Effects of size of ingestively masticated fragments of plant tissues on kinetics of digestion of NDF.

Ingestively masticated fragments were collected and sized via sieving. Different sizes of esophageal masticate and ruminal digesta fragments, and ground fragments of larger masticated pieces were incubated in vitro, and undigested NDF remaining at intervals of up to 168 h of incubation was determined. The ruminal age-dependent time delay (tau) for onset of digestion of NDF was positively correlated (P < 0.004) with the mean sieve aperture estimated to retain 50% of the fragments between successive sieve apertures (MRA). Degradation rate of potentially degradable NDF (PDF) and level of indigestible NDF were not related (P > 0.10) to MRA of masticated and ground fragments. Estimates of tau were positively related to MRA, with slopes of bermudagrass < corn silage < ruminal fragments of corn silage. It was concluded that fragment size-, and consequently, ruminal age-dependent onset of PDF degradation of a mixture of different fragment sizes results in an age-dependent rate of degradation of the more rapidly degrading of two subentities of PDF. Models are proposed that assume a tau before onset of simultaneous degradation of PDF from two pools characterized as having gamma-modeled age-dependency and age-constant rates. The ruminal age-dependent pool seems to be associated with the faster-degrading pool, and its rate parameter increases with range in MRA in the population of fragments. Conceptually, the ruminal age-dependent rate parameter for PDF degradation seems to represent a composite of several effects: 1) effects of the size-dependent tau; 2) range in MRA of the population of ingestively masticated fragments; and 3) subentities of PDF that degrade via more rapid age-dependent rates compared with subentities of PDF that degrade via age-constant rates. The estimated fractional rates of ruminative comminution of ingestively masticated fragments (0.060 to 0.075/h) were of a magnitude similar to the mean fractional rates of PDF digestion (0.030 to 0.085/h), which implies that ruminative comminution may be first-limiting to fractional rate of PDF digestion. The in vivo roles of ingestive and ruminative mastication of fragments on PDF degradation must be considered in any kinetic system for estimating PDF digestion in the rumen. These results and others in the literature suggest that the rate of surface area exposure rather than intrinsic chemical attributes of PDF may be first-limiting to degradation rate of PDF in vivo.

Models for estimating parameters of neutral detergent fiber digestion by ruminal microorganisms.

Model assumptions included number of concurrently degrading entities (or pools) and expected distributions of undegraded NDF. Degradation processes modeled included a single pool with ruminal age-constant rates (exponential distribution), a single pool with a ruminal age-dependent rate, two pools with age-constant rates, two pools with age-dependent and age-constant rates, and a continuum of pools with a gamma distribution of age-constant rates. Various sizes of ingestively masticated fragments of bermudagrass hay or corn silage were obtained via wet sieving of esophageal masticate and incubated in vitro with ruminal fluid for 0 h, every 6 h up to 48 h, and every 12 h up to 168 h. Models assuming a single pool of age-constant or age-dependent rates had larger mean residual mean squares (P < 0.05) than did the gamma mixture model or the two-pool models. Degradation rates estimated by the gamma mixture model indicated distribution of rates ranging from near exponential, age-constant distribution to a near normal bell-shaped distribution of age-constant rates for different datasets. Superior fit by the two-pool models in most datasets (83%) indicated that having two resolvable entities of potentially degradable NDF with different degradation rates was causal of a biphasic distribution of lifetimes. Increasing order of age-dependency modeled in the two-pool model improved fit and precision of estimation (standard error of estimate) for the limit parameters of time delay and indigestible NDF. Both the gamma mixture continuum of age-constant rate model and the two-pool, age-dependent models with a discrete time delay provided similar fit to data and flexibility for fitting data with lifetime distributions ranging from simple exponential to sigmodial. The two-pool, age-dependent and gamma-distributed, age-constant models were better in fitting the dominant biphasic lifetime distributions that occurred when the two pools of degrading entities were of similar size and in estimating the discrete time delay when strategic, quality data were available. Having fewer parameters (four), the gamma-distributed, age-constant model was superior when data quality was limited.

Flow paths of plant tissue residues and digesta through gastrointestinal segments in Spanish goats and methodological considerations.

A sequence of eight twice-daily meals, each marked with different rare earth elements, was fed to 24 Spanish goats (BW = 20.6 +/- 1.94 kg) to produce meal-based profiles of rare earth markers within segments of the gastrointestinal digesta on subsequent slaughter. Accumulative mean residence time and time delay of rare earths and segmental and accumulative mean residence times of indigestible NDF (IDF) were estimated for each sampled segment. Diets consisted of ad libitum access to bermudagrass hay with a limit feeding of one of four supplements: 1) minerals (basal, B); 2) B + energy (E); 3) B + CP (CP); or 4) B + E + CP for 84 d. Mean daily intake (g/kg of BW) during the 5 d before slaughter differed (P < 0.05) via diet for DM but not for IDF (8.0 +/- 0.35 g/kg of BW). Larger estimates of cumulative mean residence time for IDF vs. rare earths were suggested to be the consequence of a meal-induced bias in the single measurement of IDF pool size by anatomical site. The rare earth compartment method was considered more reliable than the IDF pool dilution method because it yielded flow estimates based on the flux of eight meal-dosed rare earth markers over 4 d and was independent of anatomical definitions of pool size. Statistically indistinguishable estimates for gastrointestinal mean residence times for IDF and rare earths conform to assumed indelibility for the specifically applied rare earths and indigestibility of IDF. The potentially digestible NDF (PDF):IDF ratio of dietary fragments (0.8) progressively decreased in the following order: caudodorsal reticulorumen (0.390) > crainodorsal reticulorumen (0.357) approximately reticulum (0.354) > mid-dorsal reticulorumen (0.291) approximately ventral reticulorumen (0.286), to that within the omasal folds and in the abomasum (0.259). Such a gradient of progressively aging mixture of plant tissue fragments is consistent with age-dependent flow paths established in the reticulorumen and flowing to the omasum and abomasum. Such heterogeneity of fragment ages within the reticulorumen is also indicated by the superior fit of marker dose site double dagger marker sampling site model assumptions. Additionally, cyclic meal- and rumination-induced variations in escape rate occur. Estimates of mean escape rates over days, needed for the practice of ruminant nutrition, must consider the complex interactions among plant tissues and the dynamics of their ruminal digestion of PDF.

Validity of specifically applied rare earth elements and compartmental models for estimating flux of undigested plant tissue residues through the gastrointestinal tract of ruminants.

The validity of using rare earth elements as flow markers of undigested residues was evaluated by comparing mean gastrointestinal residence time (GMRT) of rare earths specifically applied to cottonseed hulls (CSH) to that of the indigestible fiber of CSH. Feces were collected from five lambs fed a mineral supplemented diet of CSH containing 52 g CP/kg DM and five lambs fed a CSH plus cottonseed meal diet (CSH+CSM) containing 123 g CP/kg DM. Rare earth elements (La, Yb, and Tb) specifically bound to CSH were included in the diet for a 5-d period and then deleted from the diet for a 3-d period. Following the last fecal collection, lambs were slaughtered for collection of digesta from segments of the gastrointestinal tract. Potentially indigestible NDF (PIF) was determined in diets and digesta from each segment of the gastrointestinal tract. Mean turnover rate, time delay, and GMRT for each rare earth element was estimated by fitting an age-dependent compartment model to profiles of markers appearing in the feces (compartmental model-marker method, CMM). The GMRT also was computed by the indigestible entity pool dilution method (IEPD) as grams of PIF in sampled segment/mean intake rate of PIF proceeding slaughter, g/h. The GMRT computed by the CMM and the IEPD methods did not significantly (P < 0.05) differ (99.6 vs 94.8 h and 58.9 vs 59.5 h for CMM vs IEPD and CSH and CSH+CSM diets, respectively). Regression of GMRT estimated for rare earths vs PIF yielded a highly significant regression (P = 0.001) with a regression coefficient of 0.94 +/- 0.016. It was concluded that rare earth elements applied to specific feeds are valid flow markers for the undigested residues derived from such marked feeds.

Marker concentration patterns of labelled leaf and stem particles in the rumen of cattle grazing bermuda grass (Cynodon dactylon) analysed by reference to a raft model.

Large (>1600 microm), ingestively masticated particles of bermuda grass (Cynodon dactylon L. Pers.) leaf and stem labelled with 169Yb and 144Ce respectively were inserted into the rumen digesta raft of heifers grazing bermuda grass. The concentration of markers in digesta sampled from the raft and ventral rumen were monitored at regular intervals over approximately 144 h. The data from the two sampling sites were simultaneously fitted to two pool (raft and ventral rumen-reticulum) models with either reversible or sequential flow between the two pools. The sequential flow model fitted the data equally as well as the reversible flow model but the reversible flow model was used because of its greater application. The reversible flow model, hereafter called the raft model, had the following features: a relatively slow age-dependent transfer rate from the raft (means for a gamma 2 distributed rate parameter for leaf 0.0740 v. stem 0.0478 h(-1)), a very slow first order reversible flow from the ventral rumen to the raft (mean for leaf and stem 0.010 h(-1)) and a very rapid first order exit from the ventral rumen (mean of leaf and stem 0.44 h(-1)). The raft was calculated to occupy approximately 0.82 total rumen DM of the raft and ventral rumen pools. Fitting a sequential two pool model or a single exponential model individually to values from each of the two sampling sites yielded similar parameter values for both sites and faster rate parameters for leaf as compared with stem, in agreement with the raft model. These results were interpreted as indicating that the raft forms a large relatively inert pool within the rumen. Particles generated within the raft have difficulty escaping but once into the ventral rumen pool they escape quickly with a low probability of return to the raft. It was concluded that the raft model gave a good interpretation of the data and emphasized escape from and movement within the raft as important components of the residence time of leaf and stem particles within the rumen digesta of cattle.

The flow of forage particles and solutes through segments of the digestive tracts of cattle.

An experiment was conducted to investigate the compartmental mean residence time, (CMRT) of feed residues in segments of gastrointestinal digesta of mature Holstein steers. The objective was to evaluate assumptions that feed residues flow through ruminal digesta as sequential mixing pools having age-dependent (GN) and age-independent (G1) distributed residence times respectively (GN-->G1 flow). The basal diet was a semi-tropical hay containing 98 g crude protein and 503 g apparently digestible DM per kg DM. The hay was consumed and feed residues of different size and/or previous digestion from the hay were inserted into the reticulo-rumen (rumen) and abomasum. Marker profiles appearing at the duodenum and faeces were fitted to various compartment models to estimate CMRT. Post-abomasal CMRT did not differ among solutes or feed residues of different size and previous digestion and constituted only 5.8% of the CMRT for the entire gastrointestinal tract. Markers initially applied to orally or ruminally dosed feed residues exhibited profiles in duodenal digesta and faeces conforming to GN-->G1 flow. Previously undigested, masticated feed residues inserted into the dorsal rumen digesta had longer ruminal CMRT in the GN pool but not the G1 pool than did similarly inserted faecal small particles or normally ingested hay. These results support model assumptions of GN-->G1 flow within rumen digesta. The results support mechanisms proposed for the GN pool as the 'lag-rumination pool' and the G1 pool as the 'mass action turnover pool'. If further validated, rumen CMRT in cattle could be estimated from marker profiles in more easily obtained faeces to estimate ruminal CMRT required for feed evaluation systems.

Effects of immigration on some stochastic logistic models: a cumulant truncation analysis.

This paper uses a new cumulant truncation methodology to investigate the stochastic power law logistic model with immigration, and illustrates the model with parameter values used to describe the growth of muskrat populations in the Netherlands. This model has a stable equilibrium distribution. The incorporation of immigration into the model, therefore, simplifies the qualitative nature of the stochastic solution. The (unconditional) cumulant functions for the transient and the equilibrium population size distributions are obtained, from which the distributions are shown to be near-normal at all times for the parameter values of interest. Approximating cumulant functions, which are relatively easy to find in practice, are derived and shown to be quite accurate, except for the case of massive immigration. As the level of immigration increases, the mean value rises more rapidly initially, as expected; however, the variance and the skewness of both the transient and the equilibrium distributions are reduced.

On the cumulants of population size for the stochastic power law logistic model.

The deterministic power law logistic model is used to describe density-dependent population growth in cases where the ordinary logistic model is insufficient. This paper investigates an analogous stochastic power law logistic model. The exact (unconditional) population size distributions and the cumulant functions for this stochastic model are intractable for large population sizes. Approximating cumulant functions are derived for populations of any size, and are illustrated with examples of assumed Africanized honey bee population dynamics. Outstanding among the findings is that the approximations for the cumulant functions are very accurate for these examples. The stochastic power law logistic model is very general and may be applied to describe the growth of many other natural populations.

A general approach to non-Markovian compartmental models.

Stochastic compartmental models are usually based on continuous-time Markov processes. Markovian models assume that the compartments have exponential retention times, which is known not to hold in certain applications such as calcium clearance from bone. A number of semi-Markov models with restricted families of retention times have been proposed recently. This paper presents a general, tractable procedure for determining from suitable data the estimated retention time distributions for assumed non-Markovian phenomenological models. The procedure is based on using phase-type distributions. These distributions can describe the long tails observed in calcium clearance data, and they frequently lead to complex eigenvalues that are often overlooked in practice. The procedure is illustrated on several proposed models for describing a particular data set.

Stochastic compartment models with Prendville growth rates.

The Prendville growth mechanism, which assumes a linearly decreasing population growth rate, was generalized to a multicompartment system by Parthasarathy and Kumar. This article shows that their solution is only an approximate one. A diagnostic test is presented to indicate parameter vectors for which this first approximation is not very accurate. A second approximation, which overcomes some limitations of the first one, is developed and illustrated.

Describing the spread of biological populations using stochastic compartmental models with births.

This paper derives new models for describing the spread of biological populations in space and time from classical birth-death-migration processes. The spatial aspect is incorporated using compartmental analysis and is developed for two spatial areas (or compartments). The exact bivariate distributions for such processes are intractable; hence approximating distributions are constructed by matching cumulants. A basic Markovian model with exponential waiting times between births is investigated first. The individual effects of swarming, multiple births, and Erlang distributed waiting times, all of which enhance the biological realism, are investigated. A full model which includes all of these effects is then studied. The models are illustrated with observed data on the spread of the Africanized honey bee in French Guiana. A full model with swarming, with an average of 2.64 colonies per swarming episode, and with waiting times following an Erlang distribution with shape parameter 5 is found to provide the best description of the observed data. The methodology is very general and should have broad application for other biological population models involving dispersal and growth.

Generalized stochastic compartmental models with Erlang transit times.

This paper considers the use of semi-Markov process models with Erlang transit times for the description of compartmental systems. The semi-Markov models seem particularly useful for systems with nonhomogeneous "poorly-stirred" compartments. The paper reviews the Markov process models with exponential transit times, and illustrates the application of such models, describing the clearance of calcium in man. The semi-Markov model with Erlang transit times is then developed, and the solutions for its concentration-time curves and residence time moments are given. The use of semi-Markov models is illustrated with the same calcium data, and the results from the two models are compared. The example demonstrates that these semi-Markov models are physiologically more realistic than standard models and may be fitted to pharmacokinetic data using readily available software.

The flow of undigested corn residues through the gastrointestinal tract of cattle.

Two experiments were conducted to measure effects of source and level of roughage on the flow of corn residues through the gastrointestinal tract of cattle. In Exp. 1, steers (195 kg) were fed diets of ground corn with 0, 30 or 60% of ground Coastal bermudagrass hay (Cynodon dactylon) [L.] Pers.) at intakes of 1, 1.5 or 2% of BW in a 9 x 9 Latin square. Experiment 2 consisted of two 4 x 4 Latin squares with either rice hulls (square 1) or ground Coastal bermudagrass hay (square 2) providing 0, 7.5, 15 or 30% of the total diet fed at 1.5% of BW. After a 28-d adjustment period, a portion of the corn in one meal was replaced with cracked corn stained with brilliant green. The concentrations of stained corn residues appearing in the feces subsequent to dosing were fitted to a one-compartment, age-dependent model and compartmental mean residence time (CMRT) and time delay (tau) were estimated. In Exp. 1, increasing the level of intake of the ration from 1% to 1.5 or 2.0% of BW increased (P less than .05) CMRT by 52% and reduced (P less than .05) tau by 41%. In Exp. 2, source of roughage had no effect (P = .95) on CMRT or tau. Combined results of the two experiments indicated that increasing proportion (P) of either roughage was associated with an exponential decline in CMRT of stained corn residues (CMRT = 1211 * e-.0315P) from rations consumed at 1.5 and 2.0% of BW. No consistent effect of roughage type or proportion was noted on time delay in the two experiments collectively. These results indicate that increasing the proportion of roughage in the diet exponentially reduces residence time of corn residues in the ruminoreticulum (CMRT) without affecting residence time in the postgastric segments (tau).

Some generalized stochastic compartment models for digesta flow.

Digesta flow models have been based on linear compartment theory that assumes exponential retention times, and on a generalized theory that incorporates nonexponential (Erlang) retention times (Matis, 1987, Journal of Theoretical Biology 124, 371-376). This paper develops a new family of passage models for heterogeneous digesta by mixing the previous models with assumed parametric, usually gamma, mixing distributions. The utility of the resulting models is demonstrated with experimental data on two treatments, namely a chopped and a ground straw, given to each of four cows. Treatment differences are apparent in the preferred model form and in the means of the estimated mean residence times. The models are relatively easy to fit to data using standard estimation procedures, and they should have broad application to other compartment modeling problems with "heterogeneous particles."

Passage of chromium-mordanted and rare earth-labeled fiber: time of dosing kinetics.

Coastal bermudagrass hay was labeled with Cr by the Cr-mordant procedure and with 177Lu applied to the same fiber. Neutral detergent fiber prepared from the same Coastal bermudagrass hay was labeled with Yb, 169Yb, Tb and 160Tb by soaking overnight following by thorough washing and drying. Wood chips were similarly labeled with Sm or La, and Solka Floc was labeled with 147Nd and 141Ce. The carriers, labels and times of administration to cattle were: bermudagrass fiber with both Cr and 177Lu, bermudagrass fiber with 169Yb and Solka Floc labeled with 147Nd at 0 h; bermudagrass fiber with Yb, Solka Floc with 141Ce and wood chips with Sm at 24 h; wood chips with La at 48 h; and bermudagrass fiber labeled with 160Tb at the beginning and labeled with Tb at the end of a meal. Fecal collection followed and passage characteristics were determined with a two-compartment, age-dependent model. Markers labeling the different fiber sources had different (P less than .01) passage rates (Solka Floc greater than Coastal bermudagrass greater than wood chips), but there was no difference within fiber source for rare earth passage. There also was no difference between the passage characteristics of Cr-mordant and 177Lu. However, passage rate of particles administered at the beginning of the meal (160Tb) was 42% higher than for particles at the end of the meal (Tb).(ABSTRACT TRUNCATED AT 250 WORDS)

Compartment models for estimating attributes of digesta flow in cattle.

1. The basic assumptions involved in one- and two-compartment models with age-independent distributed residence times (exponential, G1) for describing digesta flow are reviewed as the bases for describing families of one- and two-compartment models which assume age-dependent distributions (Gn) of residence times. 2. The two-compartment, age-independent model with exponentially distributed residence times (G1G1) yielded estimates of essentially equal rate parameters when fitted to faecal values for all four cows receiving a diet of 500 g coarsely chopped, sodium hydroxide-treated straw/kg and one of four cows receiving the same diet but with ground and pelleted straw. The incorporation of progressively higher orders of age dependency (G2-G6, Gn) into the faster turnover compartment of two-compartment models (GnG1) resulted in a resolution of equal rate parameters estimated by the G1G1 model and a reduction in standard errors for the rate and the initial concentration parameters. 3. The occurrence of equal rate parameters in two-compartment models indicated an age-dependent process; a process which could equally well be described by a one-compartment, age-dependent compartment having an order of age dependency equal to the sum of these orders in the two-compartment model with equal rate parameters. 4. The age-independent models overestimated time of first appearance in the faeces of a meal's particles. The association of age dependency with the faster turnover compartment resulted in earlier estimates for first appearance of the marked particles; estimates which were more consistent with observed first appearance. 5. The faecal excretion pattern from cows fed on the ground and pelleted straw diet exhibited an age-independent distribution of longer residence times which dominated approximately 80% of the later residence times. Age-dependent, one-compartment models gave a poor fit to such data from these cows fed on ground and pelleted straw. In contrast, age-dependent, one-compartment models provided an excellent fit to data from cows fed on chopped straw; data which indicated that age-independent distributions of residence times were much delayed in appearing or were totally absent. 6. The mean residence time for the slower turnover, age-independent compartment estimated from faecal excretional of stained particles from either diet was similar to that estimated from duodenal concentrations of the stained particles. This suggests that the slower turnover model compartment was confined to preduodenal sites.(ABSTRACT TRUNCATED AT 250 WORDS)

Markers for estimating digesta flow in pigs and the effects of dietary fiber.

Effects of diluting the energy content of a corn-soybean meal diet with either alfalfa meal or corn cobs on nutrient digestibility and rate of passage of feed residues and particle markers were measured in crossbred (Yorkshire X Landrace X Chester White X Large White) barrows with a mean body weight of about 80 kg. The excretion pattern in the feces of Cr-mordanted diet and of rare earths initially bound individually to the mixed diet or to the corn or soybean meal suggested a model having a single age-dependent compartment with time delay. The compartmental turnover rate parameter (lambda 1) estimated by this model did not differ for the rare earths individually used to mark the corn-soybean meal diet (Yb), the corn (La) or the soybean meal (Sm). In contrast, lambda 1 for Cr was smaller (P less than .001) than that of the mean for the three rare earths. The residence time due to displacement flow (tau) did not differ among markers. These results were interpreted to indicate that the high specific gravity of Cr-mordanted feed slowed flow due to mixing but not due to displacement. Correlations between lambda 1 and tau were less than .71. These results suggested that the flow of rare earths initially bound to feed ingredients provides a reasonable estimate of the flow of their undigested residues through the gastrointestinal tracts of nonruminant animals. Inclusion of the fibrous feeds reduced digestibility of dry matter, cell contents, crude protein and acid detergent lignin and increased digestibility of cell walls, cellulose and acid detergent fiber. Digestibilities of cellulose and acid detergent fiber were greater with alfalfa than with corn cobs as the fiber source. Differences in digestibility of crude protein and acid detergent fiber existed due to litter in one replicate of the experiment. Variation in digestibility was not significantly related to variation in lambda 1 or tau within or among treatments and litters. This suggests that variations in lambda 1 and tau were not important causes of the observed variation in digestibility.

Use of residence time moments in compartmental analysis.

Residence time moments, particularly mean residence times, can be very useful in the modeling and kinetic analysis of physiological systems. They are defined and illustrated, and their practical utility is then explored by comparing the model formulation based on these moments with three alternative formulations: component half-lives, compartmental model rate coefficients, and multiexponential model coefficients and exponents. The residence time moment formulation is shown to have several advantages: a smaller number of parameters are usually needed to describe a biological system; they have a clear physical interpretation; their statistical "power" to detect certain treatment differences is greater; and they are computationally simpler in many situations.

On some stochastic formulations and related statistical moments of pharmacokinetic models.

This paper presents the deterministic and stochastic model for a linear compartment system with constant coefficients, and it develops expressions for the mean residence times (MRT) and the variances of the residence times (VRT) for the stochastic model. The expressions are relatively simple computationally, involving primarily matrix inversion, and they are elegant mathematically, in avoiding eigenvalue analysis and the complex domain. The MRT and VRT provide a set of new meaningful response measures for pharmacokinetic analysis and they give added insight into the system kinetics. The new analysis is illustrated with an example involving the cholesterol turnover in rats.