A dynamic model of digestion and absorption in pigs

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Animal Feed Science and Technology 143 (2008) 328–371

A dynamic model of digestion and absorption in pigs夽 Anders Bjerring Strathe a,∗ , Allan Danfær b , Andre Chwalibog a a b

Section of Nutrition, Department of Basic Animal and Veterinary Sciences, The Royal Veterinary and Agricultural University, DK-1870 Frederiksberg, Denmark Section of Nutrition and Environmental Impact, Department of Animal Health, Welfare and Nutrition, Danish Institute of Agricultural Sciences, Research Centre Foulum, 8830 Tjele, Denmark

Abstract The paper describes and evaluates the construction of a mathematical model to study the kinetics of digestion and absorption in growing pigs. The core of the model is based on a compartmental structure, which divides the gastro-intestinal tract into four anatomical segments: the stomach, two parts of the small intestine and the large intestine. Within the large intestine, a microbial sub compartment is also considered. In each of these segments, the major organic nutrients are considered: dietary protein, endogenous protein, amino acids, non-amino acid and non-protein nitrogen, lipids, fatty acids, starch, sugars and dietary fibre. Besides a chemical description of the feed, the model further requires information about daily dry matter intake and feeding frequency.

夽 This paper is part of a special issue entitled “Mathematical Methods that Predict the Effects of Feed Characteristics on Animal Performance” guest edited by Essi Evans, Daniel Sauvant and Peter Ud´en. Abbreviations: S, state variable; dS/dt, the derivative of the state variable; R, rate variable for a process; A, auxiliary variable; C, constant; GIT, gastro-intestinal tract; AS, anatomical segments; STO, stomach; SI1, first small intestinal segment; SI2, second small intestinal segment; LI, caecum and colon; MM, microbial organic mass; P, microbial protein; CHO, microbial carbohydrate; M, indexes mass conversion factors; MP, indexes molar proportions; E, indexes energy conversion factors; DM, dry matter; OM, organic matter; DP, dietary protein; EP, endogenous protein; NAPN, non-amino acid and non-protein nitrogen; AA, amino acids; ST, starch; SU, sugar; DF, dietary fibre; DDF, degradable dietary fibre; UDF, undegradable dietary fibre; LD, lipid; FA, fatty acids; VFA, volatile fatty acids; in, intake; pa, passage; hy, hydrolysis; ab, absorption; sc, secretion; v, parameter describing the maximum velocity of an enzymatic process; k, affinity constant of an enzymatic process; 0, reference state; MRT, mean retention time; OMd, protein yield per kilogram apparently digested feed OM; ENL, endogenous nitrogen loss; R, model response; P, model parameter; |S(R, Pi )|, sensitivity coefficients ∗ Corresponding author. Tel.: +45 35283065; fax: +45 35283020. E-mail address: [email protected] (A.B. Strathe).

0377-8401/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.anifeedsci.2007.05.018

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Flows of nutrients from one anatomical segment to the next are assumed to be mass action processes, whereas degradation and absorption are described as enzymatic processes and are therefore represented by saturation kinetics. The model also proposes a basic concept for integrating effects of anti-nutritional factors into a mechanistic framework. The total number of pools is 38 and their rates of change are described by differential equations on the principle of mass conservation. Parameterization of the equations describing secretion and absorption is derived from literature values, whereas parameters for degradation are empirically adjusted to ileal and faecal digestibility coefficients determined for 24 diets highly variable in chemical composition. A sensitivity analysis has revealed that the prediction of nutrient digestibility is insensitive to affinity constants and moderately sensitive to maximum velocities for nutrient degradation. Predictions of apparent protein digestibility and endogenous nitrogen loss are especially sensitive to a reference level of dietary fibre in the last segment of the small intestine. The reference quantity of organic matter, which is related to the transit in the large intestine, has a large impact on the predicted apparent digestibility of dietary fibre and energy. The model predicts coefficients of crude protein and energy digestibility with good accuracy across 65 diets from 11 studies. Lipid is predicted less accurately. Predictions of endogenous protein losses in response to increased dietary fibre are compared with results from eight studies including 39 diets and it is concluded that the predictions are moderately satisfactory. The model predicts the pattern of organic matter flow at the terminal ileum and the appearance of nutrients in the portal blood with fairly good accuracy. The validation of kinetics of nutrient absorption has highlighted the necessity for future models to include a representation of gut wall metabolism in order to validate digestive models at their most valuable terminal output. Full utilization of the model as a predictive tool can be expected when it is combined with a metabolism model. © 2007 Elsevier B.V. All rights reserved. Keywords: Pigs; Digestion; Absorption; Computer simulation

1. Introduction During three decades, development of nutritional models has been of interest for simulating animal responses to various feeding regimens. Models provide a useful tool in which to integrate quantitative knowledge and test different hypotheses on complex animal processes. Several pig models have been made with different modelling objectives. Some pig models are designed for predicting growth and chemical body composition (Whittemore and Fawcett, 1974; Pomar et al., 1991; Danfær, 2000; Birkett and De Lange, 2001) or especially fatty acid composition in the body fat (Lizardo et al., 2002) or the anatomical body composition (Halas et al., 2004). Others have focused on the development of models for predicting amino acid requirements (Moughan, 1989) or improving the understanding of the mechanism of growth (Danfær, 1991; Lovatto and Sauvant, 2003). These pig models treat of the digestive and absorptive processes rather crudely, i.e. by means of digestibility coefficients. Also mathematical models of the digestive and absorptive processes have been published. The model of Turner et al. (1987) describes nutrient transit and digestion by mass action, but the model does not include any endogenous secretions. The model objective of Usry et al. (1991) is to simulate the passage of dry matter (DM), which is obtained by dividing the small intestine into segments, 6 cm of length and then transferring the dry matter from one segment to the next by a stochastic procedure. In the model of Rivest et al. (2000), transit is portrayed

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by dividing the small intestine into 50 segments with variable lengths and then transferring protein from one section to the next at regular time intervals based on the progression of myoelectrical migrating complexes. The model divides dietary and endogenous proteins into separate pools and is used to study the differential digestion of these protein pools. These models only present some aspects of digestion or some fractions of nutrients and can therefore not be considered as complete models of digestion and absorption. The model of Bastianelli et al. (1996) includes all the major nutrient fractions and divides the gastro-intestinal tract (GIT) into four compartments. The rates of nutrient transit and degradation are represented by mass action although degradation of nutrients is enzymatic processes that can potentially be saturated. Furthermore, their model treats protein as one pool irrespective of its origin. It is well documented that EP is digested to a lesser extent than dietary protein (Low and Zebrowska, 1989; Krawielitzki et al., 1990; Souffrant et al., 1993; Grala et al., 1998a,b). Our objective was to develop a dynamic, mechanistic model of organic nutrient digestion and absorption in growing pigs sustainable for representing the kinetic features of digestion and absorption. Special consideration was given to description of the physiological effects of DF on EP losses and energy availability. This model of the GIT can generate input to a metabolic model that considers diurnal or post-prandial events since the GIT model allows the prediction of kinetic aspects of digestion and absorption.

2. Material and methods 2.1. Model structure The model is depicted as per Fig. 1. The core of the model is based on a compartmental structure describing transit and digestion in the GIT of a growing pig. The model only considers digestive and absorptive phenomena. However, these differ between various anatomical segments (AS) in the GIT and hence, the model divides the GIT into four AS, which was regarded as the minimum of segments needed for a sufficient representation (Bastianelli et al., 1996). The stomach (STO) is treated as one AS and is the site where the feed enters. The small intestine is an approximately 18 m long tube, which is divided into two segments (SI1) and (SI2) of unequal size: SI1 represents the duodenum/proximal jejunum and is separated from the rest of the small intestine due to its particular importance in endogenous secretions (bile and pancreatic) and high rates of hydrolysis (Low and Zebrowska, 1989). SI2 represents the medial and distal jejunum/ileum and is the primary site for absorption. The large intestine (LI) comprises caecum/colon and is treated as one AS due to its particular importance in fibre digestion (Bach Knudsen, 2001). In each compartment, the major class of organic nutrients and their constituting monomers are represented: dietary protein (DP), endogenous protein (EP), amino acids (AA), non-amino acid and non-protein nitrogen (NAPN), starch (ST), sugars (SU), lipids (LD), fatty acids (FA), degradable dietary fibre (DDF), undegradable dietary fibre (UDF), volatile fatty acids (VFA) and microbial organic mass (MM). The model considers a total of 38 sub-compartments or pools.

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Fig. 1. Flow chart of the model. The digestive tract is divided into four anatomical compartments: stomach (STO), duodenum/proximal jejunum (SI1), distal jejunum/ileum (SI2) and caecum/colon (LI). In each anatomical compartment the major pools of macro nutrients and constituting monomers are presented: endogenous protein (EP), dietary protein (DP), amino acids (AA), non-amino acid non-protein nitrogen (NAPN), starch (ST), sugars (SU), lipids (LD), fatty acids (FA), degradable dietary fibre (DDF), undegradable dietary fibre (UDF), volatile fatty acids (VFA) and microbial organic mass (MM). Solid arrows represent flows of matter between compartments. Absorption flows (abs) are represented by bold dotted arrows and are outputs from the individual monomer pools: AA, FA, SU, VFA and NAPN. Flows of endogenous material are represented by dotted arrows and are inputs to EP, LD and NAPN. Flows are mathematically represented by mass action or saturation kinetics, and boxes are mathematically regarded as differential equations based on the conservation of mass. Boxes represented by dotted lines are treated as zero pools.

The diet is specified in terms of its chemical composition (g/kg DM) according to the macro nutrient pools in STO (Fig. 1). The dietary fibre fraction (DF) is calculated as the difference between the total carbohydrate (organic matter–crude protein–crude fat) and starch and sugar analyzed according to Larsson and Bengtsson (1983) and Bach Knudsen et al. (1987). The digestibility of the DF fraction is estimated from faecal digestible total carbohydrate and assuming that starch and sugar is completely digested at the ileal level (Bach Knudsen and Jørgensen, 2001). Based on the 24 diets of Just et al. (1985), it was calculated that the digestibility of the DF fraction varies between 0.16 and 0.79. It was therefore assumed that the DF fraction has a potential degradability of 0.80. The dietary input in the experiments used to construct and evaluate the model is in most cases given as ingredient composition and not as chemical composition, i.e. a nutrient profile. Therefore, the dietary nutrient profile used in the individual experiments was recalculated from the ingredient composition by means of the feed table provided by The National Committee for Pig Production (2003). 2.2. Model dimensions and execution State variables are expressed as either mol C or mol N. Conversion factors for converting kilograms of matter into either mol N or mol C are similar to those used by Danfær (1990).

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Table 1 Conversion factors and combustion values for ith nutrient fractions Biochemical constituent DP

EP

AA

NAPN

ST

SU

Conversion factors for matter into nitrogen: CM, i [mol N/kg] – – – 11.420 11.420 0.112a Conversion factors for matter into carbon: CM, i [mol C/kg] 43.294 43.294 – – 37.037 35.087 Molar proportions: CMP, i [mol/mol] – 3.791 3.791 1.279b

–

–

Conversion factor for energy: CE, i [MJ/mol N] or [MJ/mol C] 2.084 2.084 2.084 – 0.470 0.479 a b

DDF

UDF

LD

FA

–

–

–

–

37.037

37.037

62.402

62.402

–

–

–

–

0.470

0.540

0.624

0.624

Average molecular weight of an amino acid—unit: kg/mol AA. Average number of moles of nitrogen per mol AA—unit: mol N/mol AA.

The conversion factors are shown in Table 1 along with combustion values for different nutrient fractions (Blaxter, 1989). Rate variables are expressed as either mol C/h or mol N/h. The differential equations describing the rates of change in each state variable are based on the principle of mass conservation and are solved by numerical integration. The model is constructed in Powersim® and is interfaced with an Excel® spreadsheet, which controls the model input, output and storage of simulation data. An iteration procedure is embedded in the spreadsheet (programmed in Visual Basic for Excel) which enables the model to perform multiple runs with simultaneous change of model input and parameter values. The numerical solver utilizes a fourth order Runge–Kutta–Fehlberg integration algorithm with variable step size where the initial time step is set at 0.01 h. This is useful for models with discontinuities, e.g. a system in which feed intake is encountered periodically (Powersim, 1996). This routine adjusts the step length to keep the error per step less than that specified by the absolute and relative error. 2.3. General parameterization The most comprehensive body of data on digestive physiology in pigs is available for a range of body weights from 20 to 70 kg. The literature on digestive physiology in pigs was reviewed for collection of data used for model parameterization. These data were organized into a database, which served as a reference for model calibration. Much of the data available on nutrient transactions in the gut are presented as average daily values or relative to intake of some biochemical constituent, e.g. digestibility coefficients. Exceptions to this are data on gastric emptying where parameters can be directly estimated by non-linear regression techniques. Therefore, the following procedure were used to derive model parameters if they could not be estimated directly. The model parameters were adjusted until model output agreed with the reference data during quasi steady state i.e. the fluctuations (caused by meal feeding) in the pool sizes within a day were identical from day to day and thus the model output calculated on a daily basis became constant. The model generally reached quasi steady state

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within 20–40 h after simulation start. During model development the simulation period was set at 96 h to ensure that the model always reached quasi steady state. Furthermore, only simulation data from the last 24 h of the simulation run was used for comparison with reference data. Comparison of simulation results with reference data was accomplished by calculating average simulation values obtained by integration of the variable in question and division by the corresponding time interval. The nomenclature used in the model, assigning names to the variables, is as follows: (1) variable (S, state variable; dS/dt, derivative of the state variable; R, rate variable; A, auxiliary variable; C, constant); (2) anatomical segment (STO, SI1, SI2, LI and MM); (3) chemical constituent (DP, EP, AA, NAPN, LD, FA, ST, SU, DDF and UDF); (4) process (in, intake; pa, passage; hy, hydrolysis; sc, secretion; ab, absorption; fr, fermentation); (5) parameter type (v, maximum velocity; k, affinity constant; 0, reference state). A complete list of all the parameters including their description, abbreviation, numerical value and unit is presented in Table 2 and the general equation format is presented in Table A.1 in Appendix A, where “i” indexes the ith chemical constituent. 2.4. Intake and transit Feed intake is modelled as a discontinuous process based on the assumptions that the ingestion phase has a constant duration in which the pig eats at a constant rate, and furthermore it eats at regular intervals during the day. Daily dry matter intake (DMI) is set at a constant value or can follow a specified time-dependent function. The model user can define duration of the ingestion phase (TFEED), meal frequency (FFEED) and DMI. 2.4.1. Gastric emptying 2.4.1.1. Stomach (STO). The stomach is represented as a single compartment, which has the capacity to store large quantities of feed during the rapid ingestion phase and releasing it slowly into the duodenum (Low and Zebrowska, 1989). The rate of gastric emptying is complex and regulated by many factors, i.e. particle size, viscosity, osmolarity, lipids, sugars, fibre and protein content (Low, 1990; Johansen et al., 1996). As shown by Potkins et al. (1991) and Johansen et al. (1996), first order mass action kinetics can be fitted to data on gastric emptying. The rate constant for gastric emptying (CSTO, pa ) can be calculated from data on evacuation half times (T1/2 ) of the stomach dry matter (CSTO, pa = ln(2)/T1/2 ). We adopted an average value of 3 h (Low et al., 1985; Rainbird and Low, 1986a,b), which corresponds to 0.231 h−1 and is close to the value 0.216 h−1 used by Bastianelli et al. (1996). T1/2 can be modified, if the model is used to simulate specific cases. Due to simplicity, it was assumed in the current version of the model that the same kinetic constant for transit can be applied to all chemical constituents. 2.4.2. Transit in the small intestine (SI1 and SI2) The mean retention time (MRT) in a specific segment is defined as the ratio of mass to flow rate of any digesta component in that segment (Faichney, 1993), and thus rate constants for passage can be calculated as CSI1, pa = 1/MRTSI1 and CSI2, pa = 1/MRTSI2 . These rate constants were assumed to apply to all nutrient fractions due to simplicity and lack of data. MRT in SI1 was set to be 15% (0.6 h) of total MRT in the small intestine,

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Table 2 Description, abbreviation, value and units of the parameters in the digestion model Segment

Abbreviation

Description

Value

Units

Stomach (STO)

TFEEDa FFEEDa DMIa CSTO, pa CSTO, EP, sc CSTO, NAPN, sc

0.25 3 1.5 0.231 0.013 0.013

h 1/day kg DM/day 1/h mol N/kg OM mol N/kg OM

Small intestine (SI1)

CSI1, pa CSI1, EPp, sc

Duration of the ingestion phase Meal frequency Dry matter intake Parameter for gastric emptying Parameter for endogenous protein secretion Parameter for non-amino acid and non-protein nitrogen secretion Parameter for passage through SI1 Parameter for endogenous protein secretion, pancreas (p) Parameter for non-amino acid and non-protein nitrogen, pancreas (p) Parameter for endogenous protein secretion, bile (b) Parameter for non-amino acid and non-protein nitrogen, bile (b) Parameter for endogenous lipid secretion, bile Maximum velocity for dietary protein hydrolysis Maximum velocity for endogenous protein hydrolysis Maximum velocity for non-amino acid and non-protein nitrogen hydrolysis Maximum velocity for starch hydrolysis Maximum velocity for lipid hydrolysis Affinity constant for dietary protein hydrolysis Affinity constant for endogenous protein hydrolysis Affinity constant for non-amino acid and non-protein nitrogen hydrolysis Affinity constant for starch hydrolysis Affinity constant for lipid hydrolysis Maximum velocity for amino acid absorption Maximum velocity for sugar absorption Maximum velocity for fatty acid absorption Affinity constant for amino acid absorption Affinity constant for sugar absorption Affinity constant for fatty acid absorption

1.670 0.070

1/h mol N/kg OM

0.030

mol N/kg OM

0.077

mol N/kg DMI

0.042

mol N/kg DMI

1.25

mol C/kg DMI

0.40

mol N/h

0.10

mol N/h

0.50

mol N/h

4.00 2.50 0.50

mol C/h mol C/h mol N

0.50

mol N

0.10

mol N

1.20 0.50 0.28 1.95 1.20 0.02 0.01 0.06

mol C mol C mol N/h mol C/h mol C/h mol N mol C mol C

Parameter for passage through SI2 Parameter for endogenous protein secretion Parameter for non-amino acid and non-protein nitrogen secretion Maximum velocity for dietary protein hydrolysis Maximum velocity for endogenous protein hydrolysis

0.294 0.37 0.246

1/h mol N/ kg OM mol N/ kg OM

0.38

mol N/h

0.15

mol N/h

CSI1, NAPNp, sc CSI1, EPb, sc CSI1, NAPNb, sc CSI1, LD, sc CSI1, DP, hyv CSI1, EP, hyv CSI1, NAPN, hyv CSI1, ST, hyv CSI1, LD, hyv CSI1, DP, hyk CSI1, EP, hyk CSI1, NAPN, hyk CSI1, ST, hyk CSI1, LD, hyk CSI1, AA, abv CSI1, SU, abv CSI1, FA, abv CSI1, AA, abk CSI1, SU, abk CSI1, FA, abk Small intestine (SI2)

CSI2, pa CSI2, EP, sc CSI2, NAPN, sc CSI2, DP, hyv CSI2, EP, hyv

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Table 2 (Continued ) Segment

Abbreviation

Description

Value

Units

Small intestine (SI2)

CSI2, NAPN, hyv

Maximum velocity for non-amino acid and non-protein nitrogen hydrolysis Maximum velocity for starch hydrolysis Maximum velocity for lipid hydrolysis Affinity constant for dietary protein hydrolysis Affinity constant for endogenous protein hydrolysis Affinity constant for non-amino acid and non-protein nitrogen hydrolysis Affinity constant for starch hydrolysis Affinity constant for lipid hydrolysis Maximum velocity for amino acid absorption Maximum velocity for sugar absorption Maximum velocity for fatty acid absorption Affinity constant for amino acid absorption Affinity constant for sugar absorption Affinity constant for fatty acid absorption

0.16

mol N/h

5.40 1.70 0.16

mol C/h mol C/h mol N

0.18

mol N

0.40

mol N

2.00 2.50 0.75

mol C mol C mol N/h

3.92 2.40 0.06 0.25 0.25

mol C/h mol C/h mol N mol C mol C

Parameter for passage through LI, reference state Reference state, e.g. quantity of organic matter Steepness parameter Parameter for endogenous protein secretion Parameter for non-amino acid and non-protein nitrogen secretion Maximum velocity for dietary protein hydrolysis Maximum velocity for endogenous protein hydrolysis Maximum velocity for non-amino acid and non-protein nitrogen hydrolysis Maximum velocity for starch hydrolysis Maximum velocity for degradable dietary fibre hydrolysis Maximum velocity for lipid hydrolysis Affinity constant for dietary protein hydrolysis Affinity constant for endogenous protein hydrolysis Affinity constant for non-amino acid and non-protein nitrogen hydrolysis Affinity constant for starch hydrolysis Affinity constant for degradable dietary fibre hydrolysis Affinity constant for lipid hydrolysis

0.033

1/h

0.250

kg OM

0.040 0.15 0.15

1/kg OM mol N/kg OM mol N/kg OM

0.08

mol N/h

0.0155

mol N/h

0.0155

mol N/h

1.20 0.30

mol C/h mol C/h

0.01 0.20

mol C/h mol N

0.50

mol N

0.50

mol N

0.50 2.50

mol C mol C

1.00

mol C

CSI2, ST, hyv CSI2, LD, hyv CSI2, DP, hyk CSI2, EP, hyk CSI2, NAPN, hyk CSI2, ST, hyk CSI2, LD, hyk CSI2, AA, abv CSI2, SU, abv CSI2, FA, abv CSI1, AA, abk CSI2, SU, abk CSI2, FA, abk Large intestine (LI)

CLI, pa, 0 CLI, OM, 0 CLI, pa, kn CLI, EP, sc CLI, NAPN, sc CLI, DP, hyv CLI, EP, hyv CLI, NAPN, hyv CLI, ST, hyv CLI, DDF, hyv CLI, LD, hyv CLI, DP, hyk CLI, EP, hyk CLI, NAPN, hyk CLI, ST, hyk CLI, DDF, hyk CLI, LD, hyk

a

Default values.

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which is approximately 4 h (Latymer et al., 1990). It was divided in this way due to its particular importance in degradation, absorption and endogenous secretion (pancreatic and bile additions). However, MRT does not provide any information on the “structural” integrity of this segment. The length of SI1 was obtained by modification of an equation derived by Rivest et al. (2000) from data on the rate of migrating myoelectric complexes and the assumption that the small intestine is 18 m long. We integrated their equation to get; length = −2.25 + 0.162 × 10−3 × (1.93 × 108 + 6.17 × 107 × MRT)0.5 and by inserting MRT = 36 min (0.6 h) into the equation, it was calculated that the SI1 compartment “structurally” represents approximately 6 m or one-third of the small intestine. It must be underlined that this is an approximate estimate, but it is still an improvement to the assumption of Bastianelli et al. (1996). Transit through SI2 was treated in the same manner as in SI1 and constitutes 85% of the total MRT. The 4 h MRT includes a delay of 1 h because the digesta is generally retained in the last part of the small intestine for 1–1.5 h (Darcy-Vrillon et al., 1980). We described this delay mathematically by a third order material delay, which is a class of exponential delays available in Powersim® . The transient response of a third order material delay produces a flow of OM, which mimics the characteristic flow of an external marker at the terminal ileum. Furthermore, it has the capability of smoothing the OM flow at the terminal ileum compared to an OM flow based directly on mass action kinetics. This in turn has a smoothing effect on the rate of VFA production (Bastianelli et al., 1996). 2.4.3. Transit in the large intestine (LI) 2.4.3.1. Large intestine (LI). Many experimental studies consider the caecum/colon as one compartment when examining the fermentative processes—see Stevens and Hume (1998) for a review. For this reason, these two anatomical segments were combined in the current modelling exercise. The OM in this compartment is the sum of digesta OM and bacterial OM. The transit through this compartment is a function of total OM in the LI. It is well documented that increased ileal flow reduces the MRT in LI (Warner, 1981; Bach Knudsen, 2001). Therefore, the fractional rate of passage was described as an exponential rather than a linear function of the OM in LI. The explicit equation form is given in Table A.1 in Appendix A. The reference quantity of OM is calculated as an average of DM pool sizes presented by Stanogais and Pearce (1985) assuming an ash content of 10%. The reference MRT in LI is assumed to be 30 h. The steepness parameter is adjusted in order to keep the OM pool size in LI and MRT of the total tract within a range of literature values. The present mathematical representation of transit is in principle analogous to that formulated by Bastianelli et al. (1996). 2.5. Degradation and absorption of nutrients The segmental digestibilities of the different nutrient fractions were estimated by use of data from experiments in which cannuals have been inserted at different sites along the GIT of pigs. The segmental digestibility is crucial for estimating degradation parameters because they cannot be derived directly by means of non-linear regression. The segmental digestibility in the proximal third of the small intestine was obtained from literature data, whereas the segmental digestibilities at SI2 (ileal) and LI (faecal) were based on the data

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set of Just et al. (1985). These authors have reported ileal and faecal digestibilities of 24 different diets highly variable in chemical composition and ingredient makeup. 2.5.1. Nutrient degradation in stomach (STO) 2.5.1.1. Stomach (STO). Although digestive enzymes are secreted to and in the stomach (salivary amylase, pepsin and ligual/gastric lipase), there seems to be no significant production of monomers (Low and Zebrowska, 1989). However, the mechanical and chemical action on dietary proteins in the stomach is very important because gastrorectomy significantly reduces protein digestibility. Some degree of fermentation does occur and can be detected by the presence of lactic acid and volatile fatty acids in the digesta (Bach Knudsen, 2001). 2.5.2. Nutrient degradation and absorption in small intestine (SI1) 2.5.2.1. Small intestine (SI1). Nutrient hydrolyses in the GIT are enzymatic processes and were therefore described by saturation kinetics. The maximum velocity of a reaction (v) is a function of enzyme capacity and availability and it is generally believed that enzyme availability in the small intestine is non-limiting (Corring, 1982). It was assumed that degradation of macronutrients in SI1 produces the monomers AA, SU and FA, but this is a simplification since it is known that degradations of protein and starch produce polymers and oligomers as well as and tri- and dimers that can be absorbed at the apical membrane. However, this was regarded as a too detailed description for the highly aggregated level of the current modelling exercise. The segmental digestibility of DP and EP in the proximal third of the small intestine is based on data from 15 N studies with growing pigs. These studies indicate that approximately 0.30 of the ingested nitrogen is digested and absorbed here (Krawielitzki et al., 1990, 1996; Grala et al., 1998a,b). Results of Grala et al. (1998a,b) further suggest that other dietary factors (DF and anti-nutritional factors) do not affect protein digestibility in the proximal part of the small intestine, whereas this seemed to be the case in the more distal parts of the small intestine. The segmental digestibility of starch was set at 0.60 based on values reported by Sambrook (1979) and Bach Knudsen et al. (1993). Dietary lipids enter the small intestine as lipid droplets with a diameter of less than 0.5 ␮m. These are subjected to the combined action of bile salts, colipase and pancreatic lipase (Tso, 1994). Although the pancreatic lipase has no specific requirement for bile salts, the increased surface area created by the action of bile salts greatly increases the rate of triacylglycerol hydrolysis catalysed by pancreatic lipase (Tso, 1994; Drackley, 2001). This effect was accounted for in the model by applying a steepness parameter (N) to the Michaelis–Menten equation. This parameter was arbitrarily set at 3. Absorption in the small intestine of SU and AA was described as saturation processes because the absorption of glucose, AA and peptides is mainly mediated by transporters and not by passive diffusion (Levin, 1994; Ferraris et al., 1990). Parameters for SU absorption were estimated from apparent disappearance rates of glucose over an isolated loop of the jejunum. The maximum velocity for this process was calculated from the absorptive capacity (7.1 and 6.8 g/(m h)) reported by Rainbird et al. (1984) and Low et al. (1986) multiplied by the length of the small intestinal compartment. We further assumed that the theoretical absorptive capacity is 25% higher (9.5 and 9.1 g/(m h)) due to the fact that some adaptation

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occurs at high SU loads (Bird et al., 1996; Croom et al., 1998). Data from Weber and Ehrlein (1998) with mini pigs supports this assumption because they measured an absorptive capacity of 9.4 g/(m h) in an isolated segment of the jejunum. Buraczewska (1981a,b) studied the ability of different parts of the small intestine in pigs to absorb AA and peptides. The experiments were carried out in pigs weighing 50–75 kg with isolated loops of the proximal, middle and terminal small intestine. It was estimated from these data that the absorptive capacity of amino acid nitrogen is 0.65 g/(m h) in SI1. Results of Weber and Ehrlein (1998) also suggest that the absorption of FA follows saturation kinetics, which they attribute to the complex metabolic processes, e.g. the binding of FA by the fatty acid binding protein and resynthesis of FA and monoglycerides within the enterocyte. The maximum velocity for FA absorption was estimated to be 3.2 g/(m h) from data with mini pigs reported by Weber and Ehrlein (1998). The affinity constants for absorption of AA, SU and FA are those reported by Bastianelli et al. (1996) recalculated according to units used in the present model. 2.5.3. Nutrient degradation and absorption in small intestine (SI2) Macronutrient degradation is described in the same way as in SI1, but the degradation parameters do not take the same numerical values. The digestion of starch is almost complete at the end of the small intestine with an average digestibility of 0.96 (Bach Knudsen and Jørgensen, 2001). The average ileal digestibility of soluble carbohydrates (sum of starch and sugar) is 0.92 in the data set used for model calibration (Just et al., 1985). The lower digestibility of starch in these data is probably due to the fact that some of the diets contained potato starch, which is classified as type B starch and is therefore less accessible for the ␣-amylase (Bach Knudsen and Jørgensen, 2001). Since the model assumes a uniform starch source, the reference digestibility was set at 0.96. As DF is indigestible to mammalian enzymes, the accumulation of DF during preceacal passage occurs and modifies the rheological properties of the digesta (Bach Knudsen, 2001). DF affects endogenous transactions not only through its abrasiveness, which increases the sloughing off of gut mucosa cells, but also through increasing digesta viscosity, which in turn prevents adequate interaction between DP and digestive enzymes (Chesson, 1993; Nyachoti et al., 1997a; Drochner et al., 2004). This results in reduced rates of proteolysis and recycling of endogenous gut protein, and hence increasing EP losses (Nyachoti et al., 1997a; Drochner et al., 2004). However, the capacity of DF to bind organic molecules is complex and the mechanism is not very well understood (Drochner et al., 2004). We assumed that these phenomena can be described by a reduced substrate affinity of the proteolytic enzymes for their substrate. This means that the substrate affinity is not constant, but regulated by the level of DF:   ASI2,DF 3 CSI2,i,hyk = CSI2,i,hyk,0 (2.5.1) CSI2,DF,0 where CSI2, i, hyk, 0 is the affinity constant for proteolysis of DP or EP corresponding to the reference level of DF in SI2 (CSI2, DF, 0 ). ASI2, DF is the actual level of DF at present time during a simulation and is expressed in proportion to OM ([mol C/kg OM]). The digestive

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capacity of the proteolytic enzymes expressed as CSI2, i, hyv is constant and unaffected by the DF level. We described LD digestion in SI2 by saturation kinetics, but the steepness parameter is excluded because it was assumed that LD solubility in this part of the small intestine is already achieved. The absorptive capacity of SU and FA in this segment was assumed to be the same as in SI1, but CSI2, i, abv for the processes differ because of the different length of the two segments. Again, we used data from Buraczewska (1981a,b) to estimate the absorptive capacity of amino acid nitrogen in SI2 and derived at an estimate of 0.88 g/(m h). The affinity constants for absorption of AA, SU and FA are those reported by Bastianelli et al. (1996) adjusted according to units used in the present model. 2.5.4. Nutrient degradation and microbial metabolism in the large intestine (LI) All degradations in LI were assumed to be caused by microbial activity. Proteins reaching the LI can be degraded to AA, which are deaminated to their corresponding keto acids and ammonium. The molar ratio of C to N was assumed to be 3.8. Carbohydrates (ST and DF) entering LI are degraded to SU. Lipids can be degraded to FA, and the glycerol moiety can be fermented. The carbon molar proportion of glycerol in lipid was assumed to be 3/51 corresponding to tripalmitin. The degradation rates are described by saturation kinetics because microbial metabolism is governed by enzymatic processes that involve breakdown of polymers at the surface of the microbes and subsequent uptake into their cells. Table 3 summarizes the parameters that are related to the microbial metabolism. The microbial population was assumed to have a fixed chemical composition based on published values on rumen microbes. The reasons for this assumption were lack of data on composition of the colonic micro flora in pigs and similarity between ruminal and colonic Table 3 Description, abbreviation, value and units of the parameters related to microbial metabolism in the digestion model Segment

Abbreviation

Description

Value

Units

Microbial (MM)

CMM CMM, P

Microbial growth efficiency Microbial protein content per kilogram organic growth Microbial carbohydrate content per kilogram organic growth Microbial lipid content per kilogram organic growth Stoichiometric coefficients for acetate fermentation Stoichiometric coefficients for propionate fermentation Stoichiometric coefficients for butyrate fermentation Stoichiometric coefficients for carbon dioxide fermentation Stoichiometric coefficients for methane fermentation

0.145 6.70

kg MM/kg OM mol N/kg MM

10.67

mol C/kg MM

7.80

mol C/kg MM

0.447

mol C/mol C

0.225

mol C/mol C

0.073

mol C/mol C

0.153

mol C/mol C

0.102

mol C/mol C

CMM, CHO CMM, LD CMM, ACE, fr CMM, PRO, fr CMM,BUT, fr CMM,CO2 ,fr CMM,CH4 ,fr

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microbes (Varel and Yen, 1997). The microbial OM was assumed to be composed of 58.7% protein (P), 12.5% lipid (LD) and 28.8% carbohydrate (CHO) as derived from the rumen module in the Nordic dairy cow model (Danfær et al., 2005). Microbial growth is usually expressed as the protein yield per kilogram apparently digested feed OM (OMd) in the rumen and can be as high as 30–35 g N/kg OMd (Nolan, 1993). However, the growth efficiency of colonic microbes in pigs seems to be much lower with an average value of 13.6 g N/kg OMd corresponding to 145 g microbial OM/kg OMd (Dierick et al., 1990). It has also been demonstrated that microbial growth efficiency in the rumen is dependent on nitrogen availability (Russell and Strobel, 1993), but it was assumed in the present model that nitrogen sources in LI (undegraded protein from ileum and endogenous urea) are abundant and in excess. Some evidence for this assumption is that pigs switch the nitrogen excretion from urine to faeces in response to increased levels of DF (Canh et al., 1997). Carbon available for microbial energy metabolism is calculated as the difference between carbon in degraded substrates (ST, DDF, DP and EP) and carbon assimilated for growth. Microbial energy metabolism yields ATP, VFA, CO2 and CH4 , which can be predicted by fermentation equations. Beever (1993) derived a rumen fermentation equation for a high forage based diet that predicts VFA molar proportions to be 70, 24 and 6% for acetate, propionate and butyrate, respectively. This is in accordance with in vivo observations by Kass et al. (1980), but these proportions may very well differ from those in the colon and caecum. Furthermore, these proportions are dependent on the type of substrate fermented (Martin et al., 2000), which is not accounted for in the model. The VFAs and ammonia were assumed to be absorbed at their rate of production. 2.6. Endogenous flows Lipids, carbohydrates and proteins in the GIT are from either exogenous or endogenous origin. Endogenous lipids, carbohydrates and proteins are digestive secretions from salivary glands, stomach, pancreas and small intestine as well as amino acids conjugated to biliary salts, mucus and desquamated cells of the GIT epithelium (Juste, 1982; Souffrant, 1991; Lien et al., 2001). Ingested hair should also be considered as endogenous protein, but was assumed to be negligible relative to the total secretions. It was assumed in the model that the inflow of organic matter into each AS is the driving variable for triggering endogenous secretions. The rate of endogenous secretion was set to be proportional to the inflow of organic matter. This simple principle has the advantage that endogenous secretions are increased if feed intake is increased or if the diet is less digestible, i.e. has an increased DF content. The parameters that define the rates of endogenous secretion and reabsorption were adjusted in such a way that the average rates across the 24 diets in the reference data set corresponded to average literature values from studies in which feed intake and meal frequency is as reported by Just et al. (1985). 2.6.1. Endogenous flows in the stomach (STO) The amount of EP entering the stomach was considered to be the sum of salivary and gastric secretions. The salivary and gastric secretion has been studied using different techniques with respect to pepsin and gastric juice secretion. These values are given as average daily

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Table 4 Reported and simulated literature values on various endogenous secretions in the GIT of the pig Source of endogenous AA-N

Data range AA-N: total N

Salivary + gastric Pancreatic Bile Small intestinal wall Large intestinal wall

0.5a 0.7c 0.65e 0.6g 0.5a

Total

–

Simulation (g N/day) Literature (g N/day) 2.0–3.3b 1.0–3.0d 1.8–3.0f 6.0–11.0h 1.0–3.0i 11.8–23.3

1.8–3.4 1.3–2.7 1.5–3.0 4.1–11.5 0.6–3.8 8.5–23.5

a

Reference: assumed. b References: Corring (1980); Auclaire (1986)—cited by Souffrant (1991) and Low (1982). c Reference: Pohland et al. (1993). d References: Corring et al. (1972); Corring (1980); Partridge et al. (1982); Zebrowska et al. (1983); Ozimek et al. (1984); Hee et al. (1985); Zebrowska (1985); Langlois et al. (1987); Hee et al. (1988); Imbeah et al. (1988); Mosenthin and Sauer (1991); Pohland et al. (1993) and Mosenthin et al. (1994). e References: Sambrook (1981); Juste (1982) and Corring et al. (1990). f Reference: Sambrook (1981); Juste (1982); Corring et al. (1990) and Jørgensen (1991). g Reference: Buraczewska (1979). h References: Souffrant (1986); Souffrant (1990) and Krawielitzki et al. (1990); Krawielitzki et al. (1996). i References: Krawielitzki et al. (1990) and Krawielitzki et al. (1996).

secretions and hardly reflect the dynamics of gastric nitrogen secretions. Auclair (1986) cited by Souffrant (1991) reported a daily secretion rate of 2–3.3 g N/day of salivary and gastric nitrogen. Low (1982) has calculated that 0.5 g AA-N appears as pepsin in the duodenum of young pigs. This alone accounts for 15–25% of the total N secretion and then we have assumed that amino acid–nitrogen represents 50% of the total N secretion. It appears that there is no significant reabsorption of endogenous material in the stomach (Krawielitzki et al., 1990, 1996). 2.6.2. Endogenous flows in the small intestine (SI1 and SI2) EP is secreted from three sources: pancreatic fluid, bile and intestinal wall. Data on pancreatic secretions of AA-N and total N are abundant compared to data on other sources of endogenous material (see Table 4). Pancreatic secretions are continuous, but modulated by feed intake and frequency; see Makkink and Verstegen (1990) for a review. The observations are often contradictory because some authors report no effect on pancreatic protein secretion in response to increased DF levels (Mosenthin and Sauer, 1991; Mosenthin et al., 1994) whereas others do report such an effect (Zebrowska and Low, 1987; Jakob et al., 2000). Due to the observed large variation in pancreatic protein secretion and the often-conflicting results, we have modelled pancreatic protein secretion simply as a function of OM inflow from STO. Bile secretion is continuous and is practically independent of feeding frequency (Jørgensen, 1991). Therefore, we assumed that the rate of bile secretion has no diurnal variation and that it can be related to DMI. Endogenous LD from other sources (e.g. epithelial cells) enters GIT as well, but data on this are scarce and for simplicity reasons, all contribu-

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tions are combined into one common estimate. Data from the review of Juste (1982) were used to derive a parameter value of 22 g LD/kg DMI to be used in the model. The contribution of the small intestine to the overall endogenous nitrogen secretion is very important because it may contribute 55% of the total N secretion (Krawielitzki et al., 1990). Secretions from the intestinal wall can be calculated by difference assuming that salivary, gastric, biliary and pancreatic secretions have already been accounted for. The total intestinal wall secretion rate is partitioned into 1/3 in SI1 and 2/3 in SI2. The total secretion (intestinal wall, biliary and pancreatic) to SI2 is on average only half of the N secretion to the entire small intestine although SI2 represents approximately two thirds of this. Buraczewska (1979) reported that N secretion in the proximal part of the small intestine is twice the N secretion in the distal part when expressed as g/(h m). This supports the partition between the two segments used in the model. The degree to which endogenous proteins are reabsorbed shows some variability (0.51–0.82) between studies, but an average of 70% at the terminal ileum can be calculated based on data from Souffrant et al. (1986, 1993) and Krawielitzki et al. (1990, 1996). The parameters for reabsorption of EP were adjusted in such a way that the simulated digestibility of EP is on average 0.70 with minimum and maximum values kept within the range of literature estimates. The description in the model accounts for effects of cell sloughing due to the physical nature of fibre (Shah et al., 1982) as well as for adsorption properties of fibre that reduce the reabsorption of EP (Bergner et al., 1981; Drochner et al., 2004). 2.6.3. Endogenous flows in the large intestine (LI) Estimation of endogenous protein secretion, reabsorption and loss at the faecal level has received less attention than at the ileal level. This is because AA absorption is completed prior to the large intestine, and nitrogen absorbed here is of no nutritional value to the pig. The endogenous nitrogen secretion in LI in the model was based on data from Krawielitzki et al. (1990, 1996). The principles as outlined above for setting the model parameters describing endogenous secretion can possibly lead to extreme values (low or high) when dietary regimes of high/low intake levels or high/low fibre levels are simulated. In order to check these principles, we conducted 48 simulations by varying the feed intake from 0.9 to 1.8 kg DM/d of the 24 reference diets (Just et al., 1985) used for model calibration. Experimental and simulated results are presented in Table 4. The model predictions are within the range of literature values, which is probably due to the large variation caused by the experimental difficulties in obtaining solid experimental data. 2.7. Model output The model outputs are as follows: (1) the nutrient and energy content of each segment at each time, (2) the ileal and faecal nutrient flows at each time, (3) the rates of absorption of AA, SU, FA, VFA and NAPN at each time, (4) MRT for the individual AS calculated by dividing the mean pool size of OM in each segment by the mean rate of OM flow from that segment during 1 h periods as well as for a 24 h period, (5) MRT for the total GIT calculated in the same way as for the individual AS, (6) ileal and faecal endogenous nitrogen losses

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(ENL) on a daily basis or related to DMI obtained by integrating the rate of passage at the ileal and the faecal level during a 24 h period, and (7) segmental digestibility values for all nutrients obtained as the integrated nutrient passage rate from each segment during a 24 h period subtracted from the daily nutrient intake and then divided by the daily nutrient intake. 2.8. Statistical analyses The model was compared with experimental data by means of four sets of parameters: (1) mean, minimum and maximum values of predicted and observed response variables, (2) intercept and slope from regression analyses (observed on predicted) as described below, (3) root mean square prediction error relative to the mean (relMSPE), and (4) coefficient of determination (R2 ). The mean square prediction error (Eq. (2.8.1)) was calculated as follows: n (Yi − Xi )2 MSPE = i=1 (2.8.1) n where Yi and Xi denote observed and predicted values, respectively. Comparisons of model simulation results with experimental results from different studies are often difficult due to large between study variations, which can be attributed to differences in animal physiological states, experimental design and analytical methods. From a statistical viewpoint, individual studies are blocks and their effects must be considered as random because the model is evaluated for its reliability in future predictions (St-Pierre, 2001). Therefore, the model validation was based on the following mixed linear regression model (Eq. (2.8.2)): Yij = β0 + Si + β1 Xij + Bi Xij + εij

(2.8.2)

where Yij is the observed value (digestibilities and endogenous losses) in the ith (1, . . ., k) study for the j’th diet (1, . . ., ni ), β0 the overall intercept across all studies, Si the random effect of study, β1 the overall regression coefficient of Y on X across all studies, Xij the predicted value by the model in the ith (1, . . ., k) study for the j’th diet (1, . . ., ni ), Bi the random effect of study on the regression coefficient of Y on X, and εij is the residual error. The random components of the mixed model are assumed to be:        σs2 σSB Si 0 ∼N and eij ∼ N(0, σe2 ). ,Σ with Σ = 0 Bi σSB σB2 When data sources were scarce, i.e. limited number of studies for comparing model predictions, simple regressions analysis were performed (Eq. (2.8.3)): Yi = β0 + β1 Xi + εi

(2.8.3)

where ei ∼ N(0, σe2 ). The probability that the intercept deviates from zero and the slope deviates from unity was calculated by means of a simple t-test. All the statistical analyses were performed in SAS 8.1 (1999).

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3. Results 3.1. Model calibration The results presented in this section cannot be considered as a true validation of the model since some of the data used for comparison with model simulations were also used for model calibration. Furthermore, the model generates many outputs and only selected outputs are presented here because of lack of corresponding literature values. The goodness of fit is presented in Table 5. It is clear that the model predicted protein and energy digestibility at both ileal and faecal level with good accuracy. This is shown by the fact that relMSPE values are less than 10% and that the slopes when regressing observed on simulated values are close to unity. LD and DF were predicted with less accuracy, but in both cases, the relMSPE is less than 25%. The model tended to overestimate the overall mean of observed values, but was able to predict the observed minimum and maximum values. The proposed framework for simulating nutrient digestibilities agrees fairly well with the reference data because they were empirically adjusted to match these. At present, no data sets are available that can be used to statistically derive the degradation parameters, which obviously limits the predictive value of the model. Available results from in vitro studies can provide information about nutrient degradation rates. However, it is questionable if degradation parameters derived from degradation profiles obtained in vitro can be implemented in the proposed framework. 3.2. Behavioural analyses An important criterion that must be satisfied is that the model has the capacity to simulate patterns of responses exhibited by the modelled system, i.e. that model responses to changes in driving variables (dietary inputs) agree with experimental data in terms of direction and approximate magnitude. The model aims at predicting endogenous nitrogen loss (ENL) in response to changes in the nutrient profile. ENL is calculated as the sum of EP and NAPN passing at the terminal ileum due to the fact that ENL is measured as nitrogen in most studies. Effects of different dietary factors on ENL were simulated by exchanging ST for DP or DF, and thus the ratios DP:ST and DF:ST seem to be the appropriate independent variables (see Fig. 2). These simulations show that replacement of ST by DP or DF increases ENL in agreement with various reviews on the subject (Boisen and Moughan, 1996; Nyachoti et al., 1997a). The effect of substituting DP for ST is primarily caused by the different degradation characteristics of protein and starch. A higher proportion of DP increases the OM flow into SI2 and thereby the endogenous secretion. However, the effect is partly counteracted because the DF concentration decreases slightly, which increases the affinity for proteolysis. This explains partly the reduced model response to variation in the ratio of DP to ST compared with the ratio of DF to ST. The curvilinear behaviour of the model in response to inclusion of DF is in accordance with observations of Taverner et al. (1981) and Mariscal-Landin et al. (1995) who reported that ENL might plateau at a given level of fibre in the diet. However, this partly contradicts

Nutrient

Mean

Observation

Prediction

Regression equation

Fit statisticsa relRMPSE (%)

R2

0.04 0.15 0.07

6.1 22.9 9.9

0.90 0.68 0.77

0.04 11.6 0.11 0.03

9.1 18.7 16.6 4.3

0.82 0.71 0.55 0.88

Observed

Predicted

Min

Max

Min

Max

Intercept

Slope

RMSPE

Ileal Protein Lipid Energy

0.70 0.48 0.65

0.72 0.55 0.69

0.35 0.18 0.42

0.86 0.85 0.85

0.37 0.23 0.46

0.86 0.78 0.83

−0.03 0.18 0.09

1.05 0.82 0.93

Faecal Protein Lipid DF Energy

0.78 0.50 0.40 0.78

0.80 0.56 0.37 0.79

0.63 0.21 0.16 0.54

0.93 0.86 0.69 0.95

0.62 0.25 0.17 0.55

0.91 0.82 0.61 0.92

0.04 0.16 0.04 −0.06

0.93 0.78 0.95 1.07

a

RMSPE, root mean square prediction error; relRMSPE, RMSPE expressed relative to the observed mean; R2 , coefficient of determination.

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Table 5 Mean, minimum and maximum values and linear regression equations describing the relationship between observed and predicted nutrient digestibilities at either ileal or faecal level for the 24 reference diets

345

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Fig. 2. Effect of variation in the (a) dietary fibre:starch ratio on predicted endogenous nitrogen loss (g N/kg DMI) and (b) dietary protein:starch ratio on predicted endogenous nitrogen loss (g N/kg DMI).

the data from Schulze et al. (1995) who observed that the ENL was linearly related to fibre level. The studies on fibre in relation to ENL have only dealt with inclusion levels below 200 g/kg DM and our results suggest that the response is not linear over a broader range of inclusion levels. The model also predicts the kinetics of nutrient and energy absorption (products of digestion multiplied by their combustion values), and thus model behaviour could be assessed in the following simulation scheme. Suppose that the model receives the same amount of digestible energy in one meal from either a high (400 g/kg DM) or a low DF diet (100 g/kg DM) with protein and lipid contents kept constant. The result of two simulations is presented in Fig. 3. When simulating the low DF diet, the model predicts a sharp raise in the rate of energy absorption with a peak 1–2 h after ingestion of the meal and then a fast decline, which is primarily caused by the rapid digestion and absorption of starch. On the other hand, when feeding the high DF diet, a smoother curve of energy availability is observed because

Fig. 3. The pattern of energy absorption after ingestion of equal amounts of digestible energy from two diets containing 100 g dietary fibre/kg DM (—) or 400 g dietary fibre/kg DM (- - -).

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of a more steady production of VFA. Furthermore, the availability of absorbed energy is higher from 3.5 to 4 h and onwards after feeding on the high fibre than on the low fibre diet. Rerat (1996) re-analyzed three successive experiments in order to compare the chronology of absorbed energy resulting from enzymatic or microbial digestion, respectively. He showed that the amount of energy absorbed per hour during the first 4 h after intake was highest for diets that contained enzymatic digestible carbohydrates, whereas the following period 5–12 h after intake was dominated by diets that contained microbial digested carbohydrates. These observations provide some evidence that the qualitative model behaviour is promising. 3.3. Sensitivity analyses The sensitivity analyses are presented in tables and graphs. A suitable dimensionless quantity for measuring sensitivity of the model response (R) to the ith parameter (Pi ) is the numerical value of S(R, Pi ), which is defined by France and Thornley (1984): |S(R, Pi )| =

dR Pi R Pi dR Pi ≈ = dPi R R dPi R Pi

(3.3.1)

where Pi represents the finite change in Pi , which causes the finite change (R) in R. For computing the |S(R, Pi )|, a 25% parameter change is chosen (Pi /Pi = ±0.25). If |S(R, Pi )| = 1, the fractional change in the parameter value produces the same fractional change in model response R. The sensitivity analysis was performed on all parameters, but only results from parameters that have a |S(R, Pi )| ≥ 0.10 are shown here and thus the model is sensitive to those presented (Tables 7–10). Parameters with |S(R, Pi )| > 0.5 have large effects on model response and are ranked as highly sensitive. Basis for the sensitivity analysis was a reference simulation with default settings for feed intake (TFEED = 0.25, FFEED = 3 and DMI = 1.5) on a “normal” grower diet (protein: 201, lipids: 47, starch: 457, sugar: 48 and dietary fibre: 184 g/kg DM). The sensitivity was evaluated in four main areas of the model: the first area is starch digestion and sugar absorption in the small intestine, where the selected model responses are time and height of the peak absorption rate and the amount absorbed during an 8-h period following a meal; the second area is lipid digestion in the small intestine, where the chosen model responses are the quantity of FA absorbed and the apparent lipid digestibility; the third area is digestion of nitrogenous substances in the small intestine, where the selected responses are time and height of the peak absorption rate, apparent ileal protein digestibility and endogenous nitrogen loss at ileum; the fourth area is metabolism in the LI, where the selected responses are MRT in LI, amount of VFA absorbed during an 8-h period following a meal as well as apparent faecal digestibilities of protein, DF and energy. When the sensitivity analysis was performed on parameters that define the endogenous transactions, then the parameters for EP and NAPN transactions were changed simultaneously because these originate from the same source. The computed sensitivity coefficients |S(R, Pi )| in relation to the SU metabolism are presented in Table 6. In general, the predicted amount of SU absorbed during an 8-h period following a meal is insensitive to a fractional change of 25% (Pi /Pi = ±0.25) in all the parameters involved in the SU metabolism. The time of peak SU absorption rate is highly

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Table 6 Sensitivity coefficients for responses in sugar (SU) absorption: time and height of peak absorption, and quantity absorbed during an 8-h period after a meal Kinetic parameter

CSI1, ST, hyk CSI1, ST, hyv CSI1, SU, abk CSI1, SU, abv CSI2, ST, hyk CSI2, ST, hyv CSI2, SU, abv CSTO, pa

Time of Peak

Height of Peak

Quantity of SU

−25

25

−25

25

−25

25

0.36 0.51 0.01 0.10 0.15 0.10 0.10 0.05

0.26 0.41 0.01 0.05 0.05 0.15 0.05 0.10

0.10 0.13 0.10 0.03 0.10 0.13 0.04 0.45

0.07 0.11 0.10 0.01 0.08 0.09 0.02 0.45

0.03 0.04 0.03 0.01 0.03 0.04 0.01 0.01

0.02 0.03 0.03 0.01 0.03 0.03 0.01 0.00

Changes in kinetic parameters: ±25% of their default values.

sensitive to the maximum velocity of starch degradation (CSI1, ST, hyv ) in the first part of the small intestine (SI1), whereas the affinity constant for the same process (CSI1, ST, hyk ) has a smaller effect. This is due to the fact that these two parameters determine the rate of starch digestion in SI1 and thus the availability of SU for absorption. Model responses (time and height of the peak absorption and cumulative absorption during an 8-h period) to parameters, which are related to sugar metabolism (CSI2, ST, hyk , CSI2, ST, hyv and CSI2, SU, hyv ) in the SI2, has sensitivity coefficients around 0.10–0.15 and thus the model is sensitive to these parameters. However, the effects are much less than in SI1, which is due to the fact that around 60% of the starch is degraded in SI1 and is available for absorption. The fractional rate of gastric emptying (CSTO, pa ) largely determines the height of peak absorption rate, which coincides very well with the general consensus that the SU load after a meal can be manipulated by delaying the rate of gastric emptying (Ellis et al., 1996; Johansen et al., 1996). The sensitivity coefficients |S(R, Pi )| concerning lipid digestion in the small intestine are presented in Table 7. The model is sensitive to changes in the parameter that defines the endogenous lipid supply (CSI1, LD, sc ). This is not surprising since the entire endogenous Table 7 Sensitivity coefficients for responses in lipid digestion: quantity of fatty acids (FA) absorbed during an 8-h period after a meal, and apparent ileal digestibility (AID) of lipid Kinetic parameter

CSI1,LD, hyk CSI1,LD, hyv CSI1,LD, sc CSI2, LD, hyk CSI2, LD, hyv CSI2, FA, abv CSI1, pa CSI2, pa

Quantity of FA

AID of lipid

−25

25

−25

25

0.06 0.06 0.46 0.14 0.18 0.05 0.06 0.24

0.04 0.05 0.46 0.20 0.13 0.02 0.05 0.23

0.14 0.13 0.26 0.33 0.42 0.12 0.14 0.56

0.10 0.12 0.26 0.45 0.29 0.05 0.11 0.53

Changes in kinetic parameters: ±25% of their default values.

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Table 8 Sensitivity coefficients for responses in protein digestion: time and height of peak amino acid absorption, apparent ileal digestibility (AID) of protein, and ileal loss of endogenous nitrogen Kinetic parameter

CSI2,DP,hyk CSI2,DP,hyv CSI2,EP,hyk CSI2,EP,hyv CSI2, DF, 0 CSI2,AA,abk CSI2,AA,abv CSI2, EP, sc CSTO, pa CSI, pa CSI2, pa

Time of peak

Height of peak

AID of protein

Endogenous nitrogen loss

−25

25

−25

25

−25

25

−25

25

0.07 0.23 0.01 0.03 0.23 0.03 0.20 0.03 0.17 0.43 0.30

0.10 0.17 0.03 0.03 0.13 0.10 0.03 0.03 0.13 0.23 0.17

0.14 0.30 0.04 0.08 0.39 0.01 0.04 0.07 0.32 0.13 0.02

0.12 0.19 0.03 0.04 0.27 0.02 0.01 0.06 0.35 0.09 0.02

0.12 0.20 0.08 0.12 0.46 0.02 0.05 0.04 0.03 0.03 0.10

0.10 0.12 0.07 0.08 0.30 0.03 0.02 0.05 0.02 0.03 0.15

0.00 0.00 0.12 0.20 0.46 0.00 0.00 0.12 0.03 0.03 0.20

0.00 0.00 0.15 0.24 0.36 0.00 0.00 0.12 0.02 0.03 0.21

Changes in kinetic parameters: ±25% of their default values.

lipid enters the SI1 segment. The model is also sensitive to changes in the maximum velocity (CSI2, LD, hyv ) and the affinity constant (CSI2, LD, hyk ) for LD digestion in the last part of the small intestine (SI2). Parameters for fatty acid (FA) absorption have generally little effect on the predicted apparent ileal LD digestibility. The model is sensitive to the kinetic constant for passage through SI2 (CSI2, pa ). Parameters that define the fat metabolism in SI2 are the most important parameters because most of the triglycerol is digested and absorbed in this segment. Furthermore, there is little knowledge on the segmental digestibility of LD in the (SI1) whereas information about ileal LD digestibilities is quite abundant (Jørgensen et al., 2000). The sensitivity coefficients |S(R, Pi )| related to nitrogen metabolism are presented in Table 8. The time and height of peak amino acid (AA) absorption rates as well as the apparent ileal protein digestibility is insensitive, whereas the endogenous nitrogen loss is sensitive to changes in the parameters (CSI2, EP, hyv and CSI2, EP, hyk ) that define the endogenous protein degradation in SI2. The time of peak AA absorption rate is sensitive to changes in the maximum velocity for AA absorption in the second segment of the small intestine (CSI2, AA, abv ), which is caused by the fact that the major part of AA available for absorption is present in this segment. The apparent ileal protein digestibility is sensitive to changes in the parameters (CSI2, DP, hyv and CSI2, DP, hyk ) that define the rate of DP degradation in SI2, which is explained by the fact that most of DP is digested and absorbed in this segment. This also causes the model to be sensitive to changes in the kinetic constant for passage through this segment (CSI2, pa ). The height of peak AA absorption is sensitive to the rate constant for passage from the stomach (CSTO, pa ), while the time of peak is sensitive to the passage rate constants from SI1 and SI2 (CSI1, pa and CSI2, pa ). All the model responses in Table 8 are sensitive to the parameter describing the relation between the affinity constant for protein degradation and dietary fibre level (CSI2, DF, 0 ). This is an internal parameter of the model, i.e. a parameter that is related to a mathematical representation and is rather conceptual. Therefore, these internal parameters of the model will be further commented on in the discussion.

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Table 9 Sensitivity coefficients for responses in large intestine (LI) transactions: mean retention time (MRT), quantity of volatile fatty acids (VFA) absorbed during an 8-h period after a meal, and apparent faecal digestibility (AFD) of protein, dietary fibre and energy Kinetic parameter

CLI, DDF, hyk CLI, DDF, hyv CLI, OM, 0 CLI, pa, 0 CMM

MRT in LI

Quantity of VFA

AFD of protein

−25

25

−25

25

−25

0.08 0.19 0.88 0.17 0.08

0.07 0.19 0.95 0.14 0.07

0.17 0.43 0.20 0.03 0.16

0.14 0.38 0.16 0.03 0.16

0.01 0.02 0.02 0.00 0.05

AFD of dietary fibre

AFD of energy

25

−25

25

−25

25

0.01 0.01 0.01 0.00 0.05

0.35 0.85 0.34 0.06 0.03

0.28 0.76 0.30 0.05 0.03

0.02 0.05 0.02 0.00 0.02

0.02 0.04 0.02 0.00 0.02

Changes in kinetic parameters: ±25% of their default values.

The sensitivity coefficients |S(R, Pi )| concerning metabolism in the large intestine are presented in Table 9. In general, the selected model responses are insensitive (|S(R, Pi )| ≤ 0.10) to changes in the parameters that define degradation rates of DP, EP and ST (not shown). This is due to the fact that these nutrient fractions are almost completely digested prior to the LI. The predicted DF digestibility is sensitive to changes in the parameter (CLI, DDF, hyk ) and highly sensitive to maximum velocity (CLI, DDF, hyv ) for this reaction. Furthermore, the digestibility of this nutrient fraction is also highly sensitive to one of the parameters that describe transit through this segment (CLI, OM, 0 ). The quantity of VFA absorbed is affected in same fashion as the DF digestibility because this quantity is calculated as the difference between carbon from nutrient degradation and carbon assimilated for microbial growth. The MRT in LI is highly sensitive to changes in the parameter CLI, OM, 0 (reference OM quantity in LI). The parameter CLI, OM, 0 can also be viewed as an internal model parameter and will be discussed later. In conclusion, all numeric values of S(R, Pi ) for the parameters investigated in the above sensitivity analyses are less than 1. This means that a 25% change in a given parameter value results in a fractional change of less than 25% in the selected response variable. On basis of the sensitivity coefficients |S(R, Pi )|, some parameters were selected for further analysis because model predictions are especially sensitive to these. We used the mean square prediction error (MSPE) between predicted and observed values from the reference data set (Just et al., 1985) to evaluate the model behaviour more closely. This reference data set was also used for parameterization of the model. Each of the parameters was varied in a grid of 21 equidistant points by varying the parameter values ±50% of their default values. The reference dataset was simulated with the model for each value of the grid, keeping the remaining parameters fixed at their default values (Foldager et al., 2002). Model behaviour was studied by plotting MSPE (Eq. (2.8.1)) as a function of the different parameter values. The purpose of the analysis was not to estimate the many parameter values in the dynamical system, but to examine whether the model is flexible enough to describe the reference data set and to gain insight on the general behaviour of the proposed dynamical system.

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Table 10 Parameters selected for further sensitivity testing Parameter

Unit

Default value

Range of variation in the simulations

Principle response variablea

CSI2, DP, hyv CSI2, DF, 0 CSI1, LD, sc CLI, OM, 0 CLI, DDF, hyv CLI, DDF, hyk

mol N/h mol C/kg OM mol C/kg DMI kg OM mol C/h mol C

0.38 22.0 1.25 0.250 0.30 2.50

0.190–0.570 11.00–33.00 0.60–1.60 0.125–0.375 0.15–0.45 1.25–3.75

AID of protein AID of protein AID of lipid AFD of dietary fibre AFD of dietary fibre AFD of dietary fibre

a

AID: apparent ileal digestibility; AFD: apparent faecal digestibility.

Based on results from the sensitivity analyses, a total of six parameters were selected for further investigation. Two of these are related to the digestion of nitrogenous substances in the small intestine (CSI2, DP, hyv and CSI2, DF, 0 ), one is related to the digestion of lipid in the small intestine (CLI, LD, sc ), two are linked to degradation of DF in LI (CLI, DDF, hyv and CLI, DDF, hyk ) and one is related to representation of the transit in LI (CLI, OM, 0 ). A description of the parameters, the range of variation and the selected response variables from the reference data set (Just et al., 1985) is given in Table 10. In Fig. 4, the MSPE of the response variables (apparent ileal digestibility of energy, protein or lipid) are plotted against the values of the selected parameters (CSI2, DP, hyv , CSI2, DF, 0 and CSI1, LD, sc ). The energy digestibility was considered as a general response variable to verify the general behaviour of the dynamical system. Each of the plots in Fig. 4 represents a total of 504 individual simulations. The MSPE of protein digestibility as a function of CSI2, DP, hyv has a minimum close to 0.38, the default value of the parameter. However, with energy digestibility as the response variable, MSPE does not have a minimum. Therefore, there is no common minimum for MSPE of protein and energy ileal digestibility when CSI2, DP, hyv is in the neighbourhood of 0.38. The qualitative behaviour of the graphs in the upper, middle and lower row of panels in Fig. 4 is quite similar, but the magnitude of response in MSPE is very different in the left side panels (digestibility of protein or lipid). The range of response in MSPE for protein digestibility is much smaller with changes of CSI2, DP, hyv than with changes of CSI2, DF, 0 , which shows that the model is much more sensitive to the latter internal parameter. Decreasing the parameter value for endogenous lipid secretion (CSI1, LD, sc ) has a tremendous effect on MSPE of lipid digestibility because of a reduced variation in the simulated digestibility at low dietary inputs of LD. The model is quite insensitive to changes in these parameters when energy digestibility is the response variable (right side panels). This is due to the fact that the simulated energy digestibility is an aggregated output variable, which masks some of the variation in the simulated digestibility of individual nutrients. In Fig. 5, MSPE of the response variables (DF digestibility and apparent faecal energy digestibility) is plotted against increasing values of the parameters CLI, OM, 0 , CLI, DDF, hyv and CLI, DDF, hyk . The upper plots in Fig. 5 show that MSPE as a function of CLI, OM, 0 has a minimum between 0.25 and 0.30 kg OM for fibre digestibility and between 0.15 and 0.20 kg OM for energy digestibility, which is not far from the default value of 0.25 kg OM.

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Fig. 4. Results of a simulation study with the parameters CSI2, DP, hyv , CSI2, DF, 0 and CSI1, LD, sc varied ±50% of their default values in a grid of 21 equidistant points. The plots to the left are the mean square prediction error (MSPE) of apparent ileal digestibility of protein (upper and middle) and lipid (lower), and the plots to the right are MSPE of ileal energy digestibility as functions of parameter values. The arrows and numbers represent the default parameter values.

The minimum of MSPE as a function of CLI, DDF, hyv occurs at the vicinity of the default value (0.30 mol C/h) for both fibre and energy digestibility. When related to CLI, DDF, hyk , MSPE has a minimum near the default value of the independent parameter (2.5 mol C/h) for fibre, but not for energy digestibility. We described feed intake as a discontinuous process and large oscillations in the nutrient pools are observed as a consequence of the variable feed intake over time.

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Fig. 5. Results of a simulation study with the parameters CLI, OM, 0 , CLI, DDF, hyv and CLI, DDF, hyk varied ±50% of their default values in a grid of 21 equidistant points. The plots to the left are the mean square prediction error (MSPE) of faecal digestibility of dietary fibre and the plots to the right are MSPE of apparent faecal energy digestibility as functions of parameter values. The arrows and numbers represent the default parameter values.

When the parameters were varied ±50% from their default values, a chaotic behaviour of the model might be expected. A close inspection of the graphs reveals that the system of differential equations is stable in the chosen range of values for the six parameters because the MSPE is not oscillating. Thus, a hypothesis of instability of the solution or chaotic behaviour can be rejected for the given simulation conditions.

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3.4. Model evaluation 3.4.1. Prediction of nutrient digestibility Digestibility values are the net result of the degradation, secretion and absorption processes that occur during passage through the GIT, and thus they represent the most frequently used criteria for determining the nutritive value of feedstuffs and mixtures. Mechanistic models of digestion and absorption must be compared at their highest aggregated level with data on digestibility to validate the consequences of the underlying hypotheses build into to the models. We have compared model predictions of faecal digestibility of DP, LD, OM and energy with data from Noblet et al. (1993). They measured the digestible, metabolizable and net energy content of 17 highly complex diets, which were used to generate estimates of the digestible, metabolizable and net energy content in 13 feedstuffs. The results of the simulations are presented in Table 11. Prediction of protein digestibility is reasonably good as the model is able to pick up the observed variation in digestibility in response to the dietary treatments, but within the observed range of values, the model overestimates protein digestibility. For lipid as for protein, the intercept of the regression line deviates significantly from zero, whereas the slope is significantly different from unity only for lipid. Predictions of lipid digestibility is not very convincing although the model is able to reproduce the general trend of curvilinear increasing lipid digestibility in response to increased dietary levels (Jørgensen and Fern´andez, 2000). However, the faecal digestibility of OM and energy is predicted very accurately with both the slope and intercept being numerically close to unity and zero, respectively. Results of the model evaluation against reported apparent digestibility values obtained from Just (1982a,b,c), Just et al. (1983), Noblet et al. (1993), Nyachoti et al. (1997b), Grala et al. (1998a,b, 1999), Leterme et al. (2000) and H¨ogberg and Lindberg (2004) are presented in Table 12. This database with reference data (Just et al., 1985) excluded contains apparent protein and energy digestibility values at the ileal and/or faecal level from 75 treatments in 10 studies. Model performance was evaluated by means of mixed model regression analysis (Eq. (2.8.2)) including the random effect of study. The model is able to predict the apparent digestibility of protein at both the ileal and the faecal level with reasonable accuracy, but the model significantly overestimates the apparent ileal protein digestibility at high levels of digestibility, which is supported by the fact that the slope (0.84) significantly (P=0.02) deviates from one when regressing observed on predicted apparent ileal protein digestibilities. The model predicts the energy digestibility with good accuracy, especially at the faecal level. The fixed slope of regression is close to and not significantly different from unity. Generally, the model predicts faecal with greater accuracy than ileal digestibility, and the digestibility of energy is predicted more accurately than that of protein. 3.4.2. Prediction of endogenous secretions Modelling digestive secretions in response to dietary factors is hampered by experimental difficulties in obtaining solid data for parameterization. Furthermore, many studies are limited to nitrogen or amino acids although it is documented that significant losses of lipids, carbohydrates and minerals take place as well (Juste, 1982; Lien et al., 2001).

Nutrient

Protein Lipid OM Energy a b * **

Mean (percentage of intake)

Predicted

Observed

Regression equation

Predicted

Observed

Min

Max

Min

Max

Intercept

82 38 80 79

75 53 81 79

76 10 72 71

85 60 87 85

70 25 72 70

81 72 86 84

15.01 28.48 1.53 −5.12

Probability that the intercept differs from zero or that the slope differs from unity. relMPSE: The root mean square prediction error relative to the observed mean. P>0.05. P>0.01.

Statistics of fit

Pa

Slope

*

0.73 0.65 0.99 1.06

**

NS NS

Pa

relMPSEb (%)

R2

NS

10.3 32.5 1.7 1.2

0.73 0.33 0.96 0.97

*

NS NS

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Table 11 Mean, minimum and maximum values and linear regression equations describing the relationship between observed and predicted nutrient digestibilities at the faecal level for 17 diets (Noblet et al., 1993)

355

356

Nutrient

na

Mean (percentage of intake)

Predicted

Observed

Regression equation

Statistics of fit Slope

Pb

relMPSEc (%)

R2

5.3 5.5

0.71 0.75

5.8 3.4

0.93 0.94

Predicted

Observed

Min

Max

Min

Max

Intercept

Pb

Ileal digestibility Protein 42 Energy 23

72 67

73 69

53 54

91 78

54 60

89 79

12.2 −7.95

NS NS

0.84 1.08

*

Faecal digestibility Protein 46 Energy 46

81 77

78 79

63 54

89 87

57 59

88 88

NS NS

1.20 0.89

*

a b c *

Number of treatments. Probability that the intercept differs from zero or that the slope differs from unity. relMPSE: the root mean square prediction error relative to the observed mean. P>0.05.

−19.3 10.9

NS

NS

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Table 12 Mean, minimum and maximum values and linear regression equations describing the relationship between observed and predicted nutrient digestibilities at the ileal and faecal level for different studies with reference study excluded

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Fig. 6. The relationship between observed ileal endogenous nitrogen losses adjusted for study effect and simulated ileal endogenous losses.

Model predictions of ileal ENL was compared with estimates from 39 different diets used in eight studies, which had either cereals or protein sources as their main ingredient or were specifically designed for investigating effects of dietary fibre (Taverner et al., 1981; Furuya and Kaji, 1992; Schulze et al., 1995; Nyachoti et al., 1997b; Grala et al., 1998a,b, 1999; Leterme et al., 2000; Jondreville et al., 2001). This validation was performed with a mixed model regression analysis because Pedersen and Boisen (2002) have shown that estimates of ENL are influenced by cannulation procedure and method of determination. Thus, it is important to consider the study effect. The relationship between observed and simulated values is presented in Fig. 6. The model predicts ENL quite well as judged by regression of ENL adjusted for study effect on simulated values (Y = 0.55 + 0.79 X; relMSPE = 11.9%; R2 = 0.68). The intercept does not differ from zero (P=0.065), but the slope differs from unity (P=0.032). Furthermore, the model predicts the apparent protein digestibility at ileum with a reasonably good accuracy, which suggests that the framework is sound. 3.4.3. Prediction of rates of absorption This aspect of the model was evaluated by comparison of literature data with simulated rates of absorption from the GIT. The simulated pattern of absorption was computed as mean flow rates at hourly time points after a meal (see Figs. 7–9 for glucose, amino acids and VFA, respectively). The predicted kinetics of glucose absorption was tested on six different diets from three studies (Bach Knudsen et al., 2000; Rerat et al., 1988, 1993) and is presented in Fig. 7 for the qualitative “best” and “worst” case. In general, the model predicts higher rates of glucose absorption than is observed in the studies. However, this could be expected because the model simulates glucose disappearance from the gut, while the experimental data are on glucose portal appearance. Transit and metabolism in the gut wall probably delays and

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Fig. 7. Patterns of glucose absorption 1–8 h after a meal. Black bars: experimental data on portal appearance; open bars: simulated data on disappearance from the gut. Left panel: data from Rerat et al. (1993); right panel: data from Bach Knudsen et al. (2000).

Fig. 8. Patterns of amino acid absorption 1–8 h after a meal. Black bars: experimental data on portal appearance; open bars: simulated data on disappearance from the gut. Left panel: data from Rerat et al. (1993); right panel: data from Rerat et al. (1988).

Fig. 9. Patterns of VFA absorption 1–12 h after a meal. Black bars: experimental data on portal appearance; open bars: simulated data on disappearance from the gut. Left panel: data from Rerat et al. (1993); right panel: data from Rerat et al. (1987).

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Fig. 10. Patterns of OM flow at the terminal ileum after a meal at 16 h. Black bars: experimental data from Johansen et al. (1996); open bars: simulated data. Left panel: barley diet; right panel: high level alfalfa diet.

reduces the amounts of glucose appearing in the portal blood. The model predicts a peak absorption rate 2 h after the meal in accordance with Rerat et al. (1993). The kinetics of AA absorption was tested against four diets from three references (Lenis et al., 1996; Rerat et al., 1993, 1988). The pattern of AA appearance in the portal blood is less clear because Lenis et al. (1996) reported a peak appearance rate during the first hour, whereas Rerat et al. (1993) reported the peak to occur 6 h after ingestion of a meal. As the GIT tissues utilize nearly 50% of the AA intake for oxidation, protein synthesis and synthesis of metabolic intermediates (Burrin et al., 2001), it is more reasonable to compare the patterns rather than the absolute values of simulated and observed absorption. The “best” and “worst” case of these comparisons is shown in Fig. 8. The simulated pattern of AA absorption is satisfactory with the peak occurring 2–3 h after the meal. The simulated pattern of VFA absorption is compared in Fig. 9 with data on VFA portal appearance from two diets with 56 and 214 g DF/kg DM, respectively (Rerat et al., 1987, 1993). The simulated shape of the time-dependent absorption curve is in good agreement with the experimental data, especially during the last 6 h of the observed time interval. However, the model overestimates the production and absorption of VFA during the 12h period by 150 and 130%, respectively. This is due partly to the assumption that VFAs are absorbed at their rate of production and partly to a potential error in the predicted DF digestibility. Metabolism of VFA for energy purposes can also explain some of these discrepancies since the colonocytes receive a major proportion of their energy supply from oxidation of butyrate (Martin et al., 2000). 3.4.4. Prediction of rate of passage When modelling digestion and absorption, an important aspect is the prediction of passage rates at different sites along the GIT after a meal. Model predictions were compared with a data set of Jørgensen et al. (1997), who investigated the diurnal variation in the chemical composition of digesta in pigs and its effect on nutrient digestibilities of diets with different DF contents. The pigs were fitted with a post-valve T-caecum cannula and fed three times daily at 8, 16 and 24 h. Results from this comparison are displayed in Fig. 10 for a barley diet and a high level alfalfa diet. The model predicts the absolute flow

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of OM at the distal ileum quite well, but the simulated peak flow of OM occurs later (at 20.00–21.00 h) than the observed peak at 18.30–19.30 h. In the study of Jørgensen et al. (1997), the peak flow of OM was observed approximately 2.5–3.5 h after the meal, whereas the model predicts the peak at 4–5 h after feeding. Hodgkinson et al. (2002) observed the maximum flow of DM 6–11 h after ingestion of a meal and similar results were obtained by Van Leeuwen et al. (1991), who reported the peak flow at the end of the ileum 3–8 h after meals in pigs fed twice daily. The predicted and observed quantities of OM passing at the terminal ileum during the 8-h collection period are 141 and 138 g, respectively, with the barley diet, and 172 and 178 g, respectively, with the alfalfa diet. The model predictions are generally within the range of literature data and therefore, the model predictions are fairly sound.

4. Discussion 4.1. Sensitivity and internal model parameters All published models have one or more parameters that are associated with hypothetical mathematical descriptions and to a lesser degree derived from the biology of the system. In fact, it can be argued that all parameters that are empirically adjusted to match a calibration data set are internal model parameters. However, maximum velocities and affinity constants can be ascribed biological meaning, which is an important aspect of mechanistic modelling. Furthermore, affinity constants applied to reaction kinetics in digestion models reflect the priorities between different pathways of a particular substrate (Gill et al., 1989) and thereby the partition of substrate between fluxes of degradation and passage. For simulation of the interaction between rate of protein degradation and level of DF in the diet, we assumed that the affinity for protein degradation is not constant, but regulated by the concentration of DF. The model is very sensitive to the parameter CSI2, DF, 0 , which is difficult to assess and therefore has been set arbitrarily. Nevertheless, this representation allowed us to simulate both variations in the true protein digestibility and increase in the ENL in response to increased levels of DF and in turn the depressive effect of DF on apparent protein digestibility (De Lange et al., 1989; Schulze et al., 1995; Grala et al., 1998a,b; Drochner et al., 2004). The fractional passage in LI was assumed to be exponential rather than linear in order to maintain the pool size of OM in LI and MRT in the total tract within the range of literature values. The mathematical representation is rather conceptual and hardly reflects the digesta mixing and propulsion that occurs in colon and caecum (Stevens and Hume, 1998). The model is especially sensitive to the parameter CLI, OM,0 , which is deduced from data published by Stanogais and Pearce (1985) and thus quite well established. The steepness parameter CLI, pa, kn was adjusted to give a representation of the MRT comparable to literature data, and fortunately the model is rather insensitive to this parameter. The above discussion highlights the problems of constructing mechanistic simulation models because one is often faced with limited data for parameterization. Instead, we used data collected at the animal level (digestibilities) to derive parameter values for relationships that exist at lower levels of the organizational hierarchy (France and Thornley, 1984).

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Some justification for this procedure can be that others have successfully applied the same methodology (Danfær, 1990; Bastianelli et al., 1996; Halas et al., 2004). 4.2. Prediction of nutrient digestibility In this paper, we have presented a rigorous test of the model against several independent data sets on nutrient digestibilties, ENL and the pattern of nutrient availability. To our knowledge, this represents the first time that a digestion model in pig nutrition has been challenged against independent data sets on nutrient digestibilities, which in itself represents an improvement to previous models (Bastianelli et al., 1996; Rivest et al., 2000). We have identified some limitations in the present framework, which we will address in the following section in anticipation of future improvements of the model. Model predictions of apparent lipid digestibility are not satisfactory and suggest that the presented framework is inadequate to describe the physiological aspects of LD digestion although the model is able to predict the curvilinear increase in LD digestibility in response to increased LD levels (Jørgensen and Fern´andez, 2000). The assumption of a uniform fat source seems too simplistic and lead to unacceptable, biased predictions. Therefore, future model development must include partition of dietary LD into multiple fractions because individual fatty acids are not digested to the same extent (Jørgensen, 1991; Jørgensen et al., 2000). One proposal would be to divide the LD pool into two pools based on the ratio of unsaturated to total fatty acids and then applying different digestion parameter values for the two pools. Adjustment of animal models to accommodate greater generality has generally relied on increasing the number of dietary fractions required as model inputs. This notion has led to the development of models that require very detailed feed descriptions (Baldwin et al., 1987; Dijkstra et al., 1992; Danfær et al., 2005), which are seldom reported in the literature (Hanigan et al., 2005). Thus, the above proposal would yield a model that relies more heavily on dietary inputs and is more difficult to parameterize due to a doubling of the number of parameters to represent LD digestion. Finally, the model assumes that dietary LD fraction (crude fat) consists solely of triglycerides which is seldom the case because crude fat may contain variable amounts of sterols, waxes, polymer fatty acids, etc., which also affects LD digestibility (Jørgensen and Fern´andez, 2000). 4.3. Prediction of endogenous nitrogen loss The model is able to satisfactorily predict the apparent ileal protein digestibility, but the predictions of ENL are less accurate. The variations in DF and DP contents are not sufficient descriptors of physico-chemical properties of different feedstuffs and feed mixtures. It has been shown recently by Leterme et al. (2000) that the effect of DF on ENL differs with the source and nature of DF and is related to the chemical composition and physico-chemical properties of the fibre fraction. Considering that the DF fraction is a very heterogeneous fraction, it is a rather crude assumption that the physiological effects could be attributed to the total DF fraction. Instead, the effects should be related to some sub-fractions of DF (e.g. cellulose, ␤-glucans, arabinoxylans, etc.). However, the effects of DF in the small intestine

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on digestive processes are difficult to predict from any of the chemical parameters currently measured (Ellis et al., 1996; Bach Knudsen, 2001). Bastianelli et al. (1996) assumed that the depressive effect of dietary fibre (DF) on apparent protein digestibility is solely due to increased intestinal wall secretions. This hypothesis has been rejected by Nyachoti et al. (2000), who discovered that protein synthesis rates are unaffected by diets that induce either a high or low EP loss. This suggests that the effects of DF and anti-nutritional factors (ANF) are more related to the reabsorption of EP (Grala et al., 1998a,b). Endogenous protein (EP) secretion is only represented as time-dependent in the model of Rivest et al. (2000) and known dietary influences are inputs and are therefore not intrinsic properties of their model. The model presented here proposes a mechanistic way of incorporating effects of DF into digestive models, which is an improvement to previous models (Bastianelli et al., 1996; Rivest et al., 2000). During the last decade there has been considerable interest in estimating basal ENL in growing pigs for calculating standardized protein and AA digestibility (Boisen and Moughan, 1996; Pedersen and Boisen, 2002; Jansmann et al., 2002). Different methodologies have been applied for estimating the basal ENL. We will not discuss this in detail, but just note that feeding an N-free diet is the most frequently used method (Jansmann et al., 2002). The DF source used in these experiments has usually been purified cellulose with inclusion percentages of 3–8% in DM (Jansmann et al., 2002). Simulation of a “basal” N-free diet (protein: 0 g/kg DM, lipids: 50 g/kg DM, starch: 750 g/kg DM, sugar: 100 g/kg DM and dietary fibre: 30 g/kg DM) yields an estimate of the basal ENL of 1.77 g N/kg DMI with the present model, which is close to estimates of Pedersen and Boisen (2002) and Jansmann et al. (2002) for N-free diets. The model predicts an increase in ENL in response to either increased DP or DF content. It is generally acknowledged that inclusion of protein in test diets used for estimating ENL leads to higher estimates of the basal ENL (Boisen and Moughan, 1996; Jansmann et al., 2002). In general, estimates of basal ENL are unaffected by inclusion of 3–8% cellulose in N-free diets, but they are affected by inclusion of “natural” fibre sources (Souffrant, 2001; Jansmann et al., 2002). The results of the model simulations can be interpreted according to the concepts of specific and basal ENL widely accepted in practical protein evaluation for pigs (Boisen, 1998; Mosenthin, 2002). However, the concept of a basal ENL can still be regarded as a rather theoretical issue. 4.4. Prediction of nutrient absorption rates For some diets, the model is able to predict the pattern of nutrient absorption 1–8 h after a meal, but the modelling exercise has also highlighted the need for future digestion models to include a minimal description of the metabolism in the portal drained viscera (PDV). As pointed out by Yen et al. (1989) and recently reviewed by Burrin et al. (2001), the PDV are metabolically very active tissues. Although they approximately represent 5% of the bodyweight, they contribute to 20–35% of the whole body protein turnover and oxygen consumption (Yen et al., 1989; Stoll et al., 1999). Data on net portal appearance of nutrients can only be fully appreciated once the metabolism in PDV has been modelled. To

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the knowledge of the authors, no such model has been constructed for growing pigs, but the issue has been addressed in human and ruminant nutrition (Fouillet et al., 2000; Danfær et al., 2005).

5. Conclusion The objective of the present work was to develop a model for simulating the kinetics of digestion and absorption in pigs with special attention to the physiological effect of dietary fibre. The present model structure contains a total of 38 nutrient pools. The model proposes a mechanistic way of integrating digestive interactions between dietary protein and fibre pools. This represents an improvement compared with previously published models. A simulation study revealed that the model is stable even with large variations (±50% of default values) of the high sensitivity parameters and with feed intake being discontinuous (meal feeding). Comparisons of model outputs with independent literature data on digestibility showed the following: • Apparent ileal and faecal protein digestibilities are predicted with good accuracy (relMSPE = 5–6%, R2 = 0.71–0.93). • Ileal endogenous nitrogen loss is predicted with reasonable accuracy (relMSPE = 12%, R2 = 0.68). • The predicted apparent faecal digestibility of lipid does not agree well with experimental data (relMSPE = 33%, R2 = 0.33), and it is concluded that the present description of this item in the model is inadequate. • Energy digestibility is predicted with high accuracy, especially at the faecal level (relMSPE = 3%, R2 = 0.94). In general, the simulated patterns of nutrient absorption (glucose, amino acids and VFA) agree well with experimental data on nutrient appearance in the portal blood when the PDV metabolism is taken into consideration. The pattern and absolute rates of organic matter flow at the distal ileum are predicted with reasonable accuracy. Full utilization of the model as a predictive tool in pig nutrition is expected when it is combined with a metabolism model.

Appendix A See Table A.1 .

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Table A.1 The general equation format presented in the model for the ith chemical constituent Equation format Stomach (STO) Differential equation dSSTO, i /dt = RSTO, i, in + RSTO, i, sc − RSTO, i, pa Rate variables RSTO, i, in a = DMI/(FFEED × TFEED) × TIMECYCLE (0, 24/FFEED, TFEED) × CM, i × Ci RSTO, i, sc = CSTO, i, sc × ASTO, OM RSTO, i, pa = CSTO, pa × SSTO, i Auxiliary variables  ASTO, OM = (SSTO, i /CM, i ) ASTO, E =



(SSTO, i /CE, i )



ASTO, OM, pa = (RSTO, i, pa /CM, i )  ASTO, OM, pa = (RSTO, i, pa × CE, i )

Description

Unit

Macronutrient pools Rate of intake

mol

Rate of endogenous secretion Rate of passage

mol/h

OM mass

kg

Energy content Rate of OM flow

MJ kg/h

Rate of energy flow

MJ/h

Macronutrient pools Monomer pools

mol

Rate of endogenous secretion Rate of digestion Rate of passage Rate of absorption

mol/h

OM mass

kg

Energy content

MJ

mol/h

mol/h

Small intestine (SI1) Differential equations dSSI1, i /dt = RSTO,i, pa + RSI1, i, sc − RSI1, i, hy − RSI1, i, pa dSSI1, i /dt = RSTO,i, pa + RSI1, i, hy − RSI1, i, ab − RSI1, i, pa Rate variables RSI1, i, sc = CSI1, i, sc × ASTO, OM, pa + CSI1, i, sc × DMI

RSI1, i, hy b = CSI1, i, hyv /(1 + (CSI1, i, hyk /SSI1, i )N ) RSI1, i, pa = CSI1, pa × SSI1, i RSI1, i, ab = CSI1, i, abv /(1 + CSI1, i, abk /SSI1, i ) Auxiliary variable  ASI1, OM = (SSI1, i /CM, i ) ASI1, E =



(SSI1, i /CE, i )

 ASI1, OM, pa = (RSI1, i, pa /CM, i )  ASI1, E, pa =

(RSI1, i, pa × CE, i )

mol

mol/h mol/h mol/h

Rate of OM flow

kg/h

Rate of energy flow

MJ/h

Macronutrient pools Monomer pools

mol

Rate of endogenous secretion

mol/h

Small intestine (SI2) Differential equations dSSI2, i /dt = RSI1,i, pa + RSI2, i, sc − RSI2, i, hy − RSI2, i, pa dSSI2, i /dt = RSI1, i, pa + RSI2, i, hy − RSI2, i, ab − RSI2, i, pa Rate variables RSI2, i, sc = CSI2, i, sc × ASI1, OM, pa

mol

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Table A.1 (Continued ) Equation format RSI2, i, hy = CSI2, i, hyv /(1 + (ASI2, i, hyk /SSI2, i RSI2, i, pa = DELAYMTR(CSI2, pa × SSI2, i ; DelayTime; Order; InialFlow)c RSI2, i, ab = CSI2, i, abv /(1 + CSI2, i, abk /SSI2, i )

)N )b

Auxiliary variable  ASI2, OM = (SSI2, i /CM, i ) ASI2, E =



(SSI2, i × CE, i )

 (R /C )  SI2, i, pa M, i

ASI2, OM, pa = ASI2, E, pa =

(RSI2, i, pa × CE, i )

ASI2, DF = (SSI2, DDF + SSI2, UDF )/Asii, OM ASI2, i, hyk d = CSI2, i, hyk × (ASI2,DF /CSI2, DF, 0 )3 ; IF [SSI2, DP , SSI2, EP ]; ELSE ASI2, i, hyk = CSI2, i, hyk Large intestine (LI) Differential equations dSLI, i /dt = RSI2,i, pa + RLI, i, sc − RLI, i, hy − RLI, i, pa dsLI, FA /dt = RSI2, FA, pa + RLI, LD, hy − RLI, FA, pa − RMM, FA dSMM /dt = RMM − RMM, pa Rate variables RLI, i, sc = CLI, i, sc × ASI2, OM, pa RLI, i, hy = CLI, i, hyv /(1 + CLI, i, hyk /SLI, i ) RLI, i, pa = CLI, PA, 0 × EXP((ALI, OM − CLI, OM, 0 )/ CLI, pa, kn ) × SLI, i  RMM e = CMM × ( (RLI, i, hy /CM, i ) + RSI2, AA, pa /CM, AA + RSI2, SU, pa /CM, SU ) Rate variables RMM, i = CMM, i × RMM Auxiliary variables  ALI, i, fr = Cfr, i × ( (RLI, i, hy ) + RSI2, i, pa − RMM, i )

 ALI, OM = (SLI, i /CM, i ) + SMM  ALI, E = (SLI, i × CE, i )  ALI, OM, pa = ALI, E, pa =

(RLI, i, pa /CM, i ) + RMM, pa



(RLI, i, pa × CE, i )

Description

Unit

Rate of digestion Rate of passage

mol/h mol/h

Rate of absorption

mol/h

OM mass

kg

Energy content

MJ

Rate of OM flow

kg/h

Rate of energy flow Dietary fibre concentration Affinity contant for digestion

MJ/h

Macronutrient pools Fatty acid pool

mol

Microbial OM pool

kg

Rate of secretion Rate of hydrolysis Rate of passage

mol/h mol/h mol/h

Rate of microbial OM growth

kg/h

Rate of Microbial biosynthesis

mol/h

Fermentation rate

mol/h

OM mass

kg

mol/kg mol

mol

Energy content

MJ

Rate of OM passage Rate of energy passage

kg/h MJ/h

a “TIMECYCLE” is the Powersim statement that controls initiation of feeding interval between feedings and duration of feeding. b The steepness parameter (N) is set at one unless otherwise documented. c “DELAYMTR” is the Powersim statement for a third order material delay—DelayTime set at 1 h. d The affinity constant is variable for proteolysis (otherwise constant)—A SI2, DF is the level of DF in SI2 at time t.

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