Mathematical model with beta-cell carrying capacity for prediabetes

To develop a minimal model for beta cell carrying capacity, we use as a basis the classical mathematical models for diabetes, exemplified by the HOMA model [7]. We add to the classical model an equation for beta cell growth, based on the model of Topp et al [9]. Following De Gaetano et al [17], we add a limit to the total beta cell function, the beta cell carrying capacity.

This carrying capacity in humans is based on a maximum to the total biomass summed over all beta cells in the body. In humans, beta cells are thought not to proliferate significantly after childhood [21], although some studies suggest adult increase in cell numbers. Beta cells increase their size (hypertrophy) to compensate for high glucose levels. However, cells cannot increase their size indefinitely. Recent data suggests that cells that grow to about twice their normal volume begin to resemble senescent cells and secrete proinflammatory signals [22]. Human data suggest that total beta cell mass can double in obesity [13,23–25] and pregnancy [26], where studies suggest the mechanism to be an increased cell number [55–57].

The carrying capacity mathematical model includes a term that forces beta cell functional mass growth to become zero when mass B reaches a carrying capacity C.

We begin with a model without carrying capacity that accounts for beta cell compensation, a simplified version of the model developed by Topp et al [9] where we exclude the glucotoxicity term relevant for glucose levels much higher than those seen in prediabetes. The model has three equations:

(1)

(2)

(3)

where m0 is a glucose source term that accounts for liver glucose production plus meal intake, Si is insulin sensitivity and k is the insulin-independent glucose removal rate. Insulin is secreted by beta cells with functional mass B (in units of insulin concentration per unit time) and removed at rate γ. The regulation of beta-cell insulin secretion by glucose is described by Hill function with halfway point h [27].The beta-cell functional mass changes with growth rate μ to achieve the glucose set point .

This model accounts for compensation on the timescale of weeks, as beta cell functional mass grows to buffer changes in parameters such as insulin sensitivity Si [10].

To develop a model for the carrying capacity, we follow [17,18] and add to the beta cell growth equation a limit for growth that is inspired by ecological carrying capacity:

(4)

As beta cell functional mass approaches the carrying capacity C, its growth rate slows down, according to a term adapted from carrying capacity in ecology and in cellular replication [28–30]. In this model, there is sufficient compensation when B is much smaller than the carrying capacity. However, when B approaches carrying capacity C, compensation begins to break down. Insulin production becomes less than needed for sufficient compensation. As a result, glucose levels increase.

We can now derive a formula for estimating the carrying capacity C from lab tests. Assuming that beta-cell mass is at steady state, Eq (4) indicates that C = G B/(G-G0). Estimating B using the well-established HOMA-B formula for beta-cell function, HOMA-B = 20 I/(G-3.5) (with G,I in unit of mmol/L, U/ml respectively), we arrive at a formula for carrying capacity that we call HOMA-C:

(5)

or in terms of insulin and glucose test values (in mM units for G):

(6)

Unlike HOMA-B that estimates current beta-cell function, HOMA-C estimates the maximal beta cell function that an individual can potentially achieve, for example if insulin resistance would rise to very high levels.

One might ask how two measurements- fasting insulin and glucose- can give rise to three estimate equation- HOMA-IR, HOMA-B and HOMA-C. The answer is that we use three equations, Eq (1), (2) and (4) at steady-state to derive three relations for IR, Beta and C as a function of insulin I and glucose G. The original HOMA model effectively had only two equations, lacking the B mass equation 4. Reduction of the two variable system is based on assuming that insulin changes to a meal are much faster than the timescale beta cell compensation. We believe this is a plausible assumption at least for beta cell compensation due to gene expression and cell biomass growth. We also note that reduction to two dimensions is not essential for the conclusions, and is done for presentation purposes and to have an analytical formula for HOMA-C (rather than a numerical computation).

Note that this formula should only be used in cases where fasting glucose is sufficiently higher than the healthy fasting glucose setpoint, which we nominally take to be 4.8mM, to avoid small values in the denominator that can increase errors. The 4.8mM glucose setpoint might be adjusted for different populations. For standard lab test error rates (SI) we estimate that a 20% accuracy in HOMA-C requires fasting glucose higher than 5.3 mM. Thus, the formula can be applied to patients with prediabetes or diabetes.

We also note that HOMA-C will not be accurate in periods where beta cell mass changes strongly over time, since we require dB/dt = 0. Thus, the formula should not be used in transient periods of weeks after changes in health or nutrition.

Carrying capacity varies widely between individuals

We used the mathematical model of Eq (1), (2) and (4) to estimate the carrying capacity using data from a cohort of n = 256 unmedicated people with prediabetes [19,20]. Of these, 8.5% had high enough glucose to be considered diabetic (G ≥ 6.9 m M/L). Each participant had 1–4 measurements over one year of both fasting insulin and glucose.

We first considered the tests performed upon enrollment in the study, considering all prediabetic patients at a single time point to examine the variation of their parameters. The distributions of glucose and insulin measurements are broad (Fig 1A). This can be explained by our model with differing values of carrying capacity for each individual (the curves in Fig 1A). Individuals with low carrying capacity fail to maintain the high insulin levels required for glucose homeostasis at high insulin resistance, in contrast to individuals with high carrying capacity.

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Fig 1. Carrying capacity estimated from insulin and glucose measurements from 256 unmedicated participants with prediabetes from (Ben-Yacov et al., 2021, Rein et al., 2022).

(A) Insulin and glucose in the unmedicated prediabetes cohort at enrollment, with outliers removed by excluding data outside of the 90% density contour. Curves denote different values of carrying capacity HOMA-C. (B) Beta cell function (HOMA-B) versus insulin sensitivity (1/HOMA-IR). Curves show different values of carrying capacity HOMA-C. Numbered circles are equivalent points in A and B for orientation. (C) HOMA-C in individuals with prediabetes and diabetes according to the criteria of [Ben-Yacov et al 2021]. (D) HOMA-C versus participant age. (E) HOMA-C versus participant BMI. (F) Distribution of the coefficient of variation (CV) of HOMA-C within an individual’s measurements. Vertical line shows the CV of the mean HOMA-C between individuals. Formulae are HOMA-B = 20 I/(G-3.5), HOMA-IR = I G/22.5 and HOMA-C = G HOMA-B/(G-4.8).

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The data can also be visualized in terms of HOMA-IR (a proxy for insulin sensitivity Si) and HOMA-B (a proxy for beta cell function) (Fig 1B). As insulin resistance rises, beta cell function rises and asymptotically reaches the carrying capacity when insulin resistance is very high. Fig 1A shows that the predicted carrying capacity HOMA-C varies by a factor of about 10 between individuals (maximal divided by minimal HOMA-C in the cohort), with a CV of about 75%. This relates to postmortem histology studies that report variation of a factor of 5 in total beta cell mass between equal aged individuals [31]. HOMA-B likewise varies between individuals by a factor of 10 (S1A Fig). Note that HOMA-B and HOMA-C are not equivalent – for example HOMA-B of 100 corresponds to individuals with HOMA-C ranging from about 500–1500.

We compared the carrying capacity between individuals with prediabetes and those with diabetes in the beginning of the trial. HOMA-C was 2.5-fold lower on average in those with diabetes (Fig 1C, p = 3 10–7). This is in line with postmortem histology studies that showed 50% lower beta cell total mass on average in individuals with diabetes [13].

HOMA-C does not vary significantly with participant age, at least in this cohort in a cross-sectional analysis (Fig 1D). This indicates that age as a risk factor for diabetes might not operate by reducing carrying capacity, but via other mechanisms such as increased insulin resistance, but longitudinal data is required to verify the age dependence of HOMA-C.

Carrying capacity shows mild variation with BMI, rising by a factor of about 2 on average across the range of BMI in the cohort (Fig 1E). This may be due to the cohort composition which includes primarily individuals with prediabetes and not full-blown diabetes. Individuals with high BMI tend to have high insulin resistance and thus transition to diabetes rather than remain prediabetic unless they have a high carrying capacity.

Repeating this analysis using all timepoints in the dataset shows essentially the same conclusions (S2 Fig). We further analyzed a larger dataset containing 20 years of blood tests (that is 7624 glucose and insulin measurements from Centers for Disease Control and Prevention, National Health and Nutrition Examination Survey [32]). HOMA-C was found to be almost 3-fold larger in the prediabetic group than in the diabetic group. It varied weakly with age (cross-sectional analysis) and only moderately with BMI (S3 Fig)

We next asked how the predicted carrying capacity changes in each individual over time. We computed the variation of HOMA-C between the longitudinal measurements of each individual over a year (Fig 1F). This intrapersonal variation is about 25% on average, as quantified by the coefficient of variation (CV = standard deviation/mean). In contrast, HOMA-C varies between individuals by about CV = 75% (Fig 1F, vertical line). Thus, carrying capacity in this dataset may be regarded as a relatively constant trait of a single individual, varying over time by about one third of the inter-person variation, although a low intrapersonal-to-interpersonal CV ratio does not by itself establish that carrying capacity is a fixed physiological trait. Other measures such as HOMA-B, S and DI show an inter-to-intrapersonal variation ratio of ~35%, 50% and 70% respectfully (S1B Fig), which are somewhat wider relative to HOMA-C, except for HOMA-B which varies in a similar magnitude. We show individual trajectories of HOMA-C, glucose, HOMA-B, HOMA-DI and HOMA-S in S4 Fig, where the CVs are presented again after screening for high fasting glucose.

We also tested how quickly insulin resistance rises in this cohort. We evaluated the timescale of insulin resistance (HOMA-IR) variation in the prediabetes cohort [19,20] over a year and find a typical timescale of about days. Thus, IR may be considered relatively constant over months (S5 Fig).

This supports the understanding of prediabetes through the model - carrying capacity does not change much over time; instead, over years insulin resistance slowly rises. Beta cell functional mass rises to compensate but comes eventually approaches the carrying capacity, and thus insulin production is insufficient to compensate for the insulin resistance. Glucose levels rise gradually until prediabetes and then T2D thresholds are crossed.

HOMA-C associates with risk of transition to diabetes more strongly than other insulin-dependent Homa indicators

To test whether low HOMA-C is a risk factor for transitioning from prediabetes to diabetes in this cohort, we considered participants with prediabetes (G < 6.9mM) at enrollment, evaluated their HOMA-C at enrollment, and asked which participants transition to diabetes over the year of follow-up. We focused this calculation on participants who are prediabetic (G > 5.6mM) and who are not on the verge of diabetes (G < 6.6mM). This was done since for the edges (nearly diabetic or not even prediabetic), we expect the transition risk within 1 year to depend mostly on their initial state and fasting glucose than on other parameters such as HOMA-C.

There were 66 participants with at least one follow-up glucose test, and the criterion for transition was a glucose test above 6.9mM. A total of 44 participants transitioned to diabetes. We analyze the risk of diabetes in each quartile of HOMA-C. The top quartile had about a 4-fold lower risk of transition to diabetes than the top quartile, meaning that 4-times more participants transitioned to diabetes in the lower quartile of HOMA-C than in the upper one.

We repeated these tests with other diabetes measures based on lab tests- HOMA-B, HOMA-S the disposition index [51] HOMA-B * HOMA-S, and fasting glucose G. HOMA-C outperforms the insulin and glucose dependent indicators, HOMA-B and HOMA-S, and is slightly less predictive than the only-glucose dependent measures. Table 1 presents a bootstrapping assessment of the mean risk factor per ratio of each parameter, as defined in Materials and Methods under ‘Bootstrapping analysis of diabetes transition risk’. This parameter was chosen to examine the dependence of the risk on the HOMA parameters since there are not many subjects in the cohort and yet we can coarsely quantify the risk dependence on parameters. The full results of the bootstrapping per quartile are shown in Fig 2.

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Table 1. A bootstrapping assessment of the mean risk factor per ratio of each parameter, defined as , where are the quartiles The mean and std were calculated for 1000 bootstraps.

https://doi.org/10.1371/journal.pcbi.1014589.t001

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Fig 2. Correlation between diabetes transition risk and 4 diabetes related parameters – HOMA-C, HOMA-B, HbA1C, HOMA-S.

The bars show, for each parameter, the frequency of transitioning to diabetes in each quartile of the parameter, e.g., the subject in the lowest quartile of HOMA-C had ~ 30% transition rate while subjects in the highest HOMA-C quartile had only ~8%. The mean risks and confidence intervals were calculated using bootstrapping. The bars are colored by quartile order. Beneath each graph are the Pearson correlation values, the risk ratio between the highest and lowest quartiles, and the average risk ratio between quartiles [q1/q2, q2/q3, q3/q4]. In this cohort, HOMA-C was the most correlated with diabetes risk among the insulin-dependent HOMA indicators HOMA-B and HOMA-S. It was similarly but slightly less correlated than only fasting glucose dependent indicators – G and disposition index HOMA-S X HOMA-B.

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We conclude that low HOMA-C estimates for carrying capacity in prediabetes predict transition to diabetes in a one-year follow-up. This is consistent with the notion that low carrying capacity reduces the ability of beta cells to compensate, though, as noted in the Discussion, this predictive association does not by itself distinguish whether HOMA-C captures information beyond a reparameterization of fasting glucose and insulin.

Integrated model for prediabetes and diabetes subtypes

The carrying capacity model provides a minimal, integrated model of beta cell function and glucose levels that describes the transitions between normoglycemic, prediabetic and diabetic conditions. In addition, it mathematically describes different diabetes subtypes.

To provide intuition, we use a graphic approach to understand the predictions of the model. This graphic approach, called nullcline analysis, is widely used in the study of dynamical systems [36]. The dynamics described in this phase plane analysis include the assumption that insulin -dynamics are much faster than Glucose dynamics, as discussed in Materials and Methods. The nullclines each describe one of the two feedback arms (Fig 3A). One arm describes the decrease in steady-state glucose due to beta cell function: the higher beta cell function the lower steady-state glucose due to insulin. This is called the G-nullcline. The other arm is the increase in steady-state beta cell functional mass B due to glucose, which underlies compensation. This is called the B-nullcline. The carrying capacity is seen in the shape of the B -nullcline, which saturates at a maximum value at the carrying capacity. These two nullclines cross at a point which defines the system’s fixed point, because it is the point where both G and B are at steady state. The fixed point determines the steady-state glucose and beta cell function mass.

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Fig 3. The carrying capacity model provides a framework to understand changes in steady state glucose and beta cell functional mass.

(A) Nullclines for the carrying capacity model. (B) Effects of raising the G nullcline as in insulin resistance. (C) Dynamics of the carrying capacity model for a step-like decrease in insulin sensitivity from to , an arbitrary magnitude to qualitatively reproduce an effect similar to the step-change the original Topp model. A key feature of the CC model, is that glucose steady state changes when changing SI – only partial compensation occurs, resulting a prediabetic state instead of a return to normoglycemia. (D) Effects of lowering the B-nullcline as a result of a reduction of carrying capacity. (E) Dynamics of the carrying capacity model for a step-like decrease in the carrying capacity. The carrying capacity C drops from to . A step like-decrease is a heuristic, over simplified input for diabetes, where the more realistic input is the slow decline in Fig 3. (F) Nullclines for the linearized Topp model without glucotoxicity. (G) Nullclines for the model with carrying capacity and glucotoxicity. (H) The dynamics of the glucotoxicity model (eq. 16) for a slow exponential increase in insulin resistance with. Kinetic parameters: , , , , , and, . B dynamics are given by Eq. 16.

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Physiological conditions and medical interventions can change these nullclines, thus shifting the fixed point and hence the steady-state value of glucose and beta cell functional mass. Some insulin conditions affect the G-nullcline, namely the steady state glucose at a given beta cell mass (Fig 3B). This includes insulin resistance, which shifts the nullcline to higher levels of glucose, due to the reduced effectiveness of insulin. This causes a rise in both steady state glucose and beta cell functional mass. Other conditions that affect the G-nullcline are excess glucose production by the liver, excess insulin removal rate, reduced insulin-independent glucose removal, and impairments in the beta-cell response to glucose. All of these conditions in the model show a rise in beta cell mass together with a rise in glucose. For example, upon a hypothetic step-like rise in insulin resistance, the model shows how glucose rises within hours. As beta cell mass increases over weeks, fasting glucose slowly declines, but there is only partial beta-cell compensation in the CC model, such that fasting glucose declines to a higher value than before, producing a prediabetic stable state (Fig 3C).

In contrast, other conditions affect the B-nullcline. This includes a reduction in beta cell health, so that a given glucose level results in lower beta cell functional mass pushing the B-nullcline to lower values (Fig 3D). Such changes also shift the fixed point. They give rise to higher glucose levels, but lower beta cell function mass. A similar effect is caused by conditions that lower the beta cell carrying capacity.

Thus changes in the G-nullcline should show increases in the steady state values of both G and B, whereas changes in the B-nullcline should show an increase of G but a decline of B. Such a drop in beta cell secretion capability causes a rapid rise in glucose followed by a decline as beta cell mass grows to compensate (Fig 3C).

Using the distinction between the two nullclines, one can analyze different subtypes of diabetes. Recently, four prevalent subtypes of non-autoimmune diabetes were proposed [37]. Two of these, mild obesity related diabetes (MOD) and severe insulin resistance diabetes (SIRD), are accompanied by a rise of insulin resistance, corresponding to an increase in the G-nullcline. Indeed, these types show a rise in both G and beta cell function B (Fig 3B).

The other two types, severe insulin deficient diabetes (SIDD) and moderate age-related diabetes (MARD) have the opposite phenotype- a rise in glucose but a decline in beta cell functional mass [37]. We propose that these two types of diabetes involve a shift in the B-nullcline, compromising beta cell compensation rather than insulin resistance as a primary etiology (Fig 3D).

A similar analysis can be made for mature onset diabetes of the young (MODY), which accounts for about 5% of diabetes cases. Most MODY cases are due to two major genetic variations. The HNF1 variants (50% of MODY) impair insulin secretion, lowering the B-nullcline and indeed show the predicted reduced B-cell function along with high glucose [38]. The other variant, GCK (30%), impairs glucose sensing in beta cells. It thus should affect both nullclines (see nullcline equations in Methods). It has higher beta-cell function than HNF1 (3-fold) and non-MODY T2D (1.5 fold) [38].

Notably, the linearized Topp model without carrying capacity or glucotoxicity has different nullclines that cannot capture these subtypes of diabetes (Fig 3F). The Topp model has the same G-nullcline as the present model, but the B-nullcline is a vertical line at G = G0. At steady state, compensation always brings glucose back to baseline. The present model differs from the Topp model by having a carrying capacity, and this is the origin of the curved shape of the B-nullcline and the existence of prediabetes steady states with glucose above baseline (Fig 3B and 3D).

Finally, we added glucotoxicity to the carrying-capacity model (see SI). Glucotoxicity is an effect, characterized primarily in rodents, in which beta cells are removed or become dysfunctional when glucose reaches very high levels [9,16,39,40]. Glucotoxicity causes the B-nullcline to have a rising and falling shape, since beta cell mass shrinks at high glucose levels (Fig 3G). The two nullclines now cross at three points: a stable fixed point at normal glucose levels and another a stable fixed point at very high glucose in which beta cells mass is zero. Between these is an unstable fixed point.

Rising insulin resistance pushes the G-nullcline up, and moves the low stable fixed point and the unstable fixed points closer to each other (Fig 3G). Glucose baseline thus rises as in prediabetes and into the range of type-2 diabetes. At a critical level of insulin resistance the system goes through a saddle-node bifurcation at which the two fixed points collide and annihilate, leaving the high glucose fixed point as the only steady-state solution. Such a process may correspond to a transition to insulin-dependent diabetes at very high insulin resistance (Fig 3H).

Carrying capacity model

The Topp model [9] describes the rates of change of blood glucose , insulin , and beta cell functional mass , adapted here as follows:

(7)

(8)

(9)

where is liver and postprandial glucose input, is insulin sensitivity, is the insulin-independent glucose removal rate. Insulin is secreted by beta cells with functional mass (in units of insulin concentration per unit time) and removed at rate

. The regulation function of insulin by glucose is a sigmoidal function with halfway point . The beta cell functional mass changes with growth rate to reach the glucose set point .

We ignored the glucotoxicity term at very high glucose levels in the original model . Ignoring this term does not change the conclusions of this study, because the glucose levels in the present study are well below the glucotoxicity threshold of mmol/L in Topp et al. [9].

This system can be simplified to a set of 2 ordinary differential equations (ODEs) by means of separation of timescales. Insulin is removed with a half-life of about 5 minutes which is faster than glucose removal on the scale of hours and beta cell functional mass growth on the scale of weeks. Insulin thus equilibrates rapidly compared to the slow timescale, so that we use a quasi-steady state assumption with . The quasi-steady-state solution is. Substituting this expression for I in the equation for glucose reduces the system to two equations:

(10)

(11)

We use the kinetic parameter values used by Topp et al. and Karin et al. [9,10].

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https://doi.org/10.1371/journal.pcbi.1014589.t002

The new model extends the Topp model by introducing a carrying capacity for beta cells, . Introducing the latter in the 2 ODE version of the Topp model, we obtain

(12)

(13)

where C is the carrying capacity.

In the limit of very slow changes in Si relative to the compensation time, which is appropriate to the present estimates, one uses to obtain the following relation between glucose and insulin sensitivity

(14)

with

The relation between functional beta cell mass and is

with

Based on this model, we define the HOMA-C formula based on the steady state assumption for B, namely dB/dt = 0. This provides . Since B is widely estimated using the HOMA-B formula, HOMA-B = 20 I/(G-3.5), we propose the following formula for carrying capacity:

with =4.8 mmol/L.

Data from an unmedicated prediabetic cohort

We analyze data from a cohort described in [19,20]. The cohort consists of unmedicated prediabetic and diabetic adults who underwent a 6-month dietary intervention and additional 6-month follow-up with several (up to four) blood tests of fasting glucose and insulin levels. We only analyzed data from patients with at least one fasting glucose and insulin measure simultaneously, leading to 256 participants (111 women aged , 145 men aged ), 8.5% being diabetic at the beginning of the dietary intervention. Patients were included in the diabetic cohort if either their blood glucose or their HbA1c levels were elevated [20].

Insulin was measured with the use of a chemiluminescent microparticle immunoassay (ARCHITECT insulin assay) with precision of less than 7% CV and plasma glucose was measured with the use of a hexokinase method (GLUC2 assay, cobas; Roche) with precision of about 2% CV.

From these data, we extract how individual insulin sensitivities and functional beta cell mass evolve over time using the HOMA-IR and HOMA-B indices [7], and where and are fasting glucose in mmol/L and insulin in U/ml.

As in Topp et al. [9], we defined the rate of change of insulin sensitivity as |. We calculated the rate of change between each measurement for each patient of the cohort as .

Data from the National Health and Nutrition Examination Survey (CDC)

We analyze data from the National Health and Nutrition Examination Survey (CDC,32). It consists of blood tests (fasting glucose and insulin) from 1999 to 2018. After filtering out the outliers for glucose, insulin and BMI, we obtained 7624 data points. Among them are 83% m prediabetic and 16% diabetic individuals (those who reported taking insulin or glucose lowering medications excluded). For prediabetic individuals, mean HOMA-C = 640.5 + /- 5.6 (mean + /- SEM) and for diabetic ones, mean HOMA-C = 215.7 + /- 5. Whether HOMA-C is predictive of the risk to transition from prediabetes to diabetes could not be tested with this cross-sectional dataset because there was no longitudinal follow-up.

Carrying capacity model nullclines

We compute the nullclines of the system, that is the curves corresponding to and . The -nullcline () is

Without carrying capacity (that is, ), as in the Topp model, the -nullcline is vertical () and there is always compensation. However the presence of a finite carrying capacity C bends the -nullcline as

When the value of Si decreases, glucose levels remain nearly constant at first because the rise of B cell mass provides compensation. However, when approaches its carrying capacity glucose levels start to rise: this is the onset of prediabetes. Note that when insulin resistance becomes very high, that is, in the limit of , the state of the system converges to and .

Model with carrying capacity and glucotoxicity

The Topp model originally contains a glucotoxicity term [9]. If we add such a term to our model with carrying capacity (Fig 3G and 3H), we obtain

(15)

(16)

When , that’s when cells are far from carrying capacity, we recover the original Topp et al equation:

(17)

Bootstrapping analysis of diabetes transition risk (Fig 2)

To evaluate how each metabolic parameter relates to the risk of transitioning from prediabetes to diabetes, we performed a non-parametric bootstrapping analysis using fasting glucose and insulin data from all participants with at least two measurements and baseline glucose ≤ 6.9 mM. Individuals who reached glucose > 6.9 mM during follow-up were classified as having transitioned to diabetes.

For each participant, we calculated the parameters HOMA-B = 20 I/ (G − 3.5), HOMA-S = 22.5/ (G I), HOMA-C = G HOMA-B/ (G − 4.8), the disposition index DI = HOMA-B × HOMA-S, and the ratio HOMA-B/ HOMA-C, using the first available fasting test.

To estimate uncertainty in the risk curves shown in Fig 2, the dataset was resampled with replacement 1 000 times. In each resample, participants were divided into quartiles according to each parameter. Within each quartile, we computed the fraction of participants who transitioned to diabetes, representing the empirical transition risk. The mean and standard deviation of this risk across bootstrap iterations were used to plot the bars and error ranges in Fig 2.

For a compact numerical comparison (Table 1), we defined a mean risk-ratio index as the average ratio of successive quartile risks, :

(18)

This value was sampled over bootstrapping iterations and its mean and standard deviation are shown in Table 1.

This index summarizes how sharply transition risk decreases across quartiles of each parameter, or increases, if the parameter positively affects transition risk.