup\u003e2\u003c/sup\u003e (2/8) with no significant difference from the highest value (\u003cem\u003eP\u003c/em\u003e\u0026thinsp;\u0026lt;\u0026thinsp;0.05). The intercept and slope in OvP had no significant difference from 0 and 1, respectively (\u003cem\u003eP\u003c/em\u003e\u0026thinsp;\u0026lt;\u0026thinsp;0.05). MAE, RMSEP, and MSE were low (1/8) with no systematic bias and regression bias. The AIC ranks third only to MM and EXPL. Due to limitations in sample quantity and diversity, the categorization of feeds in this study may not be sufficiently specified or comprehensive. In the future, collecting more extensive and diverse gas production data will enable a better exploration of the suitability of models for different feed categories.\u003c/p\u003e"},{"header":"Materials and Methods","content":"\u003cdiv id=\"Sec8\" class=\"Section2\"\u003e\n \u003ch2\u003eAnimals and diets\u003c/h2\u003e\n \u003cp\u003eAll animal procedures used in this study were approved by the Animal Care and Use Committee of China Agricultural University (Approval No.AW81404202-1-9), and the experimental procedures used in this study were in accordance with the university\u0026rsquo;s guidelines for animal research. The rumen fluid used in this experiment was collected from three Angus steers of similar weight (400.00\u0026thinsp;\u0026plusmn;\u0026thinsp;50.00 kg, mean\u0026thinsp;\u0026plusmn;\u0026thinsp;SD) with permanent rumen fistulas. The experimental animals had ad libitum access to water and were fed twice daily at 8:00 and 17:00. The diets were formulated based on the body weight and nutritional requirements to meet 1.3 times maintenance needs of beef cattle according to NASEM 2016\u003csup\u003e42\u003c/sup\u003e(Table \u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003e).\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec9\" class=\"Section2\"\u003e\n \u003ch2\u003eFeed samples preparation and nutrient analysis\u003c/h2\u003e\n \u003cp\u003eA total of 57 kinds of feedstuffs, which were commonly used for ruminants, were collected within China and categorized into three groups, with some feedstuffs comprising two varieties, resulting in a total of 68 samples. 57 feedstuffs were categorized into three groups commonly used for ruminants. Among them, there were 20 types of energy feed, totaling 24 samples, including: barley (2), beet pulp, buckwheat, corn, corn husk, dried foxnut, flaked corn, highland barley, millet, millet bran, oat, oat bran, rice, rice bran (2), rice bran meal, rye grain, sorghum (2), soybean hulls, wheat, and wheat bran (2). For protein feed, there were 18 types totaling 22 samples including: bean curd residue (2), brewer\u0026apos;s grains, cottonseed meal, DDGS (2), distiller\u0026apos;s grain, flax seed meal (2), linseed cake, mulberry leaf, palm meal, paper mulberry leaf, peanut meal, rapeseed cake, rapeseed meal, sesame cake, soy sauce residue, soybean meal, sunflower meal (2), and vinegar residue. For roughage feedstuffs, there were 19 kinds totaling 22 samples including: alfalfa, apple pomace, Chinese wildrye, corncob, dried corn stalk (2), highland barley stalk, oat grass (2), oat wheat straw, peanut seedling, peanut shell, pear pomace, ryegrass, sorghum bran, sorghum stalk, soybean straw, straw, sugarcane bagasse, sweet potato seedling, and wheat straw (2). The samples were dried in a 65\u0026deg;C air oven for 48 hours, followed by grinding through a 0.5-millimeter sieve, for chemical analysis and in vitro rumen gas production. Dry matter (DM), ash, and starch were determined following the AOAC \u003csup\u003e43\u003c/sup\u003e. Nitrogen content was determined using the Dumas combustion method with nitrogen analyzer (Rapid III, Elementar, Germany), and crude protein (CP) content was calculated as nitrogen content times 6.25. Ether extract (EE) was determined by an automatic fat analyzer (XT15, ANKOM Technology, USA).Neutral detergent fiber (NDF), acid detergent fiber (ADF), and acid detergent lignin (ADL) were determined following the method of Van Soest et al.\u003csup\u003e44\u003c/sup\u003e with automatic fiber analyzer (A2000i, ANKOM Technology, USA).\u003c/p\u003e\n \u003cp\u003e\u003cstrong\u003eIn vitro\u003c/strong\u003e \u003cstrong\u003erumen gas production\u003c/strong\u003e\u003c/p\u003e\n \u003cp\u003e\u003cem\u003eIn vitro\u003c/em\u003e rumen gas production was conducted following the method outlined by Menke et al.\u003csup\u003e45\u003c/sup\u003e. Accurately weighing 0.200g (DM) air-dried feedstuff sample, and placing at the bottom of a glass syringe. In the morning of the experiment day, prior to feeding, the contents and rumen fluid were uniformly collected from the steer. After filtering through four layers of cheesecloth, the mixed rumen fluid were placed in a preheated thermos bottle at 39\u0026deg;C. The rumen fluid was mixed with CO\u003csub\u003e2\u003c/sub\u003e-saturated rumen buffer solution at a ratio of 1:2 to prepare rumen inoculum. Then, 30 mL of the inoculum was added to the glass syringe using an automatic pipette and placed in a 39\u0026deg;C water baths shaker with 60 revolutions per minute.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec10\" class=\"Section2\"\u003e\n \u003ch2\u003eModels and curve-fitting\u003c/h2\u003e\n \u003cp\u003eTable \u003cspan class=\"InternalRef\"\u003e6\u003c/span\u003e presented 8 commonly used single-pool models describing \u003cem\u003ein vitro\u003c/em\u003e rumen gas production kinetics. The exponential model included two types: EXP0 (without lag phase) and EXPL (with lag phase), which were among the most used models in \u003cem\u003ein vitro\u003c/em\u003e rumen gas production. The Gompertz (GOM) and Logistic (LOG) models, initially proposed in microbiology, were first applied to \u003cem\u003ein vitro\u003c/em\u003e rumen gas production kinetics by Schofield et al\u003csup\u003e11\u003c/sup\u003e. Wang et al. \u003csup\u003e13\u003c/sup\u003e introduced the shape parameter d to the Logistic (LOG) model, developing a more flexible Logistic-Exponential (LE) model. France et al. \u003csup\u003e14\u003c/sup\u003eincorporated the concept of degradation by surface erosion and by complete invasion into mathematical formulas, resulting in the Mitscherlich (MIT) model. The Michaelis-Menten (MM) model originated in enzymatic kinetics, and Groot et al.\u003csup\u003e7\u003c/sup\u003e were the first to apply it to describe \u003cem\u003ein vitro\u003c/em\u003e rumen gas production kinetics.\u003c/p\u003e\n \u003cp\u003eTable \u003cspan class=\"InternalRef\"\u003e6\u003c/span\u003e provided the time (t\u003csub\u003e0.5\u003c/sub\u003e) and fractional rate of gas production (\u0026micro;\u003csub\u003e0.5\u003c/sub\u003e) for each model at V\u0026thinsp;=\u0026thinsp;Vf/2. Here, t\u003csub\u003e0.5\u003c/sub\u003ewas calculated as the t when V\u0026thinsp;=\u0026thinsp;Vf/2, and \u0026micro;\u003csub\u003e0.5\u003c/sub\u003ewas the value obtained by substituting t\u003csub\u003e0.5\u003c/sub\u003e into dV/dt/V.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec11\" class=\"Section2\"\u003e\n \u003ch2\u003eStatistical analysis\u003c/h2\u003e\n \u003cp\u003eIn R version 4.3.2 (R Core Team, Vienna, Austria), each sample was fitted using the nonlinear least squares method via the NLS function. The stats package was used to compute the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) for the fits. Gas production attributes (Vf, t0.5, \u0026micro;0.5) for each category feedstuff were compared across different models using repeated measures one-way ANOVA with Greenhouse-Geisser correction. Bar charts were plotted using the ggplot2 package. The determination coefficient (R2) for each feedstuff category was calculated separately, and comparisons were made using repeated measures one-way ANOVA with Greenhouse-Geisser correction. R2 was calculated as:\u003c/p\u003e\n \u003cdiv id=\"Equa\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equa\" name=\"EquationSource\"\u003e$${\\text{R}}^{\\text{2}}\\text{=1-}\\frac{\\text{\u0026sum;}{\\text{(}{\\text{y}}_{\\text{i}}\\text{-}\\widehat{{\\text{y}}_{\\text{i}}}\\text{)}}^{\\text{2}}}{\\sum \\text{}{\\text{(}{\\text{y}}_{\\text{i}}\\text{-}\\stackrel{\\text{-}}{\\text{y}}\\text{)}}^{\\text{2}}}$$\u003c/div\u003e\n \u003c/div\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003eHere, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\text{y}}_{\\text{i}}\\)\u003c/span\u003e\u003c/span\u003e represents the actual value of the 𝑖 -th observed data,\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\widehat{{\\text{y}}_{\\text{i}}}\\)\u003c/span\u003e\u003c/span\u003e represents the predicted value of the model for the𝑖 -th observed data, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\stackrel{\\text{-}}{\\text{y}}\\)\u003c/span\u003e\u003c/span\u003e denotes the mean value of the observed data.\u003c/p\u003e\n \u003cp\u003eUsing GraphPad Prism 9.5.0 (GraphPad Software, San Diego, CA, USA), linear regression was performed on the observed gas production values (x-axis) versus the predicted values (y-axis) for each feedstuff category. The intercepts and slopes of each model were evaluated, and fitted equations were plotted. Statistical analysis was conducted to test the significance of the regression parameters, examining the hypotheses \u0026quot;slope\u0026thinsp;=\u0026thinsp;1\u0026quot; and \u0026quot;intercept\u0026thinsp;=\u0026thinsp;0\u0026quot;.\u003c/p\u003e\n \u003cp\u003eError analysis refers to the systematic analysis and interpretation of the differences between model predictions and actual observed values. The following methods were used to assess the errors of the model: Mean Absolute Error (MAE), Root Mean Squared Error of Prediction (RMSEP), and Mean Squared Error of Prediction (MSEP). MSEP is further decomposed into system bias (SB), regression bias (RB), and random error (RE)\u003csup\u003e32,46\u003c/sup\u003e.The calculation formulas were as follows:\u003c/p\u003e\n \u003cdiv id=\"Equb\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equb\" name=\"EquationSource\"\u003e$$\\text{SB=}{\\left(\\frac{\\text{\u0026sum;}{\\text{y}}_{\\text{i}}}{\\text{n}}\\text{-}\\frac{\\text{\u0026sum;}\\widehat{{\\text{y}}_{\\text{i}}}}{\\text{n}}\\right)}^{\\text{2}}$$\u003c/div\u003e\n \u003c/div\u003e\n \u003cdiv id=\"Equc\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equc\" name=\"EquationSource\"\u003e$$\\text{RB=}{\\text{(}{\\text{S}}_{\\widehat{\\text{y}}}\\text{-r\u0026times;}{\\text{S}}_{\\text{y}}\\text{)}}^{\\text{2}}$$\u003c/div\u003e\n \u003c/div\u003e\n \u003cdiv id=\"Equd\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equd\" name=\"EquationSource\"\u003e$$\\text{RE=(1-}{\\text{r}}^{\\text{2}}\\text{)\u0026times;}{\\text{S}}_{\\text{y}}$$\u003c/div\u003e\n \u003c/div\u003e\n \u003cdiv id=\"Eque\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Eque\" name=\"EquationSource\"\u003e$$\\text{r=}\\frac{\\sum \\frac{（{\\text{y}}_{\\text{i}}\\text{-}\\frac{\\text{\u0026sum;}{\\text{y}}_{\\text{i}}}{\\text{n}}）\\text{\u0026times;(}\\widehat{{\\text{y}}_{\\text{i}}}\\text{-}\\frac{\\text{\u0026sum;}\\widehat{{\\text{y}}_{\\text{i}}}}{\\text{n}}\\text{)}}{{\\text{S}}_{\\text{y}}\\text{\u0026times;}{\\text{S}}_{\\widehat{\\text{y}}}}}{\\text{n}}$$\u003c/div\u003e\n \u003c/div\u003e\n \u003cp\u003eWhere \u003cem\u003en\u003c/em\u003e represents the number of data points, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\text{S}}_{\\text{y}}\\)\u003c/span\u003e\u003c/span\u003e denotes the variance of the observed data, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\text{S}}_{\\widehat{\\text{y}}}\\)\u003c/span\u003e\u003c/span\u003e represents the variance of the model-predicted data.\u003c/p\u003e\n\u003c/div\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eAcknowledgments\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eWe are grateful to the staff of Beef Cattle Research Center for their help with sampling and laboratory analyses.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eStudy Funding\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThis work was supported by the National Key R\u0026amp;D Program of China (2022YFA1304204) and the Government Purchase Service (16200158, 16190050).\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthor contributions statement\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eJW designed the study, analyzed the data, and wrote the manuscript. ZZ and YC performed the experiments and sample analyses.BY, SL, LR, NL, XZ and WL conducted the trial,. XY, ZD and SY assisted with the manuscript preparation. ZZ, HW and QM revised the manuscript and provided experimental guidance. All authors read and approved the final manuscript.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAdditional information\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe authors declare no real or perceived conflicts of interest.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eEthical approval\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eAll animal procedures used in this study were approved by the Animal Care and Use Committee of China Agricultural University (Approval No.AW81404202-1-9).\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eData availability\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe datasets used and/or analysed during the current study are available from the corresponding author upon reasonable request.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eDa Silva Zornitta, C.\u003cem\u003e et al.\u003c/em\u003e, Kinetics of In Vitro Gas Production and Fitting Mathematical Models of Corn Silage. \u003cem\u003eFermentation\u003c/em\u003e \u003cstrong\u003e7\u003c/strong\u003e 298 (2021).\u003c/li\u003e\n\u003cli\u003eAmanzougarene, Z. \u0026amp; Fondevila, M., Fitting of the In Vitro Gas Production Technique to the Study of High Concentrate Diets. \u003cem\u003eANIMALS\u003c/em\u003e \u003cstrong\u003e10\u003c/strong\u003e 1935 (2020).\u003c/li\u003e\n\u003cli\u003eCone, J. 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R., Avalia\u0026ccedil;\u0026atilde;o dos modelos log\u0026iacute;stico bicompartimental e de Gompertz na estimativa da din\u0026acirc;mica de fermenta\u0026ccedil;\u0026atilde;o ruminal in vitro do farelo e da torta de baba\u0026ccedil;u (Orbignya martiana). \u003cem\u003eArquivo Brasileiro de Medicina Veterin\u0026aacute;ria e Zootecnia\u003c/em\u003e \u003cstrong\u003e63\u003c/strong\u003e (2011).\u003c/li\u003e\n\u003cli\u003eCone, J. W., van Gelder, A. H. \u0026amp; Driehuis, F., Description of gas production profiles with a three-phasic model. \u003cem\u003eANIM FEED SCI TECH\u003c/em\u003e \u003cstrong\u003e66\u003c/strong\u003e 31 (1997).\u003c/li\u003e\n\u003cli\u003eGurgel, A. L. C.\u003cem\u003e et al.\u003c/em\u003e, Mathematical models to adjust the parameters of \u003cem\u003ein vitro\u003c/em\u003e cumulative gas production of diets containing preserved Gliricidia. \u003cem\u003eCi\u0026ecirc;ncia Rural\u003c/em\u003e \u003cstrong\u003e51\u003c/strong\u003e (2021).\u003c/li\u003e\n\u003cli\u003eTedeschi, L. O., Assessment of the adequacy of mathematical models. \u003cem\u003eAGR SYST\u003c/em\u003e \u003cstrong\u003e89\u003c/strong\u003e 225 (2006).\u003c/li\u003e\n\u003cli\u003eGREEN, I. R. A. \u0026amp; STEPHENSON, D., Criteria for comparison of single event models. \u003cem\u003eHydrological Sciences Journal\u003c/em\u003e \u003cstrong\u003e31\u003c/strong\u003e 395 (1986).\u003c/li\u003e\n\u003cli\u003eRussell, J. B., Sniffen, C. J. \u0026amp; Van Soest, P. 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A., Methods for dietary fiber, neutral detergent fiber, and nonstarch polysaccharides in relation to animal nutrition. \u003cem\u003eJ DAIRY SCI\u003c/em\u003e 3583 (1991).\u003c/li\u003e\n\u003cli\u003eH, M. K.\u003cem\u003e et al.\u003c/em\u003e, The estimation of the digestibility and metabolizable energy content of ruminant feed stuffs from the gas production when they are incubated with rumen liquor in vitro. \u003cem\u003eJournal of Agricultural Science (Cambridge)\u003c/em\u003e \u003cstrong\u003e93\u003c/strong\u003e 217 (1979).\u003c/li\u003e\n\u003cli\u003eKohn, R. A., Kalscheur, K. F. \u0026amp; Hanigan, M., Evaluation of Models for Balancing the Protein Requirements of Dairy Cows. \u003cem\u003eJ DAIRY SCI\u003c/em\u003e\u003cstrong\u003e81\u003c/strong\u003e 3402 (1998).\u003c/li\u003e\n\u003c/ol\u003e"},{"header":"Tables","content":"\u003cp\u003eTables 1 to 6 are available in the Supplementary Files section.\u003c/p\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"in vitro gas production, rumen fermentation, mathematical model, model evaluation","lastPublishedDoi":"10.21203/rs.3.rs-4413816/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4413816/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003e\u003cem\u003eIn vitro\u003c/em\u003e rumen gas production experiment was conducted with 57 kinds of feedstuff, which were categorized into energy feed, protein feed, and roughage, collected within China. Eight mathematical models were employed to describe the kinetics of \u003cem\u003ein vitro\u003c/em\u003e rumen gas production. The results found that for energy feeds, protein feeds, and roughages, respectively, MM or Logistic-Exponential with lag (LEL), MM, and Mitscherlich (MIT) exhibited the highest or shown no significant difference compared to the highest coefficient of determination (R\u003csup\u003e2\u003c/sup\u003e) (\u003cem\u003eP\u003c/em\u003e\u0026lt; 0.05) for all categories of feed. Furthermore, regression estimation of intercept and slope for regression estimates of intercept and slope for Observed versus Predicted of aforementioned models shown no significant difference from 0 and 1, respectively (\u003cem\u003eP\u003c/em\u003e\u0026lt; 0.05), except LEL for energy feed. Mean absolute error (MAE), root mean squared error of prediction (RMSEP), mean squared error of prediction (MSEP) of those models were relatively lower, with minimal systematic bias and regression bias. Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) rankings were higher compared with other models. Given these results, in studies where feedstuff categories are not distinguished or multiple feedstuffs categories are included, the MM model proves to be a good choice. MM or LEL was considered to better fit energy foodstuffs. The MM model was the optimal choice for fitting protein foodstuffs. MIT provided the best accuracy and moderate precision when fitting roughages.\u003c/p\u003e","manuscriptTitle":"Evaluation of different mathematical models on fitting the in vitro gas production parameters in beef cattle","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-05-24 11:27:27","doi":"10.21203/rs.3.rs-4413816/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2024-09-30T06:37:15+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2024-09-25T14:12:32+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"191318273012887514099814466032108908463","date":"2024-09-19T11:57:54+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2024-06-27T18:51:13+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"227553441554143171425788301841818953197","date":"2024-06-07T14:53:01+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"198231583797202171769303870943882509913","date":"2024-05-21T11:57:34+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2024-05-21T11:51:04+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2024-05-21T11:47:51+00:00","index":"","fulltext":""},{"type":"editorInvited","content":"","date":"2024-05-16T08:53:26+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2024-05-16T08:00:04+00:00","index":"","fulltext":""},{"type":"submitted","content":"Scientific Reports","date":"2024-05-13T14:16:58+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"8c1f12a1-26fd-47b4-a7e2-600dec5646c9","owner":[],"postedDate":"May 24th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[{"id":32309632,"name":"Biological sciences/Biological techniques/Biological models/Gastrointestinal models"},{"id":32309633,"name":"Biological sciences/Zoology"}],"tags":[],"updatedAt":"2025-03-03T15:59:21+00:00","versionOfRecord":{"articleIdentity":"rs-4413816","link":"https://doi.org/10.1038/s41598-025-90189-8","journal":{"identity":"scientific-reports","isVorOnly":false,"title":"Scientific Reports"},"publishedOn":"2025-02-24 15:56:55","publishedOnDateReadable":"February 24th, 2025"},"versionCreatedAt":"2024-05-24 11:27:27","video":"","vorDoi":"10.1038/s41598-025-90189-8","vorDoiUrl":"https://doi.org/10.1038/s41598-025-90189-8","workflowStages":[]},"version":"v1","identity":"rs-4413816","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-4413816","identity":"rs-4413816","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}