Model parameters calculated for fiber digestion and true digestion of alfalfa and tall fescue (Example 2).
The use of quantitative independent variables in experiments allows the use of regression to explore the functional relationship between treatments applied and measured responses. It provides the opportunity to not only understand the magnitude and importance of the response but also ascertain its nature. The simplest approach is to fit a polynomial. While it is often possible to obtain a very good fit using this approach, it offers in the way of providing insight into the response. At best, you can determine if the response is nonlinear and if so, if it is complex or not. The model parameters are empirical and generally cannot be interpreted as having any biological, chemical, or physical meaning—at least not directly. There are situations, however, when such a meaning can be inferred from a model fit using simple regression. In general, this is true when the relationship is truly linear or when a nonlinear model can be considered to be “intrinsically” linear; that is, it can be linearized by transforming the data in a way that can be fit using simple linear regression. A series of forage quality examples are used to illustrate these concepts in this article.
- herbage quality
- true digestibility
The use of mathematical models to describe chemical, physical, and biological processes is quite common in natural sciences . The best models are those with parameters that have chemical, physical, or biological meanings . They go beyond being descriptive and provide a deeper understanding of the process that is being evaluated. Fitting and adapting models to experimental data are as much an art as a science, and the outcome is highly influenced by decisions made by the researcher about which models to fit, what data are needed and should be used, and interpretation of fit statistics. In this article, models that describe the relationship between rumen escape protein and protein concentration, kinetics of fiber digestion, true digestion, and potential intake of herbage are developed and used to demonstrate how relatively simple models can be an effective tool for understanding biological processes and how they can be applied using experimental data. For each example, the underlying theory and assumptions are also presented and discussed.
It is important to make the distinction between the use of models for describing and understanding a biological response and their use to predict future outcomes. The use of models addressed herein relates to the former purpose and is most applicable to interpreting the results of designed experiments. That is, the experimental units on which the observations are made have been intentionally manipulated in some way that can be described quantitatively. Treatment responses for any experiment for which a quantitative treatment has been applied can be evaluated in this way. However, the inferences that can be made by using models fit to experimental data are limited to those appropriate to the design of the experiment. Their application to predicting results outside the bounds of the inference space associated with the experiment is not recommended.
Designed experiments often have unique features that both limit and extend the types of regression analyses that can be performed. They nearly always include multiple replications of individual treatments. When fitting a regression equation, this allows for the partitioning of residual error into pure error and lack of fit, thus providing a test for whether the linear model fits the response or not. It also allows statistical tests to be made about assumptions related to the distribution and homogeneity of residuals. These assessments can be used to refine the approach used in the regression and improve the value of the analysis.
This chapter is intended to demonstrate how relatively simple models can be used to describe important nutritional processes related to ruminant herbivory. It uses a series of examples to illustrate the principles and power of using simple mathematical models to better understand the functional relationship between important variables. The data used in the examples have been published previously, although the analyses employed here may be slightly different from those used in the original studies from which they were taken.
2. Linear regression; a quick review/overview
Simple linear regression is a statistical method for calculating parameters for the model:
Graphically, the model represents a straight line that intercepts the
The equation for estimating the slope (
The assumptions for linear regression are the following: (1) the independent variable X is measured without error; (2) the relationship between
Some straightforward statistics for assessing the fit of a regression equation are the coefficient of determination (
It represents the proportion of total variation in
The standard error of the estimate is calculated from the equation:
It is the square root of the residual variance of the regression. It describes how well the regression line fits the data with smaller values indicating a better fit. Smaller values indicate less departure of the actual observations from the regression line.
These five equations are all that is needed to fit and assess models that are either linear or intrinsically linear. However, there are other methods and statistics that are useful for this purpose and some of them will be described as the examples that follow are developed.
3. Intrinsically linear models
Nonlinear models that can be linearized by transforming either
The exponential model has a number of important uses. With a positive slope (
The logarithmic and power models are useful for describing responses where the rate of change gradually decreases with respect to increasing X. Many chemical and biological processes are limited and show an asymptotic response. These types of responses are generally better described by an intrinsically nonlinear model that contains an asymptote as a parameter. The Gompertz  and Mitscherlich  equations are two good examples of such models commonly used to describe biological processes. However, these models require a different approach to estimating their parameters than simple linear regression.
If a nonlinear model cannot be expressed in the form of a simple linear equation through transformation, then it is considered to be intrinsically nonlinear. There is a host of such models, and many of them can be used to describe functional responses relevant to herbivory (see Archontoulis and Miguez, 2013, for a review of 77 nonlinear models). However, fitting these models is somewhat more complicated and requires using a numerical approach that adjusts parameter values iteratively until a solution based on certain criteria is achieved. The criterion typically used is the combination of parameter estimates that results in the minimum residual sum of squares, which is why such algorithms are sometimes referred to as a nonlinear least squares approach . Convergence is then based on identifying the combination of parameter estimates that result in the lowest sum of squared deviations from the value estimated by the regression. This chapter focuses on models for which the parameters can be estimated by simple linear regression all of which are therefore intrinsically linear.
4. Herbage nutritional entities
The forgoing concepts and equations can be used to fit and assess simple linear regression equations. However, before applying them to experimental data in the following examples, a quick overview of some nutritional concepts related to herbage utilization is in order.
The nutritive entities of herbages are broadly grouped into uniform and nonuniform fractions based on the Lucas Test . Uniform fractions are those that have similar nutritional characteristics or true digestibility regardless of the feedstuff [8, 9]. These include most nutrients contained in the cytoplasm of plant cells including proteins and other nitrogenous compounds and nonstructural carbohydrates. Nonuniform fractions vary in true digestibility among different feedstuffs and even within a single feedstuff. Plant fiber is considered a nonuniform fraction. Its digestibility varies greatly among different feedstuffs and is affected by a number of genetic and environmental factors .
The Lucas test itself involves a simple linear regression model. It is performed by regressing the amount of a nutrient that is digestible against its intake. Fractions for which true digestibility is constant over a range of herbages are considered to be nutritionally uniform or ideal . The Lucas Test provided the foundation on which Van Soest  developed the detergent system for analyzing feeds. In this system, herbage or feedstuff dry matter is partitioned into cell solubles and neutral detergent fiber by refluxing a sample of the feed in a neutral detergent solution and recovering the residue by filtration. The residue remaining is fiber. The compounds removed with the filtrate are collectively referred to as cell solubles. Cell solubles have a uniformly high true digestibility regardless of the feedstuff they are contained within. They are very nearly completely available when subjected to digestion in ruminants with a true digestion coefficient of 0.98. The residue remaining after treatment with neutral detergent is the fibrous fraction and varies significantly in digestibility among feedstuffs. Both fiber and cell solubles are heterogeneous in composition and can be further partitioned into chemical constituents. Neutral detergent fiber, while structurally complex, is composed of relatively few polymers and consists almost entirely of cellulose, hemicelluloses, and lignin. True, the hemicelluloses represent a fairly diverse set of compounds, but still this is a relatively small number compared with the myriad of compounds found in plant cells. Some complex carbohydrates such as pectins and beta glucans are not recovered in neutral detergent fiber. However, these compounds are easily digested by ruminants and are considered to be part of the cell soluble fraction .
A particularly important fraction of the cell soluble fraction is protein. Proteins and other nitrogenous compounds in herbage can be converted to amino acids by rumen microorganisms that incorporate them into proteins. These proteins are eventually passed from the rumen to the lower digestive tract where they are hydrolyzed to amino acids, which are largely absorbed within the small intestine . The example that follows involves using a modification of the Lucas Test to test the hypothesis that rumen degradability of protein is proportional to the concentration of protein in the herbage.
5. Rumen degradable protein
In this application, a modified form of the Lucas Test is used to evaluate the degradability of herbage in the rumen using an
The degradability of protein in the rumen varies greatly between cool and warm-season grasses, and this may be one explanation for the observation that animals consuming warm-season grasses perform better than would be expected based on their chemical composition. The theory is that some plant proteins localized within bundle sheath cells in warm-season grasses are physically protected from degradation by the structure of the cells. These proteins bypass the rumen intact and progress to the lower intestinal tract where they are digested and absorbed as amino acids. Nitrogen in proteins degraded in the rumen is often in excess of microbial needs and that which is not needed is lost as ammonia. Thus, protecting some of the protein from ruminal degradation improves the efficiency of protein utilization .
One of the objectives of the experiment was to quantify the relationship between ruminal protein degradation and protein concentration for both species. Linear regression of rumen degradable protein (RDP) on crude protein (CP) concentration was done for several samples of both species with varying CP concentration. The linear equation for this analysis was:
The grasses used in this study were harvested at different stages of maturity and separated into leaves and stems to obtain a range of CP concentrations and were analyzed for RDP using an
Model parameters for each of the two grass species were estimated using the REG procedure in SAS (Appendix A1). The linear model described the relationship between RDP and CP very well for both species (Figure 1) based on the
Based on this comparison, it is reasonable to conclude that the two species have different rumen protein degradability and that this difference is constant and persists across a range of maturities and morphological components. These results are consistent with the observation that protein in warm-season grasses seems to be used more efficiently than that in cool-season grasses. However, the mechanism for why this is so is not clear from this study. Based on the protection theory, one might expect protein degradability to vary across tissues within a species and that does not appear to be the case. So maybe there is another explanation that would better describe what was observed in this study.
One further conclusion that can be inferred from fitting these equations is that the contribution of microbial CP to residual CP was negligible for smooth bromegrass and small (<1%), but significant for switchgrass. This is based on the test of parameter estimates included in the SAS output. The
6. Fiber digestion kinetics
In this application, we will compare the digestion of alfalfa (
To comprehend and be able to interpret the parameters of the model, an understanding of plant fiber and first-order kinetics is necessary. The next two sections provide an overview of each of these topics following that we will pick up the example in more detail.
6.1. Plant fiber
Fiber is a nutritional concept that refers to the less degradable and more variable constituents of an herbage or feedstuff. Chemically, it is comprised of plant cell walls the composition of which varies greatly among and within herbage species. Even within a single plant, the organs, cells, and tissues vary remarkably in fiber composition and digestibility . The primary chemical constituents of plant fiber are cellulose, hemicellulose, and lignin, although there are others that comprise a much smaller fraction. These constituents are aggregated and arrayed in three-dimensional space in various ways creating a network of nonliving tissues that play an important structural role in the architecture of their plant . Having a rigid cell wall is one of the defining characteristics of higher plants. Cell walls and thus fiber evolved to fulfill specific roles in plants, which do not include being a source of energy for herbivores. Their structure and function in a plant are in many ways counter to their use as a nutrient source. Even though fiber is composed of plant cell walls, it is functionally different. It is defined by its properties when subjected to digestion by an animal and has attributes that are only relevant in this context. The two terms are thus not really interchangeable .
Not all the fiber in a plant is degradable by ruminants. The degradation of plant fiber involves the hydrolysis of the principal polysaccharides by enzymes secreted by rumen bacteria. Because of the close physical and chemical interactions among plant cell wall constituents, some of the glycosidic linkages are not accessible to the hydrolases that would otherwise cleave them and render them digestible. The fraction of fiber that cannot be digested because of these interactions is indigestible and cannot be degraded within the digestive system. When determining the kinetics of fiber digestion, the indigestible (CI) portion must be considered separately and removed from that which is potentially digestible (CD) [16, 17].
The indigestible fraction is usually considered to be that which remains after being subjected to
When calculating first-order digestion parameters, it is important only to include fiber concentrations at time points where digestion is actively occurring ([CD]t ≠ [CD]0 and [CD]t ≠ [CI]; [CD]0 > [CD]t > [CI]). Including time intervals in the calculation where no change in fiber concentration has occurred biases the estimates of the parameters. Most importantly, time intervals where the fiber concentration is not different from either the initial or final concentration should be excluded from the calculations. Moore and Cherney  suggested a simple method for selecting time intervals for rate calculations. Since replicate samples are usually collected at each time point during a digestion study, it is possible to compare the mean concentration between pairs of time points using a t-test. Time intervals within the lag period can be identified as those for which the fiber concentration is not significantly different than the initial concentration. Time intervals occurring after digestion has ceased will have concentrations that are not different than the longest time point, which is usually used to determine the indigestible fiber fraction. It is entirely possible for digestion to occur throughout the sampled period, but it is more often the case that some time points will need to be excluded.
The data in Appendix 2 were collected by incubating an herbage sample in buffered rumen fluid for 0, 3, 6, 9, 12, 16, 24, 36, 48, 60, 72, and 96 h . Samples were refluxed in a neutral detergent solution following fermentation to extract undegraded fiber using the procedures described by Cherney et al. . The concentration of fiber remaining after each time interval was calculated on the basis of initial dry matter subjected to fermentation (Figure 2).
In order to calculate an unbiased estimate of digestion rate (
Based on the Levene test for homogeneity, the variances observed in neutral detergent fiber (NDF) concentration were homogeneous across each time interval and this was true for both species. This simply confirms that using the LSD procedure for comparing the mean NDF concentration remaining at each time interval was appropriate. Had they been found to be heterogeneous, it would not have been possible to use a pooled estimate of the variance for comparing means and either a data transformation to stabilize the variance or the use of different variances for each comparison would be needed. There are other tests of homogeneity that can be used to assess whether treatment variances can be considered equal. The Levene test is the default method when using GLM in SAS because it is widely used and accepted, but others are available and can be specified if desired.
The mean concentration associated with each time interval is reported in the output along with a capital letter denoting which grouping it belongs to (Appendix A2.1). Means associated with the same letter are not different from each other at alpha 0.05. By comparing the groupings, it is possible to infer the intervals during which digestion began and when it ceased. Digestion of alfalfa fiber began after the first interval so the concentration at 3 h would be included in the calculation of
This analysis (Appendix A2.1) can also be used to determine the concentration of indigestible fiber that must be subtracted from the concentration ([
The concentration of digestible fiber is the initial fiber concentration minus the indigestible fraction ([
|Fiber [C]0, g kg−1 DM||298.3||626.0|
|Indigestible fiber [CI], g kg−1 DM||148.2||232.3|
|Digestible fiber [CD], g kg−1 DM||150.1||393.8|
|Cell solubles [CS], g kg−1 DM||701.7||374.0|
|Rate of fiber digestion ||−0.111||−0.044|
|Lag time, h||1.7||5.3|
|True digestibility, g kg−1 DM||851.8||767.8|
6.2. First-order model
The rate of fiber digestion in the rumen is dependent on the concentration present and, therefore, follows first-order kinetics:
The absolute rate of change in concentration per segment of time depends on the concentration present during that segment. Higher rates of digestion for a given herbage substrate correspond to higher concentrations since the relative rate (
Eq. (10) can be written in differential form and divided by
Integrating both sides gives:
This is a convenient form of the equation because the parameters can be calculated by simple linear regression of the logarithm of concentration remaining over time. This is the exponential decay model referred to in the discussion of intrinsically linear models above. Concentrations in Eq. (12) are in log units.
The rate constant (
There are also qualitative methods for assessing the appropriateness of a model for a set of data. A residual is defined as the difference between the observed value and that estimated by the regression. Examination of residuals is a fast and easy way to visualize the fit of a model and determine if it is biased for some values of
The constant for rate of fiber digestion was 0.111 h−1 for alfalfa and 0.044 h−1 for tall fescue. These two slopes can be compared using a
Based on this comparison, it is clear that the rate of fiber digestion is quite different between the two species with the digestion of alfalfa fiber occurring at over 2.5 times the rate as that of tall fescue. For reasons we will see, interpreting the rate of fiber digestion independently of concentration and degradability can be misleading. However, at this point, for whatever reasons, we conclude that in this study the rate at which fiber was digested in alfalfa was much faster than in tall fescue.
It is possible to test multisample hypotheses about slopes when more than two equations are being compared; for example, if an additional species were being considered in this example. This can be accomplished using analysis of covariance , but the test is not easily implemented using statistical software. The calculations involved are laborious enough to consider avoiding making them altogether and instead making pairwise comparisons between slopes using the
It is common for there to be a lag period before fiber digestion begins in
The lag time (
The foregoing discussion raises the question of how many time points are needed to accurately estimate kinetic parameters. Theoretically, the answer is only two as long as the substrate concentrations are measured accurately, concentrations at the two selected time points are different than the initial concentration and final concentration, and the first-order model can be assumed. With only two time points, the calculation of
As long as fiber digestion is occurring throughout the interval defined by the two time points, then
There are advantages to using more than two points. Each additional time point decreases the leverage of the others. Since it is not possible to measure the fiber concentration with absolute accuracy, the slope of a line computed with only two points may be subject to higher error than one calculated with more points. Because a line is defined by two points, it not possible to assess if the linear model describes the relationship so it is necessary to assume that it does. However, one might argue that it is better to have more replications to estimate a mean concentration for a few values of
7. Fiber digestion model
This equation is the nonlinear form of Eq. (12) that has been adjusted for
This equation indicates that the digestible fiber concentration at any given time (
which factors to:
The units of [
Eq. (18) is one form of a common nonlinear model that is used to describe asymptotic increase. It goes by many names including the monomolecular equation that is used to describe chemical reactions involving a single molecule and the Mitscherlich equation that is used to describe crop yield responses to fertilizer. Archontoulis and Miguez  simply refer to it as “Exponential gives rise to maximum.” It is useful for characterizing a host of biological relationships that exhibit asymptotic behavior. In its simplest form, it is a two-parameter equation, but a third parameter is sometimes used as a scaling factor to reduce the pool size  affected by the proportion defined by e−
As presented in Eq. (18), the monomolecular equation cannot be linearized in a manner that lends itself to an algebraic solution for all parameters. Estimates of all parameters, however, can be obtained simultaneously using nonlinear regression. In the case of fiber digestion, the parameters are well defined chemically and biologically, and it is more straightforward and, therefore, advantageous to determine their values directly and sequentially as demonstrated in Examples 2.1 and 2.2 (Table 1).
Estimates of the digestion model parameters (Eq. (18)) are presented in Table 1. We have already concluded that the rate of digestion was over two times as fast for alfalfa as for tall fescue. However, this rate only applies to the digestible portion of fiber, which was much greater in tall fescue. The absolute rate of fiber digestion (Eq. (10)) at the onset of digestion (
In the next section, we will combine the information we developed in Example 2 with a theory developed by Van Soest and colleagues  to estimate the true dry matter digestibility of herbage.
8. True digestion model
True digestion of herbage dry matter is distinguished from apparent digestion by the contribution of the animal to fecal dry matter . That is, some of the dry matter contained within the feces arises from the animal and microbes inhabiting its digestive system. This includes microbial cells and material sloughed from the walls of the digestive tract . The animal, however, does not contribute fiber to the feces. All the fibers contained within the feces originate from the plant matter consumed by the animal. Given that all dietary constituents other than fiber are virtually digested completely, the true digestibility of the diet can be calculated by excluding nonfiber components from the fecal dry matter. Thus, the true digestibility coefficient of an herbage or diet can be calculated as:
where DMI is the dry matter intake; the amount of DM consumed and
Depending on the composition of the diet and its physical form, there may be a difference between undigested fiber and that which is truly indigestible . In diets with high passage rates, some of the fiber that may have been digested if exposed to the rumen environment for a longer time escapes undigested and is recovered in the feces. However, for diets that consist entirely or mostly of herbage, it is reasonable to assume that the indigestible fiber fraction reasonably reflects that portion of the diet that is undigested. In
The comprehensive system for feed analysis developed by Van Soest  partitions herbage dry matter into two primary fractions based on studies of nutritive uniformity using the Lucas test: cellular contents and neutral detergent fiber. Cellular contents are nutritionally uniform in that they have the same true digestibility across a range of herbages and other feedstuffs. They are virtually completely digestible and according to Van Soest have a true digestion coefficient of 0.98. Digestibility of fiber varies greatly among herbages as we have seen.
Van Soest  used these relationships to develop a summative equation for predicting true and apparent digestibility. Accordingly, true digestibility is the sum of cell contents (×0.98) and digestible fiber (C
Adding the [
Having a model that includes the principal parameters affecting herbage digestion allows assessment of how each entity and the parameters acting upon it influence herbage digestibility and by extension energy availability. Two herbages with similar true digestibility may differ greatly in how that value is achieved. They may have different concentrations of cell solubles and digestible fiber and rate of fiber digestion. Based on the model (Eq. (20)), strategies for increasing the true digestibility of herbage could include simply increasing the cell soluble concentration, increasing the concentration of digestible fiber, and/or increasing the rate at which the latter is digested. Focusing on improving any one of these parameters in isolation of the others would not necessarily lead to an improvement in true digestibility because whatever gains achieved in one could be lost from negative changes in the others. Using any or all of these strategies, the end result is a decrease in the concentration of indigestible fiber (C
9. Dry matter intake
The nutritional value of herbage depends largely on the amount of digestible energy that the animal derives from consuming it . In this application, we present a model for predicting true digestibility that relates directly to the energy concentration available to support maintenance and production. How much energy the animal actually ingests, however, also depends on the amount of herbage consumed. Digestible energy intake is the product of dry matter intake and digestibility and is often limited for herbage diets.
There are many factors that influence the amount of herbage that is consumed by an animal. Some of these are related to the animal and its body size, plane of nutrition, and psychogenic factors that influence palatability . There are chemostatic controls that regulate intake and tend to suppress it once the animal’s demand for energy has been satisfied . Intake of diets that are predominantly herbage, however, often are regulated by physical distention of the digestive tract. This latter mechanism is generally referred to as fill volume because it represents the quantity of undigested herbage than can be accommodated by the size of the digestive system.
The intake of indigestible fiber is often observed to be relatively constant across similar herbage diets that vary in digestibility suggesting a limit in the amount of indigestible material that an animal can consume . As the digestibility of the diet increases for animals of similar size and nutritional status, the amount of dry matter that can be consumed also increases because there is less undigested material to retard its passage through the digestive system. Because of this, dry matter intake is often correlated to indigestible fiber concentration and a simple fill model can be used predict it:
In a brief survey of the literature Moore et al.  found that growing beef steers consumed between 0.4 and 0.6% of their body weight of indigestible fiber when fed diets consisting of warm-season grasses. A graph showing predicted intake as a function of CI using a fill constant of 0.5 is presented in Figure 6. Using this relationship, the estimated DMI would be 3.4% BW for alfalfa and 2.2% BW for tall fescue evaluated in Section 7. These are realistic estimates and could be reasonably accurate as long as the fill constant is similar for the class of animal consuming these diets.
It should be obvious that something as complex as DMI cannot be universally predicted using a simple model with one parameter. However, that is beside the point when using the model to develop strategies for improving forage quality. There should be no disagreement that for a given animal, there is a physical limit on how much indigestible dry matter they can consume. However, assuming that it is the same for all animals or even all animals within a specific class is probably unreasonable. This does not negate the utility of the concept for understanding how indigestible fiber affects DMI and nutritive value of herbage. The model is useful in that it demonstrates why modest improvements in true digestibility usually result in disproportionate increases in digestible energy intake .
The value of using a simple model to describe biological responses is that it enables a better understanding of the response. It is one thing to say that observed values are different, another to say how they are different, and still yet another to say why they are different. Fitting a model to the response creates the possibility of accomplishing all three outcomes. It is important to realize, however, that the parameters of some models that fit a response cannot be easily interpreted. The coefficients from a quadratic equation used to fit the data from Section 5 would be difficult to interpret relative to any biological meaning or significance even though the model fits reasonably well (
The examples presented in this chapter demonstrate the utility of using simple mathematical models to explain nutritional aspects of herbivory. It should be understood that simple models cannot be expected to fully explain complex phenomena. There are too many factors involved in most biological systems to be able to do so. This does not mean that the models are not valid within the constraints they are used, but that they should not be generalized to other situations without validating their predictive performance in those situations.
A1. Rumen degradable protein
A2. Fiber digestion
A2.1. Determining time intervals to include in regression
A2.2. Calculating rate constants using linear regression
Note that this procedure requires that time intervals where no digestion occurred have been deleted from the active data set, the indigestible fiber concentration ([
The lackfit option in the model statement requests that the residual variance be partitioned into lack of fit and pure error in order to test if the model describes the response.