Mathematical Biosciences 258 (2014) 101–112

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Mathematical Biosciences

journal homepage: www.elsevier.com/locate/mbs

Digestion modeling in the small intestine: Impact of dietary fiber

M. Taghipoor a,b,∗, G. Barles a, C. Georgelin a, J.R. Licois a, P. Lescoat b,1

a Laboratoire de Mathématiques et Physique Théorique (UMR CNRS 7350), Fédération Denis Poisson (FR CNRS 2964), Université de Tours Faculté des Sciences et

Techniques, Parc de Grandmont, F-37200 Tours, Franceb INRA, UR83 Recherches Avicoles, F-37380 Nouzilly, France

a r t i c l e i n f o

Article history:

Received 23 April 2013

Revised 6 June 2014

Accepted 19 September 2014

Available online 13 October 2014

Keywords:

Modeling of intestinal digestion

Water holding capacity

Soluble dietary fiber

Insoluble dietary fiber

a b s t r a c t

In this work, the modeling of the digestion in the small intestine is developed by investigating specifically

the effects of dietary fiber. As our previous model, this new version takes into account the three main

phenomena of digestion: transit of the bolus, degradation of feedstuffs and absorption through the intestinal

wall. However the two main physiochemical characteristics of dietary fiber, namely viscosity and water

holding capacity, lead us to substantially modify our initial model by emphasizing the role of water and its

intricated dynamics with dry matter in the bolus. Various numerical simulations given by this new model are

qualitatively in agreement with the positive effect of insoluble dietary fiber on the velocity of bolus and on its

degradation all along the small intestine. These simulations reproduce the negative effect of soluble dietary

fiber on digestion as it has been experimentally observed. Although, this model is generic and contains a large

number of parameters but, to the best of our knowledge, it is among the first qualitative dynamical models

of fiber influence on intestinal digestion.

© 2014 Elsevier Inc. All rights reserved.

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. Introduction

Digestion in the small intestine can be described through three

ain phenomena: transit of the bolus along the small intestine, degra-

ation of macromolecules into smaller ones and absorption through

ntestinal wall. Taking into account these phenomena, the authors

ave presented in [17] a generic model of digestion in which the

olus includes only one category of macromolecules (carbohydrates,

roteins or lipids) and water.

However mixing these nutrients influences the digestion process

hrough interactions between molecules. To improve this model, the

ffects of such interactions have to be incorporated. As a first step, we

onsider the effect of dietary fiber because of its significant role on

igestion. It is known that one of the key properties of dietary fiber

s its water holding capacity; this leads us to investigate the role of

ater in the digestion process. To reach this target, dry matter and

ater in each substrate are separately accounted for. Then, the kinetic

f water is modeled by taking into account its correlation with the

inetic of the dry matter.

Unavoidably, this new model is rather complicated. Consequently

o describe and support our modeling approach, we proceed in three

∗ Present address: INRA, UMR1348 Pegase, F-35590 Saint-Gilles, France.

el.: +33 638748589.

E-mail address: masoomeh.taghipoor@rennes.inra.fr (M. Taghipoor).1 Present address: AgroParisTech, Life Science and Health Department, 16, rue Claude

ernard, F-75231, Paris cedex 05, France.

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ttp://dx.doi.org/10.1016/j.mbs.2014.09.011

025-5564/© 2014 Elsevier Inc. All rights reserved.

ain steps mostly written in three sections: firstly, in Section 2, the

equired biological background on soluble and insoluble dietary fiber,

heir main physiochemical characteristics and effects on the diges-

ion are introduced. Next, each biological quantity or effect has to be

ranslated into mathematical notations or equations. In Section 3, a

ist of the main assumptions used to build the mathematical model

re described. Obviously these assumptions can be discussed and it

s important to have a clear and precise formulation of each of the

roposed hypothesis. Finally, the state variables of the model with a

rief biologic description for each variable and the associated mathe-

atical equations are given in Section 4.

The aim of this presentation in three steps is to guide the reader

ll along our modeling process, from the biological background to the

nal mathematical equations descriptions. Our aim is to have a more

eadable article in that way, although it may induce a longer paper.

Once the equations of the model are determined, we performed

umerical simulations whose aim is to check the behavior of the

odel according to the proposed biological hypotheses and/or to

ompare different sets of hypothesis. Specifically, known or assumed

iological mechanisms relating dietary fiber and digestion process

re simulated. In Section 5 we provided several simulations with

omments on the hypothesis. These numerical simulations can be

onsidered as in silico experiments. Of course, since our model is

eneric, it contains a lot of parameters and some of the hypothe-

es are still to be checked (by biological experiments), we are look-

ng for qualitative behavior with very few attentions on quantitative

utputs. This is why no sensitivity on the parameters is checked.

102 M. Taghipoor et al. / Mathematical Biosciences 258 (2014) 101–112

As

Babs

BintAns

And

W

Fs

Fins

Fig. 1. Feedstuffs degradation inside the bolus (Ans , As , Bint , Babs) and water exchanges

(W to other components) conceptual chart. The arrows → and ↔ represent the feed-

stuffs transformations. The pointed arrows represent water exchanges.

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2 The macromolecules are supposed to be the large molecules that cannot penetrate

intestinal wall.3 The nutrients are the very small molecules which can go through the intestinal

wall.

Finally, in Section 6, the model is discussed and perspectives are

proposed.

2. Biological background on water and dietary fiber

Digestion modelling requires the knowledge of the physiochem-

ical properties of macromolecules concerned by this phenomenon

as well as mechanical and biochemical reactions observed for their

degradation.

Dietary fiber (DF) is usually defined as the sum of plant non-starch

polysaccharides and lignin that are not hydrolyzed by the enzymes se-

creted by the non-ruminant digestive system, but that can be partially

digested by microflora in the gut. A main effect of fiber is to regulate

intestinal degradation and absorption of nutrients as well as their

transit along the gut. Physiochemical characteristics of fiber include

viscosity, hydration, fermentability (mostly in the large intestine), ad-

sorption or entrapment of nutrients and bulking effect. Each of these

characteristics affects meaningfully the function of the gastrointesti-

nal tract [19,10]. These characteristics depend on the polysaccharides

chemistry. One way to classify dietary fiber is based on their water

solubility. Insoluble dietary fiber includes cellulose, some hemicellu-

loses and lignin. The other is soluble dietary fiber such as viscous fiber

which includes beta-glucans, pectins, gums and some hemicelluloses

[1,18].

For monogastrics, most available nutrients are degraded and ab-

sorbed in the small intestine. At the beginning of duodenum the bo-

lus consists of partially degraded feedstuffs and water. Once in the

small intestine, mechanical and chemical digestion of feedstuffs make

the nutrients available to the organism. Enzymatic hydrolysis is the

most important chemical reaction in digestion, which takes place in

aqueous solution. Enough water is required for an efficient digestion

process even though water/nutrient ratio is not precisely known. Fur-

thermore, classification of dietary fiber through their water solubility

and the impact of water holding capacity (WHC) of DF on digestion re-

veal the key-role of water on digestion. WHC is defined by B.M. Clearly

and L. Prasky [10] as the ability of fiber source for water absorption

when mixed with water and to hold it within its matrix.

2.1. Soluble dietary fiber

Soluble DF are believed to impact meaningfully digestion and ab-

sorption as well as bolus transport in the small intestine. The main

physiochemical properties of soluble DF are viscosity, water holding

capacity (WHC) and organic compound entrapment [7]. Soluble DF,

because of its high viscosity, is generally associated with slow tran-

sit through the stomach and increasing of the small intestinal transit

time [14]. In the presence of soluble DF, this delay in transit time may

also be related to the jelly consistency of the bolus which causes a

resistance to the propulsive contractions of small intestine [5]. Vis-

cosity and organic compound entrapment capacities of DF, slow down

digestion and absorption of substrates. This effect on digestion is as-

sociated with (i) the reduction of the nutrient diffusion through the

small intestine because of the viscosity and (ii)dilution of bolus com-

pounds with the addition of non-digestible material [12].

2.2. Insoluble dietary fiber

Insoluble fiber does not dissolve in water but retains it in its struc-

ture. In fact, insoluble DF add a bulk to the bolus which speeds up

the motion of food through the gut. Insoluble DF acts primarily in

the large intestine where, due to its WHC, it increases fecal bulk, di-

lutes colonic contents and decreases mouth-to-anus transit time [7].

Wilfart et al. [20] proposed that adding the insoluble DF reduces the

transit time in the small intestine. On the other hand, insoluble DF has

a positive impact on digestion. In fact, the insoluble DF improves the

access of the enzymes to the substrates by enhancing the effects of

ropulsive contractions [4]. Another reason may be the delays in the

astric emptying because of the presence of insoluble DF in the bolus

20,11]. Recent studies have shown that the inclusion of a moderate

evel of dietary fiber improves the dige stibility in chicks [9]. Therefore

o obtain an optimal efficiency in nutrient utilization, Burhalter et al.

3] proposed to increase the ratio of insoluble to soluble DF. Moreover,

he use of insoluble fiber in commercial broiler chicks improves the in-

estine morphological parameters and result in a better performance

ssumed to be connected to more efficient digestion and absorption

rocesses [11]. Two hypotheses are proposed in order to study the

nfluence of insoluble DF on nutrients digestibility in the small in-

estine: (i) insoluble DF increases the retention time in the stomach

hanging the nutrient profile of the bolus at the entry of the small

ntestine which could lead to a higher digestion and absorption. (ii)

hysical characteristics of insoluble DF change the digestion process

ostly through their capacity for absorption of water and nutrients.

hese both hypotheses are either tested in the in silico experiments

cf. Section 5.1) or included in the equations (cf. Section 4).

.3. Water

To have a better understanding of the role of dietary fiber, it is

herefore required to study more precisely the evolution of water

uring digestion in the small intestine. Water is needed for most of

igestion functions, including, in particular, feedstuffs degradation

y enzyme hydrolysis, nutrients absorption as well as bolus transit

y peristaltic waves. We have assumed that total water in the small

ntestine has three principal sources: drunk water, water included in

he foods and water in the endogenous secretions. Bolus composition

n dry matter regulates the quantity of water required for digestion.

hus, the evolution of water amount in the small intestine depends

n other substrates’ kinetics within the bolus.

Fig. 1 illustrates the exchanges between water and feedstuffs com-

artments.

. Key model assumptions

In this section, the key assumptions for the model are presented.

H1: Each component of the bolus (macromolecules,2 partially de-

graded macromolecules, nutrients3 and fiber) is represented

mathematically as a portion of dry matter and a characteristic

proportion of water.

or example, “starch in a bolus” includes both dry starch and water

sed to maintain starch molecules in aqueous solution. The same is

bserved for the “disaccharides in a bolus” and “glucose in a bolus”

M. Taghipoor et al. / Mathematical Biosciences 258 (2014) 101–112 103

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ombining smaller molecules resulting from starch hydrolysis asso-

iated with a specific level of water. In other words, a component C of

olus is represented as Cdm + WC where Cdm denotes the dry matter

f C and WC is the necessary amount of water to maintain it in a so-

ution state. Moreover, the mass of WC is assumed to be proportional

o the mass of Cdm, i.e. equal to c Cdm for some characteristic number

≥ 0 which represents the required amount of water proportion to

aintain C in solubilized phase. Despite the presence of water in the

olus, a little amount of non-solubilized dry matter may be included

n bolus, which is (of course) associated with c = 0.

H2: The bolus is represented by a homogeneous cylinder with the con-

stant length � and a variable radius r(t). The whole lateral surface

of the cylinder is assumed to be in contact with the intestinal wall.

Including insoluble DF transforms this homogeneous bolus into a

heterogeneous one by modifying the concentrations of feedstuffs

and their degraded forms.

bolus in the small intestine is a viscous solution of dry matter and

ater: its volume is assumed to be very close to the volume only

ccupied by the water in the bolus. In other words, our assumption

s that DM does not fill any meaningful volume (e.g. solubilized sugar

nd the water in which it is solubilized do fill the same volume as

he water alone). The variation of V(t) is therefore a consequence of

ll the degradation and absorption phenomena, and in particular the

bsorption of water. Once the volume V(t) is known, one can compute

he radius of the bolus at each time through the formula r(t) =√

V(t)π�

.

H3: Digestion in the small intestine is due to volumic and surfacic

transformations. Volumic degradation is the enzymatic hydrolysis

of bolus components by pancreatic and exogenous enzymes inside

the bolus while the surfacic one is the degradation by brush border

enzymes on the internal wall of small intestine.

ome additional facts should be pointed out. First the volumic degra-

ation is thought as following a law of mass action (therefore depend-

ng on the concentration of the macromolecules) and since the space

ccupied by insoluble dietary fiber is unavailable for other macro-

olecules, adding insoluble DF enhances the degradation by increas-

ng the concentration of nutrients in the homogeneous part of bolus.

Concerning the water, it has different effects on the degradation:

n one hand, increasing the ratio water/dry matter dilutes the bolus

nd decreases the volumic degradation. Indeed, if [s] denotes the

ubstrate concentration in the bolus (in terms of mass ratio), the

olumic transformation is introduced in [17] for [s] as

d[s]

dt= kvol[s]V(t)

here kvol denotes the degradation rate of the substrate by unit of vol-

me. This reaction being proportional to the mass ratio of concerned

ubstrates, it is clear that additional water decreases the degradation

ate.

On the other hand, water facilitates the contact of the macro-

olecules with the brush border enzymes, enhancing the surfacic

egradation. Indeed, its surfacic transformation in a homogeneous

olus has been represented in [17] as

d[s]

dt= ksurf [s][W]S(t).

In this equation, [W] denotes the whole bolus mass ratio of the

ater and, since S(t) = 2π r(t)� is the area of the cylinder, the product

s]S(t) can be seen as the area occupied by s on the boundary of the

ylinder. Finally ksurf is the degradation rate of this substrate by the

nit of surface. The assumption in this equation is that water enhances

his surfacic degradation through the term [W].

H4: Bolus movement along the small intestine is due to peristaltic

waves. The efficiency of these waves is proportional to the radius

of bolus r(t), and inversely proportional to the distance from the

pylorus x(t).

he following equation, introduced in [17], represents the transport

f the bolus through intestinal lumen

d2x

dt2= τ

(1 − c−1 dx

dt

)c0 + c1r

a + bx− Kvisco

[W]

dx

dt. (1)

The movement of bolus as described by this equation is due to a ho-

ogenized acceleration caused by an average effect of the peristaltic

aves. This averaging effect is described by the constant τ and we re-

er to [16] for a rigorous mathematical justification. This acceleration

s reduced by a viscosity coefficient (Kvisco[W]

), where [W] represents the

ater mass proportion. The term c represents the mean velocity of

eristaltic waves [2].

H5: Water in the bolus which is not held by the macromolecules and di-

etary fiber (through their water holding capacity) is called “avail-

able water”. It decreases the viscosity of the bolus and facilitates

its movement. Due to osmotic type equilibrium, the concentration

of this “available water” tends to reach a fixed ratio.

he “available water” provides the term [W] in the transport equation

bove: it is clear in this equation that increasing “available water”

ecreases the viscosity coefficient and facilitates its motion through

ntestinal lumen. The difference with [17] is that this available water

ill now vary in a more important way because of the exchanges of

ater in the different degradation steps and the absorption.

H6: Dietary fiber modifies the bolus evolution through its WHC by

holding water in its matrix, and therefore changing the volume

of the bolus. Soluble DF increases the viscosity of bolus, thus de-

creases the efficiency of peristaltic waves.

y their WHC, dietary fiber holds a meaningful quantity of water in

he bolus which, of course, is not part of the available water. Since

ietary fiber is not degraded, the volume and the radius of the bolus

end to be higher. The impact of soluble dietary fiber as described in

ection 1 is mainly due to the change of the consistence of the bolus,

y making it more jelly. It implies then the decreases of efficiency of

eristaltic waves.

. Model equations

A general framework for modelling the digestion in the small in-

estine has been presented in [17]. In this section, we describe the

odifications performed to take into account the effects of dietary

ber (see Section 2) (i) on the mechanistic properties of bolus in pres-

nce of soluble DF (ii) on the heterogeneity and on the WHC property

ue to the presence of insoluble DF.

.1. Physiological aspects and bolus composition

Different steps of mechanical and chemical transformations are

etailed below. The composition of the bolus (model variables) and

heir interactions are presented.

• Non degradable substrate And: The quantity And represents the

mass of macromolecules which is not degradable by endogenous

enzymes of the digestive tract.• Non solubilized substrate Ans: The quantity Ans represents the

mass of macromolecules which is not accessible to enzymatic hy-

drolysis. In presence of a sufficient quantity of water, Ans is trans-

formed into As.• Solubilized substrate Adm

s : The quantity Adms is the mass of dry

substrate in solution state. It is called solubilized substrate and it is

assumed that one unit of Adms requires Ws units of water to remain

solubilized. Recalling [H1], Ws represents the required mass of

104 M. Taghipoor et al. / Mathematical Biosciences 258 (2014) 101–112

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water to solubilize Adms . This quantity depends on the properties

of each macromolecule. The mass of Ws is assumed to be equal to

the mass of αAdms where α represents the ratio of water associated

with Adms . For simplification purposes, the mix of Adm

s and water is

represented by As .• Intermediate substrate Bdm

int: The quantity Bdm

intis the mass of dry

intermediate substrate obtained from the degradation of As by

volumic transformation [H3]. It represents the intermediate sub-

strate which cannot be absorbed by the intestinal wall before an

additional surfacic degradation. It is solubilized and Wint repre-

sents the required amount of water to maintain solubilization.

The mass of Wint is assumed to be equal to the mass of βBdmint

where β represents the ratio of water associated with Bdmint

. For

Bint = Bdmint

+ Wint , volumic transformation is represented as

As + enzymes → Bint + (Ws − Wint) (Reaction 1)

Depending on the values of Ws and Wint , the amount of Ws − Wint

of water can be released or held in the bolus.• Absorbable nutrients Bdm

abs: The quantity Bdm

absis the mass of dry

absorbable nutrients obtained from surfacic reactions (cf. [H3]).

For Babs = Bdmabs

+ Wabs, the surfacic transformation is defined as

As + enzymes → Babs + (Ws − Wabs) (Reaction 2)

Bint + enzymes → Babs + (Wint − Wabs) (Reaction 3)

Wabs is the required amount of water to maintain solubilization

and its mass is assumed to be equal to γ Bdmabs

where γ represents

the ratio of water associated with the Bdmabs

.

To the best of our knowledge, the values of α, β , γ are not available in

the literature and we make numerical experiments using our model

to describe the obtained behaviors in the different cases α < β < γor α > β > γ or α = β = γ .

• Soluble and insoluble dietary fiber: Fdmsol

and Fdminsol

represent the

dry matter of soluble and non-soluble dietary fiber respectively.

The main property of dietary fiber presented in Section 1 and

hypothesis [H5] is its water holding capacity.

Fsol = Wsol + Fdmsol

Finsol = Winsol + Fdminsol

where the mass of Wsol (Winsol) is assumed to be equal to

λsFdmsol

(λiFdminsol

) for λs and λi which represent the ratio of water as-

sociated with Fdmsol

and Fdminsol

respectively. As described in Section 1,

DF is not degradable by endogenous enzymes of the small intes-

tine.

The following diagram (1) shows the different transformations

inside the bolus.• Dry matter: Total amount of dry matter substrate in the bolus is

therefore

DM = And + Ans + Adms + Bdm

int + Bdmabs + Fdm

sol + Fdminsol.

• Water: The impact of dietary fiber on digestion is closely linked to

their WHC capacity. Though, water evolution in the bolus has to

be described to understand effects of DF on digestion.

The total water Wtot in the bolus comes from three main sources

(i) Wf eed: Water incorporated naturally in feedstuffs (e.g.: 1 g of

wheat contains 12% of water). The amount of Wf eed coming

out of the stomach is assumed to be proportional to the

ingested dry matter DM, Wf eed = Kf eed × DM.

(ii) Wsec: Water included in the endogenous secretions of saliva

and stomach which is also assumed to be proportional to

the ingested dry matter, Wsec = Ksec × DM.

(iii) Wdrink: Drunk water is assumed to be independent to the

quantity of DM.

o

Total water included in the bolus is therefore defined as the sum

of Wf eed, Wsec and Wdrink

Wtot = Wf eed + Wsec + Wdrink. (2)

A first part of this total water is used for the solubilization of

the different macromolecules and nutrients (Ws + Wint + Wabs), a

second part is bound to the fiber (Wsol + Winsol) and the remaining

part is the “available water”, W , as presented in [H5]. The evolution

of W(t) at any time t can be written as

W(t) = Wtot(t)− Ws(t)− Wint(t)− Wabs(t)− Wsol(t)− Winsol(t) .

(3)

While Wsol, Winsol are fixed, Ws, Wint, Wabs vary because of degra-

dation and absorption phenomena. The evolution of available wa-

ter W(t) is mainly due to two causes: the equilibrium related to

osmotic pressure (which can be mostly seen as an absorption phe-

nomena) and the balance of water due to degradation and absorp-

tion of nutrients.

Eq. (2) represents the initial value of Wtot , while Eq. (3) allows

to see its evolution (Wtot(t)) as a function of (i) the variation of

available water W as well as (ii) the water bound to different

types of bolus components, for 0 ≤ t ≤ Tf , where Tf represents the

required time for digestion in the small intestine.• Mass of bolus M: The total mass of bolus, M, is given by M =

DM + Wtot .• Volume of bolus V: To define the bolus volume, as explained in

[H2], the volume of each substrate in solution is assumed to be the

same as the volume filled by water associated with that substrate,

i.e. the volume of As is equal to Ws/ρw, where ρw = 1 is the water

density. The volume of bolus is therefore represented as

V(t) = Wtot(t)

ρw= π r2(t)�.

Since the length of the bolus is assumed to be fixed, the volume

evolution over time leads to compute the radius r(t) of bolus at

each time. Consequently its surface is written as S(t) = 2π r(t)�.

.2. Effects of soluble dietary fiber

As described in Section 2, soluble dietary fiber, because of its jelly

tructure, decreases the efficiency of the contractions caused by peri-

taltic waves [8,6,5], thus increases transit time of bolus. Moreover,

dding these fiber to the bolus composition decreases the degrada-

ion and absorption rate. In order to incorporate these features in the

odelling we introduce the notion of “efficient radius” associated to

notion of “efficient surface”.

efinition 4.1. The efficient radius of bolus is defined as

sol =√

(Wtot − Wsol)

2π�.

n the same way, the efficient surface of bolus is described as

sol = 2π rsol�.

When bolus contains soluble DF then rsol < r. Using both this ra-

ius (rsol) and the associated area (Ssol) in the model equations, de-

reases the acceleration of the bolus and all the surfacic effects. There-

ore this notion allows to describe (at least qualitatively) the effects

f soluble DF.

.3. Effects of insoluble dietary fiber

One of the hypotheses in the first model of digestion in [17] is

he bolus homogeneity. The mass concentration of each component

f the bolus is assumed to be its mass divided by the total mass M

f the bolus. To model the digestion in presence of insoluble DF, new

M. Taghipoor et al. / Mathematical Biosciences 258 (2014) 101–112 105

r

Fig. 2. Conceptual representation of insoluble fiber distribution in the bolus as as-

sumed in the model. The apparent volume Vapp (see Definition 4.3) is the white part of

the cylinder. r is the radius of bolus.

n

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otions are defined because of heterogeneity of bolus as described in

H2]. The volume filled by insoluble DF (the mix of Fdminsol

and Winsol), is

ssumed to be unavailable to the macromolecules of feedstuffs in the

olus. In the digestion model, this hypothesis is taken into account by

he following definition.

efinition 4.2. The apparent concentration of different substrates in the

olus is represented as

Adms ] = Adm

s

M − Finsol

, [Bdmint ] = Bdm

int

M − Finsol

, [Bdmabs] = Bdm

abs

M − Finsol

Ans] = Ans

M − Finsol

, [W] = W

M − Finsol

, [And] = And

M − Finsol

The degradation of macromolecules As and Bint as well as the ab-

orption of the nutrients Babs are affected by WHC property of the

nsoluble DF through this definition. Fig. 2 shows the regions of bolus

hich are filled by insoluble DF and therefore unreachable by the

acromolecules and nutrients.

Integrating the insoluble DF in the bolus changes also the region

eachable to volumic degradation (see Fig. 2). According to hypothesis

H2], Winsol is the volume filled by insoluble fiber.

efinition 4.3. The apparent volume of bolus Vapp is therefore defined

s

app = Wtot − Winsol.

The above considerations are taken into account in the following

teps of digestion described below.

.4. Transport of bolus

The averaged equation of transport of bolus introduced in [16]

eads

d2x(t)

dt2= τ

(1 − c−1 dx(t)

dt

)c0 + c1r(t)

a + bx(t)− Kvisco

[W(t)]

dx

dt,

here τ is the mean effect of the pulses by unit of time, c is mean

elocity of peristaltic waves, and x(t) is the position of bolus in the

mall intestine. The constants a, b, c0, c1 and Kvisco are the model

arameters and might be estimated from biological data [13]. Taking

nto account the properties of dietary fiber, this equation changes to

d2x(t)

dt2= τ

(1 − c−1 dx(t)

dt

)c0 + c1rsol(t)

a + bx(t)− Kvisco

[W(t)]

dx

dt. (4)

he bolus movement described by this equation depends on its po-

ition in the small intestine and on its efficient radius. Moreover, the

cceleration is slowed down by a viscosity term which depends on

he available water.

.5. Volumic transformation

Volumic transformation presented by (Reaction 1) in Section 4.1

s the degradation of Adms due to pancreatic and exogenous enzymes

esulting in production of Bdmint

.

Evolution of Adms by this transformation is represented by

dAdms

dt= −kvol(x)[A

dms ]Vapp

here kvol(x) takes into account the enzymatic activity which is a

unction of bolus position at each time t. Indeed, it has been assumed

hat the endogenous enzymes are in excess in the small intestine lu-

en. The term [Adms ]Vapp describes the dependence of volumic degra-

ation on the concentration of Adms at each unit of apparent volume i.e.

he volume filled by the insoluble DF is not accessible to the enzymes

nd macromolecules.

Consequently, integrating the insoluble fiber in the bolus influ-

nces the volumic transformation by increasing the substrates con-

entration via the Definition 4.2 and by changing the volume and

sing Vapp introduced in Definition 4.3.

As described in Section 4.4, integrating the soluble fiber modi-

es the velocity of bolus along the small intestine and therefore the

istance traveled at each time x(t). Consequently, it influences the

olumic degradation through the function kvol(x(t)).The volumic production of intermediate substrate Bdm

intis the result

f degradation of Adms

dBdmint

dt= kvol(x)[A

dms ]Vapp.

hese reactions take place in a solution and each degraded unit of Adms

auses the release of water Ws = αAdms , which is then partially used to

aintain the produced Bdmint

in a solubilized state through Wint = βBdmint

.

ccording to (Reaction 5), the result of volumic transformation is

he consumption or release of “available water” W . Thereby, volumic

volution of water is represented as

dms + Ws + enzymes → Bdm

int + Wint + (Ws − Wint) (Reaction 5)

dW

dt= k̃vol(x)[A

dms ]Vapp , (5)

here k̃vol = (α − β)kvol.

Soluble fiber can be hydrolyzed by exogenous enzymes eexo

cellulase, hemicellulases, . . . ) ingested by food

sol + eexo → Bint + (Wsol − Wint). (Reaction 4)

Hydrolysis of Fsol by exogenous enzymes follows the same evolu-

ion as the volumic transformation of Adms

dFdmsol

dt= −ks(eexo, x)[Fdm

sol ]Vapp

here [Fdmsol

] = Fsol/(M − Finsol). The function ks(eexo, x) = ks × eexo ×h̃(x) represents the degradation rate of Fsol, which depends on the

xogenous enzyme activity along the small intestine p̃h(x) and the

oncentration of exogenous enzymes.

This reaction produces the intermediate substrate Bdmint

dBdmint

dt= ks(eexo, x)[Fdm

sol ]Vapp.

he amount (Wsol − Wint) is released by (Reaction 4) and modifies the

volution of water

dW

dt= · · · + k̃s(eexo, x)[Fdm

sol ]Vapp (6)

here k̃s = (λs − β)ks(eexo, x).

.6. Surfacic transformation

Surfacic degradation is the last step of transformation of macro-

olecules in the small intestine. The produced nutrients by this

106 M. Taghipoor et al. / Mathematical Biosciences 258 (2014) 101–112

4

c

T

w

w

4

a

t

w

p

d

e

f

T

o

i

t

T

d

5

o

b

d

v

βp

T

c

s

p

p

a

w

a

degradation are then absorbed through intestinal wall. Surfacic degra-

dation depends on the fraction of As on the surface of the bolus rep-

resented by [As]Ssol therefore

dAdms

dt= −ksurf [As][W]Ssol

where ksurf stands for the rate of surfacic degradation of As and the

efficient surface Ssol has been defined by Definition 4.1. Moreover, it is

assumed that the brush-border enzymes are always in excess in the

small intestine. Surfacic degradation of Bdmint

follows the same process

as for Adms . Therefore for [Bint] defined by Definition 4.2, we have

dBdmint

dt= −k̃surf [Bint][W]Ssol.

where ksurf stands for the rate of surfacic degradation of Bint . Evolution

of water in the bolus is influenced by the surfacic degradation, i.e. the

quantity of water consumed (or released) by (Reaction 2). Therefore

the surfacic evolution of water is

dW

dt= · · · + ((β − γ )k̃surf [Bint] + (α − γ )ksurf [As])[W]Ssol. (7)

4.7. The equilibrium between As and Ans

Modifications of feedstuffs in the stomach by the enzymes and

water change most of the Ans into As = Adms + Ws and makes them

accessible to intestinal enzymes.

However, for some feedstuffs, the bolus may contain Ans at the

beginning of small intestine. In this case, the digestion of bolus con-

tains also the transformation of Ans into As. Mixing with bile acid for

lipids and producing the micelles, denaturing for the proteins and

adding water and solubilization for the dry starch are examples of the

transformation of Ans into As in the small intestine.

The solubilization of Ans which takes place in the presence of

enough quantity of W and results in the production of As, is a phe-

nomenon taken into account in the model. Solubilization is reversible

and lack of water may cause production of Ans releasing W in the

bolus.

Thereby, the balance is assumed to be reached when

As = μ([W])Ans

for μ which is an increasing function of [W]. If kequi stands for the rate

of turning back to equilibrium then the dynamical equilibrium may

be defined

dAns

dt= −kequi(μ([W])Ans − As) (8)

and therefore

dAdms

dt= kequi(μ([W])Ans − As).

The variation of water quantity caused by the equilibrium may be

represented as

dW

dt= · · · + αkequi(μ([W])Ans − As). (9)

4.8. Pancreatic and biliary secretions

Pancreatic and biliary secretions consist of a solution of nutrients

and enzymes which does not contain available water W . In fact, water

included in this solution is assumed to be associated with nutrients

and enzymes to keep them solubilized. Modelling details on these

secretions could be seen in [17]. Adding dietary fiber increases the

quantity of pancreatic secretions. However this point is not yet in-

cluded in the model. j

.9. Absorption through intestinal wall

Absorption of nutrients through intestinal wall depends on their

oncentration on the inner surface of intestinal wall

dBdmabs

dt= · · · − kabs[B

dmabs]Ssol

he passage of nutrient through intestinal wall releases the associated

ater, thus

dW

dt= · · · + γ kabs[B

dmabs]Ssol (10)

here kabs represents the rate of absorption through intestinal wall.

.10. Water equilibrium

Water equilibrium was already taken into account in [17]. The

ssumption was that [W] tends to reach a fixed ratio (10%), suggesting

he equation

d[W]

dt= −kw([W] − 0.1) (11)

here [W] = W(t)/M(t), M(t) representing the bolus mass. The su-

erposition of Eqs. (5), (6), (7) and (9)–(11) provides the equation

escribing the evolution of W along the small intestine.

The evolution of bolus mass is represented as

dM

dt= (α + 1)

dAdms

dt+ (β + 1)

dBdmint

dt+ (γ + 1)

dBdmabs

dt

+ (λsol + 1)dFdm

sol

dt+ dAns

dt+ dW

dt,

ach term of the above equation is replaced by its expression, there-

ore we obtain

dM

dt= M

M − W

( − kw(W − 0.1M)− kabs[Bdmabs]Ssol

). (12)

he variation of bolus volume depends on the absorption or secretion

f “available water” and endogenous secretions in the small intestine

.e.

dV

dt= dW

dt, (13)

herefore

dV

dt= −kw(W − 0.1M)+ secretions.

able 9 presents all the equations of the digestion model when adding

ietary fiber.

. Numerical simulations

A thorough examination of the effects of the different parameters

f the model on transport, degradation and absorption is carried out

y Scilab software. In these in silico experiments, the presence of

ietary fiber Fsol and Finsol in the bolus, the variations of the initial

alues of bolus and the sensitivity of the model on the parameters α,

and γ (cf. Section 4.1) are investigated. Tables 6 and 7 show the

arameters’ values used in the numerical simulations of the model.

able 8 shows the default initial conditions, however they have been

hanged to test different compositions of the bolus at the entry of the

mall intestine.

To the best of our knowledge, most of the actual values of the

arameters are unknown, even if the order of magnitude of these

arameters can be guessed. The only well-known facts are the input

nd output characteristic of the bolus and its kinetic. For example,

e have calibrated the parameters of the transport equation so that

standard bolus travels the 17 m from the pylorus till the ileo-caecal

unction in 7 h.

M. Taghipoor et al. / Mathematical Biosciences 258 (2014) 101–112 107

a

t

W

d

5

d

c

s

i

5

d

t

o

t

Table 1

The proportion of absorbed nutrients to DM does not vary significantly for different

scenarios of solubilization in the bolus at the entry of duodenum.

x = 0 End of the small intestine x = 17

Adms × DM−1 (%) Ans × DM−1 (%) Absorbed dry nutrients to DM ratio (%)

0 85 56

42 42 57

85 0 58

r

A

The mass of the bolus (M) at the entry of the small intestine is

ssumed to be fixed in all our following experiments. The bolus at

he entry of the small intestine contains And, Ans, As, Bint , Babs and W .

hen studying the influence of dietary fiber on digestion, the non

egradable substrate And is replaced by Fsol or Finsol.

.1. Influence of dietary fiber on intestinal absorption

As described in Section 1, the positive effect of insoluble DF in

igestion may be due to two main reasons: (i) modification of the

omposition of bolus due to the increase of retention time in the

tomach, (ii) modification of the bolus physical characteristics.

The effects of these two cases on the digestion model are studied

n this section.

.1.1. Influence of the modification of the bolus in the stomach

Including insoluble DF in the bolus delays gastric emptying. The

irect effect of this phenomenon is to increase the solubilization in

he bolus and to start partially the degradation. We studied the effects

f such a change in the initial conditions for our model. To this aim,

wo numerical experiments are carried out: (a) the increase in the

0

5

10

15

20

25

30

35

40

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Time (hour)

Mas

s (g

)

(a) Adms = 36, Ans = 0

0

5

10

15

20

25

30

35

40

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Time (hour)

Mas

s (g

)

(c) Adms = 0, Ans = 36

Fig. 3. In silico experiments of the ratio Adms − Ans . The equilibrium A

atio of As to Ans and (b) the increase in the ratio of Bint to As when

ns = 0.

(a) Our first experiment consists in increasing the ratio of As to

Ans in the bolus at the entry of the small intestine as a result of

increasing the retention time in the stomach. The value of ab-

sorbed dry nutrients at the end of the small intestine does not

vary meaningfully. Table 1 shows the absorbed dry nutrients

to dry matter (DM) ratio at the end of ileum x = 17 for different

Adms to DM ratio at the beginning of the small intestine.

Despite the variations in the ratio of As to Ans, the equilibrium

between As and Ans defined by Eq. (8) is reached quickly (see

Fig. 3). The sensitivity analysis shows also that the value of

0

5

10

15

20

25

30

35

40

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Time (hour)

Mas

s (g

)

(b) Adms = 18, Ans = 18

dms − Ans is reached quickly for different initial values of Adm

s .

108 M. Taghipoor et al. / Mathematical Biosciences 258 (2014) 101–112

Table 2

The relation between the absorbed dry nutrients at the end of digestion and the differ-

ent initial values of Bdmint

.

x = 0 End of the small intestine x = 17

Bdmint

× DM−1 (%) As × DM−1 (%) Absorbed dry nutrients to DM ratio (%)

0 85 57

42 42 61

85 0 64

Table 3

Intestinal transit time for the different quantities of F = Fsol or Finsol in

the bolus at the entry of duodenum

x = 0 Intestinal transit time (h)

F × DM−1 (%) Bolus containing Finsol Bolus containing Fsol

0 5 5

2 4.7 5.4

7 4.3 5.9

11 4 6.5

14 3.9 6.7

Table 4

Transit time and absorbed dry nutrients at the end of the small

intestine depend on the values of α, β and γ .

Absorbed dry

nutrients to

DM ratio (%)

Retention time

in the small

intestine (h)

α = 1, β = 2, γ = 3 45 5.2

α = 2, β = 2, γ = 2 50 4.9

α = 3, β = 2, γ = 1 54 4.6

w

v

i

r

b

t

5

g

i

a

u

B

f

5

i

a

t

o

a

absorbed dry nutrients is almost independent to the variations

in the ratio of As to Ans. However, these results depend on the

choice of equilibrium rate kequi, a small value of kequi means

a small solubilization rate. It will be interesting to study the

sensitivity of the model to the value of kequi.

(b) In the second experiment, the modification in the stomach is

assumed to result in the transformation of all Ans in the bolus

into As and additionally the production of Bint . Variations in

the ratio of As to Bint inside the bolus at the entry of the small

intestine are tested. Table 2 shows the variation of absorbed

dry nutrients at the end of the small intestine.

Numerical results show the increase in absorbed dry nutrients

when the ratio of Bint to As is increased.

5.1.2. Direct effect of DF on the function of the small intestine

Besides the modification of the bolus in the stomach, presence of

insoluble DF changes also the physiochemical characteristics of bolus

as described in Section 4.

To observe the effect of DF in the model of digestion, values of

insoluble and soluble DF was increased from 1 g to 5 g in a bolus of

120 g in separated in silico experiments.

Fig. 4 shows that the presence of insoluble DF promotes intestinal

absorption, however this increase in absorbed dry nutrients is not

meaningful. The results in this figure, show that the increasing of the

value of soluble DF decreases the quantity of absorbed dry nutrients

and increases the final total mass.

Therefore, numerical simulation shows that the positive effect of

insoluble DF on the amount of absorbed dry nutrients is mainly due

to the modification of the bolus in the stomach. However, the di-

rect effect of the insoluble DF on digestion in the small intestine as

described in Fig. 4 may not be neglected.

5.1.3. Time of intestinal transit in the presence of DF

Numerical results of transit time in presence of DF are presented

in Table 3.

These in silico results show that integrating insoluble DF in the

bolus decreases the time of intestinal transit from 5 h for a bolus

of 120 g without insoluble DF to 3.9 h for a bolus of the same mass

Fig. 4. Change in the final mass of bolus and absorbed dry nutrients fo

hich contains 5 g of DF. These results are consistent with published

alues. The experiments done by Wilfart et al. [20] have shown that

ncreasing dietary fiber content reduced or tended to reduce the mean

etention time in the small intestine.

These numerical results show that integrating soluble DF in the

olus increases the intestinal digestion time from 5 h to 6.7 h illus-

rating the effect of viscosity due to soluble DF on transit time.

.2. Water associated to dry matter

To study the influence of the quantity of associated water on di-

estion, variations of the values of α, β and γ have been tested. Their

nfluence on digestion and specifically on the absorbed dry nutrients

nd As − Ans equilibrium are represented in Table 4. Two main sim-

lations were carried out: an uniform water content for As, Bint and

abs i.e. α = β = γ and a non uniform one.

The objective of these experiments is to understand how the dif-

erent values of α, β and γ influence the digestion in our model.

.2.1. Uniform water content for As, Bint and Babs

The value of α = β = γ varied from 1 to 4 in the model presented

n section 4. Our objective is to observe its effect on the value of

bsorbed dry nutrients as well as on the final mass of bolus.

Numerical results presented in Fig. 5 show the negative effect of

his increase on the absorbed dry nutrients. Increasing the quantity

f water (α, β and γ ) associated with the dry feedstuffs (Adms , Bm

int

nd Bdmabs

) in our model, dilutes the bolus and decreases the volumic

r different amount of DF in the bolus at the entry of duodenum.

M. Taghipoor et al. / Mathematical Biosciences 258 (2014) 101–112 109

Fig. 5. Dependence of the absorbed dry nutrients and the final mass of bolus at the

end of the small intestine on the value of α, β and γ .

d

w

t

b

b

i

t

a

a

t

b

i

5

a

e

c

t

m

s

k

egradation, it decreases also the quantity of dry nutrients in contact

ith the internal surface of the bolus.

These results seem to be consistent with the in vivo data, in fact,

he more water is presented in the bolus, the less (pancreatic and

rush border) enzymes and molecules are in contact.

Fig. 6. The evolution of the equilibrium Adms − Ans de

The effect of these variations on the Adms − Ans equilibrium defined

y Eq. (8) was studied. The results are shown in Fig. 6. The equilibrium

s almost reached in the four experiments. However, the model shows

hat a lower value required for solubilization of substrates results in

rapid solubilization of Ans. In fact, when that values of α, β and γre small, the value of available water is higher, thus it can be used

o solubilize Ans. A significant production of Ans is observed in Fig. 6d

ecause of the lack of the available water at the beginning of the small

ntestine.

.2.2. Non-uniform water content for As, Bint and Babs

In the second experiment, the quantity of absorbed dry nutrients

t the end of the small intestine and the numerical results of As − Ans

quilibrium for different values of α, β and γ were observed. The

hoice of the values of As, Bint and Babs is based on the hypothesis that

he value of β is always between the values of α and γ because of the

olecule size of Bdmint

.

Even if the longest transit time was observed for α < β < γ , it

hows the lowest level of dry absorbed nutrients. This stresses the

ey-role of “available water” W on digestion.

pends strongly on the quantity of α, β and γ .

110 M. Taghipoor et al. / Mathematical Biosciences 258 (2014) 101–112

Fig. 7. The equilibrium Adms − Ans for different quantity of α, β and γ .

Table 5

Dependence of absorbed dry nutrients to the water ratio. DM stands for the amount of

dry matter in the bolus at the entry of bolus.

Amount of water

at x = 0 inside

bolus (%)

Ratio of absorbed

dry nutrients at

x = 17 (%)

Mass of Absorbed

dry nutrients (g)

Transit

time (h)

50 50 25 5.7

60 35 26 5.2

66 28 27 4.5

a

i

v

n

d

t

(

6

i

n

r

e

f

t

p

i

i

p

v

i

d

g

b

t

d

b

c

t

w

l

s

o

s

t

o

As − Ans equilibrium is observed in two cases:

(i) Firstly, it was assumed that α > β > γ . There is a little pro-

duction of Ans from As (desolubilization) because of the lack

of water at the entry of the small intestine followed by the

degradation of As.

(ii) Secondly, it was assumed that α < β < γ . There is a rapid sol-

ubilization of Ans followed by degradation of As into Bint . The

released water by degradation of As is not sufficient to main-

tain the solution state of Bint since α < β . The required water is

provided by the transformation of As into Ans as it can be seen

in Fig. 7.

5.3. The variation of the water ratio inside the bolus

We are interested by the change in the ratio of water inside the

bolus and its influence on the absorption in the small intestine. For

the sake of simplicity, it was assumed that Wf eed = Wsec = 0.

In order to study this effect, the value of dry matter in the bolus is

assumed to be fixed at 42 g while the value of water increases in each

experiment.

In the numerical simulations, the ratio of water included in the

bolus represented 55%, 60% and 66% of bolus. The ratio of absorbed

dry nutrients to the total absorption (water and nutrients) is collected.

Results are presented in Table 5.

The numerical results of Table 5 shows that increasing the value

of water in the bolus decreases the ratio of absorbed dry nutrient to the

total absorbed matter (water + dry nutrients), even though the value of

absorbed dry nutrients increases.

In fact, in our model, at the end of each experiment the value

of Wt is approximately 55% of the total mass of bolus and this

equilibrium is achieved almost quickly (because of the choice of kw).

Therefore, increasing the value of water in this model does not have

meaningful effect on the final absorption and the slight increase

n the absorbed mass of dry nutrients is due to the change of the

olume of bolus in each experiment which promotes the access of

utrients to intestinal wall for absorption, although this increase

ilutes the bolus and decreases the volumic degradation. Here again,

his is the direct result the choice of the parameters (rate of surfacic

ksurf ) and volumic (kvol) degradation).

. Discussion

This work deepened the modelling of the digestion in the small

ntestine started in [17]. The objective was to obtain a more mecha-

istic model of digestion process by including new phenomena and

evisiting the others: DM and water are considered separately, water

volution is influenced by the degradation of other molecules, the ef-

ects of DF on the digestion are taken into account which is a first step

oward a non-homogeneous model with different types of feedstuffs.

The advantages and the drawbacks of this model as well as the

erspectives are outlined below.

One of the main characteristics of this model remains its gener-

city. The main generic phenomena of digestion have been explicitly

ncluded in the model while the ones which required the specific

roperties of feedstuffs’ molecules (effects on gastric emptying, on

iscosity, . . . etc.) have been ignored or implicitly taken into account

n the model parameters. On the one hand, the different steps of

egradation, the effects of physical characteristic of bolus on its di-

estion, the interaction between DF and feedstuffs molecules have

een considered. Moreover, some other phenomena like the separa-

ion between the enzyme activities of different feedstuffs’ molecules,

ependence of the enzyme activity with respect to the dilution of

olus, the different substrates density, the impact of DF on initial

ondition have been ignored.

To our point of view, a more realistic model was obtained by in-

egrating these new phenomena. Specifically, WHC of dietary fiber

hich in turn interferes the digestion of other feedstuffs molecules,

eads us to introduce the separation between DM and water and con-

ider all the impacts of water on digestion dynamic.

However, some other phenomena are still ignored either because

f the lack of information concerning their effects or because of their

upposed limited impact on digestion at this integrated scale. In-

eractions between different categories of feedstuffs and definition

f specific enzyme activity for different bolus contents would be a

M. Taghipoor et al. / Mathematical Biosciences 258 (2014) 101–112 111

Table 6

Parameters used for

transport parameters.

Notion Value

kvisco 28

a 2.72

b 1.23

c0 0.01

c1 25

c 7.2

τ 300

Table 7

The parameters used for

degradation equations.

Notion Value

kvol 20

ksurf 10

k̃surf 10

kabs 3500

ke log(4/3)

ks 20

kw 0.5

kequi 3

α 3

β 2

γ 1

λsol 4

λinsol 6

kf eed 0.4

kdrink 2

ksec 0.4

� 0.1

Table 8

Initial conditions.

Notion Value

x0 0

v0 15

z0 0.1

Adms 6

Bdmint

0

Bdmabs

0

Ans 31

And 0

Fsol 0

Finsol 0

v

m

o

(

h

t

s

l

o

w

t

c

i

o

c

e

b

d

Table 9

System of equations for model of digestion in presence of dietary fiber. The

secretion terms are described in [17]. In this new version of the model, the

water associated to each dry matter contained in pancreatic secretions should

be taken into account in the water equation (8).

1. d2 xdt2 = τ

(1 − c−1 dx

dt

) c0+c1 rsol

a+bx− Kvisco

[W]dxdt

2. dAns

dt= −kequi

(μ([W])Ans − As

)3.

dFdmsol

dt= −kseexop̃h(x)[Fdm

sol]Vapp

4.dAdm

s

dt= −kvol(x)[A

dms ]Vapp − ksurf [As][W]Ssol + kequi

(μ([W])Ans − As

)+ secretions

5.dBdm

int

dt= kvol(t)[A

dms ]Vapp + kseexop̃h(x)[Fdm

sol]Vapp − k̃surf [Bint][W]Ssol

+ secretions

6.dBdm

abs

dt= (k̃surf [Bint] + ksurf [As])[W]Ssol − kabs[Bdm

abs]Ssol

7. deexo

dt= −keeexo

8. dWdt

= −αkequi

(μ([W])Ans − As

) + (α − β)kvol(t)[Adms ]Vapp

+ [W]((β − γ )k̃surf [Bint] + (α − γ )ksurf [As])

+ (λs − β)kseexop̃h(x)[Fdmsol

]Vapp + γ kabs[Bdmabs

]Ssol − kw(W − 0.1M)

11. dMdt

= MM−W

( − kw(W − 0.1M)− kabs[Bdmabs

]Ssol + secretions)

12. dVdt

= −kw(W − 0.1M)+ secretions

o

s

o

p

g

fi

p

e

w

D

l

p

s

o

fi

u

a

c

o

(

fi

(

o

t

t

e

e

n

a

a

A

t

m

b

d

f

I

ery interesting improvement step of the future development of the

odel.

Modelling the influence of soluble and insoluble DF on degradation

f other feedstuffs’ molecules as well as on the movement of bolus

Experiment 5.1) is the first attempt to model a more realistic non-

omogeneous bolus. DF has normally a high WHC which increases

he volume of aqueous phase in the bolus and therefore dilutes the

olution of nutrients and enzymes [12], known to influence the vo-

umic and surfacic reactions. However, these effects depend highly

n the choice of Wsol and Winsol. Another aspect of the model is the

ater equilibrium and its impact on absorption, depending again on

he choice of parameters (Was, Wint and Wabs) which play a signifi-

ant role on the final results of digestion and should be supported by

n vivo or in vitro experiments. Experiments 5.3 stresses the role of

ther parameters (kw, kequi, ksurf , kvol). Water absorption rate kw, can

hange (even dramatically) the numerical results. The model param-

ters are listed in Tables 6–8.

In the present paper, the objective is to reproduce the dynamic

ehavior of the bolus in the presence of dietary fiber rather than to

eal with its final composition. Therefore, our choice was to focus

n the understanding of each phenomenon and its mathematical de-

cription rather than on the precise effect of the parameters’ values

n the final results. To perform the numerical simulations, model

arameters have been chosen in order to roughly fit known observed

lobal behaviors, such as the transit duration or an approximate ef-

ciency of digestion obtained from the literature and/or by the ex-

ertise of biologists. This choice of parameters allows to observe the

volution of each substrate in time which is consistent qualitatively

ith the observed biological results (positive influence of insoluble

F on digestion, negative effect of soluble DF, . . . ).

To have a more predictive model, the next step would be to ana-

yze the impact of each parameter (as well as the interaction between

arameters) on the final results using a method based on decompo-

ition of variance to perform a global sensitivity analysis [15] and to

rganize these parameters in a hierarchy according to their impact on

nal results. Then one can identify the model influential parameters

sing an adequate mathematical optimization method.

One of the advantages of this model is its capacity to be reduced

nd to be adapted to new experiments enabling parameters identifi-

ation. Reduction consists in the decreasing the number of equations

f system or the number of parameters without affecting its genericity

e.g. a bolus which does not contain the DF, results in a more simpli-

ed digestion process which in turn caused decreasing the equations

parameters) involved in the digestion model).

It is also worth pointing that values of most parameters depend

n the category of considered feedstuffs. The close collaborations be-

ween biologists and mathematicians is therefore required to iden-

ify these new parameters (literature data in biology, define new

xperiments, . . . ). This highlights one of the main interest of mod-

lling which is to ask the precise questions about the modeled phe-

omenon by crossing contrasted points of view. In fact, this approach

llows to use efficiently the existing data and to find new research

reas.

cknowledgments

The multidisciplinary collaboration on this research project be-

ween the INRA Center of Nouzilly and the Laboratoire de Mathé-

atiques et Physique Théorique was initiated within and supported

y the CaSciModOT program (CAlcul SCIentifique et MODélisation

es universités d’Orléans et de Tours) which is now a Cluster of the

rench Region Centre. This collaboration also takes place in a CNRS-

NRA PEPS program “Compréhension et Modélisation du devenir de

112 M. Taghipoor et al. / Mathematical Biosciences 258 (2014) 101–112

[

[

[

[

[

[

[

[

[

l’aliment dans le tube digestif“. This work is part of the PhD thesis of

Masoomeh Taghipoor, financed by CNRS and INRA.

The author wish to warmly thank the anonymous referees for their

very careful readings of the article, for their numerous interesting

remarks and questions which lead to a meaningful improvement of

the paper.

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