Abstract
The maintenance of blood glucose homeostasis is complex and involves several key tissues. Most of these tissues are not easily accessible, making direct measurement of the physiological parameters involved in glucose metabolism difficult. The use of isotope tracer methodology and mathematical modeling allows indirect estimates of in vivo glucose metabolism through relatively noninvasive means. The purpose of this paper was to provide a mathematical synthesis of the models developed for describing glucose kinetics. As many of the models were developed using dogs, example data from the canine literature are presented. However, examples from the human and feline literature are also given in the absence of dog data. The glucose system is considered in both the steady and nonsteady states, and the models are examined by grouping them into schemes consisting of one, two, and three glucose compartments. Noncompartmental schemes are also considered briefly.
1. Introduction
Glucose is a ubiquitous cellular fuel source for all mammalian tissues. As such, the regulation of glucose metabolism has been under extensive investigation for the past century. In healthy animals, several biological mechanisms ensure that the rate of appearance of glucose in the bloodstream tightly matches that of glucose uptake by tissues, resulting in relatively constant blood glucose concentrations irrespective of physiological condition. These control mechanisms are predominantly dictated by the energy status of the animal. In postabsorptive and fasting periods the body largely relies on endogenous glucose production via liver glycogenolysis or gluconeogenesis to maintain glucose homeostasis. In the postprandial period (or fed state) glucose uptake and utilization by tissues is increased in response to the absorption of glucose in the small intestine, giving rise to the subsequent stabilization of blood glucose concentrations. Both production and uptake are principally hormonally regulated by glucagon and insulin, but substrate availability, circulating free fatty acids (FFAs), and catecholamines (released in response to stress and/or exercise) also impact glucose metabolism.
Dysregulation of glucose homeostasis, as seen in metabolic diseases including obesity and diabetes, has become a worldwide epidemic in humans [1] and companion animals [2]. These diseases share the common feature of progressive insulin resistance and hyperglycemia. The rat has been used extensively as a model for studying the etiology of these diseases (reviewed by [3]). However, the dog provides a model more relatable to the human condition due to the similarities in pathophysiology of insulin resistance between dogs and humans. In addition, the dog enables better in vivo access to the key tissues involved in metabolic disease than rodents. Therefore, it is not surprising that the early kinetic models to describe glucose metabolism were developed using dogs. Pancreatectomy and chemical destruction of the pancreas (i.e., alloxan) are the most common methods used to induce dysregulation in glucose metabolism in dogs [3]. Such models are reflective of overt diabetes, as these procedures significantly impair (or in some cases abolish) insulin secretion resulting in marked hyperglycemia. These invasive procedures cannot, however, be used to investigate the progression of the diabetes. The “prediabetes” state, characterized by insulin resistance, is most commonly studied using dietary interventions, including high fat and/or high fructose feeding. These diets have been shown to induce insulin resistance in humans and dogs; however, the etiology of insulin resistance has not been fully elucidated.
The use of stable and/or radioactive isotope tracers allows for the quantification pool sizes and flows between compartments involved in glucose metabolism in vivo at the whole-body level. Both hydrogen (2H or 3H) and carbon (13C or 14C) labelled glucose tracers are available, and selection is dependent on what aspect of metabolism is of interest. The most commonly used hydrogen tracers are labelled in the 2, 3 or 6 position. Label in the 2 or 3 positions will be lost in glycolysis and incorporated into body water, making these tracers most useful for determining plasma turnover and/or glycolytic flux. Label in the 6 position is not lost in glycolysis but in gluconeogenesis, making it ideal for assessing endogenous glucose production. Carbon tracers include labelling of the 1, 6, or 1-6 (uniformly, U) position(s). The carbon label is lost in the tricarboxylic acid (TCA) cycle as labelled CO2 enabling estimation of whole-body glucose oxidation. Carbon label may become incorporated in TCA cycle intermediates and as lactate (via the Cori cycle). This sequestration of carbon is referred to as carbon recycling and should be accounted for in kinetic calculations.
The most common techniques used to assess glucose tracer kinetics include the bolus injection and the constant infusion [4]. The bolus injection method generally involves venous injection of a bolus dose of isotope resulting in an initial increase in plasma isotope enrichment followed by an exponential decline/decay over time. The most common application is the intravenous glucose tolerance test (IVGTT), where a bolus of unlabelled glucose is administered with (or without) tracer allowing for the calculation of glucose effectiveness (GE) (the ability of glucose to stimulate its own uptake) and insulin sensitivity (IS) (the ability of insulin to stimulate glucose uptake). In the constant infusion method, isotope is infused at a constant rate until an isotopic equilibrium is reached. In this situation the rate of appearance of glucose in the plasma is equal to the rate of disappearance. This methodology is commonly used during “clamp” studies, where either unlabelled glucose, insulin, free fatty acids, or a combination thereof is also infused at a constant rate (“clamped”). A common example would be the euglycemic hyperinsulinemic clamp (EHC), where insulin concentrations are clamped at a supraphysiological level and glucose is infused at a variable rate to maintain euglycemia. The rate of glucose infusion is used as an index of insulin sensitivity. As a result of the slow turnover rate of glucose relative to its pool size, it takes several hours to reach isotopic equilibrium using the constant infusion technique. A bolus (priming) dose of isotope in combination with constant infusion can be used to reduce the time to achieve isotopic steady state. Either methodology allows for the calculation of glucose kinetics under physiological steady-state conditions. However, the constant infusion technique ( prime) enables the most accurate description of nonsteady-state kinetics. The premise being that once isotopic equilibrium is reached any changes in isotope enrichment will reflect those induced by the metabolic perturbation (i.e., exercise).
Interpretation of the kinetic data generated from isotope tracer studies requires the use of mathematical modeling. Several mathematical models of varying complexity have been published to describe in vivo wholebody glucose kinetics. The purpose of this paper was to provide a mathematical synthesis of these models. As many of the models were developed using dogs, example data from the canine literature are presented. However, examples from the human and feline literature are also given in the absence of canine data. The glucose system is considered in both the steady and nonsteady states, and the models are examined by grouping them into schemes consisting of one, two, and three glucose compartments. Noncompartmental schemes are also considered briefly. Mathematical notation is defined in Table 1. Software implementations are not reviewed as this paper is a mathematical synthesis rather than a typical review. Although reviews and accounts of whole-body glucose models have recently been published [5, 6], these are not specific to kinetic modelling (i.e., resolving isotope tracer data).
2. Models with One Glucose Compartment
2.1. Insulin-Independent Representations
The minimal scheme is shown in Figure 1 with example data given in Table 2. This scheme contains one pool (venous plasma glucose), one inflow (rate of glucose appearance, Ra), and one outflow (rate of glucose disappearance, Rd). Isotope (either stable or radioactive) is administered by constant infusion or by single dose injection, and plasma glucose concentration and enrichment (or specific activity) are monitored following isotope administration. The fundamental equations follow.
For total (plasma) glucose: For labelled glucose: Further: If multiple exits should exist, the rate of disappearance is the sum of all the exits, likewise for and multiple entrances.
If the plasma glucose pool is in steady state, such as after an overnight fast, the derivative becomes zero, and the constant can be calculated from (3) as . Likewise becomes zero for constant infusion. Equations (1) and (2a) then reduce to two simultaneous equations in two unknowns ( and ), which can readily be solved to give Under single dose injection, assuming outflow obeys mass action kinetics (i.e., ), (2b) becomes Dividing both sides by constant yields Integrating The rate constant can be estimated by regression analysis of the isotope dilution curve (see (7)), allowing determination of the rate of glucose disappearance (and hence the rate of appearance).
The model can be solved for the plasma glucose pool in nonsteady state by combining the fundamental equations using as follows that is, Substituting for and using (1) and (2a) Rearranging: that is, The derivative , which varies over time, can be estimated numerically from the isotope dilution curve. This nonsteady-state model was first proposed by Steele [17] in studying hepatic glucose output using glucose, with one modification: where is interpreted as the “effective” glucose volume, a fraction of , the “total” glucose distribution volume. The generally chosen value of is 0.65. If isotope administration is by single injection rather than constant infusion, then (12) is replaced by
For IVGTT, where a significant dose of glucose is injected following an overnight fast, (1) becomes where is assumed constant, and glucose disappearance follows mass action kinetics. Dividing by constant : where is glucose concentration in the fasted (basal) state. The differential equation can be integrated analytically, using the integrating factor method [18] to give The unknown parameters (, ) can be determined by regression analysis of the glucose concentration curve after injection. If a labelled IVGTT is performed, the resultant isotope dilution curve is described by (7). Of some interest is the partial derivative of (16) with respect to G: as it gives a measure of the effect of glucose per se on its own disposition (see next section).
Bergman et al. [19] proposed this model of glucose disappearance (among several others) in examining the feasibility of using an unlabelled IVGTT (α-D-glucose) for estimating peripheral insulin sensitivity in dogs. Bergman et al. [19] also proposed an identical model for this purpose, except that glucose utilization is a saturable, rather than mass-action process that obeys Michaelis-Menten kinetics, that is, in (16), where is maximum velocity and an affinity constant. The resulting differential equation is now nonlinear and has to be handled numerically using specialist software for integration and parameter estimation. The partial derivative of (16) becomes
2.2. Insulin-Dependent Representations
Figure 2 shows the insulin-dependent minimal model, again due to Bergman et al. [19] and promulgated by Cobelli et al. [20], for glucose disappearance during an IVGTT. It builds on work by Bolie [21], the first to introduce insulin dependence into the minimal glucose model. The model contains two pools (plasma glucose and remote insulin). The glucose pool has one inflow (rate of glucose appearance) and two outflows (rate of glucose disappearance to liver and disappearance to peripheral tissues). All glucose flows are affected by remote insulin concentration. A single dose of cold glucose with (or without) labeled glucose is administered by injection, and plasma glucose concentration, its enrichment (or specific activity), and plasma insulin concentration are monitored following dosing. The remote insulin pool has one inflow (from plasma insulin) and one outflow. The fundamental equations follow. For total plasma glucose: For labelled plasma glucose: For remote insulin: Further: If the outflows follow mass-action kinetics, then: where the ’s are relative rate parameters. The fundamental equations now yield
All flows out of the plasma glucose pool are assumed to be directly affected by remote insulin concentration and flow into the remote insulin pool by the concentration of plasma insulin. These assumptions, together with the assumption of constant glucose appearance rate, give rise to the following equations: where , , , , , are constants and is basal plasma insulin concentration. The fundamental equations can therefore be reparameterized to give (28)–(30):
Glucose effectiveness (GE or SG), following Bergman et al. [19], is defined as the enhancement of glucose disappearance due to an increase in plasma glucose concentration: Insulin sensitivity (IS or SI) is then defined as the ability of insulin to increase glucose effectiveness: For the insulin-dependent minimal model, the definitions yield in the basal steady state: These parametric formulae may be derived as follows. Differentiating (28) with respect to G: Differentiating with respect to : In steady state, (30) gives Therefore In applying the insulin-dependent minimal model, integration and parameter estimation are undertaken numerically.
There are numerous applications of the model in the literature (see Table 3) (reviewed by [22]), the most common being the modified insulin or tolbutamide (stimulates endogenous insulin secretion) IVGTT (miIVGTT and mtIVGTT, resp.). In this application, either insulin or tolbutamide is infused into the plasma to stimulate the remote insulin pool. A novel application was introduced by [23], who used a modified version of the minimal model to analyze EHC data on lean and obese cats obtained using an infusion of cold and 3H-glucose. The primary novelty is the addition of a pool of labelled water to the model to determine the rate of glycolysis and the introduction of a Michaelis-Menten type inhibition of glucose appearance by plasma insulin characterized by a parameter defined as the suprabasal concentration at which glucose appearance is inhibited by 50%.
3. Models with Two Glucose Compartments
3.1. Insulin-Independent Representations
Single pool models of glucose kinetics assume that glucose is uniformly distributed in the extracellular fluid space. In attempt to overcome this simplification of glucose distribution, a two-compartment approach was proposed by Radziuk et al. [29]. The general scheme is shown in Figure 3. It contains two glucose pools: the first encompassing plasma and the second being interstitial fluid. The external inflow to pool 1 is from hepatic glucose production/output (HGO) (appearance). There is a flow out of the system from each pool and a bidirectional flow between the two pools. Irreversible loss from pool 1 is due to insulin independent tissue uptake (i.e., brain and liver) and irreversible loss from pool 2 represents insulin stimulated glucose uptake. Isotope is administered by constant infusion or by single dose injection, and then plasma glucose concentration and isotope enrichment/specific activity monitored. The fundamental equations follow. For total glucose: For labelled glucose:
If the system is in steady state with respect to total glucose, the derivatives and are zero, and if isotope is administered by constant infusion, and become zero and equilibrates with . Equations (38)–(40) then reduce to a set of 3 simultaneous equations ((40) becomes redundant when equals ) in 5 unknowns which can be solved uniquely only for : For single dose injection, assuming mass-action kinetics, the differential equations for labelled glucose become Integration of these two equations can be accomplished through their Laplace transformation [18, 30]: where the ’s and ’s are coefficients of integration (dimensionless) and ’s are rate parameters (min−1).
Radziuk et al. [29] developed this model to describe glucose kinetics in the nonsteady state (and therefore the differential equations have to be solved numerically). Two versions of the model were used for analysis: the first version with a single irreversible loss emanating from the primary pool with a varying relative rate parameter, that is, and the second with a loss from both pools with a single varying rate parameter, that is, Cobelli et al. [31] propose a version of the model with 3 time-varying parameters: to describe the transition between basal and final state of an EHC. The irreversible loss from the primary pool is interpreted as insulin-independent utilization and the irreversible loss from pool 2 as insulin-dependent utilization. They also propose a simplified version to interpret glucose kinetics during an IVGTT: In another version of the model, Hoenig et al. [23] propose a third pool, the water distribution space, to study 3-3H-glucose and 3H-H2O kinetics in cats to estimate the glycolytic fraction of glucose disappearance. Literature data are presented in Table 4.
3.2. Insulin-Dependent Representations
Figure 4 shows the insulin-dependent version of the model, due to Caumo and Cobelli [33], for glucose disappearance and HGO during an IVGTT (example data in Table 5). The model contains two glucose pools and a remote insulin pool. Insulin-independent glucose disposal (removal) and appearance take place from the accessible (venous plasma) pool and insulin-dependent glucose disposal from the slowly exchanging interstitial pool (pool 2). Inflow to the remote insulin pool is from plasma insulin. A single combined dose of cold and labelled glucose is administered by injection, and plasma glucose concentration, its enrichment (or specific activity), and plasma insulin concentration are monitored following dosing. Pool exits are assumed to follow mass-action kinetics. The fundamental equations for the glucose pools are given by (38), (39b), and (40), and that for the remote insulin pool by which is analogous to (30). Both relative disposal rates are made time dependent, that is, where the ’s are constants and is the value of in the basal state. All the other relative rates (’s) are assumed constant, and the model is solved numerically. To aid parameter estimation, Caumo and Cobelli [33] employ the constraint that insulin-independent glucose disposal is three times insulin-dependent glucose disposal in the basal state.
4. Other Schemes
4.1. Models with Three Glucose Compartments
The general scheme for three glucose pools is shown in Figure 5. Pool 1 is the accessible venous pool encompassing plasma. The external inflow to pool 1 is glucose production (appearance). Pools 2 and 3 represent peripheral compartments in rapid and slow equilibrium, respectively, with the accessible pool. Pool 2 is associated with insulin-independent tissues, such as brain, liver, kidneys, and pool 3 with insulin-dependent tissues, including muscle and adipose tissue. There is a bidirectional flow between pools 1 and 2 and between pools 1 and 3. Irreversible losses occur from pools 2 and 3 accounting for insulin-independent and insulin-dependent glucose utilization, respectively. Isotope is administered by constant infusion or by single dose injection, and then plasma glucose concentration and isotope enrichment/specific activity monitored in plasma. The fundamental equations follow. For total glucose: For labelled glucose:
If the system is in steady state with respect to total glucose, the derivatives , , are zero, and if isotope is administered by constant infusion, , , become zero and both and equilibrate with . Equations (50)–(52) then reduce to a set of 4 simultaneous equations ((52) become redundant when ) in 7 unknowns which can be solved uniquely only for : For single dose injection, assuming mass-action kinetics, the differential equations for labelled glucose become Integration of these three equations can be accomplished through their Laplace transformation:
For further solution details, see Shipley and Clark [30, Appendix II].
This model was originally proposed by Cobelli et al. [36] to analyze EHC data (Table 6). They constrained insulin-independent glucose disposal to be three times insulin-dependent disposal in the basal state in order to facilitate parameter estimation.
4.2. Noncompartmental Schemes (Dispersal Models)
In this type of model, the glucose system is considered a region governed by diffusion and convection. The fundamental equations are statements of the general equation for diffusion, convection, and chemical reaction [37, Page 122]. To illustrate the approach, consider glucose occupying a region bounded by a regular cylinder of radius r and indeterminate length (Figure 6). An impulse dose of labeled glucose is applied at time zero at the point . The general diffusion-convection-reaction equation simplifies to where [cm2/h] is the diffusion coefficient, v [cm/h] the velocity of convection (i.e., blood flow) which is assumed to be in the direction of increasing , and [per h] is the relative rate of glucose disappearance (uptake). Instantaneous disappearance of label is Labeled glucose remaining at time t (as a fraction of the impulse dose) is therefore
Solutions to the model are obtained by solving (56) and (58) numerically. Equation (58) is a convolution integral. This model is a very simple (arguably simplistic) illustration of a dispersal model of the glucose system and is meant to be instructive rather than realistic. Though dispersal models assume less structural knowledge, their equations are clearly more complex and their solutions require advanced numerical methods. A more physiologically based example of a dispersal model for glucose kinetics is given by Radziuk et al. [29], who consider two regions. A first region, in which cold and labeled glucose appear and are sampled, is well mixed and exchanges with a second region in which diffusion and convection are allowed. Disappearance of glucose takes place in both regions. The equations of the model are more complex than the illustrative example described above. McGuinness and Mari [38] also describe a dispersal model with two regions but use a markedly different kinetic analysis. Their model, designed for EHC application, is comprised of heart-lung and periphery blocks. The impulse responses of the two blocks are modeled as multiexponential functions then converted to linear differential equations. The differential equations of the heart-lung and periphery blocks are subsequently combined to obtain the differential equations of the circulatory model.
5. Discussion
The maintenance of blood glucose homeostasis is complex and involves several key tissues. Most of these tissues are not easily (noninvasively) accessible, making direct measurement of the physiological parameters involved in glucose metabolism difficult. As such, several mathematical models have been developed to describe in vivo whole-body glucose metabolism. The application of single pool models dominates the literature, while examples of two pool models are limited, and three pool models are rare.
The wide spread use of single pool models is likely due to their simplicity both technically and mathematically. Technically, they require the insertion of a venous catheter for isotope, glucose and/or insulin infusion, and repeated blood sampling from a single site, often a vein (i.e., jugular or cephalic), and glucose and/or insulin is measured in the plasma or serum. Such methodology is relatively straightforward, inexpensive, and noninvasive in most animals and humans. Mathematically, the underlying assumption with one pool models is that blood glucose rapidly and uniformly distributes in all physiological pools (i.e., plasma, interstitial and intracellular). However, most agree that glucose cannot be adequately explained so simply [4]. Plasma glucose exchanges with the interstitial fluid pool where it is then taken up by the tissues through facilitated diffusion. In humans, plasma accounts for 4.5% of body weight and interstitial fluid ~16% of body weight, meaning that changes in whole-body plasma glucose appearance will not be immediately reflected by changes in the entire interstitial fluid pool [4]. Furthermore, the rate of diffusion from the interstitial pool to the intracellular pool differs between tissues, and exchange between these pools is bidirectional in gluconeogenic tissues like the liver. As one pool models group all these factors together, the rate of plasma glucose turnover is assumed to be reflective of all glucose pools and due to the combined abilities of glucose and insulin to stimulate glucose disposal and suppress glucose production (GE and IS). Such assumptions may be valid if the system is in steady state or a large perturbation in plasma glucose is made.
As the one-pool insulin-independent model is unable to separate the contributions of glucose and insulin to plasma glucose Rd, it is most often applied to steady-state conditions. Steady-state, or “clamp,” conditions may take several hours of continuous isotope infusion to achieve. Such experiments often aim to determine acute (i.e., AICAR infusion) or chronic (i.e., after several weeks of high fat feeding) changes in plasma glucose turnover from basal, or control, conditions (Table 2).
Large perturbations to the plasma glucose pool, as made in the IVGTT, presume a rapid exchange (within minutes) among glucose pools. In these experiments glucose Rd is assumed to be monoexponential, and the first few data time points are often excluded from model calculations. Addition of the remote insulin pool to the model allows for indirect estimates of GE and insulin mediated glucose disposal/production (IS). It was developed in healthy canines; however, its application to pancreatectomized dogs revealed that it could not accurately predict GE or IS in this population. This limitation is due to the model assumption that insulin secretion in response to glucose infusion rises above a certain basal level. Therefore, if insulin secretion is impaired, as in the case of diabetes, the measurement of GE may be overestimated and IS underestimated. In order to rectify this problem, the addition of an exogenous bolus of insulin or tolbutamide was added to the IVGTT and was shown to improve estimates. The application of the imIVGTT is widespread and is often done without the use of isotope tracers (likely due to the cost of isotopes). Without the use of isotopes, one cannot distinguish between HGO and hepatic glucose uptake. Instead it describes net hepatic glucose flux. As such, the relative contribution of HGO to IS cannot be determined and is known to vary depending on the degree of insulin resistance [39]. However, even when isotopes are used with the imIVGTT, this model can produce unrealistic HGO values [33].
The two-pool insulin-dependent model provides more meaningful estimates of HGO [33]. Unlike the single pool version, it takes into account the fact that glucose is not uniformly distributed, by separating glucose into fast and slow exchanging pools. The fast or accessible plasma pool describes tissues such as the brain, which requires a constant rate of glucose uptake independent of insulin. The other component of the fast pool is the ability of glucose to stimulate its own uptake (diffusion) independent of insulin (i.e., liver). Insulin-dependent glucose uptake is considered to take place in the slow interstitial pool, the key tissues being muscle and adipose. This model relies on the assumption that basal insulin-independent glucose disposal is three times insulin-dependent glucose disposal. In addition, Rd at time zero is fixed to at 1 mg kg−1 min−1. These values were attained experimentally by Cobelli et al. [36] in six human subjects. Despite these assumptions, model estimates correlate well with direct measures of IS and HGO. The necessity for isotope tracers is likely the reason why this model has not been universally adopted.
Two and three pool insulin-independent models of glucose turnover were developed to describe glucose turnover in nonsteady-state conditions. Application of these models requires constant infusion of isotope and frequent blood sampling over several hours. Both models require previous knowledge of parameters, either measured in a separate experiment or prior to initiation of experimental conditions. The three-pool model in particular is computationally difficult and has received limited attention.
Dispersal models, except for Radziuk et al. [29], have received virtually no consideration in the study of whole-body glucose metabolism, despite their theoretical appeal. Such models have the advantages of requiring less structural knowledge and representing key processes such as convective flow, diffusion and uptake explicitly. Their lack of adoption could be due to biologists finding them computationally challenging and mathematically obscure.
In conclusion, the use of isotope tracer methodology and mathematical modeling has been used extensively to determine indirect estimates of various aspects of in vivo glucose metabolism through relatively noninvasive means. Overall, minimal model schemes dominate the literature, likely due to their technical and mathematical ease.
Acknowledgments
Funding was provided, in part, by the Natural Sciences and Engineering Research Council of Canada (NSERC), the Canada Research Chairs program, and the Spanish Ministerio de Educacion, Cultura y Deporte.