A mathematical model of intestinal motor activity

A MATHEMATICAL MODEL OF INTESTINAL MOTOR ACTIVITY*t A. BERTUZZI, Centro

di Studio

R. MANCIXLLI$

G. RONZOXI$and S. S.ALINARI

dei Sisremi di Controllo e Calcolo Automatici de1 C.N.R.-Istituto di Automatica. Via Eudossiana. l&001% Rome. Italy

: Istituto di Fisiologia Umana. UniversitB Cattolica de1 Sacro Cuore, Via Pineta Sacchetti. 64400168 Rome, Italy 3 Istituto di Clinica Chirurgica Generale e Terapia Chirurgica-UniversitB Cattolica del Sacro Cuor:. Via Pineta Sacchetti. 64-L00165 Rome. Italy Abstract-A mathematical mod4 describing the dynamics of intestinal wall and accounting for Its large deformations is presented. The model makes reference to an isolated gut segment in a preparation well known in experimental physiology. A simpiiiied representation of the functional characteristics of the smooth muscle and of the contraction process was used at the present stage. This model should help to analyse the mechanisms controllin, o the contraction of the intestinal wall and to evaluate diagnostic and therapeutic procedures for cases of motility disorders. ISTRODUCTION

A large part of the literature on intestinal physiology concerns in ciuo or in I;irro measurements of mechanical and electrical variables which characterise intestinal motility, with the aim of analysing its control mechanisms and the transport phenomena in the intraluminal content (Daniel and Chapman, 1963). Also theoretical analyses exist, both of the fluid motion associated with peristalsis (Fung and Yih, 1968; Lew, Fung and Lowenstein. 1971) and of the electrical control activity, or slow wave, in the intestine (Sarna et nl., 1971; Linkens, 1975). The mechanical properties of the smooth muscular tissue of the intestinal wail on the other hand have been studied less than those of cardiac and skeletal muscles perhaps because of its many complicating characteristics. It has been shown however that the dynamic behaviour of muscle strips excised from the intestinal wall may be represented by a force-velocity-length relationship of the same form but with different parameter values than for the other types of muscle (Aberg and Axelsson, 1965; Gordon and Siegman, 1971). This paper is an account of a theoretical and experimental research undertaken with the aim of analysing the mechanisms, still not well understood, that coordinate and control the contractile activity of the intestinal wall. The research could result in a more careful evaluation of diagnostic and therapeutic procedures for many cases of motility alterations. The general scheme of the work is similar to that of other researches in biomechanics, and particularly to that outlined by Fung (1971a) for the study of ureteral function: this scheme is represented in Fig. 1. The Trendelenburg preparation (Trendelenburg, 1917). which makes it possible to study propulsive activity in isolated intestinal segments avoiding the complex* Received 29 Mat-cl! 1977. t This work was partially Nazionale delle Ricerche.

supported

by Consiglio

ity of in ciro behaviour, was used to measure electrical and mechanical variables in given experimental conditions. With reference to this preparation a mathematical model, which so far accounts only for mechanical aspects of motility, has been constructed. The constitutive equation, which describes the dynamics of muscle wall contraction, and its parameter values play an essential role in the model. A series of measurements on muscle strips 6 now under way to obtain these parameters; strips are excised from the same intestinal segments used in Trendelenburg experiments and prepared in the same wa>- to assure homogeneity in experimental data. Finally. the inputs to the model, which account for the development and propagation of the state of activation in the segment muscte wall, appear now as external variables without any reference to the coordination and control structures that generate them. In this paper the equations of the model and some results obtained from their numerical sdlution are discussed. A preliminary selection of experimental data has already been presented (Ronzoni et al.. 1976). THE

MATHEMATICAL

MODEL

The physical system considered for the model construction is an intestinal segment 3-7cm long in the so-called modified Trendelenburg or isometric isovolumic preparation (Mackenna and 4lcKirdy, 1972). This preparation is illustrated in Fig. 2. The segment is laid horizontally in a bath of oxygenated Tyrode solution at constant temperature. One end of the segment is ligated around the mouth of a glass tube connected to a pressure transducer; Tyrode soiution is used to fill the lumen through a tube connected to the other by a two-way tap. The other end is closed by a plug tied by a silk thread to an isometric force transducer. The total intraluminal volume and the length of the segment remain so virtually constant. while pressure and force can be measured and the

42

A. BERTWZI

er al.

1LATE0STRIPS1 j 1 PREPARATION 1

; M_E_CHANISMS ; --a-----

1 MECHnNlCi 1

I

I

L___________________!

Fig. 1. General scheme for the analysis of the intestinal function. The dashed part is to be determined.

changes in shape of the wall recorded by cinematography during spontaneous or stimulated peristaltic activity. The peculiarity of this preparation is the absence of net flow through the segment: intraluminal fluid motion, also in view of the slowness of contraction, can thus be neglected, making it possible to focus attention mainly on wall dynamics. The following basic assumptions were made in model construction. (a) The tubular segment remains axisymmetric in all configurations. (b) The intraluminal fluid produces a uniform pressure in all the occupied volume; external pressure is uniform and constant in time since no variation occurs in the bath level. (c) The intestinal wall is anatomically arranged in two muscle layers, the external one of longitudinal fibres and the internal one of circular fibres, which contract independently but in a coordinated way. The elastic material of these two layers and of the other tissues of the wall (connective, vascular and nervous) is considered to be concentrated in a surface, so that the wall is assimilated to a membrane with suitable mechanical properties in longitudinal and circumferential directions. Inertia forces in the wall are also neglected. (d) Concerning the mechanical properties of the smooth muscle tissue of each layer, reference is made to the simplified model, shown in Fig. 3(c), for the

element of wall that in Fig. 3(a) is at rest and in Fig. 3(b) is subjected to a force N. L,, and l,, are the reference (undeformed) dimensions of the element of tissue; 7and f0 (Q are the forces per unit length referred to the undeformed state (that is to IO)acting respectively on the whole element and on its parallel elastic component. A constitutive equation was assumed for the contractile component of the general form i/L,, = g(LIL0, T - Jo.?),

(1)

where g is a function, to be obtained through interpolation of force-velocity data given by suitable experiments on muscle strips, of normalized length L/L,, tension T-f, and stimulation ‘J. Stimulation was assumed in this paper as representing an average degree of mechanical activation of muscle. So y, 7 E [O,l], is a time-dependent parameter, multiplying a function f(L) not explicitly indicated which gives the tension exerted by the contractile component for i = 0. Functions f0 and f are both obtained from force-length data. Examples of curves used in this work are in Fig. 4 (see the fourth Section for analytical expressions), This simple description of muscle bebehaviour was idopted considering the slowness of the process of contraction in smooth intestinal muscle. A more complicated description (Fung, 1971b) can be inserted into the model later, particularly concerning the presence of a series elastic component in the

FT

TS

C ."B

4

C.

-&

T-i+ -L-.-A

Fig. 2. Lay-out of the modified Trendelenburg preparation; IS intestinal segment, B-bath, TS-Tyrode solution, GT-glass tube, PT-pressure transducer, T-twoway tap, FT-force transducer, C-cinecamera.

Fig. 3. (a). Element of muscular tissue at rest; L.,,l& reference dimensions. (b). Element subjected fo a force N; LJ,h dimensions in the deformed state. (c). Functional model of the element; C contractile component, PE parallel elastic component, T= N/l, force per unit undeformed length.

13

h mathematical model of intestinal motor activity

6 L/L,

4+-G

Fig. 4. Diagrams of the force-length (left) and of the forcevelocity (right) relationships assumed for the intestinal muscle. F,Mmaximum active force, kF, value of isometric force under which constant slope force-velocity characteristics are used.

Oji-x

/.

X

U

Fig. 6. Variables for the description of segment deformation.

an inclination form of g with respect to T - j;. In Fig. 5 the variables of interest for comparison with the experimental data, and which are to be considered as the model outputs, are represented. The wall configuration of the tubular segment of constant length 12,is given at time t in the indicated reference system by the vector MM’ function of x, ~E[O,L,]. It should be noted how two wall elements, both of undeformed length d. are displaced and deformed by the action of longitudinal and circular fibres. P(r) and F(r). assuming the external pressure equal to zero, are pressure and force values at the same time. Deformation and forces are considered in the model-as a consequence of the evolution of two stimulation functions that play in the segment the same role as 7 in (1): r(.~,r) for circular fibres and &,t) for longitudinal fibres. These two functions are the model inputs.

1

:

I$ and a length dl, given by:

model of Fig. 3(c) and the actual nonlinear

DERIVATION OF MODEL EQUATIONS

The equations of the model are the following: (a) two equations expressing the balance of forces acting on the wall; (b) two constitutive equations; (c) two other equations accounting for the type of preparation (isometric isovolumic). Some geometric relationships and the force equilibrium equations are obtained as follows. An element of wall of undeformed length dx is represented in Fig. 6. The section with abscissa .Kand radius r,, in resting state has abscissa 5 = .Y+ LLand radius p = r. + u in the deformed state. The configuration is thus defined by the functions u(x,r) and c(x,t), x E [O,L,] r E [0, x], which are the components of the displacement vector MM’. The element of wall takes in the deformed state

Fig. 5. Resting and deformed configuration of the intestinal segment in the preparation considered. L, segment length, r, undeformed segment radius.

25

d[, = ;-

-

d.u

=

cx cosf$

.:[l +;)’ L/

+($x.

(2)

while the volume corresponding to a given configuration is

The tract of wall and the portion of fluid it contains as represented in Fig. 7 should now be considered. Ti and T;, are the forces per unit length in the deformed state acting on the membrane respectively in longitudinal and circumferential directions. The balance of forces in axial and radial direction gives: _)

;(T;pcos$ +‘E PPzdx

- :Pp’)d.u = 0 (4)

- Tkdl, + L(Tipsin$)dx 2X

= 0.

Equations (4) can be transformed consid:ring that the forces referred to the undeformed state r, and T@ are related to T; and 2-Lby

T; =

7;1 \ ‘(1 + zu/L3x)2 + (dc/kK)”

(3

Fig. 7. Forces acting on a tract of intestinal wall in a deformed state and on the Ruid enclos&

A.

BERXZZI

et al.

I

lrxtial conflguratlon v.u,

. ~=r,g,kbx. bv

_

bu

2 -

P.F,a)

b2v

&+t9QG=

-

*

1

COS Q

P.F.p 1

g, be,

,

,I

Configuration “bu

‘bx

Stmulation 0-P

Pressure and force

P. F

d

h,lu.g

.P.F.a.Pl=O

h2iu,e

,P,F,a,pl=O

I

Fig. 8. Functional block diagram of the and that the quantity r’,p cos 4 - fPp2

is a function of time only equal to F/277 - +Pra where the force F, since ~(0,c) = v(L,, t) = 0, is given by F = ZiV,T
The two equations expressing respectively the conservation of length and intraluminal volume are obtained as follows: d 0 = -[t&, dr

=

So the following relationships are obtained:

(6) I

P

dx

’

-

where A, B, C, D are functions of (x,t) defined by equations (6) and depending only on the configuration. Constitutive equations for the wall are obtained from equation (1) with the following positions respectively for circular and longitudinal fibres: L = 2n(ro f 0); T=Ta;

Lo = 2nr0; fo=feo;

y=z

L=J(f+;r+(;rdx; 9 = 91;

T= 7$

g = gc;

i’ =

(8)

p,

Ls t%

-dx i e dxBt

o=$a

where the expressions at the second member are now functions of x and r. In this way are obtained the two following equations, in which equations (6) were substituted:

L, s0 L,

(BgI + Dg,) dx = 0 (13) (Ag, + Cg,)dx = 0.

I 0

System (13), if c, &/ax, CL,/I are known, can be solved through a minimization procedure giving P and F; so equations (9) and (13), with the conditions (10) and (1 l), make it possible to determine the evolution of c, II, P and F for given inputs a and fl and initial conditions c and ri,. The model can be represented by the block diagram in Fig. 5. NUMERICAL DISCUSSION

r$,CP+DF-I,,,n

>

(9)

In numerical analytical form as follous. The spect to r-f0

=--&,(J!l+;~+($h’+BF-,,>I+ Equations (9) substantially determine model dynamics. Boundary and initial conditions are c(0, t) = t(L.,, t) = 0

Qt 2 0

(10)

(12)

Substituting equations (9) in (12), performing an integration by parts and considering that dv/dr = A = 0 for x = 0 and x = L,, the following equations, which are of type h(v,~~~~~,d2c~~~2,~ll~~~.~2ujax~,p,~,~,~) = 0, are obtained:

(7)

L,=dx; f. =j&;

d t) - u(0, r)] = dt

= JbL’[?(rO+Ll)o+~)$+(ro+r)~~]d:.

+‘“f99,_Cp+DF 2n

model.

SOLUTION

ASD

OF RESLZTS

integration of model equations the of functions g, f0 and f was chosen function g was taken linear Gth reobtaining

i/Lo = gt + (T-fo)gz.

(14)

where g1 and g2, which are now functions of i/L0 and y only, are so defined as to obtain the families of straight lines shown in Fig. 4, that is

A mathematical model of intestinal motor activity

i -=-

LO

c:, f

fo)

(i--

5.

if;$>

d’

kF,M, (l-lb)

kF, being a threshold value defining passage from constant slope (14a) to variable slope (14b) characteristics. Functionsf, (supposing the in situ Length equal to the rest length) andf were chosen as

Expressions similar to those indicated in (ll)-(16) have already been used by authors working on muscular contraction (Vickers. 1968; Green, 1969; Yin and Fung, 1971). Using the expressions of g, and g1 indicated in (l?), and considering that gi = -.ifg2. system (9) becomes

45

Kutta procedure with automatic step size control. A normal run takes ca. 2 min on UNIVAC 1110. Figure 9 shows a typical result of integration. Diagrams of g, J, and f‘are reported for circular (continuous) and longitudinal fibres (dotted); mechanical properties are different for the two layers but are supposed uniform along the segment. Input functions I and p are represented at the four indicated times; the form of these functions was hypothesized on the basis of available experimental data (Bortoff and Ghalib, 1972). The wall configuration is represented at the same time values. The crosses indicate the positions of points of the wail with respect to the rest positions (c = u = 0: .Xe[O,L,]) of the same points indicated by dots: in this way the evolution of segment mor‘i T- to

(g/mm,

5‘f,f3 cg/mm)

2 A

I,/

,-

2 LA,

+*. *_ii

l l

+++ct*,+++.

* .+. ? .+. + + 4 +.+‘+

t.4.’

while, due to assumption (14) which makes system (13) linear with respect to P and F, expressions (18) hold:

blat, - ha,2

P=

, ait -

;

b,a,,

F=

a12a21

a:1

- biaz, -

l t .+.++.+_++l .+.+. +

+.+.+. t + *+++&+a+++

+.+.+.++.t*+?+,+..+.; +

l

;.+.

+ + +*++t+.*+++

10. ,

(18)

(71291

(b)

where L, 011 =

J0

(ABg,, + CDg,,)d.u ,.

“L, (I,?

=

(B’g,,

+

D%,,)

’

d.x

’

! 0

.

7

r *

L

a 21 =

b,

=

b2 =

I 0

(,4’g,: + C’g,Jd.u

[“C~(slh, “0

-

sd

+

(19)

D(g&

mL’CA!giJto - a,) + C&Lo

-0

-

scJ1 d.x

-

s,,,l d.x.

It should be noted that g, and gz, appearing in (19) with the pertinent subscripts, take at time t along coordinate x the expressions indicated in (14) according to the values of rfc and /3fi respectively. Equations (17) have been solved by converting them to a system of ordinary differential equations via the Method of Lines (Hicks and Wei, 1967) and integrating this system by a fourth-order Runge-

Fig. 9. Model parameters, inputs and outputs in a typical run. Parameter values: r0 = 3 mm, L, = 30 mm. ywC=IJw, = 0.2 set - ‘, K,=Kt = 0.2, vw:,,=&:,, = 0.2 set- ‘, K ,C = 0.05 g/mm, Ku = 0.1 g/mm, K1, = 3.5, KzI = 4, KSc = -2.5 g/mm, K,, = -4 g/mm, K,, = 7.5 a’mm. K,, = 12 g/mm, K sr = -3.12 g/mm, K,t = -5 g/mm. (a). Diagrams off.fO and 9 assumed for circular (continuous) and longitudinal (dotted) muscle. (b). Input stimulation represented at the four indicated time values, z = /5 (dots); configurations at the same time values as computed by the mode!. (c). Waveforms of pressure P (crosses) and of force F (dots) as computed by the model.

phology can be followed from the initial one (supposed now as corresponding to the rest state) as a consequence of stimulus propagation. Time courses of pressure and force are also shown in the same units of the experimental data reported by the literature. An examination of various results leads to the following preliminary conclusions. The propagation of stimulation in the two layers is followed by a contraction of the corresponding fibres with a delay dependent on force-velocity characteristics while the degree of contraction is mainly related to active and passive properties of elasticity. The .waveforms and the maximum values of P and F are similar to those experimentally observed (Mackenna and McKirdy, 1972); in particular the increase of F precedes the increase of P though stimulations u and b and contractions in the two layers are simultaneous and not sequential.

CONCLUSlONS

The attempt has been made to represent in the model those structural characteristics of the physical system considered which are deemed essential for the analysis of the mechanisms coordinating and controlling the contraction of the two muscle layers of the intestinal wall. The process of contraction itself, on the other hand, was represented by a functional model which does not refer to the microscopic structure of tissue. Our future research work will be centred on the study of motility control. It should be pointed out that the actual way of operating of this control has been for a long time an object of discussion in intestinal physiology; motility alterations, on the other hand, are responsible for a large part of intestinal pathology. A second goal of this work is to obtain indications for the design of devices for intestinal electrostimulation that has already been indicated in many cases as the proper therapy for motility disorders.

Fung, Y. C. (1971a) Peristaltic pumping: a bioengineering model. In Urodynamics: Hydrodynamics of the L’rerzr and Renal PelFis (Edited by Boyarsky. S.). pp. 177-198. Academic Press, New York. Fung, Y. C. (1971b) Comparison of different models of the heart muscle. J. Biomechanics 4, 289-295. Gordon, A. R. and Siegman, M. J. (1971) Mechanical properties of smooth muscle--I. Length-tension and force-velocity relations.-II. Active state. Am. J. Physiol. 221, 12431254. Green, D. G. (1969) A note on modelling muscle in physiological regulators. .Ved. Biol. Engng 7, 4137. Hicks, J. S. and Wei. J. (1967) Numerical solution of parabolic partial differential equations with two-point boundary conditions by use of the method of lines. J. Assoc. Comput. Mach. 14. 549-562. Lew, H. S., Fung, Y. C. and Lowenstein, C. B. (1971) Peristaltic carrying and mixing of thyme in the small intestine (an analysis of a mathematical model of peristalsis of the small intestine). J. Biomechanics 4, 297-315. Linkens, D. A. (1975) Outimization in the modelina of digestive tract ‘electricai signals. Proc. 7th IFIP C&f.. pp. 156169. Mackenna, B. R. and McKirdy, H. C. (1972) Peristalsis in the rabbit distal colon. J. Physiol. 220, 33-54. Ronzoni. G.. Grassetti. F., Pescatori, M., Di Bet@ G., Bertuzzi, A., Salinari. S. and Mancinelli. R. (1976) Confront0 tra i dati sperimentali e un modello matematico della motilita intestinale del coniglio (Comparison between experimental data and a mathematical model of rabbit intestinal motility), presented at the 78th Congr. Sot. It. Chir.. Roma. Sarna, S. K., Daniel. E. E. and Kingma Y. J. (1971) Simulation of slow-wave electrical acttvity of small intestine. Am. J. Physic/. 221. 166175. Trendelenburg, P. (1917) Physiologische und Pharmakologische Versuche uber die Diinndarmperistaltik. Arch. Exp. Path. Pharmak. 81. 55-129. Vickers, W. H. (1965) A physiologically based model of neuromuscular system dynamics. IEEE Trans. ManMachine Syst. 9, 21-23. Yin, F. C. P. and Fung, Y. C. (1971) Mechanical properties of isolated mammalian ureteral segments. Am. J. Physio[. 221, 1484-1493.

SOMENCLATURE

a+ bi

1;fo

~cinowledgenlanrs-The authors would like to thank Dr. G. Fusco of the Istituto di Matematica Applicata, Universiti di Roma. for his valuable help in model constructtion and Prof. >I. Petternella of the Istituto di Automatica, Universita di Roma. for his suggestions and his critical comments on the manuscript.

ki k

1, lo r0 t 11

REFERENCES

Aberg, A. K. G. and Axelsson, J. (1965) Some mechanical aspects of an intestinal smooth muscle. Acta physiol. stand. 64, 15-27. Bortoff, A. and Ghalib, E. (1972) Temporal relationship between electrical and mechanical activity of longitudinal and circular muscle during intestinal peristalsis. Am. J. digest. Dis. 17, 317-325. Daniel, E. E. and Chapman, K. M. (1963) Electrical activity of the gastrointestinal tract as an indication of mechanical activity. rim. J. digest. Dis. 8, 54-102. Fung, Y. C. and Yih. C. S. (1968) Peristaltic transport. Trans. AS.UE: J. appl. Mech. 35, 669-675.

L’ x A,

B,

F F&f L, Lo L,

N P T 4,

4

VOL

C,

D

coefficients defined in the expressions (19) active and passive force-length characteristics. g/mm force-velocity characteristic and its components as defined in equations (14) parameters in f. and f expressions (i = 1,. ,5) threshold constant in equations (14) instantaneous and rest strip width, mm segment radius, mm time, set axial wall displacement, mm radial wall displacement, mm axial coordinate, mm functions of configuration defined in equations !%e exerted by the segment g maximum active force per u’nz length, g/mm instantaneous and rest strip length, mm segment length. mm force acting on the strip, g intraluminal pressure, cmHzO force per unit length, g,mm axial and circumferential components of T; g/mm intraluminal volume, mm3

A mathematical model of intestinal motor activity maximum normalized contraction velocity. set-’ stimulation in circular and longitudinal layers stimulation abscissa in the deformed configuration mm

~7 4 Subscriprs c

I

radius in the deformed configuration. mm wall inclination. rad circular layer longitudinal layer