Although it is generally agreed that some saturable insulin action dynamics occur during most insulin sensitivity (SI) tests [1-3], the identification of these effects is often crudely handled or ignored [4]. This choice can be attributed to the assumption that saturation effects are not often encountered during the comparatively low insulin concentrations induced during frequently sampled intravenous insulin tolerance tests (FS-IVGTT). Similarly, in hyper-insulinaemic euglycaemic clamps (EIC) the very large insulin doses lead to saturation [2, 3, 5, 6] but saturation effects are still ignored, creating difficulty in comparing results across protocols or EIC insulin doses.

It is important to understand insulin action saturation effects when testing for SI, or when adjusting insulin therapy in glycaemic control [7-9]. Commonly used diagnostic tests that do not use a patient specific dosing protocol, such as the two-hour oral glucose tolerance test or dynamic insulin sensitivity test (DIST) [10-12], may be affected, at least in some cases, by saturation. In particular, differing patient specific volumes of distribution will cause inter-subject variation in concentration for the same dose, and thus, the insulin efficiency may not be measured equally across subjects leading to greater error in these critical values. For example, successive stepped clamp tests with varying glucose or insulin concentrations have yielded significantly different outcome insulin sensitivity metrics for the same individual [5, 6]. Glycaemic regulation may also be improved by understanding insulin saturation, as simple assumptions of unsaturated, proportional action may not be appropriate with very insulin resistant individuals [13, 14].

Fig. (1) shows a typical response curve for the action of drugs or hormones as a function of concentration. The linear response line shows the gradient at the theoretical zero concentration point and measures the infinitesimal increase in action caused by an infinitesimal increase in concentration from zero. The saturation line shows the theoretical maximum action line. The action will asymptotically approach this line as concentration increases and significant saturation effects become apparent.

Fig. (1). Typical response curve for any typical saturative drug or hormone. The graphs use proportional and logarithmic x-axis for the same response (linear gradient of 1 and saturation maximum of 1).

Fig. (2). The rate of change of available glucose as a function of the insulin concentration in the interstitium for the three models identified on the same data set. Note the curved shape of the saturation model captures the behaviour of Fig. (1, left). Furthermore the linear and SG free variable models almost overlay.

A typical EIC protocol with a single insulin infusion rate will define a single point on a subject-specific saturation curve. Although equivalence should be given to results along the same saturation curve, it is assumed that equivalence lies on a straight line between this point and the origin. Hence comparing EIC values from differing dosing regimens loses value. Models used during SI identification for dynamic tests frequently have linear insulin-dependent terms [11, 15, 16]. Such terms could be represented as linear lines from the origin, with higher sensitivity represented with steeper gradients, and insulin action increasing with concentration with no diminishing returns.

To define saturation effects, the glucose disposal at a series of insulin concentrations must be observed. Previous studies into the saturation of insulin action have used either stepped or multiple EIC tests [2, 6, 17, 18]. The stepped EIC is a long protocol wherein the insulin infusion rate is sequentially changed in a stepwise fashion, usually at 2 hourly intervals for 6-8 hours clamping patients at each steady-state plasma insulin concentration. Studies that used multiple tests have required subjects to have multiple EIC tests on separate days with differing insulin infusion rates. In both cases, increasing insulin doses resulted in decreasing estimates of SI for a given subject.

This article presents a method for the identification of a patient-specific insulin concentration at half maximal action (I₅₀) and the theoretical zero insulin gradient (SI_S) of glucose disposal (as a function of glucose availability). Contrary to previously presented studies, the saturation parameters will be identified using a single dynamic test. Hence, the method presented, if successful, offers the advantages of both reduced testing and reduced (single) test intensity to determine an important patient-specific value.

All tests achieved convergence for all identified parameters in all models. No re-simulated profiles were divergent from the measured data. The 10000 virtual trial in-silico analysis showed that the small positive saturation term out performs the linear assumption 49.79% of the time (in terms of fitting accuracy). In comparison, the real clinical data analysis showed that greater accuracy was achieved with a non-zero saturation term 73.9% of the time (p<0.002). Hence, the identification method was robust in the presence of noise and a false effect was not identified.

Table 2 shows the re-simulation accuracy and relative values of the respective terms and models, both as a whole study population and divided into diabetes diagnosis. Fig. (2) shows insulin action profiles from the three models for a typical NGT test participant.

The mean insulin concentration at half maximal action (I₅₀) for the total population was found as 139.3 mU·L^-1 (CV =130.5%). There were almost significant (p=0.092) differences between the NGT and T2D-IFG subgroups I₅₀ values, whereas the SI_S values for these groups were indistinguishable. The SI_L and SI_G parameters showed a difference between these groups whereas the SG term showed no such contrast. Table 3 shows that the I₅₀ values found in this analysis were within the wide range of previously published values that were either derived using multiple or stepped EIC protocols [2, 6, 17, 18].

The novel methods presented in this article allow identification of saturation parameters from relatively low-dose, 40-50 minute dynamic tests. However, the intra-subject repeatability of the derived metrics is not of sufficient resolution to promote the use of the method to identify a saturation threshold for an individual. The population trend of higher I₅₀ values in the IFG-T2DM subgroup observed in this study has also been reported previously [2, 6, 17].

The saturation model also produced, more accurate numerical data-fitting re-simulations than either the proportional model or the SG free variable model. This result indicates that the method was truly detecting small levels of saturation of insulin action for most participants. This outcome would match expectations given the relatively low dose data available for this study.

The improvement of insulin data fit was slight when considering the additional variable in Equation 6 (Q₅₀). However, there was a significant improvement in the glucose data fit obtained using the saturation model. This would imply that the saturation effect is most evident in the glucose data. Furthermore the glucose model fit error showed the increased fit accuracy possible with the added model variables. However, the saturation variable (Q₅₀) showed a significant further data fit advantage over the added variable of the SG free variable model. Thus implying the presence of saturation effects in the data and the suitability of the saturation model for analysing glucose data.

To date, studies that derive a parameter for saturation have used stepped clamp tests [2], multiple clamp tests [6], or multiple insulin-modified IVGTT tests at various insulin or glycaemic doses [3]. These tests are arduous in terms of clinical intensity and time. A typical three-step clamp protocol will take over six hours of patient and clinician time. Hence, most importantly, this study shows that saturation of insulin sensitivity identification is achievable with a single 40-50 minute test, albeit an identification with a potentially high inter-test variability.

The stepped clamp and multiple clamp studies found physiological I₅₀ values for 100% of subjects. However, none of the studies provided an intra-subject repeatability analysis so these results cannot be put further into context. In addition, their much higher insulin doses, well into saturation ranges, provide better resolution to identify this parameter, but also carry increased risk to the participant and thus added clinical burden as well.

The potential reasons behind the poor repeatability of the proposed saturation model are the decreased robustness of the solver in the presence of assay error and physiological and assay noise (such as mixing) [11, 33]. Such noise provokes the interference issues frequently observed when deriving SG and SI with the minimal model [24, 25]. These issues would be exasperated by low insulin doses.

Both the saturation model of Equations 6-7 and the SG free variable model of Equation 8 utilise two free variables to model the glucose decay. Thus, some noise-generated parameter trade-off error is expected. These issues are exacerbated when the free variables in both the saturation and SG free variable models are coupled to the functions of glucose concentration. In simpler terms, while the iterative methods identify the values that minimise the least–square simulation error, the noise in the data causes identifiability issues for multiple free variables, even though the noise-free systems are theoretically identifiable [34].

Table 2 shows that the saturable and SG free variable models produced a higher intra-subject parameter variability than the proportional model. This result is likely to be an artefact of the number of free variables used by the respective models. Indeed, it could be assumed that the saturable model produces results in accordance with what should be expected by models that utilise two free variables to model glucose decay in short duration dynamic tests.

The p-value between the in-silico analysis and the real data confirms that the reduction in error is not due entirely to noise fitting. However, the reduction in error could be attributed to the addition of a second free variable. Both the SG free variable and saturation models achieved a lower mean residual error than the proportional model. However, the reduction in error for the SG free variable model was smaller compared to the saturation model. Thus, it could be concluded that if a second variable is to be included and identified, it is more physiologically favourable to identify a saturation variable than the glucose dependant clearance rate (SG).

Improvements in assay techniques may reduce the level of noise in the data. Thus enabling the methods presented to more accurately and repeatably identify these parameters. However, it is likely that the physiological mixing effects that are present during this type of dynamic test will limit any potential improvements obtained by a more accurate assay. This issue would be further compounded by a relatively low sampling rate of 5-10 minutes in the DIST designed to reduce cost and patient/clinical burden [11]. However, it suggests that a modified test protocol with an increased sampling frequency, larger or repeated boluses may provide better insight or stability in results.

Furthermore, the dosing levels of the DIST test were purposefully lower than those used by comparable tests to avoid significant saturation effects, while still allowing detection of a signal and physiological relevance. This low dose, acting as designed, will have partially hidden the saturation effect that this study attempted to measure. The average interstitial insulin concentration reached during the DIST tests (36.6 mU·L^-1) was significantly below the average identified Q₅₀ value. In contrast, most stepped clamp studies achieve plasma insulin concentrations significantly above I_50. For example, Natali et al. [2] achieved an average plasma insulin concentration of 509 mU·L^-1 with a 200 mU·m^-2·min^-1 infusion and found an average I₅₀ of 293 mU·L^-1. This research attempted to identify the saturation effect with data that generally lies toward the linear region of the saturation curve in Fig. (1). Hence a limitation in this study is the lack of higher dose data which was unavailable to better prove the concept. Thus, the proportional model is the most appropriate model to identify metrics from the DIST test.

A significant finding of this study is shown in the p-values for SI_S and α_Q in Table 2. There is virtually no difference between the mean SI_S metrics for the two subgroups (p=0.65). However, there is an almost significant difference between the mean I₅₀ metrics (p=0.092). Physiologically, this result may imply that most people have similar insulin efficiency or sensitivity at the receptor level once insulin has been activated by the cell or at very low concentrations, and observed differences could instead be caused by saturation dynamics. In particular, a lack of receptors may limit sensitivity, and not the binding rate to them. Thus, much of the differences observed in proportional SI metrics from other studies could be affected by these saturation dynamics at cellular and receptor level, particularly if dosing is not patient specific. This hypothesis offers an interesting and physiologically justified insight, but remains to be proven by a purpose driven test.

The mechanisms by which insulin action saturation occurs have not been confirmed. However, most studies which have investigated the matter have promoted either a transportation delay of insulin to the skeletal muscle amongst insulin resistant individuals [3, 32], a lower intensity or availability of insulin receptors at the cell [6], or both [17]. If the former were the case, significant saturation effects would be more prominent in dynamic than steady state tests. The latter would show similar results across protocols. To date no such intra-subject cross-over investigations have been undertaken. However, this study has shown similar findings to the previous clamp investigations, implying that some ‘at the cell’ effect is likely.

It can be concluded by the reduction in simulation error caused by the incorporation of the saturation parameter that the proportional model (Equations 3-5) does not fully capture some of the subtle variations of the test dynamics. However, it cannot be safely concluded from the analysis presented that it was only saturation effects that caused this error reduction. It is well known that endogenous glucose production (EGP) can become suppressed in the presence of elevated blood glucose and insulin concentrations [35-37]. Although it is assumed that these suppressions have a small effect [38], they may be of similar magnitude to the saturation effects detected in this study. In particular, Equations 5, 7 and 8 assume that basal insulin-dependant and non-insulin-dependent glucose clearance is equal and opposite to EGP (as it is in the basal state) for the duration of the test and is thus cancelled out of the equations. However, the suppression of EGP would result in the opposite effect to saturation; the observed rate of glucose decay would be greater than expected during a period of elevated insulin rather than lesser as occurs with the saturation model. Tracer glucose studies with a purpose-specific test would be required to further delineate this effect, but would also add significant clinical burden and ethical considerations.

The novel techniques presented in this article allow a unique identification of a physiologically justifiable, saturation value trend from a single dynamic test at a population level. Despite the robust identification methods, the high intra-patient variability implies that significant development of the protocol is required before high accuracy in saturation parameter identification is possible from single dynamic tests at an individual level. Potential protocol improvements may include larger, and possibly, repeated boluses. The saturation model showed similar accuracy in terms of parameter variation to generally accepted two parameter models, and allowed better fits to the clinical data. However, despite the increased fitting accuracy of the saturation model, the single parameter proportional model is the most stable, and thus it should be used for individual tests.

The comparable sensitivity coefficients between IFG-T2DM and NGT sub-groups with disparate saturation thresholds implies that an ‘at-the-cell’ effect is the rate limiting factor in glucose disposal. Thus an insulin action saturation threshold may be a significant governing factor of insulin resistance.