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An adaptive control function of the rectoanal inhibitory reflex in humans - model and simulation
AN ADAPTIVE CONTROL FUNCTION ' OF THE RECTOANAL INHIBITORY REFLEX IN HUMANS - MODEL AND SIMULATION. C.P .A. VASSEUR:l,J .G. POSTAIRE:l,P .ARHAN:l:l, G'.DEVROEDE:l:l:l. , :l Department of Automatic Control, University of Lille, B.P.~6, ~2~~Q_.Y!~~!?~!nmL~~~~q (FRANCE). :l:l
Dep~rtment
of Physiology, Necker Hospital, 156, rue de ,Vaugirard (FRANCE)
Z~Ql~_~~8!§,
:l:l:l Gastrointestinal Researc~ Unit, Department of Surgery, University of Sherbrooke, ~ltl_~~!~_§h~r2rQQ~~L_q~~~ CANADA. t Summary : The behaviour of the internal sphincter induced by repetitive stimulations of the rectal wall was recorded in ten healthy human volunteers. Since the amplitude of the rectal distension tends to vary with respect to the morphology of the rectum, the output recordings were numerically processed to minimize amplitude dis-, persion. These normalized data ,formed the basis for the postulation of a mathematical model. . This model was de,v eloped under the fundamental assumption that the input signal could be decomposed into different components,such that each of them is affected by a specific adaptive phenomenon • . The characteristic paraaeters of ,this model were determined by a model reference method i.pl~mented on a hybrid 'computer. Introduction : The study of the anal mechanical activity ,shows that a distension of the rectal wa11 resl,llts in a slackening of the inter- , nalsphincter. This well known phenomenon is called the rectoanal inhibitory reflex. This control function ,of the anus appears 'as a , biological process for which the input is the distension of the rectal wa11 and the output the anal sphincter mechanical activity. The nervous pathways of this refelx are the myenteric plexus which behaves as an elementary central nervous system and the extrinsIc vegetative nerves. It was shown that periodic solicitations of the rectal wall result in a slackening of the anal sphincter ' decreasing in aaplitude from one patient to another /1/. The purpose of this paper is to model this adaptive control func- ,. ' tion of the rectoanal inhibitory reflex. Materials and methods : Investigations were carried out on 10 chil+ . , dren in the age group of 93- 16 months. Figure-:l shows the probe designed for this study ( Constructed by Marquat Inc. Paris ). It is i:luilt with a soft polyethylene t,u be (O.D.-S_.) containing 3 para11el canals (LD.,-O.S_.). Two of the canals are connected to , a distending balloon. 1:he free' extreaity of the first , one is plugge~
to a syringe used to inject !2Oml of air at a tiae.
1,6
The free extremity of the second one is connected to a pressure transducer (Statham PM6) to DISTENDING BALLOON record the balloon pressure P1 • Four centimeters below the distending balloon, the last canal is connected to an annular balloon filled with' water and PRESSURE used to measure theinterna~ TRANSDUCERS } anal sphincter mechanical activity. This activity w~s r~cor SYRINGE ded by means of the pressure P2 in the annular ballo~n (statham PM6 transducer with StaFigure-1 tham 1008 amplifier)'. The systematic source of error introduced by the viscouselasticity of the distending balloon itself had to be removed. Thus, after each experiment, the pressure in the distending balloon was recorded in air. This reco'r ding was then sys,tematically subtracted from the in vivo..,:curve. On each patient, recordings were performed for various types of I"epetitive waves and for an unique long step input . Definition of the bio-system: The input of the system is coaposed of a succession of active periods, called solicitations, where the distending balloo~ is inflated, followed by passive periods, called relaxations, whe'r e the balloon is deflated.
~
-!·-··_...,-tI'?::.·~:· .. ._.1.t;l'
P1 r " f ... ....... I- : - ---.. :l
1
1
t
:INPUT SIGNAL, .. N ' ~t
...
p,~ Figure-2: Solicitations,input and output of the biosystem. '
First the recordings of the pressure in the distendin. balloon during solicitations show that the greatest variation 6P never exceeds ~Oca. of 1 water (Figure-2). Assuming the compression to be isotheraal~ the resulting varia~ions of the volume of the balloon are below 6\. Hence, ' we ~hall suppose, in , what ,f ollows, that the' disten!;ion Po of the rectal wall reo.'
157
mains constant during solicitations.Furthermore,for periodic, solicitations, the variations of the pressure P1 in the distending balloon is periodic (Figure-2).Hence,the volume of the rectal ampulla and the subsequent distension of the rectal wall can be considered to be constant and equal for all the successive solicitations of each experiment.However for the same type of experiment, the value of the pressure may vary significantly from one patient to another . This phenomenon indicates that the rectal distensions tend to vary with the morphology of the rectum. Second when solicitation occurs, the distension D(t) of the rectal wall reaches qua$i-instantaneously a constant value DO,depending onlyon the morphology of the patient.During the relaxation period,the distension can be mode led by the following exponential law : o (t) - DO e- kt The coefficient k was determined on the basis of the evolution of the volume of the rectal ampulla observed with the help of radiologicill technics . During solicitations, the ampulla is almost spherical. Furthermore,it retains this shape when it comes back by itself to stand-still conditions (These observations are made when Qpening the .distending ballon in the atmosphere ). Such experiments were performed 10 times on each . patient . The resulting mean value of k is: X - 0.14 ! 0.02 s-l with an error probability of 10\. Preprocessing of the raw data : To derive a unique model from the ten different patients, the output recordings were scaled to the same level. The scaling of the recordings of the pressure in the annular anal balloon was performed numerically on a T6l6m6canique T1600 digital computer IZ/.For each type of experiment,'the recordings were sampled for one minute. The mean response of the internal anal sphincter was first computed over the 10 patients.Then,each response was linearly stretched or shrunk to minillize its distance to the mean curve.Once the 10 responses were modified,the new mean curve was computed to reinitialize the algorithm. This procedure stops by itself when the distance of the 10 curves to the mean curve cannot be significantly reduced by a new iteration. The normalized data formed the basis for the postulation of a mathematical model (Figure-3).
158
Figure-3 :Normalized responses of ' the internal anal sphincter ' for five types of experiments with corresponding ~olicitations. Analysis of the input :The output recordings show that the effects of inputs tend to decrease with time (Figure-3a).This adaptive control function of the rectoanal inhibitory reflex is a physiological acco·modation phenomenon. A model of this reflex is developed with the fundamental assumption that whenever a step of distension is applied to the rectal wall, the respons'e of the system to this step is affected by an adaptive phenomenon initialized simultaneously with this step.For repetitive solicitations,an adaptive phenomenon is initialized at each step. Figure-4 shows this decomposition of the input signal e(t) for repetitive solicitations.The signal e,(t) of Figure-Sd is the compo': nent · of the input e(t) affected by an adaptive phenomenon initialized at time T,.Similarly,figure-Sg shows the 'contribution eZ(t) of the second solicitation which is affected by an adaptive phenomenon iniiialized at time TZ.This decomposition procedure goes on for the third and fourth solicitations. The component e,(t) is obtained from ' the input e(t) as follows. At each local minimum met) of e(t),the signal e(t) is held constant at met) until the arrival of the next local minimum inferior to met) (Cf.Figure-Sc).The signal e,(t) is then defined ;as : . , ' 8 , (t) • in ( e , • (t), e (t) ) The sa.e procedure can then be applied to ihe signal . • ', (t) • e (t) - e , (t) starting from Tz,to find eZ(t).This iterative technique was implemented on a hybrid computer EAI-S80 /3/. The input sighal was decomposed into different,components,each of ,
them being aff~cted by an adaptive phenomenon initialized simultaneously with the associated solicitation. SOLICITATIONS (a) t ·z IT3 ~ ,INPUT
e, (t)
:;lliN~~
-
---
---,~
r-
~_t
I
(c)
t
e 1 (t) (d)
-
e', (t)
e~(t)
I
.t--..
...... ~)
r
(f)
f-
,
-
eZ (t) e 3 (t) e 4 (t)
Cg)
~ ~ ~""" ~" ~///
~//
'///77/7
t
Ch)
+
~
t
~ ~(j)
.t
I--
DECOMPOSED INPUT
t
Figure-4: Decomposition of the input signal Simulation :The adaptive phenomenon was simulated oh the basis of the data collected· in the experiment of Pigure-3a when the input is a unique long step.With this particular input, the behaviour of the anal sphincter was modelized by multiplying the input by a time decreasing exponential function. This function is initialized a~ the unit value sim~ltaneously with.the ~tep input. The introduction of a second order linear filter in series with this ' adaptive functi~n gives a satisfactory model fo~ this particular input (Cf.Figure S).The adjustment of the parameters was achieved by a model reference method. It consisted os sequential variations of the different parameters in order to minimize a mean square deviation between the normalized exponential data and the model output.
160
The model for repetitive solicitations is also derived from the normalized experiemental data, provided that the input is decomposed as shown in the preceeding paragraph. The model is then simply achieved by multiplying each component relative to each solicitation by the exponential decreasing function found earlier and ini t iali zed s imul taneous ly wi th the '··solici ta tion. By summing up the results of these multiplications, we obtain the input of the second order filter identified formerly (Cf. Figure 5). FUNCTION
K + ",2
tADAPTIVE FUNCTION
~
INPUT SIGNAL
~igure-5
n
Simulation of n successive solici tations.
Results and discussion:The fundamental parameters of the rectoanal reflex are the damping ratio ~,the natural frequency "'n of the second_order linear filter and the time constant k of the adaptive function. The static gain of this bio-system,which may vary to a large extent from one patient to another,is a less characteristic parameter of the rectoanal inhibitory reflex process. From the results (table 1) it is seen that ,the difference in amplitude between the outputs of the model and the normalized process remains below 13\ for all the 5 types of experiments. l"'p=O.15rd/S ~-0.8 Ik 2.5s- 1 Table 1 R E FER E N C E S /1/ P.ARHAN,C.FAVERDIN,J.THOUVENOT - Scand.J.Gastroenterol.7, pp 309-314 /2/ R.W.HORNBECK - Numerical Methods-Quantum Publishers Inc.1975 /3/ S.FIFER - Analogue Computation -Vol.2 - Mc.Graw Hill . Book Company Inc.1961 ACKNOWLEDGEMENTS The authors hereby express their acknowledgements to Miss ~FAVERDIN,Miss C.DORNIC and Professor B.PERSOZ for their kind aid in the various phases of this work.Sincere thanks also go to C.V.KAJAGUPALAN for . his contribution in the preparation of the manuscript.
Dep~rtment
of Physiology, Necker Hospital, 156, rue de ,Vaugirard (FRANCE)
Z~Ql~_~~8!§,
:l:l:l Gastrointestinal Researc~ Unit, Department of Surgery, University of Sherbrooke, ~ltl_~~!~_§h~r2rQQ~~L_q~~~ CANADA. t Summary : The behaviour of the internal sphincter induced by repetitive stimulations of the rectal wall was recorded in ten healthy human volunteers. Since the amplitude of the rectal distension tends to vary with respect to the morphology of the rectum, the output recordings were numerically processed to minimize amplitude dis-, persion. These normalized data ,formed the basis for the postulation of a mathematical model. . This model was de,v eloped under the fundamental assumption that the input signal could be decomposed into different components,such that each of them is affected by a specific adaptive phenomenon • . The characteristic paraaeters of ,this model were determined by a model reference method i.pl~mented on a hybrid 'computer. Introduction : The study of the anal mechanical activity ,shows that a distension of the rectal wa11 resl,llts in a slackening of the inter- , nalsphincter. This well known phenomenon is called the rectoanal inhibitory reflex. This control function ,of the anus appears 'as a , biological process for which the input is the distension of the rectal wa11 and the output the anal sphincter mechanical activity. The nervous pathways of this refelx are the myenteric plexus which behaves as an elementary central nervous system and the extrinsIc vegetative nerves. It was shown that periodic solicitations of the rectal wall result in a slackening of the anal sphincter ' decreasing in aaplitude from one patient to another /1/. The purpose of this paper is to model this adaptive control func- ,. ' tion of the rectoanal inhibitory reflex. Materials and methods : Investigations were carried out on 10 chil+ . , dren in the age group of 93- 16 months. Figure-:l shows the probe designed for this study ( Constructed by Marquat Inc. Paris ). It is i:luilt with a soft polyethylene t,u be (O.D.-S_.) containing 3 para11el canals (LD.,-O.S_.). Two of the canals are connected to , a distending balloon. 1:he free' extreaity of the first , one is plugge~
to a syringe used to inject !2Oml of air at a tiae.
1,6
The free extremity of the second one is connected to a pressure transducer (Statham PM6) to DISTENDING BALLOON record the balloon pressure P1 • Four centimeters below the distending balloon, the last canal is connected to an annular balloon filled with' water and PRESSURE used to measure theinterna~ TRANSDUCERS } anal sphincter mechanical activity. This activity w~s r~cor SYRINGE ded by means of the pressure P2 in the annular ballo~n (statham PM6 transducer with StaFigure-1 tham 1008 amplifier)'. The systematic source of error introduced by the viscouselasticity of the distending balloon itself had to be removed. Thus, after each experiment, the pressure in the distending balloon was recorded in air. This reco'r ding was then sys,tematically subtracted from the in vivo..,:curve. On each patient, recordings were performed for various types of I"epetitive waves and for an unique long step input . Definition of the bio-system: The input of the system is coaposed of a succession of active periods, called solicitations, where the distending balloo~ is inflated, followed by passive periods, called relaxations, whe'r e the balloon is deflated.
~
-!·-··_...,-tI'?::.·~:· .. ._.1.t;l'
P1 r " f ... ....... I- : - ---.. :l
1
1
t
:INPUT SIGNAL, .. N ' ~t
...
p,~ Figure-2: Solicitations,input and output of the biosystem. '
First the recordings of the pressure in the distendin. balloon during solicitations show that the greatest variation 6P never exceeds ~Oca. of 1 water (Figure-2). Assuming the compression to be isotheraal~ the resulting varia~ions of the volume of the balloon are below 6\. Hence, ' we ~hall suppose, in , what ,f ollows, that the' disten!;ion Po of the rectal wall reo.'
157
mains constant during solicitations.Furthermore,for periodic, solicitations, the variations of the pressure P1 in the distending balloon is periodic (Figure-2).Hence,the volume of the rectal ampulla and the subsequent distension of the rectal wall can be considered to be constant and equal for all the successive solicitations of each experiment.However for the same type of experiment, the value of the pressure may vary significantly from one patient to another . This phenomenon indicates that the rectal distensions tend to vary with the morphology of the rectum. Second when solicitation occurs, the distension D(t) of the rectal wall reaches qua$i-instantaneously a constant value DO,depending onlyon the morphology of the patient.During the relaxation period,the distension can be mode led by the following exponential law : o (t) - DO e- kt The coefficient k was determined on the basis of the evolution of the volume of the rectal ampulla observed with the help of radiologicill technics . During solicitations, the ampulla is almost spherical. Furthermore,it retains this shape when it comes back by itself to stand-still conditions (These observations are made when Qpening the .distending ballon in the atmosphere ). Such experiments were performed 10 times on each . patient . The resulting mean value of k is: X - 0.14 ! 0.02 s-l with an error probability of 10\. Preprocessing of the raw data : To derive a unique model from the ten different patients, the output recordings were scaled to the same level. The scaling of the recordings of the pressure in the annular anal balloon was performed numerically on a T6l6m6canique T1600 digital computer IZ/.For each type of experiment,'the recordings were sampled for one minute. The mean response of the internal anal sphincter was first computed over the 10 patients.Then,each response was linearly stretched or shrunk to minillize its distance to the mean curve.Once the 10 responses were modified,the new mean curve was computed to reinitialize the algorithm. This procedure stops by itself when the distance of the 10 curves to the mean curve cannot be significantly reduced by a new iteration. The normalized data formed the basis for the postulation of a mathematical model (Figure-3).
158
Figure-3 :Normalized responses of ' the internal anal sphincter ' for five types of experiments with corresponding ~olicitations. Analysis of the input :The output recordings show that the effects of inputs tend to decrease with time (Figure-3a).This adaptive control function of the rectoanal inhibitory reflex is a physiological acco·modation phenomenon. A model of this reflex is developed with the fundamental assumption that whenever a step of distension is applied to the rectal wall, the respons'e of the system to this step is affected by an adaptive phenomenon initialized simultaneously with this step.For repetitive solicitations,an adaptive phenomenon is initialized at each step. Figure-4 shows this decomposition of the input signal e(t) for repetitive solicitations.The signal e,(t) of Figure-Sd is the compo': nent · of the input e(t) affected by an adaptive phenomenon initialized at time T,.Similarly,figure-Sg shows the 'contribution eZ(t) of the second solicitation which is affected by an adaptive phenomenon iniiialized at time TZ.This decomposition procedure goes on for the third and fourth solicitations. The component e,(t) is obtained from ' the input e(t) as follows. At each local minimum met) of e(t),the signal e(t) is held constant at met) until the arrival of the next local minimum inferior to met) (Cf.Figure-Sc).The signal e,(t) is then defined ;as : . , ' 8 , (t) • in ( e , • (t), e (t) ) The sa.e procedure can then be applied to ihe signal . • ', (t) • e (t) - e , (t) starting from Tz,to find eZ(t).This iterative technique was implemented on a hybrid computer EAI-S80 /3/. The input sighal was decomposed into different,components,each of ,
them being aff~cted by an adaptive phenomenon initialized simultaneously with the associated solicitation. SOLICITATIONS (a) t ·z IT3 ~ ,INPUT
e, (t)
:;lliN~~
-
---
---,~
r-
~_t
I
(c)
t
e 1 (t) (d)
-
e', (t)
e~(t)
I
.t--..
...... ~)
r
(f)
f-
,
-
eZ (t) e 3 (t) e 4 (t)
Cg)
~ ~ ~""" ~" ~///
~//
'///77/7
t
Ch)
+
~
t
~ ~(j)
.t
I--
DECOMPOSED INPUT
t
Figure-4: Decomposition of the input signal Simulation :The adaptive phenomenon was simulated oh the basis of the data collected· in the experiment of Pigure-3a when the input is a unique long step.With this particular input, the behaviour of the anal sphincter was modelized by multiplying the input by a time decreasing exponential function. This function is initialized a~ the unit value sim~ltaneously with.the ~tep input. The introduction of a second order linear filter in series with this ' adaptive functi~n gives a satisfactory model fo~ this particular input (Cf.Figure S).The adjustment of the parameters was achieved by a model reference method. It consisted os sequential variations of the different parameters in order to minimize a mean square deviation between the normalized exponential data and the model output.
160
The model for repetitive solicitations is also derived from the normalized experiemental data, provided that the input is decomposed as shown in the preceeding paragraph. The model is then simply achieved by multiplying each component relative to each solicitation by the exponential decreasing function found earlier and ini t iali zed s imul taneous ly wi th the '··solici ta tion. By summing up the results of these multiplications, we obtain the input of the second order filter identified formerly (Cf. Figure 5). FUNCTION
K + ",2
tADAPTIVE FUNCTION
~
INPUT SIGNAL
~igure-5
n
Simulation of n successive solici tations.
Results and discussion:The fundamental parameters of the rectoanal reflex are the damping ratio ~,the natural frequency "'n of the second_order linear filter and the time constant k of the adaptive function. The static gain of this bio-system,which may vary to a large extent from one patient to another,is a less characteristic parameter of the rectoanal inhibitory reflex process. From the results (table 1) it is seen that ,the difference in amplitude between the outputs of the model and the normalized process remains below 13\ for all the 5 types of experiments. l"'p=O.15rd/S ~-0.8 Ik 2.5s- 1 Table 1 R E FER E N C E S /1/ P.ARHAN,C.FAVERDIN,J.THOUVENOT - Scand.J.Gastroenterol.7, pp 309-314 /2/ R.W.HORNBECK - Numerical Methods-Quantum Publishers Inc.1975 /3/ S.FIFER - Analogue Computation -Vol.2 - Mc.Graw Hill . Book Company Inc.1961 ACKNOWLEDGEMENTS The authors hereby express their acknowledgements to Miss ~FAVERDIN,Miss C.DORNIC and Professor B.PERSOZ for their kind aid in the various phases of this work.Sincere thanks also go to C.V.KAJAGUPALAN for . his contribution in the preparation of the manuscript.