non-steady-state modeling

Non-steady-state modeling of effects of timing and level of concentratesupplementation on ruminal pH and forage intake

in high-producing, grazing ewes1

R. Imamidoost and J. P. Cant2

Department of Animal and Poultry Science, University of Guelph, Guelph, Ontario, N1G 2W1 Canada

ABSTRACT: A computer model was developed topredict responses of lactating ewes to concentrate sup-plementation, whether on pasture or stall-fed, givenconcentrate once per day or in multiple feedings, andsuckling multiple lambs. The model considers effectsof concentrate supplementation on organic acid produc-tion, saliva flow, ruminal pH, and forage intake. Theuser defines ewe BW, feed composition, and concentratefeeding times and amounts. The reference ewe has freeaccess to forage and water. Upon consumption, foragesand concentrates enter into lag pools for 2.0 and 0.24h, respectively. Carbohydrates then enter ruminal poolsof degradable fiber, undegradable fiber, or nonstructu-ral carbohydrate, from which they are degraded or passto the lower gut. Rapid dissociation of organic acidsfrom carbohydrate fermentation and buffers from rumi-nation are simulated to determine ruminal pH ac-

Key Words: Forage Intake, Modeling, Ruminal pH, Sheep

2005 American Society of Animal Science. All rights reserved. J. Anim. Sci. 2005. 83:1102–1115

Introduction

The NRC (1985) provides nutrient recommendationsfor ewes suckling up to two lambs, but ewes of highlyprolific breeds produce more than two lambs per gesta-tion. To produce enough milk to raise three to fourlambs, there is a high demand for dietary nutrients,which is often met by feeding high levels of concen-trates. A ewe suckling four lambs would be expected toproduce more than 4 kg of milk/d during the first monthof lactation (Peart et al., 1975). According to the facto-rial AFRC (1993) model of energy partitioning, a 75-kgewe producing 4 kg/d milk containing 6.5% fat requires

1Financial support was provided by the Gartshore Memorial Re-search Fund, the Ontario Ministry of Agriculture and Food, andNSERC Canada.

2Correspondence—phone: 519-824-4120, ext. 56222; fax: 519-836-9873; e-mail: [email protected]).

Received March 25, 2004.Accepted January 28, 2005.

1102

cording to the Henderson-Hasselbach equation. ThepH, in turn, affects fiber degradation rates. Forage in-take continues during daylight hours until ruminalNDF exceeds 1.0% of BW, or organic acid concentrationexceeds 130 mM. A circadian pattern of organic acidconcentrations and pH of rumen contents with multipleconcentrate feedings was simulated by the model withroot mean square prediction error of 7.7 and 3.0 to 4.0%of the observed mean, respectively. However, ignoringfermentation of dietary protein may have caused anunderestimation of organic acid production rates. Themodel predicted the increase in total DMI and the sub-stitution effect on forage intake of increasing levels ofconcentrate supplementation. Simulations suggestedthat a single concentrate meal daily was best fed in theevening to minimize the substitution effect, and thatthere was no benefit in forage intake to feeding 2 kg/dconcentrate in more than two meals per day.

a daily ME intake of 9.2 Mcal/d. At a DMI of 4% of BW,the required ME content of the diet is 3.07 Mcal/kg.Barley grain itself is reported to have a ME content of3.11 Mcal/kg (NRC, 1985), which leaves little room forinclusion of forage in the diet. A common practice is toallot 500 g of concentrate/d per lamb suckled, which,when four lambs are suckling, can mean that concen-trate makes up more than 80% of the ewe’s intake.

Increasing the level of concentrate in a ewe diet in-creases milk production (Avondo et al., 1995; Zervas etal., 1999); however, high levels of concentrate feedingcan cause low ruminal pH (Mould et al., 1983; Carroet al., 2000), which can decrease forage degradability(Mould et al., 1983; Allen, 1997) and induce clinicalruminal acidosis. The level of concentrate feeding isthen a question of optimizing costs and benefits. Theacidotic cost of concentrate feeding occurs within hoursof each meal of concentrate consumed, and so is depen-dent on the size and frequency of meals. Ruminal pHcan range more than 1 pH unit over the course of aday, and it is the time spent under pH 6.0 or 5.6 that

Ruminal pH model of lactating ewe 1103

Figure 1. Diagram of nutrient flows in the model. Boxes represent state variables, and arrows represent flows. Poolsizes and nutrient flows for a ewe consuming 2.0 kg/d of mixed pasture at an instantaneous intake rate of 2.0 kg/d(all other parameters and inputs as in Tables 1 and 2) are shown inside boxes and beside arrows, respectively.

is taken to indicate acidosis (Keunen et al., 2002). Toaddress the need for feeding recommendations for high-producing ewes, a dynamic model was constructed topredict the consequences of timing and level of concen-trate intake in lactating ewes. The ewe may be pasture-or stall-fed, given concentrate once per day or in multi-ple feedings, and may be suckling multiple lambs.

Experimental Procedures

The general strategy for modeling the rumen wasbased on the lag approach of Illius and Gordon (1990),the rumination modeling of Argyle and Baldwin (1988),the concepts of Kohn and Dunlap (1998) for calculatingruminal pH, and the constraints on ad libitum intakefrom Forbes (1993).

The simulated animal is a grazing, lactating ewe withfree access to forage and a restricted level of concentratefeeding during the day. Forage and concentrate compo-sition is defined at the beginning of a model run. Theuser also may set the number of concentrate meals perday, as well as the time and amount fed in each meal.Feed ingredients are described in terms of their non-structural carbohydrate (Nc), degradable fiber (Df),and undegradable fiber (Uf) contents. It is assumed theanimal has free access to water during the day.

Upon being consumed, forages and concentrates en-ter lag pools and subsequently become available to theruminal environment for microbial degradation. Essen-tially, ruminal pH is calculated from concentrations oforganic acids (Oa) produced in fermentation of dietarycarbohydrates and of buffers entering from saliva (Fig-ure 1). Ruminal pH affects fiber digestibility. Ruminalfiber content, organic acid concentration, and daylength affect forage intake. In turn, forage and concen-trate intake and size of the forage lag pool affect buffer

production. The user defines concentrate feeding levels,whereas the forage intake is controlled by feedbackmechanisms.

Differential equations describing state variableswere written in Advanced Continuous Simulation lan-guage (Aegis Simulation, Inc., Huntsville, AL), andwere solved using a fourth-order Runge-Kutta algo-rithm with an integration interval of 0.001 d. Expres-sion of state variables in units of concentration is ob-tained by dividing the integrated pool size by ruminalvolume. Ruminal volume (in liters) is calculated fromBW using an allometric equation (Figure 2):

vol = 1.3 BW0.37 [1]

Lag Pools

According to Illius and Gordon (1990), concentratesand forages each enter a lag pool upon consumption.The feed components are not subject to degradation orpassage from the rumen during this time. The differen-tial equations for size, in kilograms, of the lag pools forconcentrates (CNlag) and forages (Frlag) are:

dCNlagdt = inCn − UinCn [2]

and

dFrlagdt = inFr − UinFr [3]

where inCn is the instantaneous concentrate intakerate set at 0 or 30 kg/d depending on concentrate mealtimes defined by the user, and instantaneous intake of

Imamidoost and Cant1104

Figure 2. Plot of ruminal volume of sheep measuredby liquid marker dilution and BW (♦; from Bergman etal., 1965; Stokes, 1983; Mees and Merchen, 1985; Judkinsand Stobart, 1988; Weston, 1988b; Gunter et al., 1990;Charmley et al., 1991; Martin et al., 2001; Lopez et al.,2003). The solid line represents Eq. [1] from the text: rumi-nal volume = a BWb, where a = 1.3 (approximate SE =0.7), and b = 0.37 (approximate SE = 0.11); root meansquare prediction error = 8.2% of the observed meanvolume.

forage (inFr) is calculated in the model as either 0 or4 kg/d (Table 1) when all conditions for forage intakeare met (see below). The user defines concentrate mealtimes and daily concentrate intake (DMICn), and themodel simulates the meals numerically as they occur.The rates of concentrate (UinCn) and forage (UinFr)release to the ruminal environment after the lag periodare calculated as a simple delay of intake according toIllius and Gordon (1990):

UinCn(t) = inCn(t − Cnlagtime) [4]

and

Table 1. Definition and numerical value of parameters used in the model

Parameter Definition Value Equation

BW Body weight 70 kg 1Cnlagtime Concentrate lag time 0.017 d 4Dawn Time of dawn 0.208 dDusk Time of dusk 0.875 dFrlagtime Forage lag time 0.083 d 5fSaBf Concentration of saliva that is buffer 0.15 M 27KaBf Dissociation constant for buffer 1.80 × 10−8 M 28, 30KaOa Dissociation constant for organic acids 1.58 × 10−5 M 28, 30kDfmax Maximum degradable fiber degradation rate constant 1.92 d−1 11kNcOa Nonstructural carbohydrate degradation rate constant 7.2 d−1 14kOaAabs Associated organic acid absorption rate constant 227 d−1 17kOaDabs Dissociated organic acid absorption rate constant 1.41 d−1 24kpass Liquid passage rate constant 2.7 d−1 16, 23, 26kPpass Particle passage rate constant 0.40 kpass 7, 9, 13MRC Maximum ruminal capacity 0.01 BWMwDf Molecular weight of degradable fiber 0.162 kg/mol 20MwNc Molecular weight of nonstructural carbohydrate 0.162 kg/mol 20

UinFr(t) = inFr(t − Frlagtime) [5]

where Cnlagtime and Frlagtime are the lag time peri-ods for concentrates and forages, respectively. There isusually a longer lag time for forages than for concen-trates due to their higher cell wall contents. Varga andHoover (1983) reported lag times of 0.24 h for barleygrain and 0.9 to 4.3 h for forages. Illius and Gordon(1990) used a 2-h lag in their model for feedstuffs with30 to 50% cell contents. For the pasture and barleygrain inputs for behavioral runs of the model, lag timesof 2 and 0.24 h, respectively, were used (Table 1).

Nutrient Pools

After the lag, nutrients become available to microor-ganisms for degradation and utilization. Feedstuffs arecharacterized as containing Df, Uf, and Nc. Fiber pools(Df and Uf) are considered for simulation of ruminalfill. In addition, Df and Nc are considered for determin-ing Oa production and ruminal pH (Figure 1). Net Oaproduction from protein is ignored assuming proteindegradation to Oa is balanced by Oa utilization for pro-tein synthesis by microorganisms.

Undegradable fiber is resistant to degradation in therumen and can be calculated from analytical NDF,NDF-insoluble CP, ADF-insoluble CP, and lignin val-ues according to Weiss et al. (1992). There is no degrada-tion of Uf in the model, so Uf (kg) is calculated frominflow from the delayed lag pool (Eq. [4] and [5]) andpassage out of the rumen (UUfpass) as

dUfdt =UinCn × fCnUf + UinFr × fFrUf [6]

− UUfpass

where fCnUf and fFrUf are the fractions of Uf in concen-trate and forage DM, respectively.

UUfpass = kPpass × Uf [7]


where kPpass is 40% of the liquid passage rate constant(Table 1).

Degradable fiber is that proportion of NDF that canpotentially be degraded by enzymatic activity of micro-organisms in the rumen. Thus, Df (kg) is simulatedbased on input from the delayed lag pools (Eq. [4] and[5]), passage to intestines (UDfpass), and degradationto Oa (UDfDfOa) as

dDfdt =UinCn × fCnDf + UinFr × fFrDf [8]

− UDfpass − UDfDfOa

where fCnDf and fFrDf are fractions of Df, calculatedaccording to Weiss et al. (1992), in concentrate andforage, respectively.

UDfpass = kPpass × Df [9]

and

UDfDfOa = kDfOa × Df [10]

where the degradation rate constant, kDfOa, rangesfrom 0 to 100% of the maximum rate constant, de-pending on ruminal pH. The lower the ruminal pH, thelower the activity of fibrolytic bacteria and rate of fiberdegradation (Mourino et al., 2001). The ruminal pH thatis optimal for growth of fibrolytic bacteria is between6.2 and 6.8 (Mould and Ørskov, 1983), and at pH 5.2,cellulolysis stops (Mourino et al., 2001). The followingequation was parameterized using PROC NLIN of SAS(SAS Inst., Inc., Cary, NC) from the in vitro gas produc-tion data of Mourino et al. (2001) to capture the transi-tion in degradation (Figure 3):

kDfOa =kDfmax

1 +

5.92pH

38[11]

where pH is calculated in Eq. [30]. Values of kDfOaestimated from serial nylon bag incubations of feed-stuffs in situ have ranged from 0.55 to 6.48 d−1 (Vargaand Hoover, 1983; Lopez et al., 1999). For the referencepasture, the value of kDfmax is set at 1.92 d−1 (Table1), so that 0 < kDfOa < 1.92 d−1.

Nonstructural carbohydrate is mainly composed ofstarch and soluble carbohydrates, which are readilydegraded by enzymatic activity of microorganisms. TheNc content of feedstuffs is calculated as the remainderafter analyzed NDF, CP, fat, and ash contents havebeen subtracted from the DM. The differential equationfor Nc (kg) is based on inflow from the delayed lag pools(Eq. [4] and [5]), outflow to the intestine (UNcpass),and degradation to Oa (UNcNcOa) as:

dNcdt = UinCn × fCnNc + UinFr × fFrNc [12]

− UNcpass − UNcNcOa

Figure 3. Rate constant for cellulose degradation byruminal microbes in vitro measured at different pH val-ues and expressed as a percentage of the value at pH 6.56(♦; from Mourino et al., 2001). The solid line represents

Eq. [11] from the text: rate constant =a

1 +

bpH

c

,where

a = 102% (approximate SE = 2), b = 5.92 (approximateSE = 0.02), and c = 38 (approximate SE = 8); root meansquare prediction error = 6.8% of the observed meanrate constant.

where fCnNc and fFrNc are the fractions of Nc in con-centrate and forage DM, respectively, and

UNcpass = kPpass × Nc [13]

and

UNcNcOa = kNcOa × Nc [14]

where kNcOa is the degradation rate constant. Starchdegradability depends on the feedstuff and the pro-cessing applied to it (Theurer, 1986). Degradation rateconstants have ranged from 0.58 to 14.0 d−1 in situ(Offner et al., 2003). A value of 7.2 d−1, measured forbarley, is used for the reference starch degradation rateconstant (Table 1).

Organic Acids

The production of Oa that dissociate into conjugatebase and H+ decreases ruminal pH, which in turn af-fects the dissociation of Oa. To model the equilibriumbetween Oa species, two pools were considered for un-dissociated (OaA) and dissociated (OaD) forms of Oa,respectively.

Organic acid in its undissociated form is produced inthe rumen by degrading Df (POaDfOa) and Nc (POaN-cOA) and is removed from the rumen by passage(UOaApass), absorption through the rumen wall(UOaAabs), and dissociation to OaD (UOaAOaD):


dOaAdt = POaNcOa + POaDfOa [15]

− UOaApass − UOaAabs − UOaAOaD

where the fluxes in Eq. [15] are all in moles per day.

UOaApass = kpass × OaA [16]

where kpass is the liquid passage rate constant (Table1). Liquid passage rate from the rumen was 2.30 to 2.69d−1 in wether lambs (Judkins and Stobart, 1987) and2.66 to 2.81 d−1 in lactating ewes (Weston, 1988b; Gun-ter et al., 1990). A value of 2.7 d−1 is used as the refer-ence liquid passage rate constant in the model (Table 1).

For absorption,

UOaAabs = kOaAabs × OaA [17]

where kOaAabs is 227 d−1 (Table 1) based on steady-state solutions of the model for inputs from 16 experi-ments in the literature (see below).

The terms POaNcOa and POaDfOa represent totalOa production from Nc and Df respectively, and arebased on total acetate, propionate, butyrate, and valer-ate productions rates as:

POaNcOa = PAcNcOa + PPrNcOa [18]

+ PBuNcOa + PVaNcOa

and

POaDfOa =PAcDfOa + PPrDfOa [19]

+ PBuDfOa+ PVaDfOa

Each mole of glucose fermented can produce 2 molof acetate, 2 mol of propionate, 1 mol of butyrate, or 1mol of valerate (Baldwin, 1995). The proportions pro-duced on a net basis from ruminal fermentation ofstarch and cellulose were calculated from literaturedata by Murphy et al. (1982). There were different pro-portionalities for high-forage and high-concentratediets. Linear extrapolations of the Murphy et al. (1982)constants are used to predict molar yields (Y) of individ-ual VFA per kilogram of Nc or Df fermented:

YAcNcOa =

2/MwNc(fDMFr × 0.5948 + fDMCn × 0.3987)

YPrNcOa =

2/MwNc(fDMFr × 0.1415 + fDMCn × 0.3020) [20]

YBuNcOa =

1/MwNc(fDMFr × 0.2050 + fDMCn × 0.1955)

YVaNcOa =

1/MwNc(fDMFr × 0.0587 + fDMCn × 0.1038)

YAcDfOa =

2/MwDf(fDMFr × 0.6579 + fDMCn × 0.7880)

YPrDfOa =

2/MwDf(fDMFr × 0.0866 + fDMCn × 0.0575)

YBuDfOa =

1/MwDf(fDMFr × 0.2280 + fDMCn × 0.0650)

YVaDfOa =

1/MwDf(fDMFr × 0.0275 + fDMCn × 0.0895)

where the molecular weight of glucose in Nc (MwNc)and Df (MwDf) is 0.162 kg/mol. The fractions of forage(fDMFr) and concentrate (fDMCn) in the daily DMI arecalculated in Eq. [32].

The production of acetate, propionate, butyrate, andvalerate is calculated by applying the yield coefficientsto rates of Df and Nc degradation (Eq. [10] and [14]) as:

PAcNcOa = UNcNcOa × YAcNcOa

PPrNcOa = UNcNcOa × YPrNcOa

PBuNcOa = UNcNcOa × YBuNcOa

PVaNcOa = UNcNcOa × YVaNcOa [21]

PAcDfOa = UDfDfOa × YAcDfOa

PPrDfOa = UDfDfOa × YPrDfOa

PBuDfOa = UDfDfOa × YBuDfOa

PVaDfOa = UDfDfOa × YVaDfOa

The OaD is produced from OaA (UOaAOaD) and re-moved from the rumen with liquid passage (UOaDpass)and absorption across the ruminal wall (UOaDabs).

dOaDdt = UOaAOaD − UOaDpass [22]

− UOaDabs

where

UOaDpass = kpass × OaD [23]

and

UOaDabs = kOaDabs × OaD [24]

with kOaDabs = 1.41 d−1 according to steady-state solu-tions of the model for inputs from 16 experiments inthe literature (see below).

Saliva Production and Ruminal pH

Bicarbonate is the most important buffer in saliva.Therefore, for the sake of simplicity, the bicarbonatesystem is the paradigm for pH buffering in the rumen.


The quantity of buffer in the rumen is determined byinflow from the saliva (PBfSaBf) and from the diet(PBfCnBf; in cases where a buffer like sodium bicarbon-ate is included in the concentrates), outflow from therumen (UBfpass), and to buffer acid production(UOaAOaD):

dBfdt = PBfSaBf + PBfCnBf − UBfpass [25]

− UOaAOaD

where

UBfpass = kpass × Bf [26]

PBfSaBf = PSa × fSaBf [27]

The term fSaBf is the concentration of buffer in saliva,and PSa is the rate of saliva production. Ruminantsaliva contains 126 mEq/L of HCO3

− and 26 mEq/L ofPO4

2− (McDougall, 1948; Bailey and Balch, 1961) for afSaBf of 0.15 M (Table 1). According to Meot et al.(1997), 18% of saliva is produced during resting, 36%during eating, and 46% during rumination. Daily salivaproduction in sheep ranges from 6 to 16 L (Carter etal., 1990). Assuming the parotid glands contribute 50%of total saliva production (Meot et al., 1997), we usedthe parotid secretion rates reported by Meot et al. (1997)to set instantaneous PSa at 7.2, 16.2, and 19.5 L/dduring resting, eating, and ruminating, respectively,following Argyle and Baldwin (1988). Rumination wasconsidered to occur when Frlag > 0.01 kg (Eq. [3]) andthe ewe was not eating.

For pH modeling, it was assumed, following Kohnand Dunlap (1998), that dissociation of 1 mol OaA(UOaAOaD) is buffered by 1 mol Bf flowing to CO2 +H2O, and that the bicarbonate and Oa systems are al-ways at the equilibrium state described by their respec-tive dissociation constants (KaOa and KaBf), and theCO2 pressure remains at a constant 0.7 atm due toexchange with the gas phase and eructation.

Accordingly, the equalities

KaOa =cOaD × cH

cOaA and [28]

KaBf =cBf × cH

0.7

yield the constraint

cOaD =KaOa × cOaA × cBf

KaBf × 0.7 [29]

where the leading “c” denotes a concentration in molesper liter.

The value of UOaAOaD in Eq. [15], [22], and [25],for which Eq. [29] holds true, is solved by a Newton-Raphson iteration method. From an initial guess,

UOaAOaD is modified in successive iterations until Eq.[29] holds true.

The ruminal pH is then calculated according to theHenderson-Hasselbach equations:

pH = pKaBf + log

cBf0.7

or [30]

pH = pKaOa + log

cOaDcOaA

where pKaBf and pKaOa are the pKa values for Bfand Oa, respectively. Both equations yield the samepH value.

Forage Intake

Instantaneous intake of forage inFr = 4.0 kg/d wheninCn ≤ 0 kg/d, cOaA + cOaD ≤ 0.13 M, actual ruminalcapacity (ARC) ≤ 0.8 × maximum ruminal capacity(MRC), and time of day is between dawn and dusk.According to Baile (1975), a typical instantaneous rateof good quality roughage intake from a manger by sheepis 15 kg/d. Intake rate measured over 12 to 15 bites of apasture sward was 6.2 to 11.5 kg/d. These rates excludenonbiting and nonchewing times that occur duringgrazing. Sixty-five-kilogram wethers on a grass swardconsumed 1 kg/d DM in 8.9 h of grazing (Thomson etal., 1985), which yields an average rate of 2.7 kg/d.Lactating ewes consumed 2.9 kg/d DMI in 16 h of theheaviest grazing times of the day for a rate of 4.4 kg/d(Bermudez et al., 1989). A value of 4.0 kg/d was chosenfor the forage intake rate in the model.

Of NDF weights measured at six time points through-out a day in grazing sheep, the maximum ranged from0.76 to 1.2% of BW depending on the pasture. Valuesfor MRC of 0.9, 1.2, and 1.38 % of BW have been usedin past intake models (Mertens, 1987; Poppi et al., 1994;Chilibroste et al., 1997). The MRC value used here is1.0% of BW (Table 1). Actual ruminal capacity is calcu-lated as:

ARC = Df + Uf + Frlag(fFrDf + fFrUf) [31]

+ Cnlag(fCnDf + fCnUf)

Although forage intake commences when ARC = 0.8MRC, it continues until ARC ≥ MRC or some othercondition on inFr is violated.

Organic acid concentration is a controlling factor toterminate feeding in many models (Forbes, 1993; Chili-broste et al., 1997). Forbes (1993) used a threshold of130 mM Oa, beyond which sheep stop eating. The sameconcept and parameter value are used in this model.

Day length was considered a limiting factor on totaldaily intake simulated from instantaneous rates. Ac-cording to Bermudez et al. (1989), there is lower intakebetween the hours of 2400 and 0800 compared with the


rest of the day. Such a decrease in feeding activity iscommon among animals that are more active duringthe day. Sheep typically commence grazing at dawnand continue for 4 h and then have a second grazingbout in the afternoon until dusk (Thomson et al., 1985).Dawn and dusk are inputs to the model and, for thereference state, we simulated a temperate summer day,with dawn at 0500 and dusk at 2100.

Proportions of forage (fDMFr) and concentrate(fDMCn) in the total daily DMI were used in Eq. [20]to calculate proportions of individual Oa produced inmicrobial fermentation:

fDMFr = DMIFr/DMI [32]

fDMCn = DMICn/DMI

where DMICn is the daily concentrate intake definedby the user (Eq. [2]) and DMIFr is the rolling averageinFr over the past day:

DMIFr = ∫t

t−1inFr [33]

In total,

DMI = DMIFr + DMICn [34]

Parameters of Oa Absorption

Short-chain fatty acids can be absorbed across theruminal epithelium in both associated and dissociatedforms (Bugaut, 1987; Kramer et al., 1996), although thetransport mechanisms are different for the two forms.First-order rate constants for OaA and OaD absorption,kOaAabs (Eq. [16]), and kOaDabs (Eq. [23]) were calcu-lated from 16 measures of BW, nutrient intake, totalconcentration of Oa in the rumen (cOaT), and ruminalpH (Mees and Merchen, 1985; Judkins and Stobart,1988; Weston, 1988b; Gunter et al., 1990; Charmley etal., 1991; Chiofalo et al., 1992; Hadjipanayiotou andPhotiou, 1995; Carro et al., 2000; Martin et al., 2001;Castro et al., 2002). Differential Eq. [2], [3], [6], [8],[12], [15], [22], and [25] were set to zero to solve themodel in steady state for each of the 16 input sets, usingthe parameter values given in Table 1. Given, from Eq.[30], that cOaA = cOaT/(1 + 10pH − pKaOa), Eq. [15] and[22], respectively, were rearranged to calculate rateconstants as

kOaAabs = [35]

POaNcOa + POaDfOa − UOaAOaDcOaT

1 + 10pH−pKaOa vol− kpass

and

Table 2. Chemical composition of mixed-grass–legumepasture and barley input for behavioral analysis of themodela

Variable Definition Value

fFrDf Degradable fiber content of forage 0.391fFrUf Undegradable fiber content of forage 0.053fFrNc Nonstructural carbohydrate content of forage 0.238fCnDf Degradable fiber content of concentrate 0.146fCnUf Undegradable fiber content of concentrate 0.027fCnNc Nonstructural carbohydrate content of concentrate 0.652

aValues are fractions of the forage or concentrate; from NRC (1996)and Weiss et al. (1992).

kOaDabs = [36]

UOaAOaDcOaT − cOaT

1 + 10pH−pKaOa

vol− kpass

The value of UOaAOaD at steady state for use in Eq.[35] and [36] was calculated from Eq. [25] as the differ-ence between production (PBfSaBf) and passage (UBf-pass) of Bf.

Model Analysis

Behavior of the model was analyzed by simulatingthree different concentrate-feeding levels: 1) free graz-ing with no supplemental concentrate (FR); 2) freegrazing with one concentrate meal of 1 kg of barleygrain at 2200 daily (CN1); 3) and free grazing with twoconcentrate meals of 1 kg of barley grain each at 1000and 2200 daily (CN2). Chemical analyses of mixed-grass–legume pasture and barley grain (Table 2) wereobtained from NRC (1996). All parameter values usedin the simulations are presented in Table 1. Simula-tions were run for 11 d, and results from d 11 are pre-sented. Initial values of the state variables have littleeffect on the d-11 results, but the values used in allsimulations were Cnlag(0) = 0.0 kg, Frlag(0) = 0.0 kg,Uf(0) = 0.15 kg, Df(0) =0.439 kg, Nc(0) = 0.129 kg, OaA(0) =0.001 mol, OaD(0) =0.062 mol, Bf(0) = 0.087 mol.

To evaluate model predictions of circadian patternsof ruminal pH and Oa concentration, and substitutionof forage intake by concentrate, three experiments fromthe literature were simulated. Inputs of BW, concen-trate meal times and sizes, forage and concentrate com-position, and milk composition were taken from thepublications, if provided. All parameter values used inthe simulations were those listed in Table 1 unlessotherwise noted. Simulations were run for 11 d, andresults from d 11 are presented. Mean square predictionerror (MSPE) was decomposed into error due to meanbias, due to deviation of the regression from one, anddue to unexplained variance (Bibby and Toutenburg,1977).

Leng and Leonard (1965) fed twelve 75-g meals ofalfalfa chaff hourly to adult sheep from 0800 to 1900


daily, and measured Oa concentration in ruminal fluidsamples taken hourly from ruminal fistulas. The experi-ment was simulated by scheduling forage meals as forconcentrate and setting the intake rate to 4.0 kg/d (DMbasis). Crude protein content of the alfalfa was 16.4%(DM basis), and all other chemical composition inputswere taken from NRC (1996).

Istasse et al. (1986) investigated the effect of fre-quency of concentrate feeding on ruminal pH in adultsheep, measured at 2-h intervals throughout a day.The sheep were fed hay ad libitum and a barley-basedconcentrate at 1.86 times the hay intake in two or fourequally spaced meals per day. The value of DMICnwas set to 1.02 and 1.09 kg/d for the two treatments,respectively, and DMIFr was predicted by the model.Ash and CP contents of the hay were given in the publi-cation and remaining feed composition inputs weretaken from NRC (1996) for mature timothy hay andbarley.

Concentrate was fed at 0, 480, and 960 g/d (DM basis)in a single meal at 0900 to 64-kg ewes nursing twins inearly lactation and ad libitum intakes from a perennialryegrass sward were measured by chromic oxide dilu-tion (Milne et al., 1981). The experiment was simulatedwith chemical composition of ryegrass and barley takenfrom NRC (1996) to represent the forage and concen-trate, respectively.

Results and Discussion

Parameterization of Oa Absorption

An important component of pH regulation in the ru-men is the absorption of H attached to OaA (Allen,1997). There has been some question as to the rate ofabsorption of Oa in the dissociated form, and it hasoften been called negligible (Dijkstra et al., 1993; Allen,1997), although Kramer et al. (1996) demonstrated thatOaD were absorbed by a system that required Cl−. Ata pH above the pKa for Oa of 4.8, the predominant formof Oa is OaD, and at pH 6.5, the concentration of OaDis 50 times that of OaA. In preliminary modeling, as-signing zero absorption to OaD forced a low ruminal pHto obtain realistic Oa absorption rates, and Bf content ofsaliva was inadequate to maintain Bf presence in therumen. Accordingly, as in Pitt et al. (1996), first-orderrate constants for both OaA and OaD absorption wereestimated by solving the model in steady state. Becauseof the high relative concentration of OaD, values forkOaDabs (−0.7 to 3.22 d−1) were two orders of magni-tude lower than for kOaAabs (82 to 1,460 d−1), whichmay account for the tendency to discount OaD absorp-tion. Pitt et al. (1996) reported only three- to ten-folddifferences in absorption coefficients for OaA and OaD,but their parameters yield a faster rate of OaD absorp-tion than of OaA absorption, which is not consistentwith observations (Ash and Dobson, 1963). Accordingto the steady-state model solutions, on average, 84% ofOa absorbed were in the OaA form and 25% of Oa

Figure 4. Plots of first-order rate constants for absorp-tion of associated (kOaAabs) and dissociated organicacids (kOaDabs) across the ruminal wall calculated fromsteady-state solutions of the model for observations ofBW, nutrient intake, total concentration of organic acidsin the rumen, and ruminal pH in peer-reviewed publi-cations.

produced passed out of the rumen with the liquid flowas opposed to being absorbed across the rumen wall.Previous estimates of passage of Oa range from 15 to40% of net production (Dijkstra et al., 1993; Allen,1997).

In two cases, a negative number was obtained forkOaDabs. In these cases, the expected rate of OaD pro-duction, calculated as the difference between produc-tion and passage of Bf (Eq. [25]), was lower than theexpected rate of OaD passage, calculated with Eq. [23]from the observed cOaT and pH. Because there is errorin prediction of rates of Bf production and liquid pas-sage out of the rumen, kOaDabs values were calculatedfrom many sets of observations and averaged.

There seemed to be an effect of ruminal pH on effi-ciency of OaA absorption, wherein kOaAabs increasedexponentially beyond pH 6.5 (Figure 4). The increasemay have been an artifact of the extremely low concen-tration of OaA relative to OaD at such pH or may havebeen a consequence of consistent deviations in otherparameter values, such as PSa or kpass, at higher pH.Alternatively, some aspect of OaA transport, such asproton donation from the ruminal epithelium (Bugaut,


Figure 5. Simulated intake of forage (inFr; solid line)and concentrate (inCn; dashed line) throughout a day bylactating ewes given ad libitum access to mixed-grass–legume pasture (Table 2) with no supplementary concen-tration (FR), 1 kg of barley grain at 2200 (CN1), or 1 kgof barley grain at 1,000 and 1 kg at 2200 (CN2). Simula-tions were run using parameter values given in Table 1.

1987), could be upregulated at high pH. Because themodel was constructed to simulate the high-producingewe experiencing periods of low ruminal pH, Oa absorp-tion parameters used (Table 1) were averages of thenine estimates from pH ≤ 6.5 (Figure 4). Means ± SEwere 227 ± 27 d−1 and 1.41 ± 0.38 d−1 for kOaAabs andkOaDabs, respectively.

Model Behavior

The simulated ewe, with 24-h access to mixed pas-ture, had two bouts of grazing during the day (Figure5), as is commonly observed (Thomson et al., 1985).Supplementing concentrate, either at morning or night,decreased the length of the subsequent grazing bout.Actual ruminal capacity increased during each mealand decreased gradually at other times (Figure 6). For-age intake and degradation rates are lower than thoseof concentrates, so ARC and Oa concentration in therumen increased more slowly than CN1 and CN2. Thefirst grazing bout of the day on FR was terminated bythe ARC limit of 0.70 kg of NDF. The second bout wasterminated at dusk at 2100. Grazing commenced atdawn at 0500 and again when ARC = 0.8 × MRC.

For CN1, the concentrate meal at 2200 caused anincrease in Oa concentration and ruminal pH dropped

Figure 6. Simulated ruminal fill (ARC), concentrationsof organic acids (Oa) and buffer (Bf) in the rumen, andruminal pH throughout a day in lactating ewes given adlibitum access to mixed-grass–legume pasture (Table 2)with no supplementary concentrate (solid line), 1 kg ofbarley grain at 2200 (dotted line), or 1 kg of barley grainat 1000 and 1 kg at 2200 (dashed line). Simulations wererun using parameter values given in Table 1. The verticaldotted lines indicate concentrate meal times.

(Figure 6). Because low ruminal pH reduces Df degrada-tion rate (Eq. [11]), ARC decreased more slowly duringthe night, and the fill restriction on forage intake washit earlier in the day (Figure 6). The break betweengrazing bouts and the duration of the second were ofthe same length as for FR. Due to Oa absorption fromthe ruminal wall and passage to the lower gut, Oa con-centration decreased gradually after the concentratemeal, and the pH returned to FR levels by 1200.

The addition of a second concentrate meal at 1000caused ruminal fill to be higher throughout the day,which led to shorter grazing times and a long breakbetween the two daily bouts (Figure 5). Although dailyforage intake was decreased with concentrate supple-mentation, total DMI and, therefore, ME intake, wasincreased in CN1 and CN2 ewes (Figure 7).


Figure 7. Simulated daily intake of forage (black bars)and concentrate (white bars) by lactating ewes given adlibitum access to mixed-grass–legume pasture (Table 2)with no supplementary concentrate (FR), 1 kg of barleygrain at 2200 (CN1), or 1 kg of barley grain at 1000 and1 kg at 2200 (CN2). Simulations were run using parametervalues given in Table 1.

Ruminal pH

Predicted Oa concentration throughout a day insheep fed 0.9 kg/d forage followed the same patternas that observed by Leng and Leonard (1965), but atapproximately half the concentration (Figure 8); 98%of the MSPE was due to a mean bias. Daily productionof Oa was measured by Leng and Leonard (1965) at 5.4mol/d, whereas the simulated value was 4.2 mol/d. Thesimulated volume of the rumen may have been higher

Figure 8. Observed (♦; from Leng and Leonard, 1965)and predicted (solid line) concentration of organic acids(Oa) in the rumen throughout a day in adult sheep given0.9 kg/d alfalfa chaff in 12 equally spaced meals between0800 and 1900. Simulations were run with inFr = 0 at alltimes, inCn = 0 or 4.0 kg/d depending on meal times,fCnDf = 0.345, fCnUf = 0.120, fCnNc = 0.277, and Eq. [32]was replaced by setting fDmFr = 1.0. All other parametervalues and were as given in Table 1. See Tables 1 and 2for abbreviations. The root mean square prediction error(MSPE) was 50.5% of the mean observed Oa concentrationwith 98% of the error due to a mean bias. A secondsimulation (dashed line) was run with BW = 50 kg andkpass = 1.2 d−1, where the root MSPE was 7.7% of theobserved mean and 76% of the error was unexplainedvariance.

than in the sheep of Leng and Leonard (1965); neithervolume nor BW was reported. Additionally, rate of liq-uid passage out of the rumen may have been too high.Increasing DMI of alfalfa chaff from 0.69 to 1.03 kg/dresulted in an increase in the rate constant for liquidpassage from 2.30 to 2.82/d (Ulyatt et al., 1984). Thevalue for kpass of 2.7 d−1 used here was obtained fromlactating ewes with an elevated consumption of waterand DM (Weston, 1988b; Gunter et al., 1990), so thesheep at maintenance of Leng and Leonard (1965) couldreasonably be expected to manifest a slower fractionalpassage. Setting BW = 50 kg and kpass = 1.2 d−1 re-sulted in predictions of Oa concentration similar tothose observed (Figure 8; root MSPE = 7.7% of theobserved mean) and a net Oa production rate of 5.1mol/d. In modeling ruminal water dynamics, Argyle andBaldwin (1988), like us, also used a constant fractionalliquid passage rate for all simulation conditions. Toimprove predictions of cOa, the rate constant for liquidpassage may need to be the dependent variable in somemathematical function, or the approach of Dijkstra etal. (1996), where kpass was inputted as a diet-specificvariable, may be warranted. The model of Dijkstra etal. (1996) predicted nonsteady Oa concentrations in therumen of dairy cows at the end of grazing and starvationperiods in three different experiments with a rootMSPE of 32% of the observed mean (Chilibroste etal., 2001).

To simplify the calculation of Oa production rate, theproduction of Oa from protein fermentation and thepartitioning of degraded carbohydrate use to microbialgrowth, maintenance, and energy spilling were not con-sidered. The consequences of the simplifying assump-tions were explored by estimating, from 16 sets ofsteady-state organic matter and nonammonia microbialand feed N intakes and duodenal flows in sheep (Ivanet al., 1996; Perez et al., 1996; Faichney et al., 1997;Rogers et al., 1997; Poncet and Remond, 2002), ratesof Oa production from fermentation of protein and ofcarbohydrates used in microbial growth and nongrowthprocesses. The stoichiometries of Oa production, ac-cording to Dijkstra et al. (1996), are 15.67 mmol/g pro-tein degraded, 6.75 mmol/g carbohydrate used in micro-bial protein synthesis from ammonia, 8.35 mmol/g car-bohydrate used in protein synthesis from peptides, and10.64 mmol/g remaining carbohydrate degraded. Onaverage, Oa production calculated from the observedcarbohydrate and N balances across the rumen, assum-ing 50% of the microbial protein flow was synthesizedfrom peptides, was 0.59 ± 0.05 mol/d higher than withthe assumption that all degraded carbohydrate and noprotein was converted to Oa (Eq. [10], [14], and [18] to[21]). On the other hand, Allen (1997) suggested a sim-ple stoichiometry of 7.44 mmol/g of OM fermented,which yielded an average ± SE Oa production that was0.21 ± 0.06 mol/d lower than with Eq. [10], [14], and[18] to [21] on the data set. Thus, although it seemsthat Oa production may be underestimated by ignoringprotein fermentation, the bias was considered too small


Figure 9. Observed (♦; from Istasse et al., 1986) andpredicted (solid line) ruminal pH throughout a day inadult sheep given ad libitum access to hay and 1.02 kg/d concentrate in meals at 0900 and 2100 (top panel) or1.09 kg/d in meals at 0300, 0900, 1500, and 2100 (bottompanel). The black horizontal bars indicate predicted for-age intake > 0. Simulations were run with fFrDf = 0.582,fFrUf = 0.124, fFrNc = 0.161, fCnDf = 0.146, fCnUf = 0.027,fCnNc = 0.652, and all other parameter values as givenin Table 1. See Tables 1 and 2 for abbreviations. The rootmean square prediction errors were 4.0% (top panel) and3.0% (bottom panel) of the mean observed pH.

and ill defined to warrant the additional complexityof predicting rates of microbial growth. Dijkstra et al.(1996) presented 36 additional algebraic equations toaccomplish the task. In construction of a nonsteady-state nutrient supply model, microbial growth equa-tions would be needed.

Using the parameter values of the reference ewe (Ta-ble 1), but setting the daily concentrate intake and mealtimes equal to those of Istasse et al. (1986) and takingfeed composition from NRC (1996), resulted in predic-tions of a circadian pattern of ruminal pH similar tothose observed (Figure 9). Root MSPE were 4.0 and3.0% of the mean observed pH for the twice and fourtimes daily feeding treatments, with 96 and 79% ofthe errors, respectively, due to unexplained variance.Predicted DMI for the whole day was 2.07 kg (85.5 g/kg of BW) on both treatments, whereas observed DMIwas 67.1 g/kg (Istasse et al., 1986). The overpredictionof forage intake may have influenced absolute pH pre-dictions but not the circadian pattern. It has long beenrecognized that fluctuations in ruminal pH that occurthroughout a day can adversely affect intake and wellbeing of the ruminant animal. Continuous pH recordingis the current state-of-the-art method in subacute rumi-nal acidosis diagnosis (Keunen et al., 2002). If analyti-cal techniques have advanced to the point of measuringpH continuously, then there is a need to simulate pHcontinuously to test hypotheses of pH regulation and

Figure 10. Observed (black bars; from Milne et al., 1981)and predicted (white bars) daily intakes of forage (DMIFr)in grazing, lactating ewes given 0, 0.48, or 0.96 kg concen-trate once daily at 0900. Simulations were run with BW =64 kg, fFrDf = 0.380, fFrUf = 0.068, fFrNc = 0.187, fCnDf =0.141, fCnUf = 0.028, fCnNc = 0.551, and all other parame-ter values as given in Table 1. See Tables 1 and 2 for abbre-viations.

its consequences. Previously, Pitt and Pell (1997) simu-lated identical, sequential periods of pH fluctuation bycalculating a steady-state pH empirically from effectiveNDF content of the diet and the deviation from thatmechanistically based on the change in Oa concentra-tions, a constant saliva concentration, and the stoichi-ometry of dissociation. Argyle and Baldwin (1988) usedan empirical regression equation to relate pH to Oaconcentrations that were simulated by solving differen-tial equations numerically. Here, we have presented afirst attempt to formulate prediction equations entirelymechanistically according to the rate:state formalism.The approach can be easily incorporated into other mod-els of ruminal digestion that also follow rate:state prin-ciples. The approach also allows for simulation of re-sponses to concentrate meals unequally distributed insize and time throughout a day.

Forage Intake

Simulation of the so-called substitution effect of con-centrate supplementation on forage intake (Figure 7)was a consequence of slower ARC decline at decreasedruminal pH (Figure 6). Freer et al. (1997) predictedsubstitution by simulating a selective intake behavior,in which grazing sheep consume the feed or herbagecomponents of highest digestibility first, followed byless and less digestible components until MRC or anME requirement is reached. Using the feedback fromlow ruminal pH resulted in predicted forage intakesthat were within 10% of values observed by Milne etal. (1981) and the magnitude of depression at 0.96 kg/d concentrate was 0.58 kg/d compared with 0.56 kg/dobserved (Figure 10). Simulation of results from a singleexperiment does not constitute a test of intake predic-


Figure 11. Simulation of effect on total DMI of time ofa single 2-kg concentrate meal (top panel) and effect offrequency of feeding the 2 kg concentrate (bottom panel)without (♦) or with (�) infusion of 1.63 mol/d of bufferinto the rumen. All other parameters and variables wereas given in Tables 1 and 2.

tion accuracy of the model. An intake accuracy testwould require a larger set of data from several differentexperiments. The response analysis in Figure 10 merelyindicates that the model responds appropriately to theperturbation of concentrate supply and can thereforebe used to predict responses in novel situations. Ofcourse, such predictions may not be correct.

Based on simulations of a single, 2-kg concentratemeal at different times throughout a day, forage intakewas greatest when the meal was at 1800 (Figure 11).A concentrate meal in the evening had the least affectbecause of the long time available for recovery beforegrazing commenced again at dawn. When the 2 kg con-centrate was fed in one to 12 equally spaced mealsthroughout the day, there was a predicted increase inforage intake up to two meals per day but a decreasethereafter (Figure 11). The average ruminal pH, calcu-lated from the area under the predicted pH curve, wasimproved by increased frequency of feeding the concen-trate, but inserting concentrate meals into the grazingtime interfered with forage intake. These two opposingforces essentially cancelled each other out beyond two

meals per day. Several experiments have demonstratedno effect of feeding frequency beyond two meals per dayon forage intake (Sutton et al., 1985; Weston, 1988a;Chestnutt and Wylie, 1995). Weston (1988a) noted thatinfusion of buffer into the rumen decreased the substi-tution effect of wheat supplementation by 0.32 kg/d.Setting PBfCnBf (Eq. [23]) equal to 1.63 mol/d to mimicthe infusion of Weston (1988a) increased the predictedforage intake by 0.25 kg/d at two concentrate meals perday (Figure 11).

Implications

A model that mechanistically predicts the nonsteadystate of acid concentrations in the rumen can be used totest hypotheses of ruminal pH regulation and voluntaryforage intake and to optimize concentrate supplementa-tion strategies for high-producing ewes. Predictingrates of absorption of organic acids from the rumensuggested that approximately 84% of acids producedwere absorbed in the undissociated form. The increasein absorption coefficient for undissociated acids at highruminal pH suggests that some aspect of absorptionremains unaccounted for and should be the subject offurther investigation. The rate:state pH model pre-sented here could be integrated with protein degrada-tion and microbial growth models to generate furtherpredictions of postabsorptive nutrient supply and per-formance responses to timing and level of concentratesupplementation.

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