## Abstract

Recently, we developed the [^{13/14}C]octanoic acid breath test to measure gastric emptying of solids. Although the method has been validated extensively, absorption, metabolism, and excretion of the label in the breath need to be corrected for. In this study a mathematical model was developed that allows for*1*) separation of the global CO_{2} excretion after ingestion of the labeled test meal into the emptying rate of the labeled test meal from mouth to pylorus and the postgastric processing of absorption, metabolism, and excretion of the label, and*2*) numerical calculation of the half-emptying time and lag phase of the emptied meal. The model was applied to the gastric emptying results obtained by simultaneous scintigraphic and breath test measurements. An excellent correlation was found between the gastric half-emptying time (*r* = 0.98) and lag phase (*r* = 0.85) determined scintigraphically and via breath test. There was also a good agreement between the two methods [mean values and confidence limits for differences: *t*
_{½}= 10 min (−20 to 41) and*t*
_{lag} = −3 min (−39 to 34)]. Moreover, the separated gastric emptying curves, lacking the influence of postgastric processing of the label, showed real patterns of gastric outflow, which changes from moment to moment.

- breath test technology
- mathematical models

recently, we developed the [^{13/14}C]octanoic acid breath test to measure gastric emptying of solids (7, 16). The rationale of a breath test is based on the firm retention of^{13/14}C-labeled octanoic acid in the solid phase of a test meal during mixing and grinding in the stomach, followed by rapid absorption from the chyme entering into the duodenum, an immediate and maximal oxidation in the liver to labeled CO_{2}, and a fast exhalation in breath. In vitro experiments showed that octanoic acid is firmly retained in a standard solid test meal in a gastric environment (7). It has been known for a long time that octanoic acid, an eight-carbon fatty acid found in dietary fats, is rapidly absorbed from the intestine and carried to the liver via the portal venous system, where it is rapidly and completely oxidized (1, 2, 4-6, 10-13,17-20, 22, 25). Therefore gastric emptying of the meal, and not the postgastric processing of the label, could be considered the rate-limiting step in the rate of labeled CO_{2} excretion in breath after ingestion of a labeled solid test meal.

Mathematical analysis of CO_{2}excretion curves made it possible to exclude the influence of endogenous CO_{2} production on the breath test results, and breath test and radioscintigraphic measurements taken simultaneously in normal subjects and dyspeptic patients allowed for highly accurate description of the gastric emptying rate of a solid test meal (7, 16). With the use of a regression model, we were able to calculate the half-emptying time and lag phase, correcting for the postgastric processing of octanoic acid.

The aim of this study was to develop a separation model in which the postgastric processing of octanoic acid could be mathematically separated from the CO_{2} excretion curve after ingestion of a standard solid test meal to obtain real-time gastric emptying curves. This approach of breath test curve analysis has two potential benefits: *1*) physiologically meaningful gastric emptying parameters can be calculated from breath test curves without correcting for postgastric processing of the label on a linear regression-estimated basis between radioscintigraphy and breath tests, and*2*) it allows for the evaluation of gastric emptying rates, instead of amounts of emptied food, as a function of time (flow curves). The classic multicompartmental analysis, however, was not used due to the specific conditions encountered in breath test technology. The multiple-chamber model is difficult to apply in clinical practice, because the dynamic exchange of CO_{2} with the rapid and slow bicarbonate pool and the loss of label via excretion in urine and feces and incorporation into bone is difficult to estimate in humans, certainly in each individual. Solution of the breath curve would require the fitting of at least four exponential functions. This can rarely be done convincingly with biological data, even if sampling takes place over long periods of time. Also, it is not possible to obtain a steady state of exchange between the different compartments (especially the slowly exchanging ones) during the 4-h period of breath sampling. Moreover, when dose is not in the subsequently measured compartment the rate constants for intercompartmental exchange cannot be explicitly calculated from the multiexponential curve for tracer in breath.

## RATIONALE FOR THE SEPARATION MODEL

To elaborate the mathematical model, three functions were introduced to describe three different processes.

*1*) The emptying rate of a labeled solid meal from mouth to pylorus is given by M(*t*).

*2*) The rate of postgastric processing (absorption, metabolism, and excretion in breath) of the label is given by D(*t*).

*3*) The global process of CO_{2} excretion after ingestion of a labeled solid test meal is given by T(*t*).

The aim of this study is to determine M(*t*) given T(*t*) and D(*t*), which can both be measured, and to describe the relation between the three functions. Therefore the following assumptions were made. *1*) The meal is ingested at once, at *time 0*. This is not true, but the time of ingestion was always restricted to 10 min, and *time 0* was taken as the time of completion of the ingestion of the meal. *2*) T(*t*), D(*t*), and M(*t*) are piecewise continuous functions, not identical to zero and positive for each time ≥ 0.*3*) The rate of metabolism of the label [D(*t*)] is proportional to the rate of gastric emptying of the label [M(*t*)]. This implies that the kinetics of metabolism of the label are independent of the rate at which the label is emptied [no saturation of D(*t*) as a function of M(*t*)], or, stated differently, that D(*t*) is invariant of M(*t*).

We first demonstrate that, in theory, under the assumptions made above, the separation model is a mathematically correct alternative to the multicompartmental model to separate a function (i.e., gastric emptying rate) from a global process when rate constants for intercompartmental exchange cannot be explicitly calculated. We then demonstrate the practical elaboration of deriving the gastric emptying rate from labeled octanoic acid breath test curves and the proportionality of D(*t*) to M(*t*).

## DESIGN OF THE SEPARATION MODEL

To simplify the rationale of the model, T(*t*), D(*t*), and M(*t*) are not considered to be continuous but are divided into discrete time intervals. The rate of^{13/14}CO_{2} excretion during a certain time interval is the result of the accumulated effect of parts that have left the stomach in the past intervals (Fig.1). For example, the rate of label recovered in breath during *time interval 3*, called T_{3}, is the result of the part of the label that left the stomach in the first time interval but was metabolized during the second time interval (simplified: during the second passage in the liver), plus the part of the label that left the stomach in the second time interval but that had already been metabolized during the first time interval (simplified: the first passage in the liver). The addition of all layers describes the total process T(*t*), i.e., a^{13/14}CO_{2} excretion curve after ingestion of a solid test meal. Mathematically it is expressed as
or, in general, as
Equation 1By decreasing the length of the time intervals to zero, the formula becomes a continuous function
Equation 2The relationship between the different rates as described in*equation 2* is mathematically known as a convolution product. A number of properties can easily be derived mathematically. However, these properties are not of interest in this study, since it is not possible in general to find the inverse relation between T and M, except for special classes of functions such as e^{x}. Such functions are used in Fourier and in Laplace transforms, but these functions do not have the form observed in our data. Therefore, we have used the discrete formalism (*Eq. 1
*) to derive a discrete calculation in practice
or, in general
Equation 3If T(*t*) and D(*t*) are known, M(*t*) can be separated from the total process T(*t*) by decreasing the length of the time intervals.

## ELABORATION OF THE MODEL

### Methods

#### Subjects and materials.

As functions of T(*t*), the^{14}CO_{2}excretion data obtained in the validation study comparing the [^{14}C]octanoic acid breath test and the radioscintigraphic technique were used (7). Briefly, in this study a standard solid test meal (250 kcal) consisting of one egg (labeled with 74 kBq of [^{14}C]octanoic acid and 110 MBq of ^{99m}Tc-labeled albumin colloid), two slices of bread, and 5 g of margarine was ingested by 16 healthy volunteers and 20 dyspeptic patients. Immediately after ingestion of the meal, each subject was seated between the two heads of a dual-headed gamma camera equipped with parallel-hole low-energy collimators and interfaced to a computer. Scanning scintigraphic information was obtained every 10 min for up to 1 h and every 15 min for another period of 1 h. Radioactivity remaining in the stomach at each scanning period was expressed as a percentage of the activity initially present. The gastric emptying rate so obtained was fitted by the modified power exponential formula of Siegel et al. (24). The half-emptying time (*t*
_{1/2 s}) and lag phase (*t*
_{lag s}) were calculated according to that formula. Breath sampling for^{14}CO_{2}followed the same time schedule as the scintigraphic imaging technique but continued for another 2 h of sampling, during which breath was collected and measured in 15-min intervals. The results were expressed as the percentage of ^{14}C recovery per hour and were further analyzed by nonlinear regression analysis to calculate *t*
_{1/2}and *t*
_{lag}. The gastric emptying parameters of both techniques were compared by correlation and linear regression analysis in this study.

To obtain the function D(*t*), 20 healthy subjects (10 women and 10 men, mean age 23 yr, range 18–28 yr) were examined. None of the subjects had a history of gastrointestinal disease or surgery and none were taking medication. After an overnight fast, a flexible tube was positioned in the second part of the duodenum under radioscopic control. The dynamics of^{14}CO_{2}appearance in breath were measured after intraduodenal administration of 74 kBq of [^{14}C]octanoic acid sodium salt (DuPont NEN, Boston, MA), dissolved in 20 ml of water. Breath samples were taken before and every 3 min during the first 30 min, every 5 min for the next 30 min, and every 15 min thereafter for up to 4 h. The^{14}CO_{2}excretion curves were evaluated by*1*) the^{14}CO_{2}peak excretion time, *2*) the^{14}CO_{2}peak excretion, and *3*) the half-emptying time of the curve [using the formula for D(*t*)].

To validate the invariance of D(*t*) from M(*t*), six healthy volunteers (3 women, 3 men; age 18–24 yr) were studied. None of the subjects had a history of gastrointestinal disease or surgery and none were taking medication. After an overnight fast, a flexible tube was positioned in the second part of the duodenum under radioscopic control, and 129.5 kBq of [^{14}C]octanoic acid sodium salt (supra) dissolved in 50 ml of water were injected into the second part of the duodenum in a bolus at three different times: 74 kBq at *time 0*, 18.5 kBq (1/4 of the initial dose) 1 h later, and 37 kBq (1/2 of the initial dose) at 2 h. Breath samples were taken every 5 min for 4 h. The kinetics of metabolism of each bolus of [^{14}C]octanoic acid were evaluated by three parameters:*1*) the time until peak excretion of^{14}CO_{2}in breath, *2*) the maximal increase of^{14}CO_{2}excretion after injection of each bolus, and*3*) the increase in area under the curve of^{14}CO_{2}in breath obtained during the first hour after injection of each bolus of [^{14}C]octanoic acid [using the formula for D(*t*)].

The study protocol was approved by the ethics committee of the University of Leuven. Informed consent was obtained from all subjects.

#### Measuring techniques and mathematics.

^{14}CO_{2}in breath was collected by blowing through a pipette into vials containing 2 ml of 1 M hyamine hydroxide and 2 ml of ethanol together with one drop of thymolphthalein solution. This amount of hyamine is neutralized by 2 mM of CO_{2}. The end point of neutralization is indicated by decoloration of the indicator. After decoloration, 10 ml of scintillation cocktail (Hionic Fluor, Packard Instruments) were added and radioactivity was determined by liquid scintillation spectrometry (Packard Tri-Carb liquid scintillation spectrometer, model 3375; Packard Instruments, Downers Grove, IL). CO_{2} production was assumed to be 300 mmol per square meter of body surface per hour. Body surface area was calculated by the weight-height formula of Haycock et al. (9). The results were expressed as the percentage of^{14}C recovery per hour as a function of time.

### Application of the Model

The function T(*t*) can be adequately described in both healthy volunteers and subjects with abnormal gastric emptying rates (1) by two classes of function:*at ^{b}
*e

^{−}

^{ct}or

*mk*βe

^{−}

^{kt}(1 − e

^{−}

^{kt})

^{β − 1}, where

*t*is time and

*a*,

*b*,

*c*,

*m*,

*k*, and β are regression-estimated constants.

The mean^{14}CO_{2}excretion curve obtained in 20 healthy volunteers after intraduodenal administration of 74 kBq of [^{14}C]octanoic acid served as the function D(*t*). As far as the function D(*t*) is concerned, no class of functions exists. Accurate fitting of this curve is done by a combination of exponential and polynomial functions
where*t* is time and*a*, *b*,*c*,*d*,* f*,*g*,* h*,*i*,* j*,*k*, and *l* are regression-estimated constants.

Using these equations for T(*t*) and D(*t*), in *Eq.3
* the curve M(*t*) is obtained. Two gastric emptying parameters were calculated numerically from the individual curves M(*t*):*1*) the gastric half-emptying time is calculated by solving the equation
and*2*) the lag phase (*t*
_{lag b}), as defined by Siegel et al. (24), which corresponds to the time of peak excretion in the function M(*t*).

#### Statistics.

The gastric half-emptying times and lag phases of the separated functions of M(*t*) were calculated numerically after integration into M(*t*) as a function of time and were compared with the scintigraphically determined half-emptying times and lag phases of the validation study (7), using correlation analysis [SAS: PROC CORR (21)]. The two tests were further compared using the Bland and Altman procedure (3). The three parameters for evaluation of the kinetics of metabolism of [^{14}C]octanoic acid after intraduodenal administration were compared for the three boluses using the Mann-Whitney-Wilcoxon test (21).

## RESULTS

### Postgastric Processing of [^{14}C]Octanoic Acid

Figure 2 represents^{14}CO_{2}excretion as a function of time in 20 healthy subjects, after intraduodenal administration of 74 kBq of [^{14}C]octanoic acid (means ± SE).^{14}CO_{2}appeared in the breath almost immediately, with a peak excretion of 33.73 ± 1.69% dose/h after 10.69 ± 0.77 min, followed by an exponential decrease of^{14}CO_{2}activity in the breath. The half-excretion time of the curves was 67.5 ± 1.37 min.

### Invariance of D(t) from M(t)

In Fig. 3, the^{14}CO_{2}excretion as a function of time is given in six subjects, after intraduodenal administration of three different boluses of [^{14}C]octanoic acid. At each bolus injection of [^{14}C]octanoic acid, peak excretion in breath was reached at 10 ± 0.83 min, with a peak of 33.05 ± 2.49% dose/h after the first bolus, 24.18 ± 1.54% dose/h after the second bolus, and 28.61 ± 2.03% dose/h after the third bolus. The increase in^{14}CO_{2}excretion 10 min after injection of the bolus was 33.05% (0.447% per injected kBq of activity) at 10 min, 8.18% (0.442%/kBq) at 70 min, and 16.59% (0.448%/kBq) at 130 min. The area under the curve during the first hour was 22.99 ± 1.20% (0.31% per injected kBq of activity) for the first injected bolus of 74 kBq, 7.17 ± 0.47% (0.39%/kBq) for the second bolus of 18.5 kBq, and 10.64 ± 05.4% (0.29%/kBq) for the third bolus of 37 kBq. The differences between the three boluses for the three parameters (*parameters 2* and *3* calculated per kBq of activity) were statistically not significant.

### Application of the Model

Figure 4 depicts the relationship between the three functions T(*t*), M(*t*), and D(*t*) in two subjects after ingestion of a [^{14}C]octanoic acid-labeled standard solid test meal. The first subject had a normal gastric emptying rate with a scintigraphically determined half-emptying time of 59 min (Fig. 4
*A*): the rate of gastric emptying accelerates very quickly before reaching a peak, followed by a gradual decline in the velocity of gastric emptying. The second subject had a delayed gastric emptying rate with a scintigraphically determined half-emptying time of 89 min (Fig.4
*B*); the acceleration and deceleration in the gastric emptying rate is less pronounced and less steep. By analyzing the gastric emptying data in this way, it was clear that gastric emptying velocity changes from minute to minute and never has a constant value.

The separated gastric emptying function M(*t*) allowed not only for evaluation of the real pattern of emptying but also for the calculation of a half-emptying time. The relationship between the gastric half-emptying times determined scintigraphically and via breath test in 16 healthy volunteers and 20 dyspeptic patients after ingestion of a dually labeled solid test meal of 250 kcal is given in Fig.5
*A*. The correlation coefficient between the two parameters was 0.98. Figure5
*B* gives the relationship between the lag phases obtained by both techniques, defined as the point of maximal gastric emptying rate according to the method of Siegel et al. (24). The correlation coefficient was 0.85. The Bland and Altman plots of gastric half-emptying times and lag phases determined scintigraphically and via breath test, given in Fig. 6, showed, first, an off-set between both methods not significantly different from zero, and second, no proportional differences between the two methods [mean and confidence limits for differences between the methods:*t*
_{1/2}, 10 min (−20 to 41) and*t*
_{lag}, −3 min (−39 to 34)].

## DISCUSSION

This study aimed to develop a mathematical model to separate one physiological function from breath test results. All breath tests are based on the administration of a substrate with a functional group containing a carbon atom with either the radioactive (^{14}C) or the stable (^{13}C) isotope of carbon. The functional group is enzymatically cleaved during passage through the gastrointestinal tract, during its absorption, or in subsequent metabolic processes. After cleavage of the target bond, the cleaved portion undergoes further metabolism to^{14}CO_{2}or^{13}CO_{2}, which mixes with the bicarbonate pool of blood and is finally expired in the breath. In this way,^{14/13}CO_{2}excretion is a reflection of the total amount or kinetic properties of the enzyme studied, given that this enzyme relates to the rate-limiting step in the whole process.

By applying this mathematical model to the [^{13/14}C]octanoic acid breath test to measure gastric emptying of solids, we were able to demonstrate that postgastric processing of [^{13/14}C]octanoic acid until^{13/14}CO_{2}exhalation occurs very rapidly, with minimal intersubject variability. This is due to very rapid absorption from the small intestine, quick transport to the liver [no mucosal esterification, no incorporation in chylomicrons (10, 18-19)], and a ready and almost complete oxidation to^{13/14}CO_{2}in the liver [no requirement for carnitine to cross the double mitochondrial membrane (4, 22)]. Therefore, gastric emptying of the meal can be considered the rate-limiting step in^{13/14}CO_{2}excretion after ingestion of a [^{13/14}C]octanoic acid-labeled solid meal. Also, an average function can be used to describe the “postgastric processing” of octanoic acid. Metabolism of octanoic acid remains unaltered not only in healthy volunteers but also in other circumstances, as has been shown for insulin-dependent diabetes mellitus (14) or after administration of octreotide (15).

The assumption of invariance of postgastric processing of [^{13/14}C]octanoic acid from the rate of emptying from the stomach was fulfilled in this study. Hence all other assumptions made were also fulfilled and the separation model could be applied by “subtracting” the shape of the postgastric processing curve on each moment from the global^{13/14}CO_{2}excretion curves after ingestion of a labeled meal, in a continuous way and according to the amount of label that has left the stomach at that moment.

The results obtained with the separation model are excellent. The model allows gastric half-emptying time and lag phase to be calculated very accurately and it also provides a method to evaluate patterns of gastric emptying velocity or flow, which changes from minute to minute. In 1990, Schulze-Delrieu (23) pointed out that radioscintigraphic gastric emptying results, expressed as a percentage of the initial amount still remaining in the stomach, represent cumulative data (i.e., mathematical integration of a velocity curve, or “distance” rather than “velocity”) and that “gastric emptying rates determined in this way do not allow any conclusions regarding the rate or pattern of actual gastric outflow and identical emptying rates may hide major differences in flow pattern.” A gastric emptying flow curve can be obtained from radioscintigraphic data by taking the first derivative of the measured curve. However, mathematical derivation is less stable than mathematical integration. This leads to inaccuracies for calculation of kinetic parameters such as the lag phase, as defined by Siegel (24), since it is mathematically easier to determine the peak of a flow curve than to determine the point of inflection of a cumulative curve. This could be the explanation for a less good correlation of the lag phases of both techniques in this study.

On the other hand, the separation model has it limits. By using fitting curves for the actual measured data of^{13/14}CO_{2}excretion, the transpyloric flow is smoothed to a general flow curve and does not display the gushes of chyme leaving the stomach in a pulsatile way.

The separation model presented has a theoretical advantage compared with the classical multiple chamber model (8), in that it makes fewer assumptions. It makes no assumptions about laws governing the flow stream of the label. Moreover, the multiple chamber model is difficult to apply in clinical practice, as discussed in the introduction. The use of the curve D(*t*), representing the postgastric processing of the label, in separating M(*t*) out of T(*t*) and D(*t*) is an appropriate solution to these problems because D(*t*) is shown to be proportional to M(*t*).

In conclusion, an accurate mathematical model was developed to separate gastric emptying flow curves from^{13/14}CO_{2}excretion curves obtained after ingestion of a [^{13/14}C]octanoic acid-labeled solid test meal, thereby also excluding the influence of endogenous CO_{2} production on breath test results. The model also has attractive prospects for other (breath) tests to separate a specific gastrointestinal function, e.g., separation of the process of intraluminal lipolysis out of the data of a mixed triglyceride breath test and separation of the assimilation of carbohydrates from gastric emptying of the given test meal.

## Footnotes

Address for reprint requests: P. J. Rutgeerts, Dept. of Medicine and Medical Research, University Hospital Gasthuisberg, B-3000 Leuven, Belgium.

- Copyright © 1998 the American Physiological Society