1Laboratorio di Biomatematica, Centro Nazionale delle Ricerche, Istituto di Analisi dei Sistemi ed Informatica "A. Ruberti"; and 2Universita' Cattolica, Cattedra di Medicina Interna II, Rome, Italy
Submitted 7 November 2003 ; accepted in final form 15 June 2005
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ABSTRACT |
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mathematical models; metabolism; kinetics; nonlinear parameter estimation; confidence regions
Even-numbered DA are metabolized to acetyl-CoA and enter the TCA cycle. In addition, the metabolism of DA produces, as intermediate, succinyl-CoA, which is both a gluconeogenetic precursor and an intermediate of TCA cycle.
The possible use of DA as alternate fuel substrate for enteral or parenteral nutrition has been suggested (6, 7, 8, 14, 15, 1720, 22, 27, 28), and their characteristics might make them useful in different pathological conditions. In dyslipidemia and late sepsis, where the tissue utilization of triglycerides administered as emulsion is impaired because of reduced clearance, DA can be more effectively administered and delivered to the tissues than conventional lipids. In decompensated diabetes mellitus or in those clinical conditions, like malnourishment or sepsis, where excessive gluconeogenesis from amino acids causes muscle mass wasting, they can help spare body protein.
However, DA with a chain length shorter than dodecanedioic acid (C12) are eliminated with urine to a high degree (7, 15, 20, 28) and are therefore not suitable in their pure form for the delivery of substantial amounts of energy in enteral or parenteral nutrition, despite their theoretical advantages.
Preliminary studies in rats (18) showed that only 3.9% of C12 administered as an intravenous bolus is lost with urine, that its half-life is short (12.47 min), and that its systemic clearance is good (0.0138 liter·kg body wt1·min1); in addition, the low C12 renal clearance indicates tubular reabsorption of dodecanedioic acid.
Within the general purpose of better understanding the absorption and distribution of this potentially useful artificial metabolic substrate, the goal of the present study is therefore that of quantifying the kinetics of per os-administered C12 in healthy volunteers.
In the course of the evaluation of C12 kinetics, a study of the degree of nonlinearity of the resulting mathematical model is conducted to verify whether the usual linearization procedure for the determination of parameter confidence regions is appropriate.
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MATERIALS AND METHODS |
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C12 was purchased from Sigma (St. Louis, MO). C12 was then purified by Real S. R. L. (Como, Italy) and was free from pyrogens and contaminants with a degree of purification, ascertained using gas-liquid chromatography and mass spectrometry, of 99.8%. A 0.4 M solution of C12 salified with NaOH was used.
Experimental Protocol
A group of six healthy volunteers underwent per os administration of sodium dodecanedioate. They had no previous history of metabolic or endocrine diseases, no active illnesses, and were currently on no medications. The experimental subjects were admitted to the Day Hospital of the Division of Metabolic Diseases of the Catholic University School of Medicine Hospital (Rome, Italy) on the morning of the experiment, after an overnight fast. A bolus P.O. dose of 3 g of the substance was administered at time 0 (around 8:30 AM). Heparinized blood samples (8 ml) were drawn from an arm vein at different times for about 4 h postadministration. Blood samples were immediately centrifuged. Plasma samples were frozen at 20°C until analysis. The protocol followed the directives of the Ethical Committee of the Institutional Health Review Board of the Catholic University School of Medicine and conformed to the principles of the Declaration of Helsinki. Informed consent was obtained in all cases.
DA Analysis
Plasma samples. One hundred micrograms of azelaic acid were added to each 1 ml of each plasma sample as an internal standard. Proteins were precipitated with 0.1 ml of 4 N HCl, and DA were extracted twice with eight volumes of ethyl acetate, maintaining the solutions at 60°C for 15 min. The combined extracts were dried in a GyroVap apparatus (Howe, mod GV1; Gio. de Vita, Rome, Italy) operating at 60°C, coupled with a vacuum pump and a gas trap from FTS Systems (Stone Ridge, NY).
High-performance liquid chromatography analysis. The extracted solutes were dissolved in 0.5 ml of acetonitrile-methanol (1:1, vol/vol) and added to 10 mg of p-bromophenacylbromide and 30 µl of N,N-diisopropylethylamine as catalyst. The mixture was heated to 60°C for 15 min. The derivatives were dissolved in a final volume of 1 ml of acetonitrile-methanol (1:1, vol/vol), and an aliquot of 10 µl was automatically injected into a liquid chromatograph (Hewlett-Packard 1050) with an HP 3396A integrator and a scanning spectrophotometer operating in the 190-to-600-nm wavelength range (light source: deuterium lamp, noise <2.5 x 105 AU peak-to-peak at 254 nm with 4 nm bandwidth, flowing water at 1 ml/min).
DA derivatives were separated on an LC-18, 4.6-mm ID, 25-cm length, 5-µm particle size, reversed phase column (Supelco, Bellefonte, PA). The high-performance liquid chromatography conditions were as follows: solvent A bidistilled water-methanol (1:1, vol/vol) and solvent B acetonitrile; after a 15-min isocratic elution with 15% acetonitrile, a gradient elution was performed from 15 to 100% of B in 80 min. The flow rate was 1 ml/min, UV detector operating at 255 nm, chart speed 0.2 cm/min, and range of absorbance from 0.300 to 1.000 absorbance units (AU).
Modeling and Statistics
After per os administration, C12 is transferred from portions of the alimentary tract (henceforth briefly referred to as "gut") to a central compartment (plasma and quickly equilibrating interstitial fluid). Loss of the substance occurs from the central compartment toward the tissues. Urinary elimination is negligible, according to previously published results (8). Fecal elimination after per os administration is also negligible. Within tissues, C12 finally undergoes oxidation in the TCA cycle.
Two three-compartment models explicitly representing gastric emptying into the bowel were evaluated and discarded because of general nonidentifiability. Four two-compartment models of C12 kinetics, including a gut compartment and a central compartment, were thus compared. They differed according to the type of transport (linear or saturable) from gut to plasma and from plasma to peripheral tissues (Table 1 gives the detailed equations for each considered model). For each model, albumin binding was then taken into account (Eqs. 3 and 4), obtaining in every case a nonlinear model for the total drug concentration.
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Given the binding relationship (3), which expresses total drug concentration in terms of free drug concentration, the free concentration may be expressed in terms of total concentration as follows:
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For all parameters, the standard errors of the estimates were first determined from the inverse Hessian matrix computed at the optimum. The evaluation of model curvature at the optimum indicated potential problems in the above procedure, and parameter confidence regions were reassessed by Monte Carlo simulation.
In general, if nonlinearity of the model at the optimum is found to be excessive for the reliable determination of parameter confidence intervals, a Monte Carlo procedure may be performed to obtain empirical parameter confidence regions. This numerical procedure is based on the assumptions that 1) the correct functional form of the model is known; 2) the point parameter estimate is sufficiently near to the true value of the parameter itself; 3) the observed variance, calculated as the mean squared residual, is close to the population variance, the expected squared error; and 4) the errors are normally distributed with zero mean and variance-covariance matrix 2I. Under these assumptions, the empirical distribution of a large number of parameter estimates obtained from randomly generated samples from the model will be close to the true distribution of the estimates. Confidence regions may thus be obtained from it.
To further explore the impact of variability in the binding parameters (n and K) onto the confidence regions of the structural model parameters (V, KM, k02, TM), the entire Monte Carlo procedure was also repeated, letting n and K vary randomly.
Parameter estimates are expressed as sample mean ± SE unless otherwise specified.
To maintain as conceptually distinct the nonlinearity of the biochemical transport and the nonlinearity of the expectation surface at the optimum, the term "saturable" has been consistently used for the former.
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RESULTS |
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The anthropometric characteristics of the experimental subjects are reported in Table 2. Each model was fitted by OLS on each one of the six subjects, and in every case the R2, Akaike criterion and Schwartz criterion were computed. Table 1 reports the average model selection criteria for all four considered models. All model selection criteria favored the SL model, which was therefore retained. This model is diagrammatically represented in Fig. 1.
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Table 3 reports the model parameter estimates obtained for each individual subject, together with the respective single-subject asymptotic coefficients of variation (CVs, computed via inversion of the Hessian at the optimum). The CVs for KM and TM were in some cases unacceptably high, leading to the conclusion that these constants could not be reliably identified on single subject data. Figure 2 shows the observed SL and LL model-predicted time courses of C12 plasma concentration for one subject.
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The bottom two lines of Table 3 report the sample average estimates obtained together with their sample standard error and relative CV. The rate of elimination of C12 from the central compartment to the periphery, reflecting essentially tissue uptake and metabolism of the substance, was on average 0.108 ± 0.009 min1. The maximal rate TM of gut-to-plasma transfer was 0.129 ± 0.017 mmol/min, whereas the mean values of KM and the C12 distribution volume were 0.667 ± 0.424 mmol and 11.890 ± 0.403 l, respectively. All CVs except for KM are below 20%, so we might be very confident of the obtained estimates.
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DISCUSSION |
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To further investigate the pharmacological kinetics of C12 in humans undergoing per os administration of the substance, several different nonlinear two-compartment models were fitted to experimental data from six healthy volunteers. All of the compartmental configurations tested included alimentary tract (gut) and central (plasma) compartments and differed for transport mechanisms between compartments. Binding of C12 to serum albumin was always taken into account. According to a series of criteria (average Akaike information criterion, average Schwartz criterion and average R2), the model including a saturable transport function of the Michaelis-Menten type from gut to plasma and a linear transfer from plasma to peripheral tissues was deemed to be the best. This does not imply in any way the absence of specific tissue receptors, particularly in the skeletal muscle. Rather, we might be in the domain of near-linearity of a possibly carrier-mediated transport.
Consistent estimates of the central volume of distribution were obtained, about 1113.5 l for subjects weighing between 54 and 85 kg, reflecting plasma volume, a quota of quickly equilibrating interstitial space, and possibly a degree of binding to nonalbumin molecules. Maximal absorption from the alimentary tract to plasma was in the range of 0.10.2 mmol/min, corresponding approximately to 300600 kcal/day (attainable, for instance, with continuous enteral nutrition). Linear transport from plasma to the periphery, at 10%/min, seems on the other hand very brisk. The estimates of the volume of distribution that we obtain in the present work (11.89 ± 0.40 l) are substantially higher than those previously obtained (6.39 ± 0.44 l) (9). This is not surprising, given the longer time allowed for distribution in the case of oral administration compared with the case of intravenous administration, therefore recruiting a larger fraction of extracellular space. On the other hand, the fact of obtaining higher elimination rates than before (0.1 ± 0.01 vs. 0.013 ± 0.0023) can be explained by supposing an essential nonlinearity of tissue uptake. Bertuzzi et al. (9) had already assumed nonlinearity but could not meaningfully estimate the relative parameters and reverted to the simpler linear elimination on a patient-by-patient basis. We also could not substantiate nonlinear plasma to tissue transfer; moreover, we observed (due to po rather than iv administration) substantially lower plasma concentrations of the substance. It is therefore very likely that the linear rate we estimated on the present data, reflecting the initial steeper portion of the nonlinear transfer curve, is higher than that estimated by Bertuzzi et al. (9) on a flatter portion of the same curve.
The conclusion above is supported by a numerical argument: let us represent the (assumed real) nonlinear tissue uptake rate by means of a Michaelis-Menten kinetic [TM02 x CFREE/(KM02 + CFREE)] and let us suppose to set the TM02 and KM02 values to be 0.44 and 3.65 mM/min, respectively. Let us now hypothesize that, at the actual observed operating concentration CFREE, we approximate the nonlinear transfer with a linear term of the form K x CFREE. We may find what the apparent value for the constant K is by solving the equation
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We can see, therefore, that the explanation that the difference in apparent linear transfer rate constants is produced by observing an inherently nonlinear transfer at two different operating concentrations is possible. If we further take into account the obvious uncertainty in the K value estimate, the conclusions arrived at in the present work seem consistent with those reported by Bertuzzi et al. (9).
The maximal caloric intake that can be provided with enteral C12 administration is comparable with the entire lipid quota of an isocaloric diet of 2,000 kcal/day. It is not likely that a physician would consider administering C12 as the only supplemental lipid substrate. However, an association of dicarboxylic and traditional monocarboxylic acids would be feasible and would exploit the metabolic benefits of both fuels. For instance, administering 420 kcal/day as C12 would substitute about 70% of the lipid component of commonly employed nutrition formulas (with 30% lipid).
In case larger amounts of dicarboxylic acids were thought to be useful, parenteral administration of the substance is likely to allow very substantial energy delivery. This form of administration is safe (8, 18), inexpensive, and practical because of C12 water solubility and good tolerability of peripheral vein infusion.
Coupled with the lack of energy and synthetic requirements for hepatic complexing and the production of precious TCA cycle intermediates from its -oxidation, C12 therefore appears to be a viable and desirable substrate for artificial nutrition.
As far as the parameter KM is concerned, it would appear that the six patients reflect two essential modes of behavior: KM near zero (subjects 14), implying an essentially all-or-none response to increasing C12 plasma concentrations, where maximal transport is already immediately present as soon as minimal C12 circulates in plasma; and substantially nonzero KM (subjects 5 and 6), where maximal transport is attained gradually with increasing C12 plasma concentrations. These differences might be due to a genetic variability of the transport system, a different expression of the transport system secondary to physiological determinants such as the diet, or the possible coexistence of more than one population of gut transporters with different affinities expressed to a different degree in the studied subjects. Further investigation in this direction is warranted, possibly obtaining a richer data set by the use of stable isotope tracers.
The speed of gastric emptying (essentially a zero-order process) could also well account for the observed divergence in KM. To investigate this possibility, we also built and fitted two models, including explicitly a "stomach" compartment, into which the bolus dose was injected and from which C12 entered the gut with zero-order or first-order transport, respectively. Although neither of the two models (presenting 5 free parameters each) could be reliably identified, still, in the second one, high stomach-to-gut transfer rates tended to be associated with high KM values, suggesting that when gastric emptying is fast, plasma uptake depends on gut concentration, whereas when gastric emptying is slow, plasma uptake of C12 from the gut proceeds at an essentially constant rate. These models, however, provide no identification advantage over the simpler four-parameter, two-compartment model. Lacking the ability to identify more than two compartments, we would retain the original model (eqs. 1 and 2) with the proviso that low (unidentifiable) KM may indeed indicate slow gastric emptying and, hence, apparent zero-order kinetics. A further observation supporting this interpretation is that forecast time-to-peak values were larger in the first four and smaller in the last two subjects (120, 120, 120, 140, 100, 80 min), again indicating that subjects with faster stomach emptying tended to have first-order gut-to-plasma transfer.
From the methodological point of view, it is instructive to note how in the present work the suitability of the standard computational procedures for the determination of parameter confidence regions was seriously challenged. It is clear that the usual linear procedure, based on the inversion of the Hessian at the optimum (26), fails when the assumption that the expectation surface can be approximated in the neighborhood of the estimate by the tangent linear subspace is not realistic.
A study of the degree of nonlinearity of the model in a neighborhood of the optimum was conducted, exploring all lifted directions from parameter space and computing the relative normal curvature.
For the present model and design points, it appears that the directions in parameter space, along which intrinsic nonlinearity is highest, are represented by simultaneous changes in the parameters V and KM. For all studied subjects we had to conclude that the maximum curvature index was so much above an acceptable threshold, and that therefore the model was so highly nonlinear that the linear inferential procedure could not be accepted. Therefore, the confidence regions obtained with the linear procedure in this case either could or could not have reflected the actual confidence regions. To obtain more reliable parameter confidence regions we employed a Monte Carlo procedure, generating the empirical distribution of the parameter estimates, assuming that the model was correct and that the point estimates of the parameters and the error variance were not too far from the true values. The results of the Monte Carlo procedure show that the model employed fits well, but that the uptake constant KM is not reliably identified from a single subject's data set. One important methodological conclusion from the present work is that the study of model curvature at the optimal parameter value should be part of all physiological model evaluation procedures.
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APPENDIX |
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Study of the degree of the nonlinearity at the optimum. In nonlinear problems the approximate confidence region for the parameter vector, as commonly obtained by inverting the Hessian at the optimum, depends on the linear approximation of the expectation surface in a neighborhood of the optimum. A generally neglected problem is that of verifying whether this linear approximation is warranted. The study of the nonlinearity of the model at the optimum, as conducted in the present work, follows the treatment by Bates and colleagues (35) and Seber and Wild (26), to which the reader is referred for more details.
Briefly, two measures of curvature of the expectation surface at the estimation point, with respect to a generic direction h in parameter space, are obtained.
These two measures, named KTh and KNh, are the tangential or "parameter effects" curvature (which depends on the parameterization used), and the normal or "intrinsic" curvature, respectively (which is invariant with respect to the parametric system used).
If Th and
Nh are the corresponding normalized curvatures and F
is the upper
-quantile of the Fp,np distribution, one possible measure of intrinsic, structural departure from linearity (in the direction h) would be the quantity
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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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REFERENCES |
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