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Mixed Meal Model in Type 1 Diabetes: ―Minimal Compartments of Triglycerides and Non-esterified Fatty Acids

In keeping with the recent recognition of the additive impact of dietary fat on postprandial blood glucose (BG) levels in type 1 diabetes mellitus (T1DM), we include a number of physiologicallyrelevant improvements to a previous conceptual minimal mixed meal model by considering independent signals for gut transit of carbohydrates and dietary fat, and explicit representation of fatty acid spillover in the compartmental representation of non-esterified fatty acids (NEFA) to represent lipid-derived insulin resistance as the complementary increase in triglycerides (TG) and NEFA levels. Simulation results of postprandial plasma lipids and BG excursion compared with clinical data in the literature demonstrate the validity and potential of the minimal mixed meal model proposed.


Introduction
Type 1 diabetes mellitus (T1DM) is a chronic disease caused by β-cell apoptosis, which results in a complete inability to secrete endogenous insulin and such characterized by extremely high blood glucose (BG) levels, especially post-meal. Currently, T1DM has no cure but a continuous management of BG levels through intensive insulin therapy-which consists of exogenous insulin administration in a basal-bolus combination-in order to prevent long-term complications. Specifically, bolus insulin requires an exact dose calculation at every mealtime according to the specific carbohydrate content of the meal in order to maintain diurnal BG levels within the normal range of 70-180 mg/dL (3.9-10.0 mmol/L) post-meal [1].
To assist T1DM patients with their BG management, recent technologies-several of them relying on mathematical modeling of glucose-insulin metabolism -include diabetes diagnostics, decision support systems for insulin dosing and smart BG sensors [2]. In recent years, BG control systems that maintain normoglycemia automatically by closed-loop continuous infusion of insulin have taken particularly large strides towards its commercial development. Some studies [3] already have had successful results for overnight freeliving conditions, whereas more recently there is focus on postprandial BG control [4] and its compliance with clinical guidelines considers patient input of the carbohydrate content of the meal. Nonetheless, BG control performance has been shown to be under-mined by consumption of high-fat mixed meal [5], thus revealing that not only glucose but also fatty acid metabolism is indispensable in the mathematical representation of postprandial glucose-insulin metabolism in T1DM.
Representative models of fatty acid dynamics include Boston model [6] and others with detailed consideration of counter-regulatory hormones [7], whereas mixed meal models that include gut transit and the ultimate impact on BG levels are, to our knowledge, limited to that of Roy and colleagues [8]. Its physiological representation, nonetheless, includes two 'remote' compartments with unknown physiological conceptualization and a questionable direct incorporation of fatty acid into the glucose compartment to represent insulin resistance. In view of this, we previously developed a minimal mixed meal model in T1DM [9] that includes an explicit representation of the two major effects of lipids on postprandial glucose-insulin metabolism given by the delaying effect of dietary fat on gut transit of carbohydrates and lipid-derived insulin resistance due to dietary fat consumption in a mixed meal [10]. More complex mechanisms involved in postprandial lipid dynamics and its intricate interaction with glucose-insulin metabolism, however, are not appropriately represented in such conceptual model, which additionally reflects on a poor ability to reproduce clinical data in patients with T1DM in particular.
In the present study, we improve our original model proposal to obtain a more accurate representation of the impact of a mixed meal on postprandial glucose and fatty acid metabolism by including relevant physiological mechanisms intertwining the two of them, as theorized by the well-known Randle cycle [11] and glucose-fatty acid cycle reversed [12]. In Section 2, we first provide an overview of the conceptual mixed meal model, and then give a detailed explanation of the improvements proposed to the mathematical representation of gut transit of dietary fat and lipidderived insulin resistance. In Section 3, model parameters are validated with simulations in Matlab utilizing clinical data in the literature. Lastly, in Sections 4 and 5, we include substantial discussion and conclusions, respectively.

Mixed Meal Model in T1DM
In the present section, we first introduce our previous conceptual mixed meal model with a minimal compartmental representation of glucose-insulin and fatty acid metabolism in T1DM [9], followed by the improvements proposed in the present study by considering physiologically-relevant mechanisms as detailed in the forthcoming subsections.

Conceptual Mixed Meal Model
The original conceptual model in T1DM [9] (Fig. 1) was based on a minimal compartmental representation of gut transit of carbohydrates as glucose equivalent according to their food-specific absorption rate, subcutaneous insulin absorption kinetics and a minimal compartmental representation of glucose-insulin metabolism in T1DM (Fig. 1, solid lines) [13,14] with single-compartments of interstitial insulin concentration X(t) and plasma glucose concentration G(t) given respectively by In the above, I(t) is plasma insulin concentration, p 1 is total non-insulin-mediated glucose uptake, p 2 indicates the rate of spontaneous decrease of tissue glucose uptake ability, p 3 expresses the rate of insulindependent glucose uptake ability in peripheral tissue and liver, G ext (t) is meal-derived glucose appearance and V 1 is volume of distribution. Such mathematical model was extended to a mixed meal models (Fig. 1, dotted lines) that includes gut transit of dietary fat and a novel minimal compartmental proposal of fatty acid metabolism specifically for exogenous and endogenous pathways represented by triglycerides (TG) and non-esterified fatty acids (NEFA), respectively.
The biphasic delaying effect in gut transit of carbohydrates by the presence of dietary fat was represented with a gastric emptying delay τ mix and a pure delay L mix as where G ext (t) is gut absorption of glucose-equivalent from carbohydrate, G EQ (t) is glucose-equivalent with maximum rate of glucose appearance and τ dg is delay of food-specific carbohydrates as detailed elsewhere [14]. The representation of gut transit of dietary fat is derived from the glucose-equivalent signal of carbohydrates, based on the assumption that foodspecific carbohydrates absorption affected dietary fat as well. Moreover, such conceptual proposal included two single-compartmental models for TG and NEFA dynamics given respectively by Here, T (t) is plasma TG levels with steady-state basal value is the antilipolytic effect of insulin and p 7 is rate of NEFA release from adipose tissue into plasma due to hormone-sensitive lipase.
Considering the original definition of insulin sensitivity S I = p 3 /p 2 [15], T (t) was particularly utilized to represent lipid-derived insulin resistance after consumption of a high-fat mixed meal as where p 30 is the original parameter of the minimal model previously proposed [13] and p 8 adjusts the degree of lipid-derived insulin resistance. The mathematical representation of p 3 (t) in eq. (6), however, does not include the most well-known effect of free fatty acids on increase in insulin resistance as postulated by the Randle cycle. Thus, in the following subsections we incorporate some physiologicallyrelevant improvements (see Fig. 1, thick lines) including (a) representation of gut transit of dietary fat independently from carbohydrates, and (b) lipid-derived insulin resistance.

Gut Transit of Dietary Fat
After consumption of a mixed meal, dietary fatwhich mainly consists of TG and cholesterol-is broken down into monoglycerides and fatty acids in the stomach by the action of gastric lipase and sent to the small intestine for its absorption through the lymphatic system independent of the rate of absorption of carbohydrates [16]. Hence, in the present model, dF where F fat (t) is gut transit of dietary fat, k F is rate of absorption of lipids and τ chy is the time-constant to represent the biphasic gut transit pattern of fat absorption.

Insulin Resistance
The mathematical representation of lipid-derived insulin resistance in the present study is improved by considering not only increase in TG but also NEFA (see Fig. 1) as an alternative source according to the theoretical explanation of fuel selection between glucose and fatty acids as proposed by the Randle cycle [11]. Thus, we include fatty acid spillover from CM-TG into the NEFA pool in the single-compartmental model in eq. (5) as where V sp is distribution space of fatty acid spillover, which is represented by given that T sp is spillover from postprandial CM-TG appearance, τ sp is the time-constant for spillover during late postprandial state and F sp (t) is spilloverderived NEFA appearance from the splanchnic bed.
Considering the above single-compartmental models for TG and NEFA dynamics, the novel proposal of lipid-derived insulin resistance is extended to considering the explicit representation of fatty acid spillover, the Randle cycle and glucose-fatty acid cycle reversed.
where variation in p 3 (t) includes the representation of glucose-fatty acid cycle reversed [12], IR T (t) and IR F (t) are increase in insulin resistance due to postprandial TG and NEFA dynamics, respectively, as given by In eq. (13), ∆T (t) = T (t) − T b is the postprandial increase in TG levels caused by dietary fat consumption since T (t) is characterized by circulating TGrich lipoprotein particles in plasma due to the presence of cycling VLDL in the liver and CM-TG appearance after a high-fat meal; whereas in eq. (14), is NEFA-derived insulin resistance due to elevation above basal levels. Hence, the physiological representation of lipidinduced insulin resistance in eq. (12) includes increase in TG in the following 7 hours after dietary fat consumption, with an extended state due to increase in NEFA (fatty acid spillover) afterwards. The compartmental diagram of the present model is shown in Fig. 2, wherein fatty acid metabolism includes fatty acid spillover, the Randle cycle and glucose-fatty acid cycle reversed. with T1DM specifically [17], k F = 0.01 min −1 in eq. (7) and τ chy = 170 min in eq. (8) are determined from a clinical test [18] wherein the difference in the half-time gastric retention between a low-fat (30 min) and TGrich (200 min) meal.

Parameters Setting
Considering that the maximum postprandial increase in TG levels is IR T,max = 1 mmol/L after a very large fat load in patients with T1DM [19], and the maximum reduction in insulin sensitivity due to postprandial increase in TG levels is one-half [20], we set parameter p 8 = p 30 /2 in eq. (12). Moreover, the degree of insulin resistance due to NEFA levels in eq. (14) is determined from previous studies [21] that indicate a dose-response relationship with a decrease in glucose uptake of 22%, 30% and 34% for lipid-infused plasma NEFA increases above basal levels of 294 µmol/L (from 401 to 695 µmol/L), 865 µmol/L (386 to 1251 µmol/L) and 1272 µmol/L (from 416 to 1688 µmol/L), respectively. Having set p 9 = p 30 , parameters in the exponential function in eq. (14) are fitted with c 4 = 0.33 and c 5 = 3.6 (see Fig. 3) for a maximum reduction of IR F,max = 34% for NEFA increases above F b .
To represent postprandial lipid metabolism in patients with T1DM, parameters are fitted to clinical data as follows. Fitting of model parameters for plasma TG compartment in eq. (4) include p 4 = 0.35 min −1 , p 5 = 6.4 × 10 −2 min −1 and V T = 370 L, whereas for NEFA dynamics in eq. (9), in particular, we follow a two-step procedure given by (a) validation of parameter p 7 = 2.5 × 10 −2 min −1 for rate of lipolysis and steady-state value F b = 1.5 mmol/L under insulindeprived conditions specifically for T1DM [22] and (b) parameter fitting of p 6 = 89 min −1 , V sp = 7.8 L and fatty acid spillover τ sp = 180 min with clinical data of postprandial NEFA dynamics in T1DM [23].

Simulation
In this section we present simulation results of the proposed mixed meal model with parameter fitting considering clinical data in patients with T1DM.

Simulation Settings
Initial settings for each simulation are set to basal conditions given by T sp (0) = T b and p 3 (0) = p 30 , with remaining parameters of glucose-insulin metabolism in T1DM specified elsewhere [13,14]. Initial conditions for lipid metabolism consider each specific study as: i) Postprandial TG [19]:

Simulation Results: Triglycerides
Based on i) above, simulation of postprandial TG in patients with T1DM after consumption of a mixed meal with 100 g of dietary fat content is shown in Fig. 4, wherein not only the peak in TG levels but the excursion throughout the postprandial state is accurately represented by the single-compartmental model given in eq. (4).

Simulation Results: NEFA
Considering the initial settings in ii) and iii), simulation of plasma NEFA dynamics by the minimal compartmental proposal in eq. (9) are given in Fig. 5 and Fig. 6 for insulin-deprived conditions [22] and singlemeal consumption [23], respectively. The former shows the adequacy in parameter fitting for rate of lipolysis under complete absence of insulin from clinical data of T1DM patients, whereas the latter demonstrates the accurately representation of postprandial NEFA excursion and specifically the increase in NEFA levels above F (0) at t > 270 min as a result of fatty acid spillover, which in comparison our previous conceptual model is unable to reproduce. Nonetheless, the peak in postprandial BG excursion does not reflect clinical data as expected, particularly during early postprandial excursion at t < 200 min arguably due to the lack of further specification of the type of carbohydrate included in the clinical test design.

Discussion
In the present study we consider a number of improvements to a previous mixed meal model in T1DM to enhance the physiological representation of glucose and fatty acid metabolism for postprandial state. Different from the original proposal [9], gut transit of carbohydrates and dietary fat are represented separately, and the mathematical representation of fatty acid metabolism includes postprandial fatty acid spillover F sp (t), the Randle cycle as IR F and glucosefatty acid cycle reversed in p 3 (t), although the limited clinical data only permits model parameter setting and model validation for plasma BG and lipid dynamics. As shown in Fig. 4, the impact of dietary fat on TG levels is accurately represented throughout the postprandial state for the carbohydrate-independent signal F chy (t), whereas NEFA dynamics in Figs. 5 and 6 demonstrate an accurate representation of the rate of lipolysis and the gradual increase in NEFA levels due to fatty acid spillover from postprandial CM-TG metabolism, respectively, which our previous conceptual model [9] is neither able to reproduce. Although carbohydrates and dietary fat from a mixed meal are digested as chyme in the same physiological chamber, the biphasic nature of gut transit of mixed meals is represented separately for each macronutrient since the rate of gastric emptying of the fat component is basically slower than that of carbohydrates [24]. Moreover, despite the existence of some evidence, there is no absolute consensus about the relationship between BG levels and gastric emptying due to conflicting results from clinical studies [25], except for an apparent normal gastric emptying at normoglycemia [26]. Such a complex mechanism has not been explicitly considered in any mathematical representation of gut transit to this date [27], and thus the model proposed in the present study is not associated with instantaneous BG levels but only the macronutrient content of the mixed meal.
The minimal compartmental representation for plasma TG and NEFA dynamics proposed in the present study have an identical structure to that of the glucose compartment of Bergman minimal model. Nonetheless, since the activation/deactivation of insulin-dependent TG and NEFA uptake depends on different enzymes, i.e., lipoprotein lipase and hormone-sensitive lipase, respectively, different parameter values for each compartment are deemed necessary. The representation of insulin-dependent NEFA uptake follows a nonlinear dose-response relationship as quantified in previous studies [28], and excessive lipolysis due to absence of endogenous insulin secretion in patients with T1DM is determined by elevated values of F b . Instantaneous variation in p 3 (t) within the physiologically-relevant range of 0.5 p 30 ≤ p 3 (t) ≤ 1.33 p 30 as proposed in the present study responds to the careful parameter setting of p 8 and p 9 , provided that the concomitant effect of TG and NEFA metabolism does not occur simultaneously. Specific conditions such as hypertriglyceridemia and high fasting NEFA levels are beyond normal postprandial lipid metabolism, and thus are not considered in our study. As the present study proposes a mixed meal model of glucose and fatty acid metabolism with a minimal compartmental approach, a direct comparison with the mixed meal model proposed by Roy and colleagues [8] becomes imminent. In Roy model, the representation of fatty acid metabolism considers two remote-compartments for insulin-dependent fatty acid uptake and fatty acid, and one accessible compartment for fatty acids concentration, although the phys-iological significance of the remote insulin and fatty acid compartments (Y and Z in their proposal) remains to be elucidated. The effect of fatty acids on glucose metabolism is directly included in the glucose compartment G(t) of the minimal model. Compared to this, in our study we emphasize the physiological mechanism involved by making a clear distinction between exogenous (meal-derived) and endogenous lipid metabolism as represented by two singlecompartment models for TG and NEFA dynamics, respectively, which are utilized for a physiologicallyrelevant representation of lipid-derived insulin resistance as the supplementary increase in plasma TG and NEFA above basal levels.
In the current understanding of the etiology of insulin resistance, it is commonly attributed to fatty acids. Nevertheless, the physiological interpretation proposed in the present study (see Fig. 2) includes not only the well-known Randle cycle [11] but also glucose-fatty acid cycle reversed [12]. Despite the apparently opposite theoretical standpoint from their respective authors, we considered an original interpretation with a complementary effect on glucosefatty acid metabolism. To the best of our knowledge, the present model is the first that explicitly considers the late-postprandial increase in NEFA levels as a result of fatty acid spillover as represented with eqs. (10) and (11) assuming a direct relationship between the amount of dietary fat consumed and degree of spillover [29].
We acknowledge that further identification of model parameters for varying proportions of mixed meals is a necessary process for the complete validation of the present proposal. Even though lipid-derived insulin resistance is the hallmark of high-fat meal consumption, it cannot be directly measured in clinical studies and thus development of alternative methods for its quantitative measurement need further investigation as well.

Conclusions
In the present study, we improved an original mixed meal model by including the mathematical representation of a number of physiological mechanisms involving glucose and fatty acid metabolism. Parameter fitting with clinical data in the literature demonstrates the potential of the minimal compartmental approach proposed in the present mixed meal model in T1DM.