Error analysis of classic colonic transit time
estimates
Michel
Bouchoucha1 and
S. Randall
Thomas2
1 Hôpital Laennec, Laboratory of Digestive Physiology,
75007 Paris; and 2 Institut National de la Santé et de
la Recherche Médicale Unité 467, Necker Faculty of
Medicine, 75730 Paris Cedex 15, France
 |
ABSTRACT |
Estimates of colonic transit times (CTT) through the three colonic
segments, right colon, left colon, and rectosigmoid, are commonly based
on radiopaque markers. For a given segment, CTT is usually calculated
from just the number of markers visible in that segment on abdominal
X-rays. This procedure is only strictly valid for the theoretical, but
unrealistic, case of continuous marker ingestion (i.e., not for a
single or once-daily ingestion). CTT was analyzed using the usual
estimate of the mean CTT of one marker and also using a new, more
realistic estimate based on the kinetic coefficients of a
three-compartment colonic model. We directly compared our compartmental
approach to classic CTT estimates by double-marker studies in six
patients. We also retrospectively studied CTT in 148 healthy control
subjects (83 males, 65 females) and 1,309 subjects with functional
bowel disorders (irritable bowel syndrome or constipation). Compared
with the compartmental estimates, the classic approach systematically
underestimates CTT in both populations, i.e., in patients and in
healthy control subjects. The relative error could easily reach 100%
independent of the site of colonic transit delay. The normal values of
total CTT are then 44.3 ± 29.3 instead of 30.1 ± 23.6 h for males and 68.2 ± 54.4 instead of 47.1 ± 28.2 h
for females.
compartment model
 |
INTRODUCTION |
COLONIC TRANSIT
TIMES (CTTs) through successive colonic segments are commonly
estimated from the distribution of radiopaque markers visible on
successive daily abdominal X-rays taken after a single ingestion of a
fixed number of markers (2, 9). This method
was useful for objective explanation of constipation and defined three
types of delay: right colon (12), left colon (7), and rectosigmoid (8) delays. In previous
work (3), to limit radiation exposure, we demonstrated
that this method was analogous to daily ingestion of markers over six
days followed by a single abdominal X-ray on the seventh morning. Under
the assumptions that marker ingestion is continuous and that marker transit has reached a steady state on the day of the X-ray
(3), the usual estimate of mean CTT of a single marker in
a segment i is represented by CTTi
(or its reciprocal, the rate constant ki)
and is simply given by
|
(1)
|
where ni is the number of
markers seen on the X-ray in segment i, N is the
number of markers ingested each day (N = 12 in the
present study), and
T is the time interval between
consecutive ingestions of markers (
T = 24 h in
the present study). We call this the classic single-film estimate of
CTTi. It is subject to several kinds of
systematic error, of which the following are particularly
characteristic, namely, 1) the steady-state assumption may
not be respected at the time of X-ray (typically on the seventh day of
marker ingestion) in patients with delayed colonic transit (constipated
states), 2) in patients with very fast transit (diarrheic states, in which some regions have no markers) the time interval between two ingestions is too great, and 3) this equation is
only strictly applicable for continuous ingestion of markers, whereas markers are actually ingested in a bolus once a day.
To overcome these limitations, we previously developed (4)
improved estimates of segmental CTT and rate constants for transit between adjacent segments based on a three-compartment model. This
method takes account of the bolus nature of the ingestion and does not
assume that marker distribution has reached a steady state. We refer to
these hereafter as compartmental estimates. Nevertheless, this model
was never verified by direct comparison of its predictions with those
of the classic techniques in the same patients.
In the present study, we use our more accurate model to evaluate
the error in classic estimates, i.e., those based on Eq. 1.
We also verified the pertinence of the model using double-marker studies and daily X-rays on a small group of patients.
 |
METHODS |
Double-Marker Test of Model Predictions
Adequacy of our three-compartment model for prediction of marker
evolution along the colon was evaluated in a study using two types of
radiopaque markers in four women and two men. These patients were given
20 markers of one type on the first day only and 10 markers of a second
type every day for 6 days. The only difference between the markers was
their shape, and their use was randomized in the different patients to
avoid systematic errors. X-rays were taken just before marker ingestion
each morning (except day 1) until the sixth or seventh day.
Markers were localized and counted in the different segments of the
large bowel according to bony landmarks (2). Three zones
of interest were defined: the right colon (ascending colon and right
part of the transverse), the left colon (descending colon and left part
of the transverse), and the rectosigmoid area.
For comparison of our compartmental approach with the classic technique
of single ingestion and daily X-rays (2), first-order rate
constants of segmental transit were estimated in two ways. First, for
the classic single-ingestion technique (as in Ref. 2; see also
Eq. 10 below), CTT in compartment i was estimated using
|
(2)
|
where
T is the time between X-rays (24 h),
N is the number of markers ingested on day 1, and
nij is the number of markers counted in region
i (right colon, left colon, or rectosigmoid segment) on day
j. The classic single-ingestion estimates of the first-order
rate constants are then the reciprocals of these
CTTi values (see Eqs. 8 and 9).
Second, estimates of the rate constants according to our
three-compartment model were made by numerical iteration as described in Compartmental estimates of the rate constants and
CTT. These two sets of rate constants were then used to
predict the number of markers that would be present in each colonic
segment, and errors were calculated for both methods according to
|
(3)
|
where nij is the number of markers
counted in region i on day j,
pij is the predicted number of markers, and
nX-rays is the number of X-rays taken on a given
patient (i.e., 5 or 6).
Retrospective Study
Population.
For the retrospective study, we used data obtained as classic
single-film estimates of CTT in patients with irritable bowel syndrome
or constipation (1,309 subjects; mean age 45.3 yrs; age range
17-80 yrs; 345 males and 964 females) and subjects used as a
control population (148 subjects; 83 males with mean age 36.5 yrs and
age range 20-61 yrs and 65 females with mean age 34.5 yrs and age
range 18-57 yrs).
Experimental procedure.
In all subjects, CTT was estimated using a previously described
technique (3). Briefly, 12 radiopaque markers within a gelatin capsule were ingested from day 1 to day 6 at 9:00 AM. A plain film of the abdomen was taken on the seventh day at
9:00 AM. Markers were counted as described in the previous section.
Segmental and total colorectal classic single-film transit times were
calculated using Eq. 1, according to the distribution of
markers counted in the different segments of bowel. Total colonic classic transit time was then taken as the sum of the three segmental transit times.
Data Analysis Using a Compartmental Model
In our previous study modeling the transit of markers through
the colon, we used a three-compartment model (Fig.
1) representing the right colon, the left
colon, and the rectosigmoid area (4). Three rate
constants, k1,
k2, and k3,
represented the net result of propagation and back-propagation between
consecutive compartments. Defining N as the number of
markers ingested daily and
n1(t), n2(t), and
n3(t) as the number of markers
situated in the right colon, the left colon, and the rectosigmoid area,
respectively, as a function of time, the problem can then be formulated
as
|
(4)
|
with initial conditions
n1(0) = N, and
n2(0) = n3(0) = 0. With a bolus
ingestion [indicated by input(t)] of N markers
every 24 h, this system has the following solution (4) for
the number of markers in compartment 1, the right colon, at
a given time t
|
(5)
|
where t is in days and imax is
the greatest integer less than or equal to t. The graphs of
this function for two arbitrary values of k1 (1 day
1 and 0.2 day
1) are shown as the
stair-step and saw-tooth curves in Fig.
2, in which the stair steps reflect the
fact that only integer values of markers can be counted. From Eq. 5, the predicted number of markers visible in compartment
1 on a film taken on the morning of the seventh day would be
|
(6)
|
Note that since day 0 is the first day of marker
ingestion, the seventh-morning X-ray comes in fact six full days after
the first ingestion. Hence the predicted value of markers at the time of the X-ray is n1(6). Using this
equation, the large dots on the two Fig. 2 graphs (i.e., the two graphs
show the behavior for different values of the true rate constant
k1) indicate the predicted numbers of markers to
be counted in compartment 1.

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Fig. 1.
Diagram of the 3-compartmental model. Three compartments,
representing the right colon, left colon, and rectosigmoid area, are
used, with 3 constants, k1,
k2 and k3, representing
the net result of propagation and back-propagation between the
consecutive compartments.
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Fig. 2.
Right colon marker kinetics. For the right colon,
compartment 1, this shows marker kinetics predicted by our
compartmental model for both continuous and once-daily ingestion of 12 markers per day for 2 values of true rate constant
k1. Large dots indicate that 6 and 37 markers
would be counted on the 7th morning's X-rays in A and
B, respectively. Classic predictions of colonic transit time
(CTT) through compartment 1 (CTT1) based on
these numbers would be 0.5 days (12 h) for A and 3.1 days
(74 h) for B, whereas the true CTT1 values are 1 day and 5 days, respectively. Thus classic predictions are
underestimates even for fast transit, when steady state is attained by
the 7th morning (A). The dotted lines show "envelope"
curves that ignore the fact that the number of counted markers is an
integer.
|
|
If ingestion were continuous instead of as a bolus (for the same number
of markers per day), then in Eq. 4 input = N,
with N a constant, and for the number of markers in the
right colon, we would have the solution
|
(7)
|
This is shown in Fig. 2 as the smooth curves running through the
middle of the saw-tooth curves.
For the left colon and rectosigmoid segment (compartments 2 and 3), for particular values of
k1, k2, and
k3, the equations for
n2(t) and
n3(t) (in the set of Eq. 4) were solved numerically using the built-in function NDSolve in
the software program Mathematica, in which
n1(t) is given by Eq. 5 and with
initial conditions n2(0) = n3(0) = 0. Figures
3 and 4
show graphs of the solutions to Eq. 4 for selected
combinations of rate constants. As explained in the Mathematica
documentation, NDSolve switches between a non-stiff Adams method and a
stiff Gear method based on LSODE.

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Fig. 3.
Left colon marker kinetics. Marker kinetics for the left
colon, compartment 2, are shown. Assuming continuous or
once-daily ingestion of 12 markers, this shows marker kinetics for
k2 = 1/day and
k2 = 5/day (k1 = 1/day in both panels). As for the right colon, the classic method
would yield underestimates of CTT2; classic
CTT2 in A and B [based on
n2(6) values of 10 and 1, respectively, shown as ] would be 0.8 days (20 h) and 0.08 days (2 h), respectively, whereas true values are 1 and 0.2 days.
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Fig. 4.
Predicted marker kinetics in the rectosigmoid area.
Marker kinetics for the rectosigmoid segment, compartment 3, are shown. Assuming continuous or once-daily ingestion of 12 markers, 3 combinations of rate constants are illustrated, as indicated on the
graphs. The classic method would yield underestimates of
CTT3; n3(6) (shown as
large dots) is 49 for A and 11 for B and
C, so classic CTT3 estimates would be 4.1 days
(98 h) for A and 0.92 days (22 h) for B and
C, whereas true values are 10 days in A and 1 day
in B and C.
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|
Compartmental estimates of the rate constants and CTT.
The three rate constants were estimated from the number of markers
counted in each compartment on the films (designated
nx1, nx2, and
nx3, respectively) using a specific software
program written in the C language (4). Two sets of
estimates were made: classic estimates kc1,
kc2, and kc3, which were
based on Eq. 1, i.e.
|
(8)
|
and compartmental estimates kr1,
kr2, and kr3, which
were based on the pulsed analysis that gives Eq. 6 and the
corresponding equations for the other two compartments:
kr1 was deduced by nonlinear fit of Eq. 6 to nx1 [i.e.,
n1(6) = nx1]; kr2 was then
deduced from nx2 and the fitted values of
kr1 and kr3 were deduced
from nx3, kr1, and
kr2. For these estimates, we assumed that
segmental CTT was at least 1 h for each compartment, a reasonable
assumption except in extreme diarrhea, which was never the case for the
patients during this study.
Using these fitted estimates of kr1,
kr2, and kr3,
compartmental segmental transit times were calculated as
|
(9)
|
In both the classic and compartmental cases, total CTT was taken
as the sum of the three segmental transit time estimates.
Relationship between the two measures.
Using the definitions given in Eqs. 8 and 9, the
relationship between classic and compartmental estimates was taken as
the ratio
|
(10)
|
 |
RESULTS |
Double-Marker Results
Figure 5 shows a comparison, in one
subject, of actual marker counts in each segment to predicted counts
using both the classic single-ingestion/multiple X-ray technique and
our compartmental analysis. The compartmental model is clearly a better
predictor of marker counts over the course of the study. This pattern
is similar to that of the other patients in this study, except that for
the rectosigmoid segment the compartmental model predictions were not
systematically better than classic predictions. For these six patients,
the means of the errors (Eq. 3) for the classic vs.
compartmental model estimates were 3.75 vs. 1.25 (P < 0.05 by paired t-test) for the right colon, 4.15 vs. 2.00 (P < 0.05 by paired t-test) for the left
colon, and 1.28 vs. 1.19 (not significant) for the rectosigmoid
segment. The finding that the compartmental model provides no
improvement over the classic method for description of rectosigmoid
kinetics probably reflects the influence of voluntary control over the
emptying of the terminal intestine.

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Fig. 5.
Predictions of daily X-ray marker counts by classic vs.
compartmental model. For the right colon (A), left colon
(B), and rectosigmoid segment (C) of 1 patient,
these 3 graphs show the actual number of markers counted in each
segment ( ) and predicted counts using the classic
1-time ingestion/daily X-ray technique ( ) and our
3-compartment model ( ).
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|
Retrospective Study Results
Patients.
In all colonic segments, the compartmental CTT estimates were
significantly higher than classic estimates (P < 0.0001; Fig. 5). The difference between the classic and compartmental
estimates in the right colon was higher than the differences in the
other segments (P < 0.0001; Fig.
6). It was minimal for CTT of 43 h in the right colon, 26 h in the left colon, 15 h in the
rectosigmoid area, and 68 h for the entire colon.

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Fig. 6.
Compartmental vs. classic estimates of CTT. In these figures,
classic and compartmental estimates of segmental and total CTT in 1,309 subjects who complained of constipation or symptoms of irritable bowel
syndrome are plotted vs. the classic estimates. A: right
colon; B: left colon; C: rectosigmoid area;
D: entire colon. In all colonic segments, compartmental
estimates of CTT were higher than classic estimates, as shown by the
absence of data under the line y = x.
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|
The offset between compartmental and classic single-film CTT seen in
Fig. 5 at low transit times (high transit rates) for the right colon is
a direct result of the bolus rather than continuous marker ingestion
regimen. For high transit rates (high k1
values), right colon transit will have time to reach a steady state by the seventh morning, but even once steady state is reached, the number
of markers present each morning before marker ingestion will be
inferior to the number that would be predicted in the theoretical case
of continuous, rather than once-daily, ingestion. Figure 2A
shows an example of this. Since the classic estimate of CTT was derived
on the assumption of continuous ingestion, it leads to a systematic
underestimate in the steady state, manifested here as the asymptotic
offset for low CTT.
Healthy subjects.
In healthy subjects, similar results were obtained (Table
1), namely, compartmental transit time
estimates were greater than classic estimates in both genders.
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Table 1.
Classic one-film and compartmental estimates of total and
segmental colonic transit time in healthy subjects
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|
 |
DISCUSSION |
The double-marker study shows that the three-compartment model is
an improvement over classic techniques for characterization of colonic
transit. In addition, the retrospective study shows that the classic
implementation of the single-film method underestimates CTT mainly for
long transit times, and the error is greatest for the right colon. The
underestimate of CTT by classic techniques has been pointed out using
scintigraphic measurements (11) and also using radiopaque
markers (3).
In the initial description (2), the measurement of CTT was
merely the measurement of the mean transit time of a single marker,
defined as the integral
|
(11)
|
where n(t) is the number of markers present
in the colon at time t.
This was applied to a protocol of a single ingestion of 20 radiopaque
markers followed by 7-12 daily X-rays. The impossibility of
continuously recording marker movement led Arhan et al.
(2) to propose the approximation
|
(12)
|
where p is the number of daily films and
ti the time of film number i, counted
from the time of marker ingestion (t = 0).
The alternative method of daily ingestion, single film, using six daily
ingestions of radiopaque markers and an X-ray on the seventh morning,
corresponds to the mean of six measures of CTT performed using the
method of daily films (3). However, this daily-ingestion
single-film protocol has its own limits: 1) CTT is
considered as a discrete and not a continuous phenomenon; 2) there is a limited number of marker ingestions; and 3) CTT
is underestimated, especially when transit is delayed.
All recent studies of CTT using radiopaque markers (1,
5, 10) used the approximation defined in
Eq. 12. Nevertheless, the absence of markers in one film
could signify that the last marker was expelled any time during the
previous day. Although propulsion of colonic contents is not a
continuous phenomenon (6), modeling it with first-order
kinetics allows derivation of a continuous law of variation of the
number of markers at one site (4). We can then use this
law to describe the number of markers at a given site at any time. For
example, if N is the number of markers ingested and
k1 the transfer coefficient from the right colon
to the left colon, the number of markers on day 1 in the
right colon is Ne
k1. Then after six daily
ingestions of N markers, the predicted number of markers in
the right colon on day 7 at the time of the film,
n1(6), given by Eq. 6,
is set equal to the observed number, nx1, and we
solve for kr1. The compartmental estimate of
right colon transit time, CTTr1, is then the
reciprocal of kr1. In the same fashion, the
reciprocals of kr2 and
kr3 (calculated by numerical fits using
nx1, nx2, and
nx3 and the upstream fitted kri
values) give the compartmental estimates CTTr2
and CTTr3.
In the three colonic segments and for the colon as a whole, and for all
the patients in the present study, Fig. 6 shows classic and
compartmental estimates of CTT vs. the classic single-film estimates.
Figure 7 shows the relative errors, based
on Eq. 10. The considerable scatter seen here for the
compartmental estimates in compartments 2 and 3 is easily understood; consider two patients for whom the number of
markers counted in the left colon is identical but whose films show
different numbers of markers in the right colon, i.e.,
nx2 is identical in both patients, but
nx1 is different, a common observation easily
understood to result from different relative motility patterns of the
two segments. By the classic method, one would erroneously conclude
that CTT2 is identical in these two patients, whereas the
more realistic method, based on the three-compartment model, easily
distinguishes the two. Obviously, the dispersion is increased further
downstream in the rectosigmoid segment for exactly the same reason.

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Fig. 7.
Compartmental vs. classic estimates. The ratio
CTTr:CTTc is plotted
against classic CTT estimates. A: right colon; B:
left colon; C: rectosigmoid area; D: entire
colon. For the right and the entire colon, this ratio is higher for
lower values of colonic transit time (rapid transit) and higher values
of colonic transit time (slow transit). For the left colon and the
rectosigmoid area, a high ratio could be observed for all values of
segmental CTT.
|
|
This approach thus overcomes the usual errors due to studies over a
limited number of days of marker ingestion, since CTT using the present
definition accounts for the kinetics of marker transit. Moreover, using
the previous methodology of daily films, counting of markers was
frequently stopped after 8 or 10 days in constipated patients with high
delayed transit. The method used in the present study overcomes this
problem and furnishes improved estimates of CTT without prolonging
marker ingestion. The large errors resulting from the classic approach
are evident in the four examples illustrated in Fig.
8 associated with Table 2.

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Fig. 8.
Values from actual patients (all showing a total of ~70
markers). We selected 4 constipated patients with similar total marker
counts (~70) but in whom the marker distributions were quite
different, indicating colonic motility blocks in different segments.
Table 2 gives actual segmental marker counts, the resulting classic
estimates of CTT (from Eq. 1), compartmental rate constants
(calculated as best fits to our 3-compartment model), and the
corresponding compartmental estimates of CTT (reciprocals of
ki, converted from days to hours). Besides
noting the flagrant classic underestimates of segmental CTT, we can
make the following observations on closer inspection. For the case of
right colon block (A), all 3 compartments are far from the
steady state, which accounts for the quite dramatic classic
underestimates of segmental CTT, e.g., the classic estimate of
rectosigmoid CTT is 2 h here, whereas the more realistic
compartmental estimate is almost 2 days. For the left colon block
(B), only the right colon has attained steady state. The
classic method suggests identical transit times for the right colon and
rectosigmoid segments, since they show equal numbers of markers, but
the compartmental analysis reveals the much slower transit in the
rectosigmoid. For the 2 cases of rectosigmoid block (C and
D), both right and left colon have attained steady state,
but the rectosigmoid has not. We thus see that the kinetic analysis
gives a much clearer notion of colonic transit over time, even from a
single, 7th-day X-ray.
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Table 2.
Classic one-film and compartmental estimates of total and segmental CTT
in four constipated patients with similar total marker
counts
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|
In conclusion, our improved kinetic estimate of CTT could be useful in
clinical or pharmacological analysis, yielding more accurate values of
CTT in constipation and better evaluation of the action of prokinetic
drugs. The problems of classic methodology hindered the determination
of mechanisms of drug action. Our improved estimates allow distinction
between patients having similar classic total CTT but with large
differences in segmental transfer coefficients, as seen in Fig.
2.
 |
FOOTNOTES |
Address for reprint requests and other correspondence: M. Bouchoucha, Université Paris V, Hôpital Laennec,
Laboratoire de Physiologie Digestive, 42 rue de Sèvres, F-75007
Paris, France (E-mail:
michel.bouchoucha{at}lnc.ap-hop-paris.fr).
The costs of publication of this
article were defrayed in part by the
payment of page charges. The article
must therefore be hereby marked
"advertisement"
in accordance with 18 U.S.C. §1734 solely to indicate this fact.
Received 6 July 1999; accepted in final form 29 March 2000.
 |
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Am J Physiol Gastrointest Liver Physiol 279(3):G520-G527
0193-1857/00 $5.00
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