AMERICAN JOURNAL OF PHYSIOLOGY Vol. 229, No. 3, Sep tember 1975. Printed in U.S.A.
Stochastic point
model
of contractions
at a
in the duodenum ROBERT B. J. SINGERMAN, EN20 0. MACAGNO, JOHN R. GLOVE& AND JAMES CHRISTENSEN Gastroenterology Research Laboratory, Department of Internal Medicine, and Iowa Institute of Hydraulic University of Iowa Colleges of Medicine and Engineering, Iowa City, Iowa 52242
SINGERMAN,
ROBERT
MACAGNO, JOHN R. model of contractions at a point in the duodenum. Am. J. Physiol. 229(3) : 613-617. 1975.Contractions at one point in the human duodenum were studied as a time series. Manometric records were made over long time periods from the duodenum in fed human subjects. A 5-s grid was superimposed on the time axis of the records. Each 5-s interval was treated as a slow-wave cycle within which either a contraction or a no-contraction could occur. The resulting series of alternating runs of contractions and no-contractions was tested for the existence of trends. Trends were found indicating possible temporal dependence. A Markov-type model was used to try to generate data similar to the real data. Success was achieved by a model that assumed a probability of contraction dependent on the three previous slow-wave cycles. The frequency distributions obtained from the real and generated data were compared using Chi-square goodness-of-fit tests and found to be statistically similar. The correlations in time found for the contractions might be due to a time dependency in the controls for contraction over four successive slow-wave periods, 20 s in humans.
&OVER,
AND JAMES
intestinal
motility
B.
J.,
ENZO
0.
Research,
of fluid flow require, for a particular solution, a description of the specific boundary conditions involved. In this case, an analytical model consists simply of the equations of motion and continuity for an incompressible fluid. The boundary conditions in this case are the wall movements, the position of the walls as a function of space and time. The work reported here is part of a long-term project to constitute an analytical model of small-intestinal flow. As a requirement for that goal, we need a quantitative description of boundary conditions, or wall movements. In the course of seeking this description, we observed that, in the fed state, there is no obvious way to predict the occurrence of contractions at a single point in the duodenum from one slow-wave cycle to the next. That is, the choice at a single point in the duodenum between a contraction cycle and a rest cycle (4) appears to be random (1, 5-7). We proposed that this choice may be, for any single slowwave cycle at any single point in the duodenum, completely independent of the occurrence of contractions in previous slow-wave cycles at that same point. This hypothesis was tested by standard analytical methods and, as described below, found to be unsupported. We therefore examined further the demonstrated temporal dependence of the occurrence of contractions to discover the degree of temporal dependence. In this process, we examined the hypothesis that this temporal dependence can be modeled by a Markov-type process. This hypothesis was found to be acceptable. This study makes use of data gathered and described previously (4). The experimental procedure is detailed in earlier papers (4, 10).
CHRISTENSEN. Stochastic
THE ABSORPTIVE FUNCTION of the small intestine is probably dependent on the nature of the fluid flow in the lumen. Both transport (net flow in the longitudinal direction) and mixing (localized circulation or eddies) probably influence absorption by determining the degree of wall contact of the luminal contents. The small intestine induces these flows through its own movements: the fluid flows because of movement of the walls that confine it. The specific flow that results is, in part, a function of the specific wall motions. Thus, a description of wall motion is critical to a description of fluid flow. Most contractions in the duodenum are probably narrow ring contractions, although peristaltic, or moving-ring contractions, have been described (3). Whatever the nature of duodenal contractions, they are mainly manifest manometrically as single monophasic pressure peaks, appearing independently among pressure sensors separated by as little as 1 cm (4). It seems reasonable, then, to assume that these pressure peaks reflect ring contractions which are narrow (that is, short in the longitudinal direction) and do not appear to migrate within the resolution provided by sensors spaced at l-cm intervals (4). In the terminology of fluid mechanics, analytical models
METHODS
Experimental data. Pressure peaks were recorded from the third part of the duodenum after a milk meal in nine healthy human subjects. A bundle of four polyvinyl tubes, ID 1 mm, OD 3 mm, was passed by mouth to the third part of the duodenum. Latex rubber sleeves, covering lateral holes in the four tubes, served as balloons. The tubes were filled with water, and the proximal ends were attached to pressure transducers. The outputs from these transducers were simultaneously recorded on magnetic tape and on an inkwriting polygraph. The probes detected a series of pressure peaks, corresponding to contractions assumed to arise in description of the circular muscle layer. A quantitative 613
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614
SINGERMAN,
this series of peaks, necessary for statistical analysis, was made by examining times between contractions by a computer technique (10) which could identify pressure peaks in the transducer output. The frequency distribution of the resulting data, a time series of intercontractile periods, was plotted previously (4). This time series, although defined over a continuous time domain, was discretized and plotted in 0.3-s intervals. Since the electrical slow waves of the duodenum have a very regular period of 5 s, they tend to divide time into sequential 5-s intervals. In each interval, at a single point in space, the muscle exhibits either a contraction or a nocontraction. The frequency distribution of intercontractile times reflected this property. The distribution exhibited multiple peaks that were grouped around integer multiples of the electrical slow-wave period, 5 s (4). Times within the range of the first peak, centered at 5 s, represent two consecutive slow-wave cycles with contractions. Times in the second peak, centered at 10 s, represent two contraction cycles with a single no-contraction cycle between them. The remaining peaks represent two contraction cycles with two, three, or four no-contraction cycles between them. Thus, duodenal contractions at a single point may be represented as a series of 5-s intervals, the slow-wave cycles, within each of which a contraction may or may not occur. Run test. In approaching an analysis of the temporal distribution and dependence of contractions (c) and nocontractions (E) at a single point and in a regular series of 5-s intervals, the most accessible statistic is the percent of c’s an E’S. With each time interval representing one slowwave cycle, the percent of c’s corresponds to the percent of slow-wave cycles during which a contraction occurred, that is, percent activity. For the simplest hypothesis, each occurrence would be an independent event. The probability of c would then be assumed to be independent of previous events and constant in time. The frequency of occurrence of c would then approximate the probability P, of c:
P = P(c) =
no. c’s no. c’s + no. E’s
g2
GLOVER,
AND CHRISTENSEN
n(n - 2) 4( n1)
=
(3)
where n is the total number of events (i.e., n/2 of each type). The mean of this distribution is the number of runs expected if the assumption of independence is true. A result different from the expected value would indicate that the data are not described by the above distribution, that the assumption of independence is false. All experimental results were lumped together to form one data set. This set of data, a total of 7,572 times, was analyzed in groups of 100. The groups were taken sequentially, the first 100 times constituting the first group, the second 100 times the second group, and so on. For each group the median of the 100 times was computed. The times were then divided into two mutually exclusive subgroups by assigning the value 0 to all times less than the median and 1 to all times greater than the median. This assured an equal number of events of each type. This series of O’s and l’s was tested for independence. We hypothesized that no trend existed, that the series of observations were independent observations of the same random variable. We next defined what range of values of r to allow for acceptance of the hypothesis. A lower bound, rl , and an upper bound, r2 , were defined such that P(r
< r1> = ~r/2
(4)
P(r
> r2) = CY/~
(5)
and
When r -< rl or r > r2, the hypothesis was rejected. The value of cy, the level of significance of the test, was set at 0.05. The probability of finding r -< r 1 or r > r2 , indicating rejection of the hypothesis, would be 0.05. From the approximate distribution of r for n/2 = 50 we have, after solving equations 4 and 5 for r 1 and r2
P(r 2 41) = 0.025
(6)
P(r > 61) = 0.025
(7)
and
The simple assumption described above is one where the data represent results which are independent observations of the same random variable and, therefore, the probability of c or c does not change from one observation to the next. If these conditions of independence are not met, and the probability of c is not constant, trends will appear in the data. Since a contraction and a no-contraction are mutually exclusive events, a run may be defined as a sequence of similar occurrences which is preceded and followed by the opposite occurrences. The run test, based on the number of runs in a data set, can be used to test the hypothesis of independence or the existence of trends. For the case where there are equal numbers of each of the two mutually exclusive events, and the probability of occurrence of each is constant, the distribution of the number of runs, r, is approximated by a normal distribution (2) with mean and variance Y =;+1
MACAGNO,
(2)
Thus, we postulated that there is no trend and computed the number of runs. Any value of r such that r < 41 or r > 61 led to the rejection of the hypothesis at a level of significance of cu = 0.05. Trend test. In this test, a second indicator of independence, the data were considered as a series of intercontractile times, tl, t2, . . . , t, . A “reverse” in such a series occurs whenever t i > t j for i < j. If we define h ij , a counting parameter, as hij the total number
=
1;
ti
>
tj,
i < j
0;
ti
2804)
=
0.025
(13)
and
Thus, we hypothesized no trend and computed the number of reverses, R. Any value of R such that R < 2,145 or R > 2,804 led to the rejection of the hypothesis at a level of significance of cy = 0.05. Stochastic model. The preceding tests showed that contractions are not always independent observations of the same random variable (see RESULTS). To account for this indication of trends, a characteristic that will be assigned to the data set as a whole, temporal dependence of the occurrence of contractions was assumed. Probability of contraction was thus assumed to be conditional on the contractile history of the point being considered. An attempt to model contractile activity of a single point as a simple series of independent events having failed, we are led to the development of a more complex model. To construct such a stochastic model, some idea of the conditional probabilities was necessary. To this end, a 5-s time grid was superimposed on the computerized version of contractilepatterns(seeFig.l).Thisshiftedattentionfrom the exact location of a contraction to the probability of its occurrence within the current 5-s period. This suppresses one of the random features relating to the occurrence of a contraction, that is, the location of a contraction within the 5-s period. For the purpose of computing conditional probabilities, however, this was done. By marching along the grid, we put the data in a binary form. If a contraction occurred, a 1 was assigned to that period; if a no-contraction occurred, a 0 was assigned. Occasionally, one 5-s period would span two or three, but never more than three, intercontractile times. This gives the impression of more than one contraction per 5-s period. However, comparison of the intercontractile times to the original polygraph records indicates these are most likely due to experimental artifact, such as the recognition of false peaks due to slight irregularities in the pressure signal. If these, in fact, are true representations of wall motion, they were judged beyond the scope of the present study. This. coupled with their rare incidence of occurrence, in less than ‘1 % of slow-wave cycles, led us to ignore these special events and assign a 1 to any such slow-wave cycle. A Markov-type model that assumes dependence on a given number of previous slow-wave cycles was used. For a zeroeth-order model, the probability of a contraction, P(c), is the unconditional probability of a contraction or the percent activity. For the first-order model, information
for the previous slow-wave cycle is needed. Thus, the frequency of occurrence of combinations of the events “01” and “ 11” were counted. This gives, respectively, the probability of a contraction given a preceding no-contraction, P(c 1E), and the probability of a contraction given a preceding contraction, P(c 1 c). The second- and third-order models require similar information about the preceding two and three slow-wave cycles, respectively. These conditional probabilities are summarized in Table 1. The occurrence of duodenal contractions lends itself to an idealizati& in a model that divides time up- into 5-s units. A decision as to whether or not a contraction occurs for any given 5-s period is made. To illustrate how the stochastic model was constructed, a hypothetical case will be illustrated using a simple generator of random numbers. A die could be tossed once for each 5-s period. If the probability of a contraction were one-third, then whenever, for example, a 1 or 6 was thrown, a contraction would be assigned, and for 2, 3, 4, or 5, a no-contraction. It is at this point that the variable location of a contraction within the 5-s period will be reinstated. For the periods in which a contraction does occur, it is located within a 5-s period by the generation of a second random number. If first-order dependence is assumed, the current probability of a contraction must first be established. Suppose that P(c 1 E) = 36 and P(c 1 c) = s. Then, for each slow-wave cycle, what occurred during the previous cycle would be noted and the die range divided appropriately. If a no-contraction occurred previously, one could assign 1 to a contraction II (A) .- -,
-1 lllllllllllllllllll
lllllllllll~llll~lllt~lI1l’fllI1ll’l’
I
I I
i-5.0 Ill
I
I
Seconds Ill
1111111~
II I
I
1. Comparison of actual intraluminal-pressure with computer version of idealized contractions posed 5-s time grid (B). FIG.
1. Conditional
TABLE
Model Order
probabilities Conditional
Event*
0 e
1
C
12
3
0.26 0.23 0.34 0.32 0.36
ccc
0.18
ccc
0.24 0.35 0.30 0.36 0.37 0.34
ccc
0.41
ccc ccc
pendence
Probability of Contraction Given Conditional Event
0.19
CCC
=
for occurrence of contractions
ik CC cc
CCC
‘C
(A)
(cl record (C) and superim-
cc
CCC
contraction.
(6)
Occurrence of a contraction, e = occurrence of For zeroeth-order model, for which temporal is assumed, there is no conditional event.
a noinde-
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616 and 2, 3,4,5, or 6 to a no-contraction. When the conditional event was a contraction, one could use 1 or 6 for a contraction and 2, 3, 4, or 5 for a no-contraction. A similar procedure for higher-order models was employed. For the actual model, two basic schemes to generate quasirandom numbers were employed, subroutines RANDU and GAUSS. RANDU generates quasirandom real numbers approximately uniformly distributed between 0 and 1.0, and quasirandom integers between the limits 0 and Z3? GAUSS generates quasirandom numbers approximately normally distributed with a specified mean and variance. Specific details for both of these programs may be found in System/360 Scientific Subroutine Package, Version III, Programmer’s Manual. Subroutine RANDU was used to make the yes-or-no decision as to the occurrence of a contraction, and GAUSS was used to locate contractions within the slow-wave cycle. After a no-contraction, the model scheme moved to the next time period, after incrementing the intercontractile period by 5 s. For slow-wave cycles in which a contraction was assumed to occur, it was located somewhere within the 5-s period of the current slow-wave cycle. The exact location of the contraction was assumed to be a random variable. Thus, for each cycle in which a contraction was assumed to occur, a second random number between 0 and 5 s was generated to locate the contraction. The data set of intercontractile times generated by this model was analyzed in a manner like that reported earlier (4). The resulting frequency distributions were compared to the distributions of the real data (4) as a measure of the performance of this model.
SINGERMAN,
MACAGNO,
c-l I
8.0
GLOVER,
I 7 1
CHRISTENSEN
-REAL DA7-A -- - GE/VERATED DATA, THlffD OhPER
LA
6.0
AND
4.0 2.0 0
0
5
10
15
20
25
3’0
FIG. 2. Frequency
distribution of intercontractile times. Horizontal axis, divided into 0.9-s intervals, is marked at 5.0-s intervals. Vertical axis shows percentages of total numbers of intercontractile periods. Solid line represents real data. Dashed line represents data generated by third-order model.
-
REAL
DAnI
- - - G0V..RAE.. DABI, THIRD ORDER
n’ ”
1
2
3
4
1
2
3
4
I
5
6-30
--i--
: --6-30
1
8o B 4
me---
RESULTS
Temporal dependence of contractions. The data were divided into 75 groups, each group consisting of 100 intercontractile times. For both the run test and the trend test, independence was hypothesized. For the run test, the hypothesis was rejetted for nine of the groups and for the trend test it was rejected for 19 of the groups. The total number of groups for which the assumption of independence was rejected, counting those groups for which it was rejected by both tests only once, was 24. This implies that, based on the present analysis, the duodenum may be capable of operation in at least two modes. One of these modes would be where contractions at a single point were occurring independently of any pievious contractile activity at that point. The second mode would then be the case where a dependency on the contractile history does exist. Generation of intercontractile data. In the generation of simulated data with the stochastic model and comparison of these data with the real data, the frequency distribution of real intercontractile times was truncated after the fifth peak, the last distinguishable peak. In modeling the numbers of sequential rest cycles and sequential contractions, the frequency distributions used were those presented in the previous report (4). These distributions are reproduced in Figs. 2 and 3. An approximately normal distribution was used to locate contractions within the low-wave cycle. The selection of a normal distribution was arbitrary, but seemed reasonable in the absence of any data about the actual kind of distribution present. This normal distribiution was arbitrarly
6040on
O
FIG. 3. A: frequency distribution of numbers of sequential rest cycles. These are sequential slow-wave cycles without contractions. Horizontal axis shows number of rest cycles occurring in a sequence. Vertical axis shows percentage of total number of groups containing that number of rest cycles. Solid line represents data generated by third-order model. B: frequency distribution of numbers of sequential contractions. These are sequential slow-wave cycles occupied by contractions. Axes and lines are same as in A.
centered in the slow-wave cycle with the mean at 2.5 s. For the initial attempt, a variance of 1 .O was tried. This led to’a satisfactory model. Any random time used to locate a contraction that fell outside the range of the slow-wave cycle (less than 0 or greater than 5 s) was ignored. For an n (2.5, 1.0) distribution this amounts to about 1.2 % of the times generated. The frequency distribution of intercontractile times of the generated data was compared to that of the real data. It was hypothesized that the real and generated distributions were the same and the Chi-square goodness-of-fit test (level of significance a! = 0.05) was used. For this test, there are four degrees of freedom. Thus, for the hypothesis to be ac-
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CONTRACTIONS
IN
DUODENUM
617
cepted,
the computed Chi-squared value, x2, must satisfy 9.488. The hypothesis was not accepted until x2 < x2*.05 = the third-order model was tried, for which x2 = 8.740. A true third-order Markov process requires that the probability conditional on the previous three slow-wave cycles is unchanged by increasing the dependency further. That is, if Ci is the event that occurs in the ith slow-wave cycle, with Co being the event for the first cycle, then
P[Ci
(
Ci-1,
Ci-2,
Ci-3
1 =
P[Ci
1 Ci-1,
Ci-2
l
l
l
3 CO]
(l4)
As this requirement was not verified, we have called this a Markov-type model (8). The frequency distributions of sequential rest cycles and contractions for the data generated with the third-order model were computed and compared to the real frequency distributions using similar tests. For these tests, there are five degrees of freedom, and at a level of significance a = 0.05, the region of acceptance is x2 < Chi-squared values for x20.05 = 11.070. The computed the sequential rest cycles and sequential contractions were 8.192 and 6.267, respectively. Graphs of the real and generated data for these three frequency distributions appear in Figs. 1 and 2. DISCUSSION
The subjects were fasted 1 Z-18 h and fed a standard meal of 0.5 pint of whole milk at the beginning of each recording session (4). Thus, the data represent a potentially nonstationary phenomenon. In the dog fasted for 10 h or more, a cycle of three modes of contractile activity occurs. A period of no activity is followed by a period of intermittent activity, which leads to a prolonged burst of contractions (9). This pattern is promptly interrupted by feeding, but there is no certainty that a steady situation develops. The data under investigation here may therefore be representative of more than one basic process. Also, the analysis showed regions where no trends were indicated and regions where trends were indicated. Yet, for purposes of modeling, the data were treated as homogeneous, and temporal dependence was assumed to exist throughout the data set. This technique may, therefore, mask a more complex process. The fact that a model which exhibits trends fits the data indicated only that trends exist at least in some of the groups tested, but not necessarily throughout.
Any improved model that is to describe these data will have to include a more detailed analysis, one that does not lump all the data together. The assumption that spikes are normally distributed within the slow-wave cycle is not conclusive. There is no evidence that this model, although satisfactory, is unique. The quasirandom number generators used in this study are not exact, but these generators are consistent with the accuracy of the data. What is obtained is a working model capable of reproducing certain quantitative measures of duodenal contractions in individuals recently fed after a relatively long period of fast. Before more information can be obtained from the model, a more detailed analysis of the real data will be required. The model is an indication of the type of behavior that duodenal contractions may exhibit. It quantifies this behavior in a manner suitable for use as boundary conditions for either physical or analytical models. Also, the model generates data that share certain predefined characteristics with the real data. These are characteristics we have judged as important relative to the pumping and mixing mechanisms of the duodenum. This is a first step in determining both qualitative characteristics (the nature of the temporal dependence) and quantitative features (conditional probabilities) of duodenal behavior. At any one point in the duodenum, it is reasonable to suppose that the choice between contraction and no-contraction (in a single slow-wave cycle) is affected by physiological control systems. The data indicate that when contractions have occurred in three successive slow-wave cycles, the probability of contraction in the next, fourth cycle, is high, whereas three successive no-contractions result in a relatively low probability of contraction in the fourth cycle. Thus, these controls would act over a period of at least four slowwave cycles, 20 s in man. This work was supported by National Institutes of Health Research Grant AM 08901 and Research Career Development Award AM 20547 and in part by Veterans Administration research funds. Requests for reprints should be sent to: J. Christensen, Division of Gastroenterology, Dept. of Internal Medicine, University Hospitals, Iowa City, Iowa 52242. Received
for publication
26 July
1974.
REFERENCES 1. BECK,
2. 3.
4.
5.
I. T., R. D. MCKENNA, G. PETERS, J. SIDOROV, AND H. STROWEZYASKI. Pressure studies in the normal human jejunum. Am. J. Digest. Diseases 10 : 436-448, 1965. BENDAT, J. S., AND A. G. PIERSOL. Measurement and Analysis of Random Data. New York : Wiley, 1966. CHRISTENSEN, J. Medical progress. The controls of gastroinintestinal movements: some old and new views. New Engl. J. Med. 285: 85-98, 1971. CHRISTENSEN, J., J. R. GLOVER, E. 0. MACAGNO, R. B. SINGERMAN, AND N. W. WEISBRODT. Statistics of contractions at a point in the human duodenum. Am. J. Physiol. 221: 1818-1823, 1971. FOULK, W. T., C. F. CODE, C. G. MORLOCK, AND J. A. BARGEN.
A study duodenum
of the and
Gastroenterology
motility upper
patterns and the basic rhythm part of the jejunum of human 26 : 601-6 11, 1954.
in the beings.
6.
7. 8.
9.
10.
G., J. D. WAYE, L. A. WETNGERTEN, AND H. De The patterns of simultaneous intraluminal pressure changes in the human proximal small intestine. Gastroenterology 47 : 258-268, 1964. KE~ENTER, J., AND N. G. KOCK. Motility of the human small intestine. Acta Chir. Stand 119 : 430-438, 1960. PARZEN, E. Stochastic Processes. San Francisco: Holden-Day, 1962, p. 187-275. REINKE, D. A., A. H. ROSENBAUM, AND R. BENNETT. Patterns of dog gastrointestinal contractile activity monitored in vivo with extraluminal force transducers. Am. J. Digest. Diseases 12 : 113-141, 1967. SINGERMAN, R. B. J., AND J. R. GLOVER. Computer Recognition of Contractions in the Small Intestine. Iowa Institute of Hydraulic Research Report. no. 134, 1971. FRIEDMAN, JANOWITZ.
Downloaded from www.physiology.org/journal/ajplegacy by ${individualUser.givenNames} ${individualUser.surname} (129.186.138.035) on January 18, 2019.
Stochastic point
model
of contractions
at a
in the duodenum ROBERT B. J. SINGERMAN, EN20 0. MACAGNO, JOHN R. GLOVE& AND JAMES CHRISTENSEN Gastroenterology Research Laboratory, Department of Internal Medicine, and Iowa Institute of Hydraulic University of Iowa Colleges of Medicine and Engineering, Iowa City, Iowa 52242
SINGERMAN,
ROBERT
MACAGNO, JOHN R. model of contractions at a point in the duodenum. Am. J. Physiol. 229(3) : 613-617. 1975.Contractions at one point in the human duodenum were studied as a time series. Manometric records were made over long time periods from the duodenum in fed human subjects. A 5-s grid was superimposed on the time axis of the records. Each 5-s interval was treated as a slow-wave cycle within which either a contraction or a no-contraction could occur. The resulting series of alternating runs of contractions and no-contractions was tested for the existence of trends. Trends were found indicating possible temporal dependence. A Markov-type model was used to try to generate data similar to the real data. Success was achieved by a model that assumed a probability of contraction dependent on the three previous slow-wave cycles. The frequency distributions obtained from the real and generated data were compared using Chi-square goodness-of-fit tests and found to be statistically similar. The correlations in time found for the contractions might be due to a time dependency in the controls for contraction over four successive slow-wave periods, 20 s in humans.
&OVER,
AND JAMES
intestinal
motility
B.
J.,
ENZO
0.
Research,
of fluid flow require, for a particular solution, a description of the specific boundary conditions involved. In this case, an analytical model consists simply of the equations of motion and continuity for an incompressible fluid. The boundary conditions in this case are the wall movements, the position of the walls as a function of space and time. The work reported here is part of a long-term project to constitute an analytical model of small-intestinal flow. As a requirement for that goal, we need a quantitative description of boundary conditions, or wall movements. In the course of seeking this description, we observed that, in the fed state, there is no obvious way to predict the occurrence of contractions at a single point in the duodenum from one slow-wave cycle to the next. That is, the choice at a single point in the duodenum between a contraction cycle and a rest cycle (4) appears to be random (1, 5-7). We proposed that this choice may be, for any single slowwave cycle at any single point in the duodenum, completely independent of the occurrence of contractions in previous slow-wave cycles at that same point. This hypothesis was tested by standard analytical methods and, as described below, found to be unsupported. We therefore examined further the demonstrated temporal dependence of the occurrence of contractions to discover the degree of temporal dependence. In this process, we examined the hypothesis that this temporal dependence can be modeled by a Markov-type process. This hypothesis was found to be acceptable. This study makes use of data gathered and described previously (4). The experimental procedure is detailed in earlier papers (4, 10).
CHRISTENSEN. Stochastic
THE ABSORPTIVE FUNCTION of the small intestine is probably dependent on the nature of the fluid flow in the lumen. Both transport (net flow in the longitudinal direction) and mixing (localized circulation or eddies) probably influence absorption by determining the degree of wall contact of the luminal contents. The small intestine induces these flows through its own movements: the fluid flows because of movement of the walls that confine it. The specific flow that results is, in part, a function of the specific wall motions. Thus, a description of wall motion is critical to a description of fluid flow. Most contractions in the duodenum are probably narrow ring contractions, although peristaltic, or moving-ring contractions, have been described (3). Whatever the nature of duodenal contractions, they are mainly manifest manometrically as single monophasic pressure peaks, appearing independently among pressure sensors separated by as little as 1 cm (4). It seems reasonable, then, to assume that these pressure peaks reflect ring contractions which are narrow (that is, short in the longitudinal direction) and do not appear to migrate within the resolution provided by sensors spaced at l-cm intervals (4). In the terminology of fluid mechanics, analytical models
METHODS
Experimental data. Pressure peaks were recorded from the third part of the duodenum after a milk meal in nine healthy human subjects. A bundle of four polyvinyl tubes, ID 1 mm, OD 3 mm, was passed by mouth to the third part of the duodenum. Latex rubber sleeves, covering lateral holes in the four tubes, served as balloons. The tubes were filled with water, and the proximal ends were attached to pressure transducers. The outputs from these transducers were simultaneously recorded on magnetic tape and on an inkwriting polygraph. The probes detected a series of pressure peaks, corresponding to contractions assumed to arise in description of the circular muscle layer. A quantitative 613
Downloaded from www.physiology.org/journal/ajplegacy by ${individualUser.givenNames} ${individualUser.surname} (129.186.138.035) on January 18, 2019.
614
SINGERMAN,
this series of peaks, necessary for statistical analysis, was made by examining times between contractions by a computer technique (10) which could identify pressure peaks in the transducer output. The frequency distribution of the resulting data, a time series of intercontractile periods, was plotted previously (4). This time series, although defined over a continuous time domain, was discretized and plotted in 0.3-s intervals. Since the electrical slow waves of the duodenum have a very regular period of 5 s, they tend to divide time into sequential 5-s intervals. In each interval, at a single point in space, the muscle exhibits either a contraction or a nocontraction. The frequency distribution of intercontractile times reflected this property. The distribution exhibited multiple peaks that were grouped around integer multiples of the electrical slow-wave period, 5 s (4). Times within the range of the first peak, centered at 5 s, represent two consecutive slow-wave cycles with contractions. Times in the second peak, centered at 10 s, represent two contraction cycles with a single no-contraction cycle between them. The remaining peaks represent two contraction cycles with two, three, or four no-contraction cycles between them. Thus, duodenal contractions at a single point may be represented as a series of 5-s intervals, the slow-wave cycles, within each of which a contraction may or may not occur. Run test. In approaching an analysis of the temporal distribution and dependence of contractions (c) and nocontractions (E) at a single point and in a regular series of 5-s intervals, the most accessible statistic is the percent of c’s an E’S. With each time interval representing one slowwave cycle, the percent of c’s corresponds to the percent of slow-wave cycles during which a contraction occurred, that is, percent activity. For the simplest hypothesis, each occurrence would be an independent event. The probability of c would then be assumed to be independent of previous events and constant in time. The frequency of occurrence of c would then approximate the probability P, of c:
P = P(c) =
no. c’s no. c’s + no. E’s
g2
GLOVER,
AND CHRISTENSEN
n(n - 2) 4( n1)
=
(3)
where n is the total number of events (i.e., n/2 of each type). The mean of this distribution is the number of runs expected if the assumption of independence is true. A result different from the expected value would indicate that the data are not described by the above distribution, that the assumption of independence is false. All experimental results were lumped together to form one data set. This set of data, a total of 7,572 times, was analyzed in groups of 100. The groups were taken sequentially, the first 100 times constituting the first group, the second 100 times the second group, and so on. For each group the median of the 100 times was computed. The times were then divided into two mutually exclusive subgroups by assigning the value 0 to all times less than the median and 1 to all times greater than the median. This assured an equal number of events of each type. This series of O’s and l’s was tested for independence. We hypothesized that no trend existed, that the series of observations were independent observations of the same random variable. We next defined what range of values of r to allow for acceptance of the hypothesis. A lower bound, rl , and an upper bound, r2 , were defined such that P(r
< r1> = ~r/2
(4)
P(r
> r2) = CY/~
(5)
and
When r -< rl or r > r2, the hypothesis was rejected. The value of cy, the level of significance of the test, was set at 0.05. The probability of finding r -< r 1 or r > r2 , indicating rejection of the hypothesis, would be 0.05. From the approximate distribution of r for n/2 = 50 we have, after solving equations 4 and 5 for r 1 and r2
P(r 2 41) = 0.025
(6)
P(r > 61) = 0.025
(7)
and
The simple assumption described above is one where the data represent results which are independent observations of the same random variable and, therefore, the probability of c or c does not change from one observation to the next. If these conditions of independence are not met, and the probability of c is not constant, trends will appear in the data. Since a contraction and a no-contraction are mutually exclusive events, a run may be defined as a sequence of similar occurrences which is preceded and followed by the opposite occurrences. The run test, based on the number of runs in a data set, can be used to test the hypothesis of independence or the existence of trends. For the case where there are equal numbers of each of the two mutually exclusive events, and the probability of occurrence of each is constant, the distribution of the number of runs, r, is approximated by a normal distribution (2) with mean and variance Y =;+1
MACAGNO,
(2)
Thus, we postulated that there is no trend and computed the number of runs. Any value of r such that r < 41 or r > 61 led to the rejection of the hypothesis at a level of significance of cu = 0.05. Trend test. In this test, a second indicator of independence, the data were considered as a series of intercontractile times, tl, t2, . . . , t, . A “reverse” in such a series occurs whenever t i > t j for i < j. If we define h ij , a counting parameter, as hij the total number
=
1;
ti
>
tj,
i < j
0;
ti
2804)
=
0.025
(13)
and
Thus, we hypothesized no trend and computed the number of reverses, R. Any value of R such that R < 2,145 or R > 2,804 led to the rejection of the hypothesis at a level of significance of cy = 0.05. Stochastic model. The preceding tests showed that contractions are not always independent observations of the same random variable (see RESULTS). To account for this indication of trends, a characteristic that will be assigned to the data set as a whole, temporal dependence of the occurrence of contractions was assumed. Probability of contraction was thus assumed to be conditional on the contractile history of the point being considered. An attempt to model contractile activity of a single point as a simple series of independent events having failed, we are led to the development of a more complex model. To construct such a stochastic model, some idea of the conditional probabilities was necessary. To this end, a 5-s time grid was superimposed on the computerized version of contractilepatterns(seeFig.l).Thisshiftedattentionfrom the exact location of a contraction to the probability of its occurrence within the current 5-s period. This suppresses one of the random features relating to the occurrence of a contraction, that is, the location of a contraction within the 5-s period. For the purpose of computing conditional probabilities, however, this was done. By marching along the grid, we put the data in a binary form. If a contraction occurred, a 1 was assigned to that period; if a no-contraction occurred, a 0 was assigned. Occasionally, one 5-s period would span two or three, but never more than three, intercontractile times. This gives the impression of more than one contraction per 5-s period. However, comparison of the intercontractile times to the original polygraph records indicates these are most likely due to experimental artifact, such as the recognition of false peaks due to slight irregularities in the pressure signal. If these, in fact, are true representations of wall motion, they were judged beyond the scope of the present study. This. coupled with their rare incidence of occurrence, in less than ‘1 % of slow-wave cycles, led us to ignore these special events and assign a 1 to any such slow-wave cycle. A Markov-type model that assumes dependence on a given number of previous slow-wave cycles was used. For a zeroeth-order model, the probability of a contraction, P(c), is the unconditional probability of a contraction or the percent activity. For the first-order model, information
for the previous slow-wave cycle is needed. Thus, the frequency of occurrence of combinations of the events “01” and “ 11” were counted. This gives, respectively, the probability of a contraction given a preceding no-contraction, P(c 1E), and the probability of a contraction given a preceding contraction, P(c 1 c). The second- and third-order models require similar information about the preceding two and three slow-wave cycles, respectively. These conditional probabilities are summarized in Table 1. The occurrence of duodenal contractions lends itself to an idealizati& in a model that divides time up- into 5-s units. A decision as to whether or not a contraction occurs for any given 5-s period is made. To illustrate how the stochastic model was constructed, a hypothetical case will be illustrated using a simple generator of random numbers. A die could be tossed once for each 5-s period. If the probability of a contraction were one-third, then whenever, for example, a 1 or 6 was thrown, a contraction would be assigned, and for 2, 3, 4, or 5, a no-contraction. It is at this point that the variable location of a contraction within the 5-s period will be reinstated. For the periods in which a contraction does occur, it is located within a 5-s period by the generation of a second random number. If first-order dependence is assumed, the current probability of a contraction must first be established. Suppose that P(c 1 E) = 36 and P(c 1 c) = s. Then, for each slow-wave cycle, what occurred during the previous cycle would be noted and the die range divided appropriately. If a no-contraction occurred previously, one could assign 1 to a contraction II (A) .- -,
-1 lllllllllllllllllll
lllllllllll~llll~lllt~lI1l’fllI1ll’l’
I
I I
i-5.0 Ill
I
I
Seconds Ill
1111111~
II I
I
1. Comparison of actual intraluminal-pressure with computer version of idealized contractions posed 5-s time grid (B). FIG.
1. Conditional
TABLE
Model Order
probabilities Conditional
Event*
0 e
1
C
12
3
0.26 0.23 0.34 0.32 0.36
ccc
0.18
ccc
0.24 0.35 0.30 0.36 0.37 0.34
ccc
0.41
ccc ccc
pendence
Probability of Contraction Given Conditional Event
0.19
CCC
=
for occurrence of contractions
ik CC cc
CCC
‘C
(A)
(cl record (C) and superim-
cc
CCC
contraction.
(6)
Occurrence of a contraction, e = occurrence of For zeroeth-order model, for which temporal is assumed, there is no conditional event.
a noinde-
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616 and 2, 3,4,5, or 6 to a no-contraction. When the conditional event was a contraction, one could use 1 or 6 for a contraction and 2, 3, 4, or 5 for a no-contraction. A similar procedure for higher-order models was employed. For the actual model, two basic schemes to generate quasirandom numbers were employed, subroutines RANDU and GAUSS. RANDU generates quasirandom real numbers approximately uniformly distributed between 0 and 1.0, and quasirandom integers between the limits 0 and Z3? GAUSS generates quasirandom numbers approximately normally distributed with a specified mean and variance. Specific details for both of these programs may be found in System/360 Scientific Subroutine Package, Version III, Programmer’s Manual. Subroutine RANDU was used to make the yes-or-no decision as to the occurrence of a contraction, and GAUSS was used to locate contractions within the slow-wave cycle. After a no-contraction, the model scheme moved to the next time period, after incrementing the intercontractile period by 5 s. For slow-wave cycles in which a contraction was assumed to occur, it was located somewhere within the 5-s period of the current slow-wave cycle. The exact location of the contraction was assumed to be a random variable. Thus, for each cycle in which a contraction was assumed to occur, a second random number between 0 and 5 s was generated to locate the contraction. The data set of intercontractile times generated by this model was analyzed in a manner like that reported earlier (4). The resulting frequency distributions were compared to the distributions of the real data (4) as a measure of the performance of this model.
SINGERMAN,
MACAGNO,
c-l I
8.0
GLOVER,
I 7 1
CHRISTENSEN
-REAL DA7-A -- - GE/VERATED DATA, THlffD OhPER
LA
6.0
AND
4.0 2.0 0
0
5
10
15
20
25
3’0
FIG. 2. Frequency
distribution of intercontractile times. Horizontal axis, divided into 0.9-s intervals, is marked at 5.0-s intervals. Vertical axis shows percentages of total numbers of intercontractile periods. Solid line represents real data. Dashed line represents data generated by third-order model.
-
REAL
DAnI
- - - G0V..RAE.. DABI, THIRD ORDER
n’ ”
1
2
3
4
1
2
3
4
I
5
6-30
--i--
: --6-30
1
8o B 4
me---
RESULTS
Temporal dependence of contractions. The data were divided into 75 groups, each group consisting of 100 intercontractile times. For both the run test and the trend test, independence was hypothesized. For the run test, the hypothesis was rejetted for nine of the groups and for the trend test it was rejected for 19 of the groups. The total number of groups for which the assumption of independence was rejected, counting those groups for which it was rejected by both tests only once, was 24. This implies that, based on the present analysis, the duodenum may be capable of operation in at least two modes. One of these modes would be where contractions at a single point were occurring independently of any pievious contractile activity at that point. The second mode would then be the case where a dependency on the contractile history does exist. Generation of intercontractile data. In the generation of simulated data with the stochastic model and comparison of these data with the real data, the frequency distribution of real intercontractile times was truncated after the fifth peak, the last distinguishable peak. In modeling the numbers of sequential rest cycles and sequential contractions, the frequency distributions used were those presented in the previous report (4). These distributions are reproduced in Figs. 2 and 3. An approximately normal distribution was used to locate contractions within the low-wave cycle. The selection of a normal distribution was arbitrary, but seemed reasonable in the absence of any data about the actual kind of distribution present. This normal distribiution was arbitrarly
6040on
O
FIG. 3. A: frequency distribution of numbers of sequential rest cycles. These are sequential slow-wave cycles without contractions. Horizontal axis shows number of rest cycles occurring in a sequence. Vertical axis shows percentage of total number of groups containing that number of rest cycles. Solid line represents data generated by third-order model. B: frequency distribution of numbers of sequential contractions. These are sequential slow-wave cycles occupied by contractions. Axes and lines are same as in A.
centered in the slow-wave cycle with the mean at 2.5 s. For the initial attempt, a variance of 1 .O was tried. This led to’a satisfactory model. Any random time used to locate a contraction that fell outside the range of the slow-wave cycle (less than 0 or greater than 5 s) was ignored. For an n (2.5, 1.0) distribution this amounts to about 1.2 % of the times generated. The frequency distribution of intercontractile times of the generated data was compared to that of the real data. It was hypothesized that the real and generated distributions were the same and the Chi-square goodness-of-fit test (level of significance a! = 0.05) was used. For this test, there are four degrees of freedom. Thus, for the hypothesis to be ac-
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CONTRACTIONS
IN
DUODENUM
617
cepted,
the computed Chi-squared value, x2, must satisfy 9.488. The hypothesis was not accepted until x2 < x2*.05 = the third-order model was tried, for which x2 = 8.740. A true third-order Markov process requires that the probability conditional on the previous three slow-wave cycles is unchanged by increasing the dependency further. That is, if Ci is the event that occurs in the ith slow-wave cycle, with Co being the event for the first cycle, then
P[Ci
(
Ci-1,
Ci-2,
Ci-3
1 =
P[Ci
1 Ci-1,
Ci-2
l
l
l
3 CO]
(l4)
As this requirement was not verified, we have called this a Markov-type model (8). The frequency distributions of sequential rest cycles and contractions for the data generated with the third-order model were computed and compared to the real frequency distributions using similar tests. For these tests, there are five degrees of freedom, and at a level of significance a = 0.05, the region of acceptance is x2 < Chi-squared values for x20.05 = 11.070. The computed the sequential rest cycles and sequential contractions were 8.192 and 6.267, respectively. Graphs of the real and generated data for these three frequency distributions appear in Figs. 1 and 2. DISCUSSION
The subjects were fasted 1 Z-18 h and fed a standard meal of 0.5 pint of whole milk at the beginning of each recording session (4). Thus, the data represent a potentially nonstationary phenomenon. In the dog fasted for 10 h or more, a cycle of three modes of contractile activity occurs. A period of no activity is followed by a period of intermittent activity, which leads to a prolonged burst of contractions (9). This pattern is promptly interrupted by feeding, but there is no certainty that a steady situation develops. The data under investigation here may therefore be representative of more than one basic process. Also, the analysis showed regions where no trends were indicated and regions where trends were indicated. Yet, for purposes of modeling, the data were treated as homogeneous, and temporal dependence was assumed to exist throughout the data set. This technique may, therefore, mask a more complex process. The fact that a model which exhibits trends fits the data indicated only that trends exist at least in some of the groups tested, but not necessarily throughout.
Any improved model that is to describe these data will have to include a more detailed analysis, one that does not lump all the data together. The assumption that spikes are normally distributed within the slow-wave cycle is not conclusive. There is no evidence that this model, although satisfactory, is unique. The quasirandom number generators used in this study are not exact, but these generators are consistent with the accuracy of the data. What is obtained is a working model capable of reproducing certain quantitative measures of duodenal contractions in individuals recently fed after a relatively long period of fast. Before more information can be obtained from the model, a more detailed analysis of the real data will be required. The model is an indication of the type of behavior that duodenal contractions may exhibit. It quantifies this behavior in a manner suitable for use as boundary conditions for either physical or analytical models. Also, the model generates data that share certain predefined characteristics with the real data. These are characteristics we have judged as important relative to the pumping and mixing mechanisms of the duodenum. This is a first step in determining both qualitative characteristics (the nature of the temporal dependence) and quantitative features (conditional probabilities) of duodenal behavior. At any one point in the duodenum, it is reasonable to suppose that the choice between contraction and no-contraction (in a single slow-wave cycle) is affected by physiological control systems. The data indicate that when contractions have occurred in three successive slow-wave cycles, the probability of contraction in the next, fourth cycle, is high, whereas three successive no-contractions result in a relatively low probability of contraction in the fourth cycle. Thus, these controls would act over a period of at least four slowwave cycles, 20 s in man. This work was supported by National Institutes of Health Research Grant AM 08901 and Research Career Development Award AM 20547 and in part by Veterans Administration research funds. Requests for reprints should be sent to: J. Christensen, Division of Gastroenterology, Dept. of Internal Medicine, University Hospitals, Iowa City, Iowa 52242. Received
for publication
26 July
1974.
REFERENCES 1. BECK,
2. 3.
4.
5.
I. T., R. D. MCKENNA, G. PETERS, J. SIDOROV, AND H. STROWEZYASKI. Pressure studies in the normal human jejunum. Am. J. Digest. Diseases 10 : 436-448, 1965. BENDAT, J. S., AND A. G. PIERSOL. Measurement and Analysis of Random Data. New York : Wiley, 1966. CHRISTENSEN, J. Medical progress. The controls of gastroinintestinal movements: some old and new views. New Engl. J. Med. 285: 85-98, 1971. CHRISTENSEN, J., J. R. GLOVER, E. 0. MACAGNO, R. B. SINGERMAN, AND N. W. WEISBRODT. Statistics of contractions at a point in the human duodenum. Am. J. Physiol. 221: 1818-1823, 1971. FOULK, W. T., C. F. CODE, C. G. MORLOCK, AND J. A. BARGEN.
A study duodenum
of the and
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in the beings.
6.
7. 8.
9.
10.
G., J. D. WAYE, L. A. WETNGERTEN, AND H. De The patterns of simultaneous intraluminal pressure changes in the human proximal small intestine. Gastroenterology 47 : 258-268, 1964. KE~ENTER, J., AND N. G. KOCK. Motility of the human small intestine. Acta Chir. Stand 119 : 430-438, 1960. PARZEN, E. Stochastic Processes. San Francisco: Holden-Day, 1962, p. 187-275. REINKE, D. A., A. H. ROSENBAUM, AND R. BENNETT. Patterns of dog gastrointestinal contractile activity monitored in vivo with extraluminal force transducers. Am. J. Digest. Diseases 12 : 113-141, 1967. SINGERMAN, R. B. J., AND J. R. GLOVER. Computer Recognition of Contractions in the Small Intestine. Iowa Institute of Hydraulic Research Report. no. 134, 1971. FRIEDMAN, JANOWITZ.
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