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Equations to describe gas production profiles, obtained using manual or automated systems for in vitro fermentation of ruminant feeds, were derived from first principles by considering a simple three-pool scheme. The pools represented were the potentially degradable and undegradable feed fractions, and accumulated gases. The equations derived and investigated mathematically were the generalized Mitscherlich, generalized Michaelis-Menten, Gompertz, and logistic. They were obtained by allowing the fractional rate of degradation to vary with time. The equations permit the extent of ruminal degradation (hence the supply of microbial protein to the duodenum) to be evaluated, thus linking the gas production technique to animal production. Rumen: Gas production: Mathematical models

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Five lactating dairy cows with a permanent cannula in the rumen were given (kg DM/d) a normal diet (7.8 concentrates, 5.1 hay) or a low-roughage (LR) diet (11.5 concentrates, 1.2 hay) in two meals daily in a two-period crossover design. Milk fat (g/kg) was severely reduced on diet LR. To measure rates of production of individual volatile fatty acids (VFA) in the rumen, 0.5 mCi 1-(14)C-acetic acid, 2-(14)C-propionic acid, or 1-(14)C-n-butyric acid were infused into the rumen for 22 h at intervals of 2 to 6 d; rumen samples were taken over the last 12 h. To measure rumen volume, we infused Cr-EDTA into the rumen continuously, and polyethylene glycol was injected 2 h before the morning feed. Results were very variable, so volumes measured by rumen emptying were used instead. Net production of propionic acid more than doubled on LR, but acetate and butyrate production was only numerically lower. Net production rates pooled across both diets were significantly related to concentrations for each VFA. Molar proportions of net production were only slightly higher than molar proportions of concentrations for acetate and propionate but were lower for butyrate. The net energy value (MJ/d) of production of the three VFA increased from 89.5 on normal to 109.1 on LR, equivalent to 55 and 64% of digestible energy, respectively. Fully interchanging, three-pool models of VFA C fluxes are presented. It is concluded that net production rates of VFA can be measured in non-steady states without the need to measure rumen volumes.

Previous attempts to apply statistical models, which correlate nutrient intake with methane production, have been of limited value where predictions are obtained for nutrient intakes and diet types outside those used in model construction. Dynamic mechanistic models have proved more suitable for extrapolation, but they remain computationally expensive and are not applied easily in practical situations. The first objective of this research focused on employing conventional techniques to generate statistical models of methane production appropriate to United Kingdom dairy systems. The second objective was to evaluate these models and a model published previously using both United Kingdom and North American data sets. Thirdly, nonlinear models were considered as alternatives to the conventional linear regressions. The United Kingdom calorimetry data used to construct the linear models also were used to develop the three nonlinear alternatives that were all of modified Mitscherlich (monomolecular) form. Of the linear models tested, an equation from the literature proved most reliable across the full range of evaluation data (root mean square prediction error = 21.3%). However, the Mitscherlich models demonstrated the greatest degree of adaptability across diet types and intake level. The most successful model for simulating the independent data was a modified Mitscherlich equation with the steepness parameter set to represent dietary starch-to-ADF ratio (root mean square prediction error = 20.6%). However, when such data were unavailable, simpler Mitscherlich forms relating dry matter or metabolizable energy intake to methane production remained better alternatives relative to their linear counterparts.

The extent to which dietary components are fermented in the rumen is a function of both rate of fermentation and residence time in the rumen. The latter, usually expressed as the mean retention time (i.e. the reciprocal of the fractional outflow rate; MRT), can be determined from the decrease in the concentration of a non-absorbable marker in rumen digesta after an intraruminal dose of marker. This technique, whilst generally satisfactory for solute markers, is less reliable for particulate markers because of difficulties in obtaining representative samples of rumen digesta. The faecal marker excretion technique (Grovum & Williams, 1973) overcomes the problem of representative sampling and also has the added advantage that fistulated animals are not necessarily required. It is based on the fact that the pattern of marker excretion in the faeces after an intraruminal dose of marker reflects the cumulative effects of marker residence time in the various sections of the digestive tract. Provided a satisfactory mathematical description of the excretion curve can be achieved and the component parts identified, or at least that part relating to the rumen, the MRT in the rumen can be obtained. Blaxter et al. (1956) suggested that the ruminant gut is essentially composed of two mixing compartments and a tubular compartment, and that digesta flow can be described by a model consisting of two exponential terms and a time delay. Subsequently, Grovum & Williams (1973) used this model to describe the change in marker concentration in sheep faeces following an intraruminal dose of marker and showed that the longer MRT was associated with the rumen. However, other workers (e.g. In their paper on the theoretical considerations and computer simulation of digesta passage, Grovum & Phillips (1973) concluded that 'a poor fit between the observed concentration of marker and the predicted values for the two-pool model may indicate that a model with two compartments is not descriptive of passage of marker through the gut. Thus new models and new equations can be sought that are more appropriate.' In the present paper an alternative model to describe faecal marker excretion is proposed and was derived by considering digesta flow as a multicompartmental exponential process. It consists of a multiplicative equation containing an exponential term and a doubleexponential term :, available at https://www.cambridge.org/core/terms. https://doi

Extensive proliferation and death of cells in the mammary gland occur during pregnancy and lactation. In this study, a mechanistic model was developed that yielded a single equation to describe the pattern of mammary growth of mammals throughout pregnancy and lactation. The model contains a single pool, which is the cell population of the mammary gland; one influx, representing cell proliferation; and one efflux, representing cell death. The parameters of the equation lend themselves to direct physiological interpretation. The model fitted data on mammary gland DNA adequately and can be related to current knowledge on factors and inhibitors of mammary gland growth. A unique definition of the parameters of the model can be difficult because of the high degree of variation among animals, an improper number of observations, or timing, as indicated by analyses of simulated data. The model can also be applied to the study of the entire lactation curve. The widely applied gamma equation and the equation that was developed in this study were compared using weekly production data from dairy cows. The new model performed well, particularly when a sharp peak in milk production occurred. The model has the advantage of providing, for the first time, a simple biological description of the lactation curve that can be used to discriminate changes in lactational performance that are associated with experimental treatments.