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INNOVATIVE METHODOLOGY
1Department of Paediatrics, University of Cambridge, Cambridge, United Kingdom; 2Department of Medicine, Division of Endocrinology; and 3General Clinical Research Center Analytical Core Laboratory, Albert Einstein College of Medicine, Bronx, New York
Submitted 8 October 2006 ; accepted in final form 11 May 2007
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ABSTRACT |
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 TOP ABSTRACT METHODSRESULTSDISCUSSIONAPPENDIXGRANTSREFERENCES |
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glucose kinetics; gut absorption; endogenous glucose production; glucose tracer; mathematical modeling
The assessment calls for the administration of at least two glucose tracers (19). One tracer is infused intravenously to quantify glucose disposition. The second tracer is included in the meal to trace the appearance of oral glucose. Recent work (1, 19) indicates that a three-tracer study design is preferable to minimize the estimation error caused by the misspecification of the model of glucose kinetics. It has been shown (3, 12) that administering glucose tracer in the format mimicking the endogenous glucose production (EGP) to achieve a constant specific activity or tracer-to-tracee ratio (TTR) reduces the model misspecification error when the EGP is calculated. The same principle also applies to the estimation of the glucose appearance from the meal (Ra meal), which is mimicked by the third tracer.
Moving on from the one-compartment Steele's model (16) to the Radziuk/Mari two-compartment model (2CM) (12, 13) further improves the accuracy of calculations (1, 19). The main reason for this is the reduction of the model misspecification error. Thus, the three-tracer study design with a 2CM structure appears to be the gold standard method.
Whereas the experimental design has evolved and the model structure has become more complex, the computational method has remained virtually unchanged. It consists of smoothing unlabeled and labeled glucose measurements and inserting them or their ratios into closed-form formulae, which iteratively calculate the EGP, Ra meal, and glucose disposal (Rd). The calculations are straightforward but are based on assumptions requiring critical appraisal. First, it is assumed that the unlabeled and labeled glucose concentrations are "smooth". This may be valid for the unlabeled glucose trace, but, because the administration of the labeled glucose in the three-tracer study design is either piecewise constant or piecewise linear, this assumption is difficult to justify for the labeled glucose concentrations. Second, the closed-form formulae transform the measurement error, and this transformation may affect the accuracy of calculation specifically for the meal absorption, given the way the meal-mimicking tracer is delivered. Finally, due to the sequential nature of computation, the Rd is calculated from the EGP and Ra meal, and thus any error in the EGP and Ra meal will propagate into the estimation of the Rd. For example, if a temporal underestimation of the Ra meal occurs due to a large measurement error at a given time, this error will be compensated by a temporal underestimation of the Rd to maintain the mass balance irrespective of whether physiological assumption of Rd smoothness is upheld.
In the present work, a new calculation method is described. The method assumes that the underlying metabolic fluxes, the EGP and Ra meal, and the glucose clearance are smooth. The method avoids the transformation of the measurement error and avoids the sequential nature of the calculations, making all estimation in "one go" to avoid the propagation of the estimation error from the EGP and the Ra meal into the Rd. The method can be implemented in a spreadsheet. A sample experimental study demonstrates the utility of the approach and contrasts it with the traditional calculations. Additionally, a comparison of the traditional and the new methods is made, adopting a simulation study.
Glossary
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METHODS |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONAPPENDIXGRANTSREFERENCES |
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 | (1) |
 | (2) |
 | (3) |
The following equalities are used on the basis of the assumptions of tracer indistinguishability:
 | (4) |
 | (5) |
The concentration of the total glucose in the plasma sample is determined by a direct measurement. The concentration of tracer species S = EM, M, and MM in the sample is determined as
 | (6) |
The concentration of the native glucose originating solely from the EGP is obtained with the use of a model-independent formula
 | (7) |
The time-invariant model parameters are assumed to be identical across all glucose species with k21 = 0.05/min, k12 = 0.07/min, and V1 = 160 ml/kg (11). The forcing functions uEM and uMM are assumed to be known with absolute accuracy. The unknowns to be estimated are the EGP (= uEN), the Ra meal (= uM), and the Rd (= k01,T Q1,T).
Traditional solution: model transformation. Following the work by Steele (16), the traditional solution (1) utilizes data-determined time derivatives of glucose concentration and concentration ratios while transforming Eqs. 1–3 to calculate directly the required quantities.
The time variant k01,EM associated with the EGP-mimicking tracer can be derived from Eqs. 1 and 3 as
 | (8) |
 | (9) |
 | (10) |
 | (11) |
 | (12) |
 | (13) |
A similar approach is used to calculate the Ra meal from Eq. 11. In this case, the concentration ratio GMM/GM, its time derivative, and the concentration GM are obtained by smoothing and interpolating smoothed data to evaluate the concentration ratios between measurements. The recursive calculations of Q2,MM and QM are also required.
The calculation of the Rd using Eq. 13 utilizes the concentration GT and its time derivative. These are again smoothed and interpolated at regular time intervals. The recursive calculations of Q2,T are needed.
Overall, the calculations of the EGP, the Ra meal, and Rd require the evaluation of the following indexes at regular time points: GEM/GE, GE, GMM/GM, GM, and GT. This involves smoothing and interpolating between sampling times. Time derivatives of GEM/GE, GMM/GM, and GT are required. The calculations approximate surrogates of masses Q2,EM, Q2,E, Q2,MM, QM, and Q2,T.
Although numerically efficient, the calculation procedure suffers from three shortcomings. First, smoothing of the concentration ratios GEM/GM and GMM/GM distorts the characteristics of the measurement error. This is of particular concern for GMM/GM following meal digestion because concentrations of both GMM and GM are small and the ratio is greatly influenced by the measurement error. Basu et al. (1) suggested overcoming this problem by omitting the first GMM/GM ratio obtained at 10 min from data processing.
Second, smoothing regularizes the concentrations and concentration ratios but not the underlying clearance rates k01,S and metabolic fluxes such as the EGP the Ra meal, which, on physiological basis, should conform to the regularity assumption. Due to experimental constraints, infusion rates uEM and uMM are normally piecewise constant, and when a step change occurs the resulting plasma concentration of GEM and GMM demonstrate a local shape change, with the time derivative of GEM and GMM presenting a step change. However, these shape changes will be smoothed out by the smoothing algorithm when evaluating GMM/GM and GEM/GE and particularly their time derivatives, which enter Eqs. 9 and 11. These two equations also utilize the discontinuous infusion rates uEM and uMM and may return, in consequence, jagged EGP and Ra meal. This will also cause the Rd to be jagged as both the EGP and Ra meal enter Eq. 13. In summary, smoothing plasma concentrations and ratios of plasma concentrations when using discontinuous tracer infusions result in a jagged and thus nonphysiological EGP, Ra meal, and Rd.
Finally, numerical errors are introduced through an approximate solution of Eqs. 1 and 2. The recursive calculation of Q2,S surrogate, which assumes a piecewise linear Q1,S between Ti and Ti +1, whereas in fact Q1,S is a of sum of two exponentials plus a linear component (see Eq. A9 in the APPENDIX), introduces a numerical error, which would be of particular importance during dynamic conditions. An additional numerical error is introduced by assuming a constant value of Q2,S surrogate during the calculation interval. A short calculation interval (i.e., 5 min) can reduce the impact of these numerical errors.
Eqs. 9 and 11 are excellent in demonstrating the benefit of using uEM and uMM, which mimic the EGP and Ra meal. Then the calculations become virtually model independent. However, due to intersubject variability, constant ratios GEM/GM and GMM/GM are difficult to achieve on individual basis, and other methods to calculate the EGP, Ra meal, and Rd may be more appropriate.
New calculation method: model retention combined with maximum likelihood and regularization. The method utilizes the maximum likelihood (ML) theory (7) combined with the regularization of Tikhonov et al. (18) to achieve a theoretically coherent computational framework. Under a mild assumption of the constancy of k01,S between the measurement points, this approach also avails itself for implementation in a spreadsheet without the need for numerical approximations.
Briefly, the method estimates the time-variant k01,EM using the plasma measurements of the EGP-mimicking tracer. Then, the k01,EM estimate is used to calculate the EGP from the plasma measurements of the endogenous glucose. A similar approach is used to estimate the Ra meal. First, k01,MM is estimated from plasma measurements of the meal absorption-mimicking tracer. Then, the Ra meal is estimated from the plasma measurements of the tracer included with the meal. Finally, k01,T is estimated from the total plasma glucose concentration, utilizing the EGP and Ra meal estimates and yielding the Rd.
The detail derivation follows. Let 
EM,k, k = 1...M, represent the measurement of the EGP-mimicking tracer at time tk, tk +1 > tk, with the measurement error normally distributed, uncorrelated, with a zero mean and a standard deviation 
EM,k. The logarithm of the likelihood function LLF(EM) defining the objective function for the estimation of k01,EM(t) consists of two components:
 | (14) |
 | (15) |
 | (16) |
The second component, LLFreg(EM), evaluates regularity of kEM(t). The variable
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EM represents the weighting factor balancing the relative importance of the LLFreg against the LLFfit component. The function 
(t) allows smoothness of k01,EM(t) to be reduced at the time of an experimental perturbation such as the meal intake, utilizing the knowledge of the experimental design in the calculation process. Assuming piecewise constant fractional clearance k01,EM(t), the differences k01,EM,k +1 – k01,EM,k are thus normally distributed, uncorrelated, with zero mean and a standard deviation kEM
 | (17) |
The variables k are defined as
 | (18) |
, > 1, is suitably chosen. In the present study, = 
was used.
The LLFreg(EM) is then defined as
 | (19) |
 | (20) |
 | (21) |
EM in Eq. 20, setting the derivative to zero, and solving for EM gives the solution at which the minimum of Eq. 20 is attained
 | (22) |
from Eq. 22 into Eq. 19 and removing constants, our minimization problem becomes
 | (23) |
EM is absent in Eq. 23. The regularity of k01,EM is fully determined by the standard deviations of the measurement error EM,k. A high-measurement error will enable only overly smooth k01,EM to be determined, and a low-measurement error facilitates a refined estimation of k01,EM.
The estimation of the EGP follows a similar path. The difference is that the EGP is continuous piecewise linear defined by a sequence, uE,k, k = 1...M:
 | (24) |
 | (25) |
E,k represents the kth measurement of the endogenous glucose and E,k the standard deviation of the associated measurement error. Q20,E is again obtained assuming steady-state conditions:
 | (26) |
 | (27) |
 | (28) |
 | (29) |
The estimation of the EGP, Ra meal, and Rd is set up as a sequence of five optimization problems. The five optimization problems estimate 1) three nonnegative sequences, k01,EM, k01,MM, and k01,T, k = 1...M – 1; 2) two sequences, uE and uM, k = 1...M; and 3) two initial conditions, Q10,EM and Q10,E. The measurements EM,k, E,k, MM,k, M,k, T,k, k = 1...M, and associated standard deviations of the measurement error, EM,k, E,k, MM,k, M,k, and T,k, are utilized in the estimation process. The results of the optimization problem for k01,EM feed into the optimization problem for the EGP. Similarly, the results of the optimization problem for k01,MM feed into the optimization problem for the Ra meal. Finally, the optimization problem to estimate k01,T utilizes the EGP and Ra meal.
These five optimization problems can be replaced by one large optimization problem, summing up the five objective functions defined for the individual subproblems; i.e., the overall likelihood function LLF is defined as
 | (30) |
 | (31) |
The calculations were implemented in a spreadsheet utilizing the "solve" function to solve the nonlinear minimization problems. A sample spreadsheet demonstrating the calculations can be obtained from the corresponding author free of charge by academic institutions and adopted for noncommercial projects.
Sample experimental study. A 30-yr-old healthy female (weight 62 kg, body mass index 25.7 kg/m2) was studied after overnight fast. The subject received a primed, constant, continuous infusion of 45.5 nmol·kg–1·min–1 [1,2,3,4,5,6,6-2H7]glucose (the EGP-mimicking glucose tracer; Cambridge Isotope Labs) for 120 min prior to the digestion of a liquid mixed meal (600 kcal, 75 g of glucose, 32.9 g of protein, and 20 g of fat) containing 3.11 g of [2-2H1]glucose (Omacron). Following meal digestion, the infusion of [1,2,3,4,5,6,6-2H7]glucose was piecewise linear in a fashion anticipating the time-varying profile of the EGP. A piecewise linear infusion of [U-13C;1,2,3,4,5,6,6-2H7]glucose (the meal absorption-mimicking glucose tracer; Cambridge Isotope Labs) was commenced 10 min after meal digestion and reached a peak of 5.43 nmol·kg–1·min–1. Figure 1 shows the time profile of the EGP-mimicking glucose tracer and the meal absorption-mimicking glucose tracer.
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Data analysis: traditional solution. The traditional analysis adopted smoothing and Eqs. 9, 11, and 13 to calculate the EGP, the Ra meal, and the Rd. Data processing involved the use of the CODE program, implementing stochastic regularization technique (9) to obtain GEM/GE, GE, GMM/GM, GM, and GT at 5-min intervals. A constant coefficient of variation for each quantity was adopted to determine the extent of smoothing. The ratio GMM/GM at 10 min was excluded from the analysis as suggested previously (1). Time derivatives were also calculated at 5-min intervals as the difference between interpolated values divided by the 5 min. The recursive technique proposed by Mari (12) was used to obtain the surrogate values of Q2,EM, Q2,E, Q2,MM, QM, and Q2,T.
Data analysis: new method based on maximum likelihood.
The five optimization problems defined by Eqs. 23 and 25 were, in the first instance, solved sequentially. Then, Eq. 31 was used to obtain the final estimate of the EGP, Ra meal, Rd, k01,EM, k01,MM, and k01,T. The measurement error associated with the total glucose was assumed to be multiplicative with 1.5% CV. The SD of the measurement error associated with the endogenous glucose was assigned values identical to that associated with the total glucose, i.e., E,k = T,k, reflecting the calculation methods of the endogenous glucose. The standard deviations of the measurement error associated with the TTRs of [1,2,3,4,5,6,6-2H7]glucose, [U-13C;1,2,3,4,5,6,6-2H7]glucose, and [2-2H1]glucose were set to 1.32 x 10–5, 2.40 x 10–6, and 9.68 x 10–4 (unitless), respectively. The standard deviations EM,k, MM,k, and M,k were calculated from T,k and the standard deviations associated with the respective TTR using the error propagation technique, adopting Eq. 6. The optimization problem defined by Eq. 31 was implemented in a spreadsheet.
Evaluation using simulation study. A comparison of the two approaches employed synthetic data sets generated by a glucoregulatory model. The synthetic profiles were analyzed with the traditional and new methods to reconstruct EGP, Ra meal, and Rd profiles. The differences between the actual and reconstructed EGP, Ra meal, and Rd profiles were summarized using the root mean square error (RMSE).
Two categories of three-tracer synthetic experiments were generated with six experiments in each category. In the first category, the time-varying infusions of the EM and MM tracers were identical in all six experiments and were based on the information from the literature (1). In the second category, the time-varying infusions of the EM and MM tracers were optimized using individual EGP and Ra meal profiles. This allowed errors because of the experimental nature and due to the model misspecification to be separated.
Generation of synthetic data sets. The synthetic data sets were generated with a glucoregulatory model consisting of a submodel of gut absorption, a submodel of insulin secretion and kinetics, and a submodel of glucose kinetics and insulin action.
The submodel of the gut absorption used a two-compartment structure, as described by Hovorka et al. (8). The submodel of insulin secretion assumed a linear relationship between plasma glucose and insulin secretion (10). A one-compartment model of the insulin kinetics was adopted (8). A submodel of the glucose kinetics assumed a two-compartment structure with three insulin action compartments representing insulin action on the glucose distribution/transport, disposal, and EGP suppression (8).
Six parameter sets were randomly generated from prior distributions (8, 10) to represent six synthetic subjects. Following the ingestion of 75-g carbohydrate meal, these parameter sets resulted in the time to peak of the gut absorption of 68 ± 9 min (mean ± SD) and the maximum suppression of the EGP of 81 ± 6% at 84 ± 30 min postmeal.
Two categories of three-tracer simulated experiments were executed. In the first category, the EM and MM tracers were given as piecewise constant infusions, using shapes defined by Basu et al. (1). In the second category, the EM and MM tracers were given again as piecewise constant infusions, with the time resolution defined by Basu et al., but the profiles were obtained by sampling the individual EGP and the individual gut absorption at midpoints of the time resolution scheme. In principle, the second-category experiments should make the estimation of the EGP, Ra meal, and Rd more accurate as the EM and MM tracer infusions more closely follow the EGP and the gut absorption, except for the piecewise approximation of the EGP and gut absorption profiles.
The model-derived total plasma glucose concentration and the TTR of the EM, MM, and M tracers were sampled with the sampling schedule described in Sample experimental study. A measurement error at the extent described in Sample experimental study was added to the measurements.
Data analysis. The noisy measurements were subjected to the analysis by the traditional and the new method, as described in Sample experimental study. This provided estimates of the EGP, Ra meal, and Rd.
Error assessment.
The difference between the actual and the estimated EGP, Ra meal, and Rd profiles was summarized by evaluating the RMSE at a 10-min interval, i.e.
 | (32) |
Statistical analysis. A two-way analysis of variance assessed the difference in the RMSE of the EGP, Ra meal, and Rd with the method (new vs. traditional) being one factor and the experimental category (literature-based vs. individually optimized EM and MM tracer infusions) being the other factor. A paired t-test was used to evaluate the difference between the RMSE associated with the Ra meal calculated with the traditional method comparing the effect of the literature-based vs. individually optimized EM and MM infusions. A similar, paired t-test was used to evaluate the Ra meal obtained by the new method, the Rd obtained by the traditional method, and the Rd obtained by the new method, as an interaction existed between the method and the experimental category when evaluating the RMSE associated with the Ra meal and Rd.
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RESULTS |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONAPPENDIXGRANTSREFERENCES |
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The concentrations of the three tracers are shown in Fig. 3. The plots also show the reconstructed tracer concentrations using the two computational approaches. The fit to the EGP-mimicking tracer is nearly identical, reflecting a nearly identical EGP estimate by the two methods. Differences in the fit to the data between the two approaches are observed with the meal-mimicking tracer and the tracer included with the meal in the second part of the study.
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The optimization of the EM and MM tracer infusions reduced the RMSE associated with the EGP by both methods. The optimization of the tracer infusions also reduced the RMSE associated with the Ra meal and Rd using the new method (P < 0.05, paired t-test). Unexpectedly, the optimization resulted in an increase of the RMSE associated with the Ra meal and Rd using the traditional method (P < 0.001 and P < 0.05, paired t-test). On a visual inspection (data not shown), the increase in the RMSE was attributable to a marked mismatch between the actual and estimated Ra meal in the first 30 min. The traditional method overestimated the actual Ra meal with an accelerated time to peak of the gut absorption at 20 min, whereas the true time to peak was 70 min. Further overestimation to a lesser extent and a jagged Ra meal profile followed from 60 min onward. The overestimation of the Ra meal propagated into the overestimation of the Rd, as the mass balance had to be maintained.
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DISCUSSION |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONAPPENDIXGRANTSREFERENCES |
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The traditional approach divides the calculation process into two stages. First, data are smoothed and interpolated. Second, the smoothed data, assumed to be error-free, enter recursive formulae. The computational complexity resides within the smoothing stage, which includes a nonlinear optimization problem (2, 5). The new approach contains only one stage. The computational complexity is comparable to the smoothing stage of the traditional approach. The nonlinear optimization problem is integrated by placing smoothness assumption on the underlying metabolic indexes. Yet the computations can be implemented in a spreadsheet.
The new method has further benefits when used during a piecewise constant infusion of the EGP-mimicking and meal-mimicking tracers. Step changes in the infusion rate result in "jumps" in time derivative of the tracer concentration, but the jumps are filtered out by the smoothing stage of the traditional approach. However, as the piecewise constant infusions also enter the recursive formulae, a jump (discontinuity) is introduced in the calculated EGP and Ra meal. The Ra meal is particularly affected, as the meal-mimicking tracer infusion varies extensively. Optimally, the discontinuity in the infusion rate and the measured concentration should both enter the recursive formulae and cancel out each other.
With the traditional approach, the calculation of the Rd and Ra meal is sequential. The estimated Ra meal enters the formula to calculate the Rd; see Eqs. 12 and 13. The drawback is that any error in the Ra meal will be accompanied by a proportional error in the Rd. This is apparent in Fig. 2, middle and bottom [Mari method (12)]. The oscillations in the Ra meal from 210 min until the end of the experiment are followed by oscillations in the Rd of a similar magnitude.
The propagation of the estimation error is a consequence of the sequential nature of the calculations. Errors in both the EGP and Ra meal will introduce a comparable error in the Rd, although the extent of the former is expected to be smaller given the lower anticipated extent of the error associated with the EGP.
The new approach eliminates the propagation of the error. The calculation is no longer sequential, and all fluxes and fractional clearance rates are calculated in one go; see Eq. 31. Conceptually, this can be visualized as information sharing. All five measurements, the three tracer concentrations, the endogenous glucose concentration, and the total glucose concentration influence the calculation of the EGP, Ra meal, and Rd. The assumption of smoothness of the Rd forces the Ra meal to reduce its oscillatory pattern (see Fig. 2, middle), ML method, albeit at the expense of a small misfit to the [U-13C;1,2,3,4,5,6,6-2H7]glucose (Fig. 3, middle) but an improved fit to the [2-2H1]glucose (Fig. 3, bottom). Although less oscillatory, the Rd provides a better fit to the total glucose; see Fig. 4, bottom.
Another drawback of the traditional approach is the transformation of the measurement error. The first 20–30 min following meal ingestion provide low values of GMM and GM. The measurement error will represent a considerable proportion of the two measurements and will impact on the ratio GMM/GM. The traditional approach solves this problem by excluding the ratios GMM/GM during the first 10 min from smoothing. However, in principle, this problem perseveres, although to a lesser extent, for other GMM/GM ratios and also GEM/GE ratios. In our particular example, the effect is minimized by the adoption of highly precise GC-MS and LC-MS measurement techniques. The new computational approach avoids using the ratios and utilizes solely measured concentrations for which the measurement error is determined on a coherent basis.
The fractional turnover rate k01 differs when estimated using the EM and MM tracers; see Fig. 6. The differences in k01 estimates reflect the extent of model misspecification and, presumably, to a smaller extent, the effect of the measurement error on parameter estimation. Conceptually, if a model is incorrect, different inputs, such as in our case different EM and MM tracer infusions, will provide different parameter estimates. If the model is a faithful representation of the reality, the parameter estimates that are not influenced by the input bar the effect of the measurement error.
In principle, glucose cycling and nonequivalent tracer loss could also contribute to the observed differences in k01. However, our assessment of glucose cycling indicates that this is unlikely. The [1,2,3,4,5,6,6-2H7]glucose was monitored as the C1–C5 fragment containing four deuteriums (C2–C5). The enrichment of the [1,2,3,4,5,6,6-2H7]glucose molecule was five- to 10-fold less (1:5–10) than the enrichment of the [2-2H1]glucose for most of the experiment, being close to 1:3 only at the very beginning and the extreme end of the meal. As expected, in test infusions we found no evidence of tracer recycling of the [1,2,3,4,5,6,6-2H7]glucose onto [2-2H1]glucose because the fragment would have added an M + 2 or M + 3. Similarly, with the [U-13C;1,2,3,4,5,6,6-2H7]glucose, we found no measurable evidence of recycling onto the [1,2,3,4,5,6,6-2H7]glucose in test infusions at the expected ratios during the study.
Although less transparent, the new method also benefits from the constant enrichments, i.e., a zero time derivative of the respective ratios. To illustrate this point, let us consider the calculation of the EGP by the Mari approach (12). Eq. 9 includes the time derivative of the ratio of the tracer to endogenous glucose, and the minimization of the derivative is achieved by mimicking the EGP with the EM tracer. The derivation of the Mari formula to calculate the EGP involves the substitution of the formula calculating k01, Eq. 8, into Eq. 1 and associated algebraic manipulations. The point here is that k01 is an "intermediate" product, parts of which conveniently drop out in the Mari model. The benefit is a simplification of the formula and a clear exposition of the effect of minimizing the time derivative. The drawback is that physiological information about the smoothness of k01 is lost and is replaced by assuming smoothness of the ratio MM tracer to the endogenous glucose.
In the new method, the intermediate step to calculate k01 is retained. This may confound the understating that the minimization of the time derivate also reduces the effect of model misspecification, but this principle holds here, too. Unlike algebraic manipulations adopted by the Mari model, the new method uses a numerical estimate of k01. The benefit is the use of the prior information on k01 smoothness. It therefore follows that, if the infusions of the EM and MM tracers exactly mimic the EGP and the gut absorption, the new method will provide accurate estimates of the EGP and the gut absorption despite the different k01 values estimated from the EM and MM tracers.
The Mari method and, similarly, the new method assume that the fractional transfer rates k21 and k12 are time variant and set to physiologically feasible values and identical to those adopted by Basu et al. (1). However, it is known that these fractional rates are stimulated by insulin with a greater effect of insulin on k21 than on k12 (4), pinpointing one source of model misspecification that will contribute, when the EM and MM tracers differ from the EGP and the gut absorption, to the computational errors in the EGP, Ra meal, and Rd. However, these errors will be smaller than those incurred when using a single compartment model (1).
The volume of distribution V1 was set at value of 160 ml/kg, comparable to the 146 ml/kg used by Mari (12) and slightly higher than 130 ml/kg used by Basu et al (1). It is identical to the value obtained during modeling of the intravenous glucose tolerance test with a two-compartment model (11).
The theoretical expectations of superiority of the new method were confirmed in a simulation study. The new method outperformed significantly the traditional method when estimating the EGP, Ra meal, and Rd with the greatest RMSE improvement associated with the calculation of Ra meal. Furthermore, the new method benefited from the optimized infusion profiles of the EM and MM tracers. The optimized infusions resulted in halving the RMSE associated with Ra meal. Smaller relative improvements were observed in the RMSE associated with the Rd and EGP.
Unexpectedly, the optimized tracer infusions resulted in a deteriorated RMSE associated with the Ra meal and subsequently the Rd using the traditional method. This is attributable to a marked overestimation of the Ra meal in the first 30 min, although additional overestimation followed from 60 to 120 min with jagged profile throughout. This jagged profile was not observed with the EGP, suggesting that the calculations of the Ra meal and Rd, but not the EGP, suffer from the piecewise constant infusion of the tracers with the traditional method.
The reasons for the overestimation of the Ra meal in the first 30 min may originate in the effect of the measurement errors on the calculations. With the optimized MM tracer infusion, the infusion rate of the MM tracer is considerably smaller in the initial part of the experiment. This smaller infusion rate will result in a lower TTR of the MM tracer. Given that the measurement error associated with the TTR of the MM tracer is additive, the signal-to-noise ratio at the early part of the experiment will be lower with the optimized infusion. These highly noisy measurements are used as a numerator when evaluating and smoothing the GM/GMM ratio. It appears that the smoothing process of highly noisy measurements is the culprit of the problem.
The computations associated with the new approach can be executed in a spreadsheet, given the explicit solution of the two-compartment model; see APPENDIX. The solution utilizes an assumption that the fractional clearance rate k01,S is constant between time instances. Figure 6 shows the time evolution of the fractional clearance rates, indicating jumps in the fractional clearance rate associated with k01,MM. The jumps could be reduced by using a finer time resolution.
The new computational approach can be used in different experimental scenarios. This includes the two tracer study designs to calculate the EGP, Ra meal, and Rd or the glucose clamp technique to calculate the EGP and Rd. Eq. 31 needs to be suitably modified to represent the reduced experimental complexity. All other components remain unchanged.
In conclusion, a new computational approach has been developed, increasing feasibility of the EGP, Ra meal, and Rd calculated from data collected during the three-tracer experiment. The new approach uses more physiologically justified assumptions and treats coherently the measurement error.
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APPENDIX |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONAPPENDIXGRANTSREFERENCES |
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The solution is recursive. It assumes that k21 and k12 are time invariant. Given a possibly nonequidistant time sequence tk, k= 1...M, it is assumed that k01(t) is nonnegative, piecewise constant, i.e., k01(t) = k01,k for tk tk+1, k01,k > 0. It is further assumed that u(t) is continuous, piecewise linear, u(t) = uk + (uk+1 – uk)/(tk+1 – tk)(t–tk) for tk 
t < tk+1.
Given the amounts Q1(tk) and Q2(tk), it can be shown that amounts Q1(tk+1) and Q2(tk+1) can be obtained as
 | (A1) |
 | (A2) |
 | (A3) |
 | (A4) |
 | (A5) |
 | (A6) |
 | (A7) |
 | (A8) |
 | (A9) |
 | (A10) |
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GRANTS |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONAPPENDIXGRANTSREFERENCES |
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FOOTNOTES |
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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. Section 1734 solely to indicate this fact.
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TOPABSTRACTMETHODSRESULTSDISCUSSIONAPPENDIXGRANTSREFERENCES |
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