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J. Anim. Sci. 2005. 83:1591-1601
© 2005 American Society of Animal Science

ANIMAL NUTRITION

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

W. C. Ellis*,1, M. Mahlooji* and J. H. Matis{dagger}

* Departments of Animal Science and and {dagger} Statistics, Texas A&M University, College Station 77843


   Abstract
Top
 Abstract
Introduction
 Materials and Methods
 Results
 Discussion
 Implications
 Appendix
 Literature Cited
 
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.

Key Words: Digestion • Mastication • Neutral Detergent Fiber • Plant Tissues • Rumen


   Introduction
Top
 Abstract
 Introduction
Materials and Methods
 Results
 Discussion
 Implications
 Appendix
 Literature Cited
 
Structural carbohydrates may be defined as NDF OM and consist of two or more subentities of potentially degradable NDF (PDF) and undegradable NDF (UDF). Kinetic models are required to estimate rate of PDF degradation, and the limit parameters of time delay and level of UDF (Waldo et al., 1972Go; Mertens et al., 1993aGo,bGo). Considering the anatomical and chemical complexity of plant tissues (Ellis et al., 1988Go; Weimer, 1996Go), it is doubtful that all subentities of PDF are homogeneous with respect to their degradation kinetics. Multiple age-constant degradation rates may be estimated for multiple subentities if the data are of sufficient quantity and quality and if the fractional degradation rates of individual subentities differ sufficiently. However, fitting such multiple-entity models to data becomes very difficult as the number of parameters associated with individual entities in the model increases (Bates and Watts, 1988Go). A statistically tractable alternative is to assume degradation of PDF occurs for a mixture of subentities with ruminal age-constant (independent of ruminal age) degradation rates or by some distribution of ruminal age-dependent degradation rates. Gamma, age-dependent distributions of lifetimes (Matis et al., 1989Go) have been proposed to model age-dependent mechanisms such as mastication, whereby PDF becomes accessible for microbial adhesion and hydrolysis (Akins and Amos, 1975; Weimer, 1996Go; Morrison and Miron, 2000Go). The current report evaluates fit and estimation of parameters by various models assuming age-dependency for degradation of one or two subentities and a gamma distribution of subentities with age-constant degradation rates. A companion paper evaluates the effects of fragment size on the kinetics of NDF degradation, model selection, and degradation mechanisms (Ellis et al., 2005Go).


   Materials and Methods
Top
 Abstract
 Introduction
 Materials and Methods
Results
 Discussion
 Implications
 Appendix
 Literature Cited
 
This experiment was conducted before requirements for institutionally approved animal use protocols were approved (Mahlooji, 1985Go). However, the animal use protocols used then were virtually identical to those subsequently approved by an Institutional Animal Use Committee at Texas A&M University.

Forages and Tissue Fragments
Coastal bermudagrass hay (120 g of CP/kg of DM and 710 g of NDF/kg of DM) or corn silage (Deswysen et al., 1988) forage were fed ad libitum to two esophageally and ruminally cannulated steers (Ellis et al., 1984Go) for at least 7 d before collection of masticated fragments as esophageal extrusa and, in the case of corn silage, as ruminal digesta. Sufficient extrusa or ruminal digesta were collected for experimentation from two meals via the open esophageal fistula and the ruminal cannulas of two steers. An aliquot of the extrusa or digesta composited over days and animals was fractionated into several ranges of fragments using a wet sieving apparatus (Fritsch GmbH, Idar-Oberstein, Germany). Material passing and retained on a column of successively smaller sieves was collected and freeze-dried to prevent adhesion of particles. Fragments of bermudagrass retained on sieves larger than 1,600 µm were separated into leaf and stem tissues based on their aerodynamic properties in a vertical column of variable air speed with entrapping structures near the top of the column. Fragments of corn silage retained on sieves larger than 4,760 µm were separated into leaf, stem, and cob tissues based on visual inspection.

In Vitro Digestion
Fragments were incubated as described by Goering and Van Soest (1970)Go, with individual incubating vessels being removed at 0 h, every 6 h up to 48 h, and at every 12 h up to 168 h. To provide representative sampling, digestion was conducted with a minimum of six of the larger fragments (>1,600 µm). Different sizes of incubation vessels were used to accommodate the volume of 42 mL of medium and 10 mL of inoculum per 0.5 g of DM contributed by the six larger fragments. To accommodate differences in bulk in the larger fragments, multiples of this base sample amount and volume were used as follows: fragments <1,600 µm were incubated in 80-mL polyethylene tubes using the base volume; bermudagrass fragments escaping a 3,360-µm sieve and stem fragments retained on a 1,600-µm sieve (3,360/1,600 fraction) were incubated in 125-mL Erlenmeyer flasks with two times the base volume; 4,760/ 3,360 fragments of corn leaf were incubated in 300-mL Erlenmeyer flasks with three times the base volume; and 4,760/3,380 cob fragments were incubated in 500-mL Erlenmeyer flasks with eight times the base volume. Six replicates of fragments were incubated for each removal time and results were averaged by removal time to minimize sampling error for fragments >500 µm. Otherwise, the digestions were conducted as described by Goering and Van Soest (1970)Go.

Analysis of Undegraded NDF
Residues from the 80-mL tubes were analyzed for undigested NDF OM (UNDF) by the addition of 100 mL of NDF solvent (Goering and Van Soest, 1970Go) to facilitate transfer to a reflux flask, followed by 0.5 g of sodium sulfite. Following reflux for 1 h, UNDF was obtained by filtration through medium-porosity sintered glass filter crucibles. The UNDF was determined as the weight loss on ashing (550°C for 5 h) of the crucible and its contents of undegraded material. The UNDF was computed as the NDF solvent-insoluble OM remaining expressed as a fraction of the initial NDF OM before digestion.

Residues from larger fragments incubated in Erlenmeyer flasks were filtered through an 11-cm weighed filter paper (Whatman No. 41, Whatman, Maidstone, U.K.) in a Buchner funnel. After being air-dried, each sample and its filter paper were weighed, and fragments were removed and ground in a micro-Wiley mill to pass a 490-µm screen. Aliquots (0.5 g) of the ground fragments were then analyzed for UNDF as described for smaller fragments.

Derivation of Models
The so-called "sums of exponentials" models are widely used in modeling natural systems (e.g., in compartmental analysis) to describe the quantity of substance, such as UNDF, remaining in a kinetic system in flux (Bates and Watts, 1988Go). The single exponential model, applicable to a single compartment or "pool" of material with a constant digestion rate, k, is a common default model. It is denoted as E and has the form Pt = P0 exp(–kt), where P0 = initial pool size and Pt = pool size at time t. The bi-exponential model is obtained by assuming a system of two separate, parallel pools, one of size P1 with rate k1 and the other of size P2 with rate k2. The model is denoted E/E, with solution Pt = P1 exp (–k1t) + P2 exp (–k2t). With total P = P1 + P2, fractional pool sizes of f1 = P1/P and f2 = P2/P, one can write the model as Pt = [f1 exp(–k1t) + f2 exp (–k2t)]. It is apparent from this formulation that Pt declines as the weighted average of the two exponential terms, where the weights are the relative pool sizes (f1 and f2).

In this case, a question concerning possible generalizations of the two-pool, bi-exponential model arises. One extension would be to assume three or more distinct pools of PDF (i.e., a model such as E/E/E). Conceptually, this would assume three or more rates, each with some fraction of the P; however, the fitting of such models to data becomes very difficult because of the increase in the number of assumed rate and pool size parameters (Bates and Watts, 1988Go). A statistically tractable alternative is to assume a model in which the PDF has a gradient of (numerous) constant rates defined by some probability distribution.

The gamma distribution has been used successfully to model particle size distributions (Allen et al., 1984Go). Often, the gamma distribution is also suitable to model digestion rates (Matis et al., 1989Go). Its probability density function is f(k) = k{alpha}–1exp(–k/ß)/[ß{alpha} {Gamma}({alpha})], where k denotes a specific rate, {alpha} and ß denote positive-valued parameters, and {Gamma}({alpha}) is the gamma function. As illustrated in Allen et al. (1984)Go, the gamma distribution, with its "shape" parameter {alpha} and its "scale" parameter ß, has considerable flexibility for describing possible distributions for the rate k. For example, as illustrated subsequently, it could be nearly bell-shaped (normal) when {alpha} is large, or it could be nearly exponential when {alpha} is close to 1. Under the assumed gamma distribution, the mean rate is {alpha}ß, the variance of the rates is {alpha}ß2, and the modal (most probable) rate is ({alpha}–1)ß, provided {alpha} > 1.

According to this conceptualization, the gamma distribution defines in this continuum of sub-pools the relative frequency or, alternatively, the subpool sizes, for any specified range of rates. For the ith subpool, with digestion rate ki, the function of material remaining over time is exp(–kit). The overall fraction remaining in all subpools is the mixture of such exponential terms (i.e., it is a weighted average of the exponentials) with the weights specified by the gamma distribution, f(x). It is shown in the Appendix that the average (or integral) is Pt = P0 [1+ßt]– {alpha}. This function is called the gamma mixture of exponentials model, and is denoted as E(G).

Age-Dependent Models
Another possible extension would be to assume an age-dependent distribution of rates for a single entity or for one of two subentities of PDF. Models have been derived for serial degradation of two entities via an age-dependent pool (GN, where N = integer of two or more) and an age-constant distribution of rates (Matis et al., 1989Go; Ellis et al., 1994Go). A single age-dependent degrading entity model is denoted G2, G3, and so on, to indicate the order of age dependency (N). Models for concurrent degradation of two entities by age-dependent (GN) and age-constant degradation (G1 or exponential, E) are indicated by GN/E.

Limit Parameters
Regression models for UNDF have two limit parameters that represent the time delay to onset of detectable degradation ({tau}) and the quantity of UNDF that cannot be further degraded. Model expressions are summarized in Table 1Go.


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  		Table 1. Digestion models for NDF assuming different lifetime distributions for potentially degradable entities (PDF) and undegradable NDF (UDF)
 
Model Fit to Data
Various models were fitted to the data using the nonlinear least squares regression procedure, PROC NLIN of SAS (SAS Inst., Inc., Cary, NC), to estimate the common parameters UDF and {tau}, and the model-specific parameters C0, P1, P2, k, {lambda}, k2, {alpha}, and ß as appropriate. Proportion P2 was estimated as 1-P1 to minimize the number of estimated parameters. A wide range of initial starting parameters was used in the initial grid search step for the iterative phase of PROC NLIN. Secondary parameters were computed from model parameters as given in Table 2Go. The SAS programs used for fitting each model and for estimating the probability density of rates will be sent by the corresponding author to any person requesting as such.


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  		Table 2. Computation of derived parameters for various modelsa
 
Statistics
Each data set was fitted to each model by PROC NLIN. Fit of various models to the data was evaluated using residual mean square (RMS at convergence) computed by adjustments for the number of estimated parameters of each model ([RMS = residual sum of squares at convergence by PROC NLIN]/[number of observations – number of estimated parameters for each specific model]). With 20 observations per data set, the number of parameters for the single-pool models, E and GN, is three; the number for the two-pool models, E/E and GN/E is five; and the number for the gamma mixture model E(G) is four. Clearly, because these numbers of parameters differ, it would not be appropriate to compare residual sums of squares. However, the residual degrees of freedom (dfE) are 17, 15, and 16, respectively, for the three classes of models. Hence, when the RMS are calculated as residual sums of squares/dfE, the number of parameters in the respective models is taken into account.

Standard nonlinear least squares programs, such as PROC NLIN in SAS, are based on a Taylor series linearization. The assumed parameter values are treated as variables in a multiple linear regression analysis, and the estimated regression coefficients in the linear regression analysis are the estimated changes in the parameter values for the next iteration (Neter et al.,1996Go). This tie-in with multiple regression is relevant because the RMS criterion is a simple, widely accepted procedure for comparing multiple regression models with differing numbers of variables, and in fact is a standard approach in SAS for determining the number of variables in variable selection procedures (Neter et al., 1996Go). This is particularly true if the differences in the number of parameters are small in comparison to the number of observations. Due to the linkage between nonlinear regression with multiple parameters and linear regression with multiple variables, the use of RMS in the latter justifies its use in the former.

Wilcoxon’s signed rank procedure (Ott, 1977Go) was used to test the hypothesis of equality of RMS distributions as estimated by each model for various fractions. In cases where the null hypothesis of equal EMS distributions of two models was rejected, the model with a smaller median RMS was considered superior. The PROC GLM of SAS was used to evaluate sources of variation in digestion parameters statistically as dependent variables vs. attributes of the fragments of plant tissues as independent variables.


   Results
Top
 Abstract
 Introduction
 Materials and Methods
 Results
Discussion
 Implications
 Appendix
 Literature Cited
 
Distributions of PDF Degradation Rates
When {alpha} was bound as >0, fitting the E(G) model yielded a wide range in estimated shape and scale parameters, {alpha} and ß, respectively. The probability density of PDF digestion rates (estimated from {alpha}; Appendix) for some data sets are illustrated in Figure 1Go. A benchmark distribution, a special case with {alpha} = 1, yields the exponential distribution illustrated in Figure 1AGo. Its intercept is 1/ß = 8.4 and its mean is {alpha}ß= 0.119/h. Some fitted models yielded {alpha} < 1, (Figure 1BGo), which represents a very skewed distribution with an intercept approaching infinity. Most data sets yielded fitted curves with {alpha} ≥ 1, as exemplified in Figure 1C and DGo. The modes are ({alpha} – 1)/ß, for 0.12 and 0.0029, respectively, with mean, {alpha}ß, of 0.41 and 0.0030, respectively. Note that the distributions become more symmetrical (normal-like) as {alpha} increases. The area under each curve is 1.0, as required of a probability distribution. When bounded to {alpha} > 0, probability densities for UDF commonly included zero because the intercept, C0, approached infinity. Therefore, {alpha} was subsequently bounded to {alpha} > 1 (i.e., equal to or greater than an exponential distribution, {alpha} = 1; Table 1Go).



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  		Figure 1. Some observed gamma distributions of potentially degradable NDF (PDF) digestion rates. An exponential distribution of PDF rates is represented as a special case with the shape parameter {alpha} = 1 (A). A more skewed distribution of rates has an {alpha} < 1 (B) and distributions become more symmetrical as {alpha} becomes larger than 1 (C and D).

 
Fit of Models
Incorporation of limit parameters, {tau} and UDF, individually and collectively decreased (P < 0.05) RMS in each model (data not shown). Fit of the complete model (with limit parameters) to the data sets was evaluated as the residual mean square of observed vs. expected values for the 30 profiles of UNDFt. Partition of variance associated with forage type (bermudagrass or corn silage), form (rumen, masticated or ground masticated fragments), size of masticated fraction, and model indicated significant (P < 0.001) differences due to forage type, form, fragment size, and model, but no significant (P > 0.75) interactions involving model. Relationships between parameters and size of forage fragments are presented and discussed in an accompanying report (Ellis et al., 2005Go).

Based on RMS (Table 3Go), single-pool models provided an inferior fit (P = 0.12 to 0.53) to the gamma mixture model and the two-pool, age-constant rates model (P = 0.06 to 0.33), which provided an inferior fit to the two-pool, age-dependent, age-constant rate models (P = 0.001 to 0.04). Mean estimates of UDF also differed (P < 0.05) by model: G3 > G2 = G1= E(G) > all other models. Because of a lack of strategic data during the early (<12 h) phases of degradation (Figure 2DGo), mean estimates of {tau} ranged from 5.8 to 11.6 and did not differ (P > 0.05) via model.


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  		Table 3. Mean (n = 30) and residual mean square (RMS) of estimates of undegradable NDF (UDF) and time delay ({tau}) estimated by various models
 


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  		Figure 2. Fit of models to observed lifetime profiles of undegraded NDF (UNDF) illustrating the biphasic distribution of lifetimes (A) and superior fit of two-pool age-independent (E/E) and age-dependent (G2/E and G5/E) vs. single-pool models (Models E, and G2) to total profile of lifetimes (A and C) and to early lifetimes (<10 h; B and D). Note that gamma distribution of age-constant rates model (E(G)) provided intermediate fit to biphasic distribution of lifetimes (A).

 
Due to large variation in the distribution of probability density rates among datasets, fit to data by individual models was best characterized by parameters derived from fitting a selected dataset. Fit of the models to a dataset for bermudagrass leaf and stem fragments >1,000 µm are illustrated graphically in Figure 2Go, and model-estimated parameters are summarized in Table 4Go. Fitting the E(G) model to this individual dataset yielded an {alpha} = 1.8 ± 0.9 (Table 4Go), which is indicative of a probability density distribution for PDF lifetimes approximating that of an exponential distribution ({alpha} = 1.0, Figure 1AGo). Although having a distribution of lifetimes approximating that expected for Model E, fit of Model E(G) to this dataset provided a numerically improved fit compared with either the E or G2 model (RMS of 0.5 vs. 1.1 and 1.1 respectively, Table 4Go). The observed lifetime distributions for this dataset were clearly biphasic (Figure 2AGo) and more closely conformed to expected lifetime distribution for two-pool models (Figure 2C vs. 2AGo). Lack of flexibility due to the rigid single-pool and age-constant rate assumptions of model E and single-pool assumptions of model G2 resulted in a lack of flexibility in the form of lifetime distributions to mimic observed midrange (50 to 120 h) lifetimes. Consequently, least squares fitting of models E and G2 resulted in an underestimation of midrange values and an overestimation of more-terminal values (140 to 180 h) critical to estimating parameters for UDF (Table 4Go). By assuming a continuum of pools of entities with different age-constant degradation rates, the E(G) model provided improved fit compared with Models E and G2 (RMS, Table 4Go), but slightly inferior fit compared with two-pool models (Table 4Go).


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  		Table 4. Degradation parameters estimated by models assuming a single or two distributions of NDF entities having an age-constant degradation rate (E), a gamma distribution of a mixture of age-constant rates (E(G)), age-dependent rates (G2 and G3), two age-constant rates (E/E) or age-dependent and age-constant (G2/E, G3/E, G4/E, and G5/E) rates. Degradation parameters are for masticated fragments of bermudagrass leaf and stem >1,000 µm
 
The two-pool model with age-constant degradation rates (Model E/E) had a slightly poorer fit compared with two-pool models with age-dependency in one pool (G2/E, G3/E, G4/E, and G5/E, Table 4Go). The greatest effect of including age dependency in two-pool models was upon estimates of the limit parameters {tau} and UDF. Increasing orders of age-dependency (G2, G3, G4, and G5) resulted in 1) decreased {tau}, 2) increased UDF, and 3) decreases in standard error of estimates. For proportions of PDF in the faster degrading pool (f1), the age-constant rate of degradation of the slower degrading entity (k2), and of UDF. We propose that the observed improvements in precision of parameter estimation associated with increasing order of age dependency resulted from increasing contrast of form between the age-constant and age-dependent distributions of UNDFt modeled.

When included in one pool of the two-pool models, age dependency was associated with the faster-degrading pool and resulted in a sigmodial distribution of composite lifetimes for two entities of PDF (Figure 2DGo). Increasing the order of age dependency modeled resulted in a more protracted profile of expected lifetimes (Figure 4 of Ellis et al., 1994Go) and thereby partitioned more lifetimes to the age-dependent degradation process rather than to a discrete {tau} (Matis et al., 1972Go; Figure 10 of Ellis et al., 1994Go).

As order of age dependency in the two-pool models was increased, the mean lifetime of the associated entity 1 (MLT1) was relatively constant (7.6 to 10.3 h), whereas the mean lifetime of the age-constant degrading entity 2, k2, and UDF increased and became relatively constant (27.3, 30.8, and 27.6) with GN of G3 or more (Table 4Go). Increased precision in estimating model parameters due to order of age dependency seems to be associated with greater precision in estimating rate parameters associated with the more slowly degrading entity in pool 2 (k2) and, consequently, providing more stable estimates of UDF (Table 4Go). However, if a discrete {tau} is postulated, data in Table 4Go suggest that GN should not exceed 3.


   Discussion
Top
 Abstract
 Introduction
 Materials and Methods
 Results
 Discussion
Implications
 Appendix
 Literature Cited
 
The current data were obtained for various fragment sizes of ingestively masticated forages. Therefore, the results represent the array in sizes of plant tissue fragments entering the ruminal digesta and subject to initial ruminal microbial colonization of NDF sites and erosive degradation of NDF. Fitting the gamma mixture model to the data indicated a wide variation in the probability density of PDF degradation rates among such fragments, varying from the exponential distribution commonly assumed (Gardner et al., 1999Go). Lopez et al. (1999)Go reported wide variations among ground forage samples in the lifetime distributions of PDF based on fit to model forms including an exponential, diminishing returns, and sigmodial distribution of lifetimes.

Improved fit of two-pool models conformed to expectations of a composite of lifetimes of two concurrently degrading subentities of PDF with different degradation rates. When age dependency was modeled in one of the two pools, it was the more rapidly degrading entities of PDF that were associated with the age-dependent pool. The age-dependent rate process involves a continuum of rates beginning at zero when t = 0 and increasing at some increasing rate until its asymptotic rate {lambda} is reached as t approaches infinity (see Figure 4 of Ellis et al., 1994Go). The biphasic distribution of composite lifetimes may result from differences in degradation rates of faster vs. slower degrading entities. Fitting the E/E model to data for bermudagrass leafs and stems >1,000 µm yielded estimates of mean lifetime of 7.6, 113.5, and 121 h, respectively, for Pools 1, 2, and the sum of the two pools (Table 4Go). Modeling increased order of age dependency in Pool 1, resulting in a slower and more protracted distribution of PDF lifetimes (Figure 2AGo; see also Figure 4 of Ellis et al., 1994Go) and decreased estimates of mean lifetime for Pool 2 until G3 was modeled (Table 4Go). Effects of modeling further increased order of age dependency to G3 resulted in decreased magnitude of estimates and precision (lower standard error of estimate) of degradation rate of the slower degrading entity, k2, and precision and estimates of UDF (Table 4Go). Increasing order of age dependency beyond G3 did not result in further improvements in estimating these parameters. Thus, assuming age dependency up to G3 provided improved contrast between the two pools and thereby improved precision in resolving associated parameters. Fit of G3/G1 order of age dependency minimized RMS in a more extensively replicated in situ study by Carrette-Carreon (2001)Go and Carrette-Carreon et al. (2001)Go.

One major limitation of the two-pool models is their requirement for extensive data to reliably estimate six parameters; CO, P1, {lambda}, k2, {tau}, and UDF (Table 1Go). A major reason for developing model E(G) was to minimize the number of parameters (C0, {alpha}, ß, {tau}, and UDF). As indicated in Table 4Go, model E(G) resulted in improved fit to lifetime distributions of PDF approximating an exponential distribution yet yielded dissimilar parameters from model E and parameters similar to those estimated by a two-pool, age-dependent, age-constant model (4G/E). Representing a continuum of rates, model E(G) lacks the flexibility to fit the biphasic distribution of lifetimes arising as the composite of two pools of different degradation rates (Figure 2AGo).

There is considerable biological rationale for age-dependency mechanisms related to fibrolytic microbial colonization and onset of PDF hydrolysis from structural PDF entities of ingestively masticated fragments. Colonization of specific tissues occur by successive waves of fibrolytic bacteria with different site preferences for adhesion required for PDF hydrolysis (Akins and Amos, 1975). Weimer et al. (1990)Go clearly demonstrated that early phases of PDF hydrolysis by mixed ruminal microbes were not age constant. Van Milgen et al. (1991)Go also observed improved fit by two-pool models similar to model E/E used here and to variable fractional degradation rate models such as the age-dependent model GN/E (Van Milgen and Baumont, 1995Go).

Quality of Data
Typically, five strategically located observations per parameter to be estimated are required or 25 observations for a model containing five parameters. The current data set had only 20 observations. Thus, quantity of data, and especially initial and terminal values for degradation were critical. Clearly, more frequent sampling between 0 and 16 h would have improved the precision of partitioning lifetimes of PDF between {tau} and the age-dependent degradation rate, {lambda} (Figure 2DGo).

Precision of estimating UDF is critical because the value of initial PDF/DM (C0) is determined by the difference between initial NDF and the quantity of NDF not degraded by further exposure to agents of digestion (i.e., UDF). Overestimation of C0 results in underestimation of UDF and overestimation of PDF as illustrated for model G2/E in Figure 3AGo. Figure 3BGo illustrates a dataset with equivalent fit by models E(G), G2/E, and G5/E, and lifetime distributions suggestive of a third pool of very slowly degrading PDF based on lifetime distributions beyond 110 h. Alternatively, the suggestive third pool could be due to a confounding of UDF with PDF due to model underestimation of UDF and the determination of PDF as NDF-UDF. Samples well beyond 168 h are needed to resolve UDF more confidently, regardless of the model involved.



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  		Figure 3. Different estimates of the limit parameter indegradable NDF (UDF) results in different estimates of potentially degradable NDF (PDF) and, consequently, undegraded, potentially degradable NDF (UPDF) in some data sets (A), but not others (B). A possible third pool of very slowly degradable entities (distinguishable beyond approximately 100 h) may be artifact of underestimation of UDF resulting in confounding of PDF and UDF estimated as PDF = NDF – UDF.

 
Lopez et al. (1999)Go compared the utility of eight models in estimating degradation characteristics of 87 Mediterranean forages ground to pass a 2-mm sieve. Large variations in the form of lifetime distributions for NDF, based on fit to different models, were observed for different forages as observed here for degradation of NDF from masticated forage fragments fitted to the mixture model. Lopez et al. (1999)Go concluded that, statistically, differences among the models they evaluated were of little nutritional significance and recommended using the simple exponential model with a discrete time delay (model E used here). It should be noted that the data used by Lopez et al. (1999)Go were critically limiting in number of observations per forage (seven) and in range to reliably estimate UDF (72 h) and time delay.

Which Model?
Although the age-constant pool of degrading entities dominates the lifetime distributions of some datasets here and in the results of Lopez et al. (1999)Go, it cannot be safely assumed that contributions of other pools do not make a significant contribution. With the possible exception of {tau}, proper fitting of models E(G) or GNG1 to quality data should yield similar estimates of comparable parameters even if the data are strictly exponentially distributed (Table 4Go and Figure 3Go). If strictly exponentially distributed, the GNG1 model will yield estimates of a very small fractional pool (<0.01) of rapidly degrading entities (P1 in Table 1Go) with a very rapid (>5) age-dependent degradation parameter ({lambda}). Proper fitting involves specifying a wide array of all initial parameter values in the grid search of PROC NLIN.

Additionally, rate of ruminal escape of undegraded fragments seems to involve two-sequential, age-dependent and age-constant processes (Ellis et al., 1994Go; Huhtanen et al., 1995Go) that must be considered in integrating the kinetics of ruminal degradation and escape of PDF. Thus, the practical need for models of NDF degradation cannot be assessed at this time.

Substitution of NDF in the initial DM in models could be substituted for C0 when loss of DM through porous in situ bags is avoided (Van Hellen et al., 1976Go). Also, UDF could be determined by independent, long-term ruminal incubations (Lippke et al., 1986Go). Due to the small extent of degradation relative to and associated with a large error of estimation, extensive replication of early time intervals (1 to 12 h) is critical, as illustrated in Figure 2DGo.

Quality of data is of paramount importance, and it is often commonly limiting in reported data to obtain precise estimates of degradation parameters needed to assess the nutritional significance of degradation parameters. Where number and quality of observations are limiting, the E(G) model is recommended because of its fewer parameters compared with the GN/G1 models (5 vs. 4).

Lopez et al. (1999)Go used fit to profiles of UNDFt for statistical comparisons among models. As emphasized here, evaluation of digestion models must consider the collective effect of all the digestion parameters including effect of both rate ({lambda} and k2 or {alpha} andß) and limit parameters (initial NDF, UDF, and {tau}).

Integration of Degradation and Flow
Effects of using a more complex model than model E in its many forms cannot be definitely assessed for the diversity of methodological assumptions that have been used. Extent of NDF degradation is the concurrent and dynamic result of age-dependent and age-constant degradation processes occurring in and through sequential, age-dependent and age-constant flow pools. Integration of such age-dependent and age-constant processes of NDF degradation and age-dependent and age-constant processes of NDF flow may be difficult but are possible (Carrette-Carreon, 2001Go). Such integration yields complex, nonlinear forms (Figure 25 of Ellis et al., 1984Go).


   Implications
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 Abstract
 Introduction
 Materials and Methods
 Results
 Discussion
 Implications
Appendix
 Literature Cited
 
Mechanistically, ruminal degradation of structural carbohydrates involves complex interactions among the physical and chemical attributes of masticated forage tissue fragments and fibrolytic bacteria, especially during early phases of degradation. Models assuming a gamma mixture of degradation rates or two pools of an age-dependent and an age-constant rate are inferred to have biological rationale and are recommended as preferred models.


   Appendix
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 Abstract
 Introduction
 Materials and Methods
 Results
 Discussion
 Implications
 Appendix
Literature Cited
 
The fraction remaining in a single-pool model with decay rate k is represented by exp(–kt). Consider the continuum of pools conceptualization, where k is a random variable, with gamma (mixing) distribution f(k) = k{alpha}–1exp(–k/ß/[ß {alpha} {Gamma}({alpha})]. The fraction remaining in the aggregate is then the integral of the exponential decay times the mixing distribution (i.e., {int} exp(–kt)f(k)df) which conceptually is the weighted average of all the exponential decay functions. The integration is not difficult mathematically. Fortuitously, it is also the form of the so-called "moment-generating function" of the gamma distribution, used for a different purpose but available in standard texts describing statistical distributions (e.g., see Johnson et al., 1994Go). The solution is (1 + ßt)–{alpha} for {alpha}, ß >0. When multiplied by the combined initial pool size, P, this gamma mixture model, E(G), for remaining P after elapsed time t is Pt = P0(1 + ßt)–{alpha}.

1 Correspondence: 2471 TAMU (phone: 979-845-5063; fax: 979-845-5292; e-mail: w-ellis{at}tamu.edu).

Received for publication December 17, 2003. Accepted for publication March 30, 2005.


   Literature Cited
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 Abstract
 Introduction
 Materials and Methods
 Results
 Discussion
 Implications
 Appendix
 Literature Cited
 


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W. C. Ellis, M. Mahlooji, C. E. Lascano, and J. H. Matis
Effects of size of ingestively masticated fragments of plant tissues on kinetics of digestion of NDF
J Anim Sci, July 1, 2005; 83(7): 1602 - 1615.
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