modelos prod de gas in vitro

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ABSTRACT: Eleven 1-pool, seven 2-pool, and three

3-pool models were compared in fitting gas production

data and predicting in vivo NDF digestibility and effec-

tive first-order digestion rate of potentially digestible

NDF (pdNDF). Isolated NDF from 15 grass silages har-

vested at different stages of maturity was incubated in

triplicate in rumen fluid-buffer solution for 72 h to es-

timate the digestion kinetics from cumulative gas pro-

duction profiles. In vivo digestibility was estimated by

the total fecal collection method in sheep fed at a main-

tenance level of feeding. The concentration of pdNDF

was estimated by a 12-d in situ incubation. The param-

eter values from gas production profiles and pdNDF

were used in a 2-compartment rumen model to predict

pdNDF digestibility using 50 h of rumen residence

time distributed in a ratio of 0.4:0.6 between the non-

escapable and escapable pools. The effective first-order

digestion rate was computed both from observed in

vivo and model-predicted pdNDF digestibility assum-

ing the passage kinetic model described above. There

were marked differences between the models in fitting

the gas production data. The fit improved with increas-

ing number of pools, suggesting that silage pdNDF is

not a homogenous substrate. Generally, the models

predicted in vivo NDF digestibility and digestion rate

accurately. However, a good fit of gas production data

was not necessarily translated into improved predic-

tions of the in vivo data. The models overestimating

the asymptotic gas volumes tended to underestimate

the in vivo digestibility. Investigating the time-related

residuals during the later phases of fermentation is im-

portant when the data are used to estimate the first-or-

der digestion rate of pdNDF. Relatively simple models

such as the France model or even a single exponential

model with discrete lag period satisfied the minimum

criteria for a good model. Further, the comparison of

feedstuffs on the basis of parameter values is more un-

equivocal than in the case of multiple-pool models.

Key words: effective digestion rate, fiber, gas production, mathematical model, rumen digestion

2008 American Society of Animal Science. All rights reserved. J. Anim. Sci. 2008. 86:26572669

doi:10.2527/jas.2008-0894

INTRODUCTION

The in vitro gas production technique has widely

been used during the last decade among ruminant

nutritionists to study feed degradation. Mathematical

description of the data has been accomplished by fit-

ting a variety of nonlinear models (France et al., 1993,

2000; Schofield et al., 1994; Cone et al., 1996). Most

of the new models are non-first-order models [i.e., the

digestion rate (kd) is variable during fermentation]. Alimitation of the non-first-order kinetic parameters is

that they cannot be used in steady-state rumen mod-

els. To overcome these problems, Pitt et al. (1999) de-

rived equations to estimate the effective first-order kd

from non-first-order models assuming that rumen is a

1-compartment system with random passage of feed

particles.

However, the feed particles are retained selectively

in the rumen (Allen and Mertens, 1988), and without

the mechanisms of selective retention, it is impossible

to attain observed in vivo digestibility of potentially di-

gestible NDF (pdNDF) with realistic passage kinetic

parameter values (Ellis et al., 1994; Huhtanen et al.,2006). Huhtanen et al. (2008) demonstrated that using

digestion kinetic parameters derived from gas produc-

tion data fitted with a 2-pool Gompertz model and indi-

gestible NDF (iNDF) concentration estimated by 12-d

in situ incubation in a 2-compartment rumen model

predicted the in vivo NDF digestibility of grass silages

accurately and precisely. Furthermore, they proposed a

method for estimating the effective first-order kd from

Prediction of in vivo neutral detergent fiber digestibility and digestion

rate of potentially digestible neutral detergent fiber:

Comparison of models

P. Huhtanen,*1 A. Seppl, S. Ahvenjrvi, and M. Rinne

*Department of Animal Science, Cornell University, Ithaca, NY 14853; and MTT Agrifood Research Finland,

Animal Production Research, FI-31600 Jokioinen, Finland

1Corresponding author: [email protected]

Received January 22, 2008.

Accepted May 30, 2008.

2657

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the model-predicted pdNDF digestibility. The model

predicted the in vivo digestion rate accurately and pre-

cisely. The problem of the 2-compartment Gompertz

model and other non-first-order models is that although

they fit the data better than simpler models such as

the first-order exponential model, the parameter val-

ues of these models do not always permit unequivocal

comparison of feedstuffs.

The objective of the present study was to evaluatethe importance of choice of models in fitting gas pro-

duction data to estimate the in vivo NDF digestibility

and effective first-order digestion rate of grass silages.

MATERIALS AND METHODS

All sheep in the digestibility trials and ruminally

fistulated cattle as rumen fluid donors for the in situ

incubations were managed according to legislation doc-

umented in the Finnish Animal Welfare Act (247/96),

the order of using vertebrate animals for scientific pur-

poses (1076/85), and the European convention for the

protection of vertebrate animals used for experimental

and other scientific purposes, appendix A and B, imple-

mented under the auspices of the local animal use and

care committee.

Feeds

In vitro gas production was measured from isolated

neutral detergent residue of 15 primary growth timo-

thy (Phleum pratense)-meadow fescue (Festuca praten-

sis) silages harvested at different stages of maturity

over 3 yr. Details of silages, isolation of NDF, estima-

tion of iNDF concentration, and in vivo digestibility of

the silages are described in more detail by Huhtanen

et al. (2008) and chemical analyses by Nousiainen et

al. (2003a).

The concentration of pdNDF (g/kg of DM) was calcu-

lated as NDF (g/kg of DM) iNDF (g/kg of DM). BecauseiNDF by definition is indigestible, amount of iNDF in-

gested in feeds was also used for output. Thus, in vivo

digestibility of pdNDF was calculated as follows: [NDF

intake (g/d) NDF output (g/d)]/pdNDF intake (g/d).The concentration of digestible NDF (dNDF; g/kg of

DM) was calculated as follows: [NDF intake (g/d) fe-cal NDF (g/d)]/DMI (kg/d).

In Vitro Gas Production Measurements

In vitro gas production measurements were made by

an automated system described in detail by Huhtanen

et al. (2008). In brief, the system consists of 39 serum

bottles with a volume of 120 mL. The system is based

on pressure transducers (142PC05D, Honeywell Inc.,

Minneapolis, MN) and solenoid valves (11151-SV-

24Q70, Pneutronics, Hollis, NH). The contents of fer-

mentation bottles are stirred intermittently with 15 s

of stirring and 30-s pauses using a magnetic stirring

plate. Solenoid valves are adjusted to open when the

differential pressure in the bottle reaches 2.8 kPa to

release accumulated gas (approximately 1.1 mL). The

system allows using a relatively large sample size (500

mg), which will decrease the relative contribution of

gas from the inoculum and decrease the random errors

associated with a small sample size (e.g., those related

to attachment of feed particles on the walls of fermen-

tation bottles). Cumulative gas production was record-

ed every 15 min for the 72-h incubation periods, result-

ing in 288 recordings per sample. Gas production fromeach sample was determined in 3 replicated runs.

Models and Curve-Fitting

For each NDF residue, a mean gas curve was calcu-

lated from the 3 replicates. Eleven different 1-pool, 7

different 2-pool, and three 3-pool models were fitted to

the data by the NLIN procedure (SAS Inst. Inc., Cary,

NC). The 1-pool models are presented in Table 1. The

models included 3 exponential equations EXP0, EX-

PDLag, and EXPLPool. Subscripts 0, DLag, and LPool de-

note absence of lag, discrete lag time, and lag pool. The

concepts of discrete lag and lag pool are explained indetail by Mertens (1993a). Groot et al. (1996) proposed

an empirical multiphase model, in which B (units, h)

describes the time at which 50% of the asymptotic value

of gas was produced and C is a dimensionless switching

parameter that together with B determines the shape

of the curve. The France et al. (1993) equation is a gen-

eralization of the Mitscherlich model, which allows ac-

commodating either a sigmoidal or nonsigmoidal shape.

The generalized Michaelis-Menten (Gen M-M) model

allows the rate of digestion to decrease continuously

or to increase first and then decrease and includes a

lag time. The logistic (LOG) model assumes that gas

production is proportional both to the microbial mass

and the digestible substrate (Schofield et al., 1994).

Gompertz (GOMP) and Richards (RICH) models are

extensively used to describe growth functions. Two dif-

ferent forms of the Gompertz equations as derived by

Schofield et al. (1994; GOMPS) and France et al. (2000;

GOMPF) were used. A probability function Weibull

(Ross, 1976;WEIB) was also included in the analysis.

The compartmental interpretation of most of the func-

tions is described in detail by France et al. (2000).

The models EXPDLag (EXPDLag2), Groot (Groot2), lo-

gistic (LOG2), Gompertz (GOMP2S and GOMP2F), Rich-

ards (RICH2), and Weibull (WEIB2) were also extendedto accommodate 2 cell wall pools with fast and slow

degradation rates. The goodness of fit was markedly

improved for the other 2-pool models except for the

EXPDLag2 model. The improvement in goodness of fit

was marginal for this model, and in many cases, the

rates were similar for both pools. For the Gen M-M, the

fit was not markedly improved by including the second

pool or the model did not converge, lacked robustness,

or resulted in local optima or biologically meaning-

less parameter values. Three-pool models were used

for Gompertz (GOMP3S), Groot (Groot3), and logistic

(LOG3) models.

Huhtanen et al.2658



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Rumen Digestion Models

Digestion kinetic parameters derived from the gas

production data were used in a dynamic mechanisticmodel described by Huhtanen et al. (2008) to predict

ruminal pdNDF digestibility for a single batch of feed.

The amount of pdNDF digested in the rumen was esti-

mated using a model incorporating the selective reten-

tion of feed particles in the rumen (Allen and Mertens,

1988). Large forage particles entering the rumen need

to be comminuted below a threshold size before they

can escape from the rumen. Specific gravity has also

been shown to influence the probability of particles to

escape from the rumen. Marker kinetic studies (e.g.,

Pond et al., 1988; Lund et al., 2006) have demonstrated

that the release of feed particles from the nonescapable

pool to the escapable pool is a time-dependent process;therefore, a gamma time-related function (G2G1) was

used to describe this process. The rate constant (kr)

was calculated as: kr = (t 2)/[1 + (t )], where =

the rate constant (= 2/residence time in the compart-

ment) and t = time. The passage rate from the escap-

able pool was assumed to be a first-order process. The

mean residence time in the rumen was assumed to be

50 h. Distribution of 0.4:0.6 of rumen residence time

between the nonescapable and escapable pools was as-

sumed (Huhtanen et al., 2006; Lund et al., 2006).

Model-predicted pdNDF digestibility was used to

calculate the effective first-order digestion rate (ef-

fective kd) using the equation described by Huhtanen

et al. (2006): effective kd = {[kp + kr] + [(kp + kr)2 +

4 pdNDF digestibility krkp/(1 pdNDF digestibil-ity)]0.5}/2. Values of 0.05 and 0.0333 were used for kp

and kr, respectively. Pitt et al. (1999) demonstrated

that the function derived to estimate effective kd is in-

dependent of intake, which allows using this param-

eter in continuous rumen models.

Digestibility of cell wall carbohydrates in the rumen

is competition between the rates of digestion and pas-

sage. The total mass disappearing by digestion is the

integral over time from t > lag to , and total extent

of digestion is defined as the total amount degraded

divided by the total amount entering the system. For

the simple exponential model, the digestibility (D) with

a kd and 1-compartment rumen model with first-orderpassage (kp) can be evaluated by a simple algebraic

equation [D = kd/(kd + kp)], but for most models, the

integrals are nonanalytical (France et al., 2000) and

must be evaluated numerically.

Except for the EXP0, EXPDLag, and EXPLPool models,

kd is or can be time-dependent. The kd at each time

point can be estimated as the mass digested as a pro-

portion of the mass in the rumen. If the function of cu-

mulative gas production is described as F(t) and the de-

rivative of the function F(t) = dF/dt describes the mass

disappearing at time = t, the kd at each time point (t)

can be described as follows: kd = F(t)/F(t). Estimated

gas production parameters from different models wereused to compute the value of F at different time points.

With the models including discrete lag time, the diges-

tion began at t = lag time, but the release from the non-

escapable to the escapable pool commenced at time t =

0. In the model ExpDLag, the retention in the lag pool

was included in the rumen nonescapable pool.

The amount of dNDF predicted by the model was

calculated as pdNDF (g/kg of DM) model-predicted

pdNDF digestibility. Total NDF digestibility was calcu-

lated as model-predicted dNDF/NDF. Single effective

first-order digestion rate was calculated from in vivo

pdNDF digestibility similarly as from model-predictedpdNDF digestibility described earlier.

The rumen models were constructed using Powersim

software (Bergen, Norway). Simulations were run for

120 h with a 0.0625-h integration step and using the

Euler integration method.

Statistical Analysis

To evaluate the performance of the models, the good-

ness of fit was compared using the proportion of vari-

ance accounted for (R2) and residual mean squares

(RMS) and ranking the models according to RMS.

Table 1. Mathematical models to describe the cumulative gas production from in vitro

fermentation

Model1 Equation2 Domain

EXP0 Vt = V [1 e(kt)] t 0

EXPDLag Vt = V {1 e[k (t L)]} t L

EXPLPool Vt = V {1 1/[kd kL] [kd

e(kLt) kLe(kdt)]} t 0

Groot Vt = V/[1 + (BC/tC)] t 0

France Vt = V (1 e {k[t L] [c ( t L)]}) t L

Gen M-M Vt = V (t T)c

/[(t L)c

+ Kc

] t LLOG Vt = V/[1 + e[2 4k (t L)]] t 0

GOMP Vt = V1 exp {exp[1 k1 (t L1)]} t 0GOMPFR Vt = V(1 exp{b[e

c(t L) 1]/c}) t L, c > 0WEIB Vt = V

{1 e[k(t L)]}n t LRICH Vt = V

{1 e [k(t L)]}1/n t 0

1Gen M-M = generalized Michaelis-Menten; LOG = logistic; GOMP = Gompertz; WEIB = Weibull; RICH= Richards.

2Vt = volume of gas at time t; V = asymptotic gas volume; k and kd = rate parameters; L = lag parameter;kL = rate parameter from lag pool; B, b, C, c, r, and n = adjustable parameters.

Gas production models 2659



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These evaluations were made separately for the 11

one-pool models, the 6 two-pool models, and the 3

three-pool models.

Root mean squared errors (RMSE) between the ob-

served in vivo and model-predicted dNDF and NDF

digestibility values and in vivo derived and model-

predicted effective kd values were calculated as follows:

RMSE = (observed predicted)2/n. Mean squareprediction error (MSPE) was divided to components re-

sulting from mean bias, slope bias, and random varia-

tion around the regression line (Bibby and Toutenburg,

1977). The significance of the deviation of the intercept

from 0 and the slope from 1 was analyzed by t-test.

RESULTS

Feeds

The chemical and morphological composition and

in vivo digestibility of the silages are shown in Table

2. Due to differences in maturity, the silages showed

a large variation in all parameters. The silages were

well-preserved as indicated by low pH (4.10) and low

concentrations of VFA (22 g/kg of DM) and ammonia N

(47 g/kg of total N).

Fitting of the Gas Production Profiles

The proportion of variance explained by the models

was in general very high, as for all 15 gas production

curves, it was greater than 0.997 (Table 3). The RMSE

were 1.1 to 4.3% of the final gas volume. The Gen M-M

and Groot models showed the greatest average R2values,

whereas LOG was the worst according to this criterion.

The differences between the models were much greater

when evaluated on the basis of RMS. The average RMS

across the 15 curves was smallest with the Gen M-M

model followed by the Groot model. Models LOG and

EXP0 clearly had the greatest RMS. The GOMPS modelhad a smaller RMS than LOG and EXP0, but it was

always greater than for the other models. Although the

mean RMS was greater for the Groot model compared

with Gen M-M, the smallest RMS was observed more

often with the Groot model (8/15) than with Gen M-M

(4/15). Only the fit to models LOG (11/15) and EXP0

(4/15) resulted in the largest RMS.

Table 2. Composition and digestibility of the silages

Item Mean SD Minimum Maximum

Composition, g/kg of DM

OM 927 8.6 909 940

CP 161 38.3 112 239

NDF 586 82 402 669

iNDF1 81 47.6 17 158

Lignin 36 10.3 20 55

Morphological composition, g/kg of DM

Leaves 346 137 192 625

Stems 596 93 375 714

Heads 57 50 0 129

Digestibility

OM 0.725 0.073 0.613 0.840

NDF 0.737 0.085 0.597 0.869

pdNDF2

0.844 0.035 0.791 0.907

1Indigestible NDF.2Potentially digestible NDF.

Table 3. Goodness of fit of the 1-pool models

Model1

R2

Residual mean squares

Mean Median Minimum Maximum

EXP0 0.99771 23.8 19.6 6.94 57.2

EXPDLag 0.99966 3.46 2.98 1.30 7.83

EXPLPool 0.99960 4.24 3.34 0.83 11.5

France 0.99969 3.20 2.43 1.12 7.53

Gen M-M 0.99982 1.83 1.47 0.84 3.35

GOMPS 0.99893 10.3 9.52 6.15 15.6

GOMPF 0.99966 3.46 2.99 1.28 7.83

Groot 0.99979 2.13 1.66 0.85 4.11

LOG 0.99730 25.8 25.8 17.0 34.4

RICH 0.99968 3.26 2.41 1.13 6.76

WEIB 0.99967 3.34 2.75 1.21 7.72

1Gen M-M = generalized Michaelis-Menten; LOG = logistic; GOMP = Gompertz; WEIB = Weibull; RICH

= Richards.

Huhtanen et al.2660



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Extending the models for a second pool assuming

that the 2 pools describe the gas produced from rapidly

and slowly digestible NDF decreased RMS markedly

except for EXPDLag (Table 4). In many cases, fitting the

gas production profiles to the 2-pool EXPDLag model re-

sulted in similar rates for the both pools. The R2 values

were above 0.9999 with all other 2-pool models, and

RMS was below 1.00 in all cases. For the best mod-

els, the RMSE was only 0.32 to 0.70% of the final gasvolume. Including a third pool into the Groot, GOMPS,

and LOG further decreased RMS.

Asymptotic Gas Volume

Model-predicted asymptotic gas volumes together

with observed endpoints are shown in Figure 1. The

variation between the models in predicted gas volume

was relatively large, ranging from 116.2 mL (LOG)

to 130.1 mL (Gen M-M). The differences between the

models were highly significant (P< 0.001). Asymptotic

extent of gas production was closely associated with po-

tential NDF digestibility (pdNDF/NDF), with R2

valuesranging from 0.757 to 0.940 (Table 5). One- and 2-pool

Groot and Gen M-M models had the lowest R2 values

between potential NDF digestibility and asymptotic

gas volume. Including the third pool in the Groot mod-

el clearly improved the relationship between potential

NDF digestibility and asymptotic gas volume.

Some examples of time-related residuals are shown

in Figure 2. The residuals were usually greatest during

the first 20 h of incubation, but there were differences

also during the later phases of incubation.

Prediction of NDF Digestibility

The relationships between the predicted and ob-

served NDF digestibility are shown in Table 6. The R2

values were greater than 0.96 for all other models ex-

cept for the WEIB model, indicating that the precision

of this model was less than that of the other models.

The models EXP0, Gen M-M, Groot, and Groot2 under-

predicted in vivo NDF digestibility, whereas the LOG

model slightly overpredicted it. In most cases, the vari-

ation resulting from mean bias or random variation

across the regression line had the greatest contribution

to the total MSPE.

The relationship between the effective kd calculated

from model-predicted and in vivo-determined pdNDF

digestibility is shown in Table 7. Most models predict-ed the in vivo-derived effective kd accurately (bias from

0.005 to 0.005) and precisely R2 > 0.80. None of themodels was the best according to all criteria (small bias,

slope close to 1, intercept close to 0, and high R 2), but

the models France, RICH, LOG2, GOMP2S, GOMP2F,

and WEIB2 performed well according to these criteria.

DISCUSSION

The main objective of the present study was to in-

vestigate the importance of curve-fitting to the gas pro-

duction data. The parameter values of gas production

kinetics were used in a mechanistic rumen model toestimate in vivo NDF digestibility and effective first-

order digestion rate of pdNDF. Our previous study

(Huhtanen et al., 2008) demonstrated that in vivo NDF

digestibility could be predicted accurately and precise-

ly by a model in which digestion kinetic parameters

were estimated from gas production profiles by the

2-pool Gompertz equation and iNDF concentration by

a 12-d in situ incubation. The model described the cu-

mulative gas production profiles very well as indicated

by small mean square errors and confidence intervals

of the parameter values. However, a disadvantage of

this and other models with a time-dependent fraction-

al digestion rate, especially with multiple pool models,is that parameter values are strongly correlated and

do not permit unequivocal comparison of feedstuffs.

Steady-state rumen models (e.g., Danfr et al., 2006)

need first-order digestion rates, which preclude the use

of kinetic parameters estimated using models includ-

Table 4. Goodness of fit of the 2- and 3-pool models

Model1

R2

Residual mean squares

Average Median Minimum Maximum

Two-pool

ExpDLag2 0.99969 3.16 2.98 1.30 7.47

GOMP2S 0.99997 0.26 0.22 0.15 0.71

GOMP2F 0.99994 0.62 0.70 0.22 1.02

Groot2 0.99995 0.47 0.49 0.38 0.61

RICH2 0.99996 0.39 0.15 0.10 0.94

LOG2 0.99992 0.70 0.69 0.32 0.95

WEIB2 0.99999 0.15 0.17 0.04 0.24

Three-pool

GOMP3S 0.99999 0.07 0.05 0.02 0.20

Groot3 1.00000 0.04 0.03 0.02 0.12

LOG3 0.99999 0.11 0.12 0.05 0.14

1Gen M-M = generalized Michaelis-Menten; LOG = logistic; GOMP = Gompertz; WEIB = Weibull; RICH

= Richards.




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ing time-dependent digestion rates. Pitt et al. (1999)

proposed equations to estimate effective first-order

digestion rates for models including digestion lag and

time-dependent digestion rate for 1-compartment ru-

men models. This approach was extended by Huhtanen

et al. (2008) for 2-compartment rumen models. The ef-

fective first-order kd was computed from the model-predicted pdNDF digestibility by solving the equation

proposed by Allen and Mertens (1988).

Gas Production Models

Different versions of simple exponential equations

have been applied to gas production studies. Sometimes

a parameter describing a rapidly or immediately de-

gradable fraction has been included in models describ-

ing cumulative gas production, similarly as for the in

situ data (rskov and McDonald, 1979). As discussed

by France et al. (2000), this often results in negative

values at zero time. However, negative gas volumes

at zero time are biologically meaningless, and the use-

fulness of such parameter values in rumen models is

questionable.

Including a discrete lag time in the simple first-

order equation improved the fit of models markedly.The discrete lag time can be attributed to hydration,

removal of digestion inhibitors, or attachment of mi-

crobes with the substrate. This model assumes that

digestion begins instantaneously after the lag time.

Although the discrete lag time does not completely de-

scribe the lag phenomena, it provides a simple quan-

titative measure of lag effects (Mertens, 1993a). Allen

and Mertens (1988) proposed a model that describes di-

gestive processes by a 2-step mechanism: the first step

is a first-order process involving a change of substrate

from unavailable to available form (lag pool), and the

Figure 1. Model-predicted asymptotic gas volumes for 15 gas production curves (SE = 1.74 mL). Observed value

is the mean of the 5 data points during the last hour of incubation. Gen M-M = generalized Michaelis-Menten;

LOG = logistic; GOMP = Gompertz; WEIB = Weibull; RICH = Richards.

Huhtanen et al.2662



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second step represents the first-order digestion of the

substrate. Biologically, the first step can be interpreted

to be related to the same processes as the discrete lag.

Although the lag pool model is theoretically more cor-

rect than the discrete lag model, it did not fit the databetter than the discrete lag model. Mertens (1993a)

postulated that a discrete lag time was a necessary ad-

dition to the model to adequately describe the digestion

processes.

France et al. (1993) proposed a generalized Mitscher-

lich model, which can accommodate both sigmoid and

nonsigmoid shapes. Further, the fractional digestion

rate can increase or decrease. An increase in digestion

rate with time may reflect an increase in substrate

availability, microbial attachment, and increased mi-

crobial numbers. France et al. (2000) discussed sev-

eral factors (e.g., build-up of fermentation products,

decreased availability of growth factors) that could

result in a decreased fractional digestion rate as the

degradation proceeds. The France model fit the data

consistently better than the ExpDLag model, but the re-

ductions in RMS were rather small, suggesting that

the France model was not able to describe the digestion

kinetics of isolated NDF very accurately.

In the Gen M-M model, the kd increases first and de-

creases later during fermentation (c >1.0) or decreases

continuously (c 1.0). The average value for the c pa-rameter across the 15 gas production curves was 1.69

(range 1.46 to 2.30; i.e., in all cases, the kd increased

first and declined thereafter, probably reflecting in-

creased microbial number, substrate accessibility, and

microbial attachment at the beginning of incubation,

and exhaustion of the rapidly degradable substrate to-

ward the end of fermentation). Compared with othergas production models, the Gen M-M is more flexible

(France et al., 2000), and it decreased RMS compared

with the previous models. Groot et al. (1996) proposed

a modified Gen M-M equation without a discrete lag.

This equation has been extensively used by the Dutch

workers (Cone et al., 1996, 1997). Omitting the lag pa-

rameter from the model increased the RMS, but it was

still much smaller than the RMS of the other 1-pool

models.

In the RICH and WEIB models, the parameter n in-

fluences the pattern of changes in kd with time. When

the parameter value is 1.0 (RICH) or 1.0 (WEIB), thekd is constant (i.e., the model is exponential). In the

WEIB model, the kd either first increases rapidly and

thereafter slowly (n > 1.0) or first decreases rapidly

and thereafter slowly (n < 1.0). The effects of param-

eters n and c on the gas production curve in the WEIB

and France models are very similar, and consequent-

ly, the RMS values of the 15 gas curves were strongly

correlated (R2 = 0.995). In the RICH model, the kd ei-

ther increases (n > 1.0) or decreases (n < 1.0) withincreased digestion time; the changes are more rapid

at the early stages of incubation. The parameter n in-

creases the flexibility in curve-fitting with both WEIB

Figure 2. Time-related residuals after fitting of the 2-pool Gompertz (GOMP2S), logistic (LOG1), Michaelis-

Menten (Gen M-M), and France models to a single gas production curve. Residual mean squares for fitting the

gas production curves were 0.16, 23.79, 1.31, and 2.17 for the models GOMP 2S, LOG1, Gen M-M, and France. The

biases (observed predicted) of NDF digestibility predictions were 0.003, 0.023, 0.046, and 0.000 units, respec-tively.




9/15

and RICH models improving the fit compared with the

EXPDLag model. The effects of the n parameter may be

attributed to the changes in hydration, microbial at-

tachment and microbial growth, or to chemical and

structural changes in cell walls with the advancing fer-

mentation time.

The 1-pool Gompertz model derived by France et al.

(2000) fit the data better than the model derived bySchofield et al. (1994). Gompertz and logistic equa-

tions as given by Schofield et al. (1994) have an inher-

ent error in that predicted gas volumes are positive at

time zero, which is not biologically possible, because

the initial gas volume should be zero. In the Gompertz

model derived by France et al. (2000), the fractional

digestion rate increases with time when the parameter

value of c >0, which was interpreted as increased mi-

crobial activity per unit of feed. However, in the pres-

ent study using isolated NDF, the value of parameter c

was only marginally different from zero, and the model

approached ExpDLag.The 2-pool models fit the data markedly better than

the corresponding 1-pool models except the ExpDLag

model, in agreement with the study of Schofield et al.

(1994). The RMS with 1-pool models was positively

(R2 = 0.65 to 0.85) correlated with the in vivo NDF di-

gestibility, suggesting that 1-pool models did not ad-

equately describe the NDF digestion kinetics of early

harvested silages. The 2-pool models assume that the

pdNDF is comprised of rapidly and slowly degradable

fractions as described by Schofield et al. (1994) and

Schofield and Pell (1995). The proportion of the rap-

idly degradable pool of the total gas produced increased

with improved in vivo NDF digestibility, but the pro-

portion of the rapidly degradable pool varied markedly

between the models (0.39 in GOMP2, 0.26 in GOMP2FR,

0.15 in Groot2, 0.55 in LOG2, 0.23 in RICH2, and 0.22

in WEIB2).

The Gompertz model derived by Schofield et al.

(1994) fit all 15 cumulative gas production profiles

better than the model derived by France et al. (2000)when 2-pool models were used despite the positive

intercept at time zero with the Schofield et al. (1994)

model. However, including the second pool markedly

decreased the positive intercept, and it took only 1.5 h

to reach zero residual. Including the second pool to the

Gompertz model derived by France et al. (2000) pre-

sented some biological problems despite the fact that

the fit was markedly improved. Predicted asymptotic

gas volume was smaller than observed gas volume at

70 to 72 h in many cases (9/15). This was probably due

to high values of the parameter c, especially for the

rapidly degradable pool but also for the slowly degrad-able pool. For example, the fractional digestion rates

of the slow pool increased up to 0.15 to 0.20 per hour

toward the end of fermentation period.

Large variation between the models in the propor-

tions of the rapidly and slowly degradable pools despite

very good fit to the data may indicate that there are not

2 easily distinguishable pdNDF pools described by the

cumulative gas production profiles but rather several

cell wall fractions differing in the rate of digestion. De-

pending on the model structure, the division between

the rapidly and slowly degradable pools can be rather

arbitrary. Differences in kd have been reported between

Table 5. Relationships between the potential NDF digestibility (pdNDF/NDF) and model-predicted asymptotic

gas production (mL)

Model1 Intercept SE P-value Slope SE RMSE2 R2

One-pool

EXP0 20 11.2 0.10 125 12.8 3.23 0.871

EXPDLag 14 10.3 0.21 125 11.8 2.96 0.890

EXPLPool 8 9.3 0.41 131 10.6 2.67 0.916

France 1 9.5 0.94 140 10.8 2.73 0.922

Gen M-M 42 12.1 0.004 101 13.9 3.49 0.789GOMPS 9 9.5 0.37 146 10.9 2.74 0.927GOMPF 9 10.5 0.38 130 12.0 3.02 0.892

Groot 47 12.2 0.002 94 14.0 3.53 0.757

LOG 14 9.3 0.16 150 10.7 2.69 0.933RICH 6 9.8 0.53 133 11.2 2.82 0.909

WEIB 5 10.0 0.66 135 11.4 2.88 0.909

Two-pool

GOMP2S 3 9.0 0.74 144 10.3 2.60 0.933GOMP2FR 11 9.1 0.25 151 10.4 2.62 0.937Groot2 6 15.6 0.73 154 17.9 4.50 0.840LOG2 14 9.1 0.16 154 10.4 2.62 0.940RICH2 8 9.5 0.39 150 10.9 2.74 0.936WEIB2 2 9.6 0.85 142 11.4 2.88 0.917

Three-pool

GOMP3 4 9.3 0.70 145 10.6 2.68 0.929Groot3 5 10.7 0.65 149 12.3 3.09 0.913LOG3 10 9.3 0.39 151 10.7 2.70 0.934

1Gen M-M = generalized Michaelis-Menten; LOG = logistic; GOMP = Gompertz; WEIB = Weibull; RICH = Richards.

2RMSE = root mean square error.

Huhtanen et al.2664



10/15

Table 6. Relationships between the predicted and observed in vivo NDF digestibility, root mean square error

(RMSE), mean bias (observed predicted), and distribution of the mean squared prediction error (MSPE)

Model1 Intercept Slope R2 RMSE Mean bias

Distribution of MSPE

Bias Slope Random

One-pool

EXP0 0.064a 0.966 0.985 0.042 0.040 0.930 0.005 0.065

EXPDLag 0.059a 0.928b 0.989 0.013 0.007 0.264 0.235 0.502

EXPLPool 0.065a

0.919b

0.990 0.013 0.006 0.212 0.302 0.486France 0.025 0.974 0.983 0.012 0.005 0.178 0.032 0.790

Gen M-M 0.143a 0.858b 0.985 0.048 0.044 0.854 0.083 0.063

GOMPS 0.017 0.962 0.989 0.015 0.011 0.561 0.048 0.391GOMPF 0.041

a 0.941 0.988 0.012 0.005 0.186 0.180 0.633

Groot 0.152a 0.834b 0.989 0.040 0.036 0.769 0.166 0.065

LOG 0.020 0.954 0.989 0.018 0.015 0.682 0.050 0.268RICH 0.038 0.953 0.988 0.010 0.003 0.095 0.147 0.758

WEIB 0.049 1.063 0.856 0.030 0.002 0.006 0.027 0.967Two-pool

GOMP2S 0.014 0.995 0.989 0.013 0.010 0.579 0.001 0.420

GOMP2F 0.022 1.025 0.986 0.010 0.003 0.107 0.039 0.854Groot2 0.033 1.008 0.961 0.042 0.039 0.850 0.000 0.150

LOG2 0.018 1.025 0.986 0.010 0.000 0.001 0.042 0.957RICH2 0.009 1.021 0.984 0.012 0.006 0.254 0.019 0.727WEIB2 0.013 0.989 0.984 0.011 0.004 0.146 0.007 0.847

Three-pool

GOMP3S 0.016 0.986 0.986 0.012 0.006 0.271 0.009 0.720

LOG3 0.015 1.023 0.986 0.010 0.002 0.036 0.036 0.928Groot3 0.033 0.975 0.984 0.018 0.014 0.630 0.015 0.356

aIntercept significantly (P< 0.05) different from 0.bSlope significantly (P< 0.05) different from 1.

1Gen M-M = generalized Michaelis-Menten; LOG = logistic; GOMP = Gompertz; WEIB = Weibull; RICH = Richards.

Table 7. Relationships between the digestion rates of potential NDF digestibility [pdNDF (1/h)] calculated from

the model-predicted and in vivo-determined digestibility of pdNDF, root mean square error (RMSE), mean bias

(observed predicted), and distribution of the mean squared prediction error (MSPE)

Model1 Intercept Slope R2 RMSE Mean bias

Distribution MSPE

Bias Slope Random

One-pool

EXP0 0.003 1.206 0.823 0.014 0.013 0.902 0.019 0.079

EXPDLag 0.011 0.858 0.864 0.005 0.002 0.160 0.108 0.732

EXPLPool 0.013a 0.823b 0.872 0.005 0.002 0.100 0.178 0.722

France 0.006 0.939 0.846 0.005 0.002 0.160 0.020 0.820

Gen M-M 0.017a

0.933 0.879 0.014 0.014 0.905 0.003 0.092

GOMPS 0.009 0.795b 0.861 0.007 0.005 0.399 0.140 0.461

GOMPF 0.010 0.872 0.858 0.005 0.002 0.107 0.093 0.800

Groot 0.020a

0.832 0.883 0.012 0.011 0.810 0.037 0.153

LOG 0.012 0.740 0.864 0.009 0.007 0.493 0.168 0.338RICH 0.008 0.897 0.882 0.004 0.001 0.060 0.078 0.862

WEIB 0.032a

0.491b

0.303 0.012 0.001 0.010 0.269 0.721Two-pool

GOMP2S 0.001 1.082 0.862 0.006 0.004 0.505 0.023 0.472GOMP2F 0.005 1.055 0.851 0.005 0.001 0.047 0.019 0.934Groot2 0.004 1.182 0.742 0.014 0.013 0.873 0.014 0.112

LOG2 0.001 1.027 0.840 0.005 0.001 0.063 0.004 0.933RICH2 0.004 1.114 0.818 0.006 0.003 0.284 0.048 0.668WEIB2 0.002 0.998 0.833 0.005 0.002 0.144 0.000 0.856

Three-pool

GOMP3S 0.003 0.993 0.840 0.005 0.003 0.233 0.000 0.766

LOG3 0.004 1.075 0.835 0.005 0.001 0.066 0.030 0.903Groot3 0.005 1.011 0.814 0.008 0.005 0.562 0.000 0.438

aIntercept significantly (P< 0.05) different from 0.

bSlope significantly (P< 0.05) different from 1.1Gen M-M = generalized Michaelis-Menten; LOG = logistic; GOMP = Gompertz; WEIB = Weibull; RICH = Richards.




11/15

stems and leaves (Chaves et al., 2006) and also between

individual cell wall polysaccharides (Mertens, 1993b).

In vitro NDF digestion of various forages also demon-

strated a fast- and a slow-digesting pool (Van Soest et

al., 2005). Including a second pool for the exponential

models did not improve the fit much, in contrast to the

other 2-pool models. With the exponential model, the

small improvement in the fit is likely because the data

generated by summing 2 exponential functions producea curve shaped very much like a single-pool exponen-

tial function (Schofield et al., 1994).

Including a third pool in the model further decreased

the RMS, and the fit was almost perfect, with the av-

erage RMSE being 0.20, 0.15, and 0.27% of the mean

asymptotic gas production for models GOMP3S, Groot3,

and LOG3, respectively. However, none of the models

were able to identify a gas pool that could result from

the microbial turnover. The proportion of the third gas

pool characterized with the longest lag time and slow

rate decreased with improved in vivo NDF digestibility.

This is in contrast to what could be assumed from the

greater energy supply and increased microbial growthfrom early-harvested silages. Most likely, this pool rep-

resents the gas production both from slowly digestible

fiber and microbial turnover.

Cone et al. (1997) suggested that the 3 phases of gas

production can be related to the fermentation of solu-

ble and insoluble substrates, and microbial turnover.

In their study with sugars, the gas production from the

pool assumed to represent microbial turnover was be-

low 10% of the total gas volume (i.e., much less than

the gas production from the third pool in the present

study). Our data indicate that despite the very good fit

of the models, multipool models do not accurately and

consistently identify feed entities characterized by dif-ferences in degradation kinetics. The pools estimated

by the multipool models should therefore be viewed

as purely mathematical constructs that may or may

not correspond to chemical entities (Schofield, 2000).

Furthermore, the declining robustness with increased

number of pools is inherent in the nonlinear curve-fit-

ting procedure, and when the number of parameters is

increased, the possibility of encountering a local mini-

mum increases.

Relationship Between Potential NDF

Digestibility and Gas ProductionThe close relationship between the extent of poten-

tial NDF digestibility and total volume of gas produced

indicates that isolated NDF used in the present study

was a relatively homogenous substrate. Differences in

VFA pattern in fermentation or changes in partitioning

of carbon between VFA production and microbial syn-

thesis would influence the amount of gas produced per

unit of fermented substrate. However, the large varia-

tion between the models in the predicted asymptotic

gas volumes suggests that it is not possible to estimate

pdNDF from the gas volume. The models LOG, GOMP,

and LOG2 predicted lower asymptotic gas production

than the observed values (mean 70 to 72 h). Because

some gas production still occurred at 72 h (mean 0.12

mL/h), these models were not able to describe the gas

production profiles correctly. On contrary, the mea-

sured value at 70 to 72 h was proportionally only 0.927

to 0.938 of that predicted by the models EXP0, Groot,

Gen M-M, and Groot2. Assuming a digestion rate of

0.05 or 0.06 per hour, proportionally 0.973 to 0.987 ofpdNDF will be digested during 72 h. This together with

the distinct time-related pattern of changes in the re-

siduals after 40 to 50 h of incubation (discussed later)

suggests that these models overestimated the asymp-

totic value of gas production. Overestimation of the as-

ymptotic gas volume by the EXP0 model is related to

its inability to describe the lag phenomenon. The kd de-

creases at later stages of fermentation in the Groot and

Gen M-M models, which although theoretically sound

may lead to overestimation of the asymptotic gas vol-

ume as the time-related pattern of residuals suggests.

To at least partly overcome this problem, the length of

the incubation period may be extended. Another solu-tion is to increase the number of pools as lower asymp-

totic gas volume with the model Groot3 (123.2 mL) com-

pared with Groot1 (128.1 mL) and Groot2 (128.8 mL)

indicates. However, rather than attempting to estimate

the potential extent of NDF digestion from asymptotic

gas production, extended ruminal in situ incubation is

likely to be a more reliable method as indicated by the

close empirical relationship between iNDF concentra-

tion in grass or legume silages and in vivo OM digest-

ibility (Nousiainen et al., 2003b; Rinne et al., 2006).

Prediction of NDF Digestibility

As indicated earlier, using the gas production kinet-

ic data in a rumen model predicted the in vivo NDF

digestibility both accurately and precisely. The good

performance of the model can be attributed to the fol-

lowing 3 factors: (1) separation of NDF into iNDF and

pdNDF, (2) using digesta passage models including se-

lective retention of feed particles in the rumen, and (3)

an accurate and precise prediction of kd by a gas pro-

duction system (Huhtanen et al., 2008). Because the

digestible neutral detergent solubles can be estimated

accurately and precisely by the Lucas test (Huhtanen

et al., 2006), especially if forage-type specific equations

are used, this model can be used to estimate both the

D-value (concentration of digestible OM in DM) and ki-

netic parameters needed in the mechanistic dynamic

models.

Good performance in the curve-fitting was not al-

ways translated to accurate and precise predictions of

the in vivo NDF digestibility values. The France and

RICH models were the best 1-pool models in predicting

NDF digestibility; these models had a small MSPE and

bias, the intercept was not significantly different from

0, and the slope was not different from 1. Although the

use of kinetic parameters estimated by the 1-pool LOG

Huhtanen et al.2666



12/15

and GOMP models resulted in accurate and precise es-

timates of NDF digestibility, much poorer fit and un-

derestimation of the asymptotic gas volume preclude

acceptance of these models. The Gen M-M and Groot1

models fit the data much better than the other 1-pool

models, but they were less accurate in predicting the

NDF digestibility than the other 1-pool models except

for EXP0. This may result from the time-related pat-

terns of the residuals (observed predicted) of gasproduction curves. Investigating the residuals fromapproximately 50 h of incubation onwards showed a

clear time-related pattern (Figure 2); with advancing

incubation time, the residuals decreased consistently,

suggesting an overestimation of the asymptotic gas

volume, and consequently underestimation of the pa-

rameters describing kd. The LOG model showed an op-

posite pattern of time-related residual, which resulted

in underestimated extent and overestimated rate of di-

gestion. In the present study, the mean extent and rate

of gas production of the 15 silages were strongly and

negatively correlated (R2 = 0.95; data not shown). Be-

cause the extent of digestion was determined separate-ly using 12-d in situ incubation, biased estimates of the

extent of gas production lead to biased rate estimates

and, consequently, biased digestibility estimates.

Both the Gen M-M and Groot1 models resulted in a

significant slope bias (i.e., the digestibility of low di-

gestibility silages was underestimated more than that

of high-digestibility silages). It is possible that the as-

sumption of decreased kd with increased fermentation

time with these models was not correct, or that the effect

was too strong. In agreement with our data, Dhanoa et

al. (2000) observed a negative bias with the model Gen

M-M compared with the France model in the extent of

rumen digestion, whereas LOG tended to overestimatethe extent of digestion. Dhanoa et al. (2000) compared

several models in estimating the extent of rumen di-

gestion from gas production profiles using measured

values for potentially digestible fraction but assumed

the rumen as a 1-compartment system.

Surprisingly, the EXP0 model was as accurate in pre-

dicting the in vivo NDF digestibility as the models Gen

M-M and Groot despite the fact that the fit of gas pro-

duction data was much better with the latter 2 models.

This may be related to the time-related pattern of re-

siduals during late stages of incubation with the Gen

M-M and Groot models. It seems that large residuals

during early stages of fermentation (


13/15

in vivo-derived estimates. The estimated effective kd

can be used in steady-state rumen models, and it re-

sults in the same pdNDF digestibility as estimated for

a single feed batch using time-related digestion rates

estimated from the gas production curves.

Pitt et al. (1999) presented a formal derivation for

several models with variable digestion rates for esti-

mating the effective kd including the effects of digestion

lag. They assumed rumen as a single-compartment sys-tem without selective retention of particles, and both

passage and lag time had an influence on the estimat-

ed effective kd. However, in the 2-compartment system,

the proportion of pdNDF disappearing by passage is

smaller during the early residence time compared with

a 1-compartment system, but for later residence times,

the reverse is true. Due to the slow disappearance via

passage during early residence times, a digestion lag

time would have a smaller effect on the digestibility

in the 2-compartment compared with 1-compartment

system (Ellis et al., 1994).

Conclusions

Two-pool models could be fit better to gas produc-

tion data than 1-pool models, indicating that pdNDF

in grass silage cannot be considered as a homogenous

substrate. The best fit was not always translated into

improved accuracy of the in vivo predictions. There was

no evidence of benefits from using 2-pool gas produc-

tion models in terms of predicting the in vivo param-

eters despite markedly better fit of the data. It seems

that large residuals during early stages of fermenta-

tion (


14/15

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