Global identifiability of nonlinear models of biological systems.

A prerequisite for well-posedness of parameter estimation of biological and physiological systems is a priori global identifiability, a property which concerns uniqueness of the solution for the unknown model parameters. Assessing a priori global identifiability is particularly difficult for nonlinear dynamic models. Various approaches have been proposed in the literature but no solution exists in the general case. In this paper, we present a new algorithm for testing global identifiability of nonlinear dynamic models, based on differential algebra. The characteristic set associated to the dynamic equations is calculated in an efficient way and computer algebra techniques are used to solve the resulting set of nonlinear algebraic equations. The algorithm is capable of handling many features arising in biological system models, including zero initial conditions and time-varying parameters. Examples of usage of the algorithm for analyzing a priori global identifiability of nonlinear models of biological and physiological systems are presented.

Role of tissue-specific blood flow and tissue recruitment in insulin-mediated glucose uptake of human skeletal muscle.

BACKGROUND: Conflicting evidence exists concerning whether insulin-induced vasodilation plays a mechanistic role in the regulation of limb glucose uptake. It can be predicted that if insulin augments blood flow by causing tissue recruitment, this mechanism would enhance limb glucose uptake. METHODS AND RESULTS: Twenty healthy subjects were studied with the forearm perfusion technique in combination with the euglycemic insulin clamp technique. Ten subjects were studied at physiological insulin concentrations (approximately 400 pmol/L) and the other 10 at supraphysiological insulin concentrations (approximately 5600 pmol/L). Four additional subjects underwent a saline control study. Pulse injections of a nonmetabolizable extracellular marker (1-[3H]-L-glucose) were administered into the brachial artery, and its washout curves were measured in one ipsilateral deep forearm vein and used to estimate the extracellular volume of distribution and hence the amount of muscle tissue drained by the deep forearm vein. Both during saline infusion and at physiological levels of hyperinsulinemia we observed no changes in blood flow and/or muscle tissue drained by the deep forearm vein. However, supraphysiological hyperinsulinemia accelerated total forearm blood flow (45.0+/-1.8 versus 36.5+/-1.3 mL x min(-1) x kg(-1), P<0.01) and increased the amount of muscle tissue drained by the deep forearm vein (305+/-46 versus 229+/-32 g, P<0.05). The amount of tissue newly recruited by insulin was strongly correlated to the concomitant increase in tissue glucose uptake (r=0.789, P<0.01). CONCLUSIONS: Acceleration of forearm blood flow mediated by supraphysiological hyperinsulinemia is accompanied by tissue recruitment, which may be a relevant determinant of forearm (muscle) glucose uptake.

Global identifiability of linear compartmental models--a computer algebra algorithm.

A priori global identifiability deals with the uniqueness of the solution for the unknown parameters of a model and is, thus, a prerequisite for parameter estimation of biological dynamic models. Global identifiability is however difficult to test, since it requires solving a system of algebraic nonlinear equations which increases both in nonlinearity degree and number of terms and unknowns with increasing model order. In this paper, a computer algebra tool, GLOBI (GLOBal Identifiability) is presented, which combines the topological transfer function method with the Buchberger algorithm, to test global identifiability of linear compartmental models. GLOBI allows for the automatic testing of a priori global identifiability of general structure compartmental models from general multi input-multi output experiments. Examples of usage of GLOBI to analyze a priori global identifiability of some complex biological compartmental models are provided.

A compartmental model of zinc metabolism in healthy women using oral and intravenous stable isotope tracers.

A mathematical model of zinc metabolism in six healthy women (average age: 30 +/- 11 y) was developed by using stable isotopes of zinc. After equilibration on a constant diet containing 7.0 mg Zn/d, an oral tracer highly enriched in 67Zn and an intravenous tracer highly enriched in 70Zn were administered simultaneously. Multiple plasma and 24-h urine samples were collected for the next 7 d with complete fecal collections for 11 d. Tracer-trace ratios in plasma, urine, and feces were calculated from isotope ratios of 67Zn to 66Zn and 70Zn to 66Zn measured by using inductively coupled plasma-mass spectrometry. An a priori identifiable model composed of seven compartments was developed to describe the kinetics of both tracers as well as that of naturally occurring zinc. The parameters of the model were fitted to the data by using the SAAM-CONSAM modeling software and were estimated with good precision. Several important, not directly measurable zinc variables were estimated (mean +/- SEM) from the model including the fractional absorption from the gastrointestinal tract (0.279 +/- 0.043), the rates of endogenous secretion (2.79 +/- 0.49 mg/d) and excretion (2.01 +/- 0.35 mg/d), the fractional turnover rate of the plasma pool (131 +/- 20/d), and the sizes (7.2 +/- 1.2 and 77.1 +/- 6.4 mg) and fractional turnover rates (22.3 +/- 7.1 and 1.49 +/- 0.18/d) of the fast and slow tissue pools equilibrating with the plasma, respectively.

Roles of glucose transport and glucose phosphorylation in muscle insulin resistance of NIDDM.

Insulin resistance for glucose metabolism in skeletal muscle is a key feature in NIDDM. The quantitative role of the cellular effectors of glucose metabolism in determining this insulin resistance is still imperfectly known. We assessed transmembrane glucose transport and intracellular glucose phosphorylation in vivo in skeletal muscle in nonobese NIDDM patients. We performed euglycemic insulin clamp studies in combination with the forearm balance technique (brachial artery and deep forearm vein catheterization) in five nonobese NIDDM patients and seven age- and weight-matched control subjects (study 1). D-Mannitol (a nontransportable molecule), 3-O-[14C]methyl-D-glucose (transportable, but not metabolizable) and D[3-3H]glucose (transportable and metabolizable) were simultaneously injected into the brachial artery, and the washout curves were measured in the deep venous effluent blood. In vivo rates of transmembrane transport and intracellular phosphorylation of D-glucose in forearm muscle were determined by analyzing the washout curves with the aid of a multicompartmental model of glucose kinetics in forearm tissues. At similar steady-state concentrations of plasma insulin (approximately 500 pmol/l) and glucose (approximately 5.0 mmol/l), the rates of transmembrane influx (34.3 +/- 9.1 vs. 58.5 +/- 6.5 micromol x min(-1) x kg(-1), P < 0.05) and intracellular phosphorylation (5.4 +/- 1.6 vs. 38.8 +/- 5.1 micromol x min(-1) x kg(-1), P < 0.01) in skeletal muscle were markedly lower in the NIDDM patients than in the control subjects. In the NIDDM patients (study 2), the insulin clamp was repeated at hyperglycemia, (approximately 13 mmol/l) trying to match the rates of transmembrane glucose influx measured during the clamp in the controls. The rate of transmembrane glucose influx (62 +/- 15 micromol x min(-1) x kg(-1)) in the NIDDM patients was similar to the control subjects, but the rate of intracellular glucose phosphorylation (16.6 +/- 7.5 micromol x min(-1) x kg(-1)), although threefold higher than in the patients during study 1 (P < 0.05), was still approximately 60% lower than in the control subjects (P < 0.05). These data suggest that when assessed in vivo, both transmembrane transport and intracellular phosphorylation of glucose are refractory to insulin action and add to each other in determining insulin resistance in skeletal muscle of NIDDM patients. It will be of interest to compare the present results with the in vivo quantitation of the initial rate of muscle glucose transport when methodology to perform this measurement becomes available.

A model to measure insulin effects on glucose transport and phosphorylation in muscle: a three-tracer study.

We studied five healthy subjects with perfused forearm and euglycemic clamp techniques in combination with a three-tracer (D-[12C]mannitol, not transportable; 3-O-[14C]methyl-D-glucose, transportable but not metabolizable; D-[3-3H]glucose, transportable and metabolizable) intra-arterial pulse injection to assess transmembrane transport and intracellular phosphorylation of glucose in vivo in human muscle. The washout curves of the three tracers were analyzed with a multicompartmental model. A priori identifiability analysis of the tracer model shows that the rate constants of glucose transport into and out of the cells and of glucose phosphorylation are uniquely identifiable. Tracer model parameters were estimated by a nonlinear least-squares parameter estimation technique. We then solved for the tracee model and estimated bidirectional transmembrane transport glucose fluxes, glucose intracellular phosphorylation, extracellular and intracellular volumes of glucose distribution, and extracellular and intracellular glucose concentrations. Physiological hyperinsulinemia (473 +/- 22 pM) caused 2.7-fold (63.1 +/- 7.2 vs. 23.4 +/- 6.1 mumol.min-1.kg-1, P < 0.01) and 5.1-fold (42.5 +/- 5.8 vs. 8.4 +/- 2.2 mumol.min-1.kg-1, P < 0.01) increases in transmembrane influx and intracellular phosphorylation of glucose, respectively. Extracellular distribution volume and concentration of glucose were unchanged, whereas intracellular distribution volume of glucose was increased (approximately 2-fold) and intracellular glucose concentration was almost halved by hyperinsulinemia. In summary, 1) a multicompartment model of three-tracer kinetic data can quantify transmembrane glucose fluxes and intracellular glucose phosphorylation in human muscle; and 2) physiological hyperinsulinemia stimulates both transport and phosphorylation of glucose and, in doing so, amplifies the role of glucose transport as a rate-determining step of muscle glucose uptake.

Bicarbonate kinetics in humans: identification and validation of a three-compartment model.

A model of bicarbonate kinetics is crucial to a correct interpretation of experiments for measuring oxidation in vivo of carbon-labeled compounds. The aim of this study is to develop a compartmental model of bicarbonate kinetics in humans from tracer data by devoting particular attention to model identification and validation. The data base consisted of impulse-dose studies of 14C-labeled bicarbonate in nine normal subjects. The decay curve of specific activity of CO2 in expired air (saRCO2) was frequently sampled for 4-7 h. In addition, endogenous production of CO2, VCO2, was measured by indirect calorimetry. A model of data, i.e., an exponential model, analysis of decay curves of saRCO2 showed first that three compartments are necessary and sufficient to describe bicarbonate tracer kinetics. Compartmental models were then used as models of system. To correctly describe the input-output configuration, labeled CO2 flux in the expired air, phi RCO2 (= saRCO2.VCO2), has been used as measurement variable in tracer model identification. A mammillary three-compartment model with a respiratory and a nonrespiratory loss has been studied. Whereas there is good evidence that respiratory loss takes place in the central compartment, whether nonrespiratory loss is taking place in the central compartment or in one of the two peripheral compartments is uncertain. Thus three competing tracer models were considered. Using a model-independent analysis of data, based on the body activity variable, to calculate mean residence time in the system, we have been able to validate a specific model structure, i.e., with the two irreversible losses taking place in the central compartment. This validated tracer model was then used to quantitate bicarbonate masses in the system. Because there is uncertainty about where endogenous production enters the system, lower and upper bounds of masses of bicarbonate in the body are derived.

A minimal input-output configuration for a priori identifiability of a compartmental model of leucine metabolism.

To develop a model describing the structure and function of a metabolic system using data from an input-output experiment, it is useful to design a pilot tracer study first which contains a predicted maximal amount of information. Having postulated a physiologically reasonable model structure from the pilot data, two questions arise. First, are the model parameters a priori uniquely identifiable? That is, assuming an error-free model structure and data, can the parameters be uniquely identified from the information content of the pilot experiment? Second, if the model parameters are uniquely identifiable, is the pilot experiment a minimal one? That is, is the pilot experiment necessary and sufficient, in the sense of information content, among feasible experiments to guarantee a priori unique identifiability? The purpose of this paper is to determine a minimal input-output configuration for the a priori unique identifiability of a compartmental model describing the metabolism of leucine, an essential amino acid. The original pilot tracer experiment was a two-stage experiment consisting first of a two input-five output experiment followed by a single input-single output experiment. Here we show to guarantee a priori unique identifiability of the leucine model that the single input-single output experiment is not necessary, and that two of the outputs of the multi-input-multi-output experiment are not required.

Transmembrane glucose transport in skeletal muscle of patients with non-insulin-dependent diabetes.

Insulin resistance for glucose metabolism in skeletal muscle is a key feature in non-insulin-dependent diabetes mellitus (NIDDM). Which cellular effectors of glucose metabolism are involved is still unknown. We investigated whether transmembrane glucose transport in vivo is impaired in skeletal muscle in nonobese NIDDM patients. We performed euglycemic insulin clamp studies in combination with the forearm balance technique (brachial artery and deep forearm vein catheterization) in six nonobese NIDDM patients and five age- and weight-matched controls. Unlabeled D-mannitol (a nontransportable molecule) and radioactive 3-O-methyl-D-glucose (the reference molecular probe to assess glucose transport activity) were simultaneously injected into the brachial artery, and the washout curves were measured in the deep venous effluent blood. In vivo transmembrane transport of 3-O-methyl-D-glucose in forearm muscle was determined by computerized analysis of the washout curves. At similar steady-state plasma concentrations of insulin (approximately 500 pmol/liter) and glucose (approximately 5.15 mmol/liter), transmembrane inward transport of 3-O-methyl-D-glucose in skeletal muscle was markedly reduced in the NIDDM patients (6.5 x 10(-2) +/- 0.56 x 10(-2).min-1) compared with controls (12.5 x 10(-2) +/- 1.5 x 10(-2).min-1, P < 0.005). Mean glucose uptake was also reduced in the diabetics both at the whole body level (9.25 +/- 1.84 vs. 28.3 +/- 2.44 mumol/min per kg, P < 0.02) and in the forearm tissues (5.84 +/- 1.51 vs. 37.5 +/- 7.95 mumol/min per kg, P < 0.02). When the latter rates were extrapolated to the whole body level, skeletal muscle accounted for approximately 80% of the defect in insulin action seen in NIDDM patients. We conclude that transmembrane glucose transport, when assessed in vivo in skeletal muscle, is insensitive to insulin in nonobese NIDDM patients, and plays a major role in determining whole body insulin resistance.

Effect of insulin on system A amino acid transport in human skeletal muscle.

Transmembrane transport of neutral amino acids in skeletal muscle is mediated by at least four different systems (system A, ASC, L, and Nm), and may be an important target for insulin's effects on amino acid and protein metabolism. We have measured net amino acid exchanges and fractional rates of inward (k(in), min-1) and outward (kout, min-1) transmembrane transport of 2-methylaminoisobutyric acid (MeAIB, a nonmetabolizable amino acid analogue, specific for system A amino acid transport) in forearm deep tissues (skeletal muscle), by combining the forearm perfusion technique and a novel dual tracer ([1-H3]-D-mannitol and 2-[1-14C]-methylaminoisobutyric acid) approach for measuring in vivo the activity of system A amino acid transport. Seven healthy lean subjects were studied. After a baseline period, insulin was infused into the brachial artery to achieve local physiologic hyperinsulinemia (76 +/- 8 microU/ml vs 6.4 +/- 1.6 microU/ml in the basal period, P < 0.01) without affecting systemic hormone and substrate concentrations. Insulin switched forearm amino acid exchange from a net output (-2,630 +/- 1,100 nmol/min per kig of forearm tissue) to a net uptake (1,610 +/- 600 nmol/min per kg, P < 0.01 vs baseline). Phenylalanine and tyrosine balances simultaneously shifted from a net output (-146 +/- 47 and -173 +/- 34 nmol/min per kg, respectively) to a zero balance (16.3 +/- 51 for phenylalanine and 15.5 +/- 14.3 nmol/min per kg for tyrosine, P < 0.01 vs baseline for both), showing that protein synthesis and breakdown were in equilibrium during hyperinsulinemia. Net negative balances of alanine, methionine, glycine, threonine and asparagine (typical substrates for system A amino acid transport) also were decreased by insulin, whereas serine (another substrate for system A transport) shifted from a zero balance to net uptake. Insulin increased k(in) of MeAIB from a basal value of 11.8.10(-2) +/- 1.7.10(-2).min-1 to 13.7.10(-2) +/- 2.2.10(-2).min-1 (P < 0.02 vs the postabsorptive value), whereas kout was unchanged. We conclude that physiologic hyperinsulinemia stimulates the activity of system A amino acid transport in human skeletal muscle, and that this effect may play a role in determining the overall concomitant response of muscle amino acid/protein metabolism to insulin.

Glucose transport in human skeletal muscle. The in vivo response to insulin.

Transmembrane glucose transport plays a key role in determining insulin sensitivity. We have measured in vivo WBGU, FGU, and K(in) and K(out) of 3-O-methyl-D-glucose in forearm skeletal muscle by combining the euglycemic clamp technique, the forearm-balance technique, and a novel dual-tracer (1-[3H]-L-glucose and 3-O-[14C]-methyl-D-glucose) technique for measuring in vivo transmembrane transport. Twenty-seven healthy, lean subjects were studied. During saline infusion, insulin concentration, FGU (n = 6), K(in), and K(out) (n = 4) were similar to baseline. During SRIF-induced hypoinsulinemia (insulin < 15 pM, n = 4) WBGU was close to 0, and FGU, K(in), and K(out) were unchanged from basal (insulin = 48 pM) values. During insulin clamps at plasma insulin levels of approximately 180 (n = 4), approximately 420 (n = 5), approximately 3000 (n = 4), and approximately 9500 pM (n = 4), WBGU was 14.2 +/- 1.3, 34.2 +/- 4.1 (P < 0.05 vs. previous step), 55.8 +/- 1.8 (P < 0.05 vs. previous step), and 56.1 +/- 6.3 mumol.min-1.kg-1 of body weight (NS vs. previous step), respectively. Graded hyperinsulinemia concomitantly increased FGU from a basal value of 4.7 +/- 0.5 mumol.min-1.kg-1 up to 10.9 +/- 2.3 (P < 0.05 vs. basal value), 26.6 +/- 4.5 (P < 0.05 vs. previous step), 54.8 +/- 4.3 (P < 0.05 vs. previous step), and 61.1 +/- 10.8 mumol.min-1.kg-1 of forearm tissues (NS vs. previous step), respectively.(ABSTRACT TRUNCATED AT 250 WORDS)

Domain of validity of classical models of leucine metabolism assessed by compartmental modeling.

Whole-body modeling of in vivo leucine (an essential amino acid) metabolism is fundamentally difficult due to the complexity of the system. This has favored the use of two simple kinetic models, the so-called primary and reciprocal pool models, to interpret tracer data, but their domain of validity is uncertain. We define here the error of these two approaches by using comprehensive compartmental models of leucine metabolism as true representations of the leucine system. Of particular interest is the comparison of the two simple models with an 11-compartment model characterized by a rich intracellular compartmentation that has recently been proposed as a sound physiological description of the system. Formulas are derived that define in structural terms the error of the primary and reciprocal pool models.

Compartmental model of leucine kinetics in humans.

The complexity of amino acid and protein metabolism has limited the development of comprehensive, accurate whole body kinetic models. For leucine, simplified approaches are in use to measure in vivo leucine fluxes, but their domain of validity is uncertain. We propose here a comprehensive compartmental model of the kinetics of leucine and alpha-ketoisocaproate (KIC) in humans. Data from a multiple-tracer administration were generated with a two-stage (I and II) experiment. Six normal subjects were studied. In experiment I, labeled leucine and KIC were simultaneously injected into plasma. Four plasma leucine and KIC tracer concentration curves and label in the expired CO2 were measured. In experiment II, labeled bicarbonate was injected into plasma, and labeled CO2 in the expired air was measured. Radioactive (L-[1-14C]leucine, [4,5-3H]KIC, [14C]bicarbonate) and stable isotope (L-[1-13C]leucine, [5,5,5-2H3]KIC, [13C]bicarbonate) tracers were employed. The input format was a bolus (impulse) dose in the radioactive case and a constant infusion in the stable isotope case. A number of physiologically based, linear time-invariant compartmental models were proposed and tested against the data. The model finally chosen for leucine-KIC kinetics has 10 compartments: 4 for leucine, 3 for KIC, and 3 for bicarbonate. The model is a priori uniquely identifiable, and its parameters were estimated with precision from the five curves of experiment I. The separate assessment of bicarbonate kinetics (experiment II) was shown to be unnecessary. The model defines masses and fluxes of leucine in the organism, in particular its intracellular appearance from protein breakdown, its oxidation, and its incorporation into proteins. An important feature of the model is its ability to estimate leucine oxidation by resolving the bicarbonate model in each individual subject. Finally, the model allows the assessment of the domain of validity of the simpler commonly used models.

Accessible pool and system parameters: assumptions and models.

Quantitative assessment of substrate metabolism from in vivo tracer kinetic data requires a model of the system, i.e., a hypothesis on the structure and functioning of the system. Some fundamentals of modeling important for studying intermediary metabolism in the steady state will be discussed. Accessible pool and system parameters are defined. Although the calculation of accessible pool parameters is structure-free, that of system parameters requires the use of non-compartmental or compartmental structures. Assumptions, bases for choice, and relative merits of these two modeling strategies are discussed. Glucose and leucine metabolism serve as prototypes to illustrate the theoretical points.

A compartmental model to quantitate in vivo glucose transport in the human forearm.

Glucose transport is a critical step in the control of glucose disposal that, until presently, has not been quantitated in vivo in humans. We have employed the perfused forearm and euglycemic insulin-clamp techniques in combination with a dual-tracer injection to measure basal and insulin-mediated glucose transport in six normal subjects. L-[3H]glucose, which is not transported, was used to trace extracellular glucose kinetics; 3-O-[14C]-methyl-D-glucose, transportable but not metabolizable, was used to monitor glucose movement across the cell membrane. After bolus intra-arterial injection of the two tracers, plasma samples were obtained every 15-30 s for 10 min from a deep forearm vein to determine the washout curves. A linear compartmental model was developed that accounts for blood flow heterogeneity. It consists of three parallel, two-compartment chains merging into the sampling compartment to which cellular compartments are appended. A priori identifiability analysis was performed. The uniquely identifiable parameterization includes the transport rate constants of glucose into and out of the cell. The model was identified using nonlinear least-squares parameter estimation. Transport parameters are estimated with very good precision, and their reproducibility is satisfactory. The model also allows the estimation of the mean arteriovenous transit times of both the extracellular and the transported tracer. The compartmental model provides a novel approach to investigate glucose transport in vivo in humans.