Modeling the acute effects of exercise on insulin kinetics in type 1 diabetes

Abstract

Our objective is to develop a physiology-based model of insulin kinetics to understand how exercise alters insulin concentrations in those with type 1 diabetes (T1D). We reveal the relationship between the insulin absorption rate $\left(k_a^I\right)$ from subcutaneous tissue, the insulin delivery rate $\left(k_d^I\right)$ to skeletal muscle, and two physiological parameters that characterize the tissue: the perfusion rate (Q) and the capillary permeability surface area (PS), both of which increase during exercise because of capillary recruitment. We compare model predictions to experimental observations from two pump-wearing T1D cohorts [resting subjects (n = 17) and exercising subjects (n = 12)] who were each given a mixed-meal tolerance test and a bolus of insulin. Using independently measured values of Q and PS from literature, the model predicts that during exercise insulin concentration increases by 30% in plasma and by 60% in skeletal muscle. Predictions reasonably agree with experimental observations from the two cohorts, without the need for parameter estimation by curve fitting. The insulin kinetics model suggests that the increase in surface area associated with exercise-induced capillary recruitment significantly increases $k_a^I$ and $k_d^I$, which explains why insulin concentrations in plasma and skeletal muscle increase during exercise, ultimately enhancing insulin-dependent glucose uptake. Preventing hypoglycemia is of paramount importance in determining the proper insulin dose during exercise. The presented model provides mechanistic insight into how exercise affects insulin kinetics, which could be useful in guiding the design of decision support systems and artificial pancreas control algorithms.

Introduction

Persons with type 1 diabetes (T1D) must inject insulin to survive. The preferred method of delivery is an injection into the subcutaneous tissue (SC). After injection, insulin is slowly absorbed into the bloodstream, and then eventually delivered to insulin sensitive tissues, where its availability facilitates glucose metabolism [1, 2].

Though the SC injection method is safe and mostly painless, the rate of insulin availability in insulin sensitive tissue is slow, and also highly variable from one injection to the next and from person to person. The problem with the variability is that if insulin becomes available too quickly, hypoglycemia can occur, causing dizziness and/or fainting. On the other hand, if insulin becomes available too slowly, then hyperglycemia can occur, causing adverse long-term health effects. The difficulty in accounting for the long and variable time-delay from insulin injection to availability in tissue is one of the main reasons why robust and safe automated insulin dosing algorithms are difficult to develop.

Predicting insulin availability becomes even more difficult during exercise because not only do glucose metabolism rates change because of increased glucose demand, but insulin absorption rates from subcutaneous tissue (SC) $\left(k_a^I\right)$ and delivery rates to skeletal muscle (SM) $\left(k_d^I\right)$ have also been shown to increase [3–9]. It has been postulated that the observed changes are caused by an increase in blood flow rate [10–12]. Increased $k_d^I$ has also been associated with an increase in microvascular blood volume (MBV) [7–9] indicating the importance of capillary recruitment—a simultaneous increase in capillary surface area and blood flow.

These experimental studies have been illuminating and have shown correlation between blood flow, MBV, $k_a^I$ and $k_d^I$, however many lack a rigorous treatment of the underlying transport phenomena. Further, most models of insulin kinetics in the literature are data-driven where experimental data is used to identify kinetic parameters [13]. These approaches are expedient and very useful from an engineering standpoint, but by design they do not include descriptions of the underlying transport phenomena and physiology of the problem. A model that connects underlying physiology to insulin kinetics can be useful in determining how rate-limiting steps are altered during exercise, which can aid in the development of insulin dosing strategies for decision support systems and artificial pancreas control algorithms. In this study, we provide a thorough treatment of the underlying insulin transport phenomenon.

The focus of this study is to derive a physiology-based pharmacokinetic (PBPK) model of insulin kinetics to elucidate the relationship between exercise intensity E and insulin absorption $\left(k_a^I\right)$ and delivery rates $\left(k_d^I\right)$. We draw on concepts from the research area of microcirculation and nutrient delivery [14]. Drug and nutrient delivery is typically characterized by two parameters: the capillary tissue perfusion rate (Q) and capillary permeability surface area (PS). These parameters characterize mass-transfer between different regions in the body and will be modeled at the microscale, where diffusion is handled directly through solution of the convection-diffusion equation. Insulin distribution within different fluid regions of the body will be modeled at the macroscale using a compartmental approach.

Because the objective of this work is to understand how the underlying physiology affects insulin kinetics, we develop the model and choose parameters without performing parameter estimation on our current clinical data that was collected in the Mayo Clinic. Rather, we simulate the model using independent parameters based on literature-data, and then compare simulation predictions to our Mayo Clinic data to evaluate whether we have correctly included the underlying physical mechanisms. We evaluate the model by comparing predictions of plasma insulin concentration (I) to measurements from two cohorts of subjects with T1D.

To the authors’ best knowledge, this is the first attempt to include directly the underlying insulin transport phenomenon and exercise-induced capillary recruitment in a broader model of insulin kinetics. The model provides insight into why those with T1D commonly experience hypoglycemia during exercise. Although the model is not being developed for direct use in control systems for the artificial pancreas, insights gained from the model can potentially help in designing decision support systems and artificial pancreas algorithms.

Methods

Model development

Insulin kinetics is modeled in five steps: (1) insulin pump infusion into the interstitial fluid (ISF) of SC, (2) diffusion into SC absorbing capillaries and convection into the circulatory system (CS), (3) mixing and distribution in the CS, (4) convection into SM delivering capillaries and diffusion into the SM ISF, and (5) clearance by the liver/kidneys in the CS and clearance by insulin degradation in the SM. One notable assumption is that the insulin hexameric dissociation step in the SC was not modeled. The logic behind this assumption will be discussed in coming sections. The model is shown schematically in Fig. 1. Absorbing and delivering capillaries are part of the SC and SM domain, respectively.

Fig. 1.

Physiological representation of model. The model consists of three distinct domains: the subcutaneous domain (SC, tan color), the circulatory system domain (CS, red color), and the skeletal muscle domain (SM, purple color). Arrows show insulin clearance directly from the CS by the liver and kidneys. Clearance also occurs in the SM tissue by insulin degradation. The human silhouette illustrates the relative size of each domain. The model is multiscale. The SC and SM domains contain both ISF and capillaries. The dynamics of the capillaries are modeled at the microscale and governed by the convection-diffusion equation. The dynamics of the fluid volumes in the SC, CS, and SM are modeled at the macroscale and are governed by compartmental models

The human silhouette of Fig. 1 illustrates the relative size of each domain. The SC domain is defined as only the SC tissue in contact with infused insulin. The CS domain is all the blood that is confined to the larger vessels and organs. The SM domain is all the skeletal muscle tissues and also includes other SC tissues that are far from the infusion site and so are not a part of the SC domain. It is worth mentioning that both capillaries and ISF are contained in the SC and SM domains. For brevity, many details of the model derivation are omitted here. Details can be found in the Supplementary Material.

The final model equations are

$$\frac{d\mathcal{C}}{dt}=-k_a^I\mathcal{C}+\mathcal{U}(t),\mathcal{C}\left(t_o\right)={\mathcal{C}}_o$$ (1)

$$\frac{dI}{dt}=\frac{k_a^I\mathcal{C}}{V_{CS}^I}-\frac{V_{SM}^{tiss}{k}_d^I\cdot (I-S)}{V_{CS}^I}-{\text{r}}_{C{S}_N}^I\cdot I,I\left(t_o\right)=I_o$$ (2)

$$\frac{dS}{dt}=\frac{V_{SM}^{tiss}{k}_d^I\cdot (I-S)}{V_{SM}^I}-{\text{r}}_{S{M}_N}^I\cdot S,S\left(t_o\right)=S_o$$ (3)

with functions defined as

$$k_a^I=Q_a^I\left(1-e^{\frac{-{PS}_a^I}{Q_a^I}}\right)$$ (4)

$$k_d^I=Q_d^I\left(1-e^{\frac{-P_d^I}{Q_d^I}}\right)$$ (5)

$$Q_a^I(E)=(1-h)\cdot \left(Q_{a_{rest}}+{\lambda }_{a}E(t)\right)$$ (6)

$$Q_d^I(E)=(1-h)\cdot \left(Q_{d_{rest}}+{\lambda }_{d}E(t)\right)$$ (7)

$$P{S}_a^I(E)=P{S}_{a_{rest}}^I\cdot \left(1+R_{a}tanh(\gamma E(t))\right)$$ (8)

$$P{S}_d^I(E)=P{S}_{d_{\mathit{\text{rest}}}}^I\cdot \left(1+R_{d}tanh(\gamma E(t))\right).$$ (9)

The model represents the movement of insulin through three distinct domains; the SC ($\mathcal{C}$ is the mass of insulin or insulin on board (IOB) in the SC), the CS (I is the concentration of insulin in blood plasma), and the SM (S is the concentration of insulin in the SM ISF). This system of three ODEs represents a significant simplification from the initial system of PDEs that were derived from first principles.

To derive the governing equation for $\mathcal{C}$ (Units) (Eq. 1), a.k.a. IOB, we started with the spherically symmetric diffusion equation and then integrate the local insulin concentration over the volume of SC in the vicinity of the injection site to obtain $\mathcal{C}$. To do this integration we assume that the tissue in the vicinity of the subcutaneous injection site is continuous and homogeneous, that insulin in the SC ISF is well-mixed, and that penetration of insulin through the capillary walls is driven only by diffusion. The insulin infusion rate from the pump is $\mathcal{U}$(t) (U/min) and the parameter $k_a^I$ (1/min) is the rate of insulin absorption (denoted by subscript a) from the SC tissue.

To derive the governing equation for I (μU/mL) (Eq. 2), plasma insulin in the CS, we assume that insulin in the CS is well-mixed and confined to the blood plasma (not in red blood cells). Insulin clearance is assumed to be linear. ${\text{r}}_{C{S}_N}^I$ is the volume-normalized plasma insulin clearance rate ${\text{r}}_{C{S}_N}^I={\text{r}}_{CS}^I/V_{CS}^I$ (1/min). $V_{CS}^I$ (dL) is the volume of distribution of insulin in the CS domain. $V_{SM}^{\mathit{\text{tiss}}}$ (mL) is the volume of SM tissue. $k_d^I$(1/min) is the rate of insulin delivery (denoted by subscript d) to the SM tissue.

We include S, the insulin in the ISF of the SM, because it is well-known that insulin action occurs in a compartment remote from plasma insulin, identified as the ISF [2]. To derive the governing equation for S (Eq. 3) we assume that insulin is well-mixed and that the SM has uniform tissue properties. Linear insulin clearance is also assumed, which on the whole-body scale is a reasonable assumption [15]. However, it is possible that clearance in the SM may deviate from linear, but limited data is available to quantify a non-linear clearance relationship. The volume-normalized insulin clearance rate is ${\text{r}}_{S{M}_N}^I$. $V_{SM}^I$ is the volume of distribution of insulin in the SM.

To couple our three governing equations we apply mass conservation to relate fluxes across domain boundaries. The four convective fluxes, shown by black arrows into and out of the CS domain in Fig. 1 are responsible for transporting insulin into and out of the capillary beds of the SC and SM domains. The amount of insulin absorbed from the SC is equal to the SC arteriovenous difference times the SC tissue perfusion rate $\left(Q_a^I\right)$. Similarly, the amount of insulin delivered to the SM is equal to the SM arteriovenous difference times the SM perfusion rate $\left(Q_d^I\right)$. $Q_a^I$ and $Q_d^I$ are estimated from measurements found in literature, so in order to calculate the fluxes between compartments, we only need to estimate the arteriovenous difference.

To estimate the arteriovenous difference we need to estimate how much insulin enters or exits through the capillary walls on each pass through the capillary bed. Insulin diffusion across the capillary walls is modeled using the cylindrically symmetric convection-diffusion equation. We hypothesize, as others have [16], that insulin is absorbed from the SC tissue into the capillaries by permeating the capillary wall through endothelial pores and junctions. It is delivered to the SM tissue in the same manner. We further assume that the absorption and delivery processes are driven only by diffusion in the radial direction of a cylindrical capillary. We recognize that there may also be a convective component of insulin transport radially through the capillary walls via capillary filtration. However, in vivo measurements of insulin transport in human tissue currently cannot directly measure the relative contribution of convective and diffusive transport, and hence measured PS parameters are reflections of the sum of both diffusive and convective fluxes.

The assumption of pure diffusion allows us to model the physics of absorption from SC and delivery to SM identically, the only difference being the tissue properties and the sign of the flux. When solving the convection-diffusion equation, we assume a uniformly perfused and homogeneous capillary bed. In the real case, properties and Q are heterogeneous, and thus this assumption will prevent us from capturing some non-linear effects. However, because capillaries have a low permeability to insulin the effects of heterogeneity will be small [17]. After making several other assumptions (detailed in the Supplementary Material), the convection-diffusion equation is solved to obtain the insulin concentration b as it depends on the capillary axial coordinate. The solution shows that the arteriovenous difference is equal to the concentration in each adjacent compartment times the factor $\left(1-e^{-\frac{PS}{Q}}\right)$. Multiplying this factor times the tissue perfusion rate reveals that mass flux between two compartments is governed by the rate parameter k

$$k=Q\left(1-e^{-\frac{PS}{Q}}\right).$$ (10)

The tissue specific rates of k, $k_a^I$ and $k_d^I$, are defined in Eqs. 4 and 5. Note that the solution is only dependent on the ratio of parameters PS and Q. The ratio of these two parameters indicates whether the absorption or delivery rate is limited by convection (flow limited) or by diffusion (surface area limited). Because of the low permeability of capillaries to insulin, this problem is surface area limited.

Presently, we have independently derived k for the purposes of modeling insulin absorption. A similar solution was previously derived for the purposes of studying the microcirculation and nutrient delivery by Renkin [14] and further discussed in [17]. To our knowledge this is the first time this solution has been used in a broader pharmacokinetics model of insulin absorption and delivery.

To model exercise, we allow the capillary dynamics to depend on the exercise intensity E. An increase in E causes ‘capillary recruitment’—a monotonic increase in both Q and PS described by Eqs. 6, 7, 8, and 9. In these equations subscript _a denotes the insulin ‘absorbing’ SC capillaries, and subscript _d denotes the insulin ‘delivering’ SM capillaries. The effects of exercise-induced capillary recruitment on the rate k, which governs mass-transfer between the compartments, are plotted in Fig. 2.

Fig. 2.

Insulin permeability surface area (PS^I, Eqs. 8 and 9) and kinetic rate (k^I, Eqs. 4 and 5) are plotted versus exercise intensity E. Subscript a corresponds to absorbing capillaries of the SC. Subscript d corresponds to delivering capillaries of the SM. PS^I rapidly increases at low levels of E because of capillary recruitment, and reaches a plateau value at only E = 0:25. The plateau exists because there is a finite reservoir of capillaries in the tissue. $P{S}_d^l$ plateaus at a higher value than $P{S}_a^I$ because the delivering SM capillaries have a higher recruitment factor R than the absorbing SC capillaries. The absorption rate $k_a^I$ and delivery rate $k_d^I$ both increase very similarly to PS^I because insulin kinetics are surface area limited, not flow limited

Equations 6 and 7 relate Q to exercise intensity E in a linear manner with slope λ_a and λ_d, consistent with measurements from [18, 19]. Hematocrit level h is included because insulin resides mainly in the plasma, and does not enter red blood cells. Equations 8 and 9 relate PS^I to E in a saturable manner, stemming from the idea of a finite reservoir of capillaries that can be recruited during exercise [16]. We describe this saturating process with a tanh function which is not physiological, but is none-the-less a compact and reasonable description. γ is a free parameter that serves to modulate how quickly the function reaches its plateau value. R_a and R_d are the recruitment factors that describe the proportional increase in capillary surface area induced by exercise and are estimated through a proxy for capillary recruitment, microvascular blood volume (MBV). Figure 2 shows the relationship between PS and E.

The initial conditions, $\mathcal{C}$_o, I_o, and S_o, are found by evaluating the system at rest (E = 0) and steady-state. This yields the relationships

$$0=-k_a^I(0){\mathcal{C}}_o+\mathcal{U}\left(t_o\right)$$ (11)

$$0=\frac{k_a^I(0){\mathcal{C}}_o}{V_{CS}^I}-\frac{V_{SM}^{tiss}{k}_d^I(0)\left(I_o-S_o\right)}{V_{CS}^I}-{\text{r}}_{C{S}_N}^I\cdot I_o$$ (12)

$$0=\frac{V_{SM}^{tiss}{k}_d^I(0)\left(I_o-S_o\right)}{V_{SM}^I}-{\text{r}}_{S{M}_N}^I\cdot S_o.$$ (13)

These are linear equations, and can be algebraically solved for $\mathcal{C}$_o, I_o, and S_o.

There are several potentially important unmodeled phenomena. Clearance of insulin in the SC tissue was not included. Hexameric to dimeric/monomeric dissociation in the SC tissue was not modeled because modern rapid acting insulins dissociate rapidly and the rate-limiting step of absorption is typically penetration of the endothelial walls [20]. Tissue heterogeneity was not modeled because of limited data available in the literature, and because adding heterogeneity would greatly increase the complexity of the model. Insulin absorption through the lymph vessels was not considered in the model because an order of magnitude analysis (not shown) revealed that lymph absorption could account for only a small fraction of total absorption. This secondary importance of lymph absorption was confirmed in sheep [21]. Lastly, changes in volume of distribution V^I and clearance rate r^I during exercise were not considered—the volume-contraction effect is estimated to be less than 10% [22], and measurements of the change in r^I during exercise are difficult to obtain and not yet validated.

Inputs and parameters

Our objective is to develop a model to understand how exercise-induced changes in physiology may affect insulin kinetics. To achieve this, we do not use the presently collected data from our human studies at the Mayo Clinic to estimate parameters. Rather, we independently choose parameters based on data found in literature. Once we have chosen our parameters we then compare our model predictions to the present Mayo Clinic data to evaluate whether the physiology included in the model can generally capture the observed behavior. All the parameters are summarized in Table 1. The model assumes that the injected insulin is either lispro or aspart and that the kinetic parameters for each type of insulin are identical. This assumption should be valid because dissociation times [20] and the kinetic profiles of the two insulins are nearly identical [23].

Table 1.

Parameters used in the insulin kinetics model

Parameter	Description	Unit	Value	Source
$V_{{CS}_N}^{\mathit{I}}$	Normalized volume of distribution of insulin in circulatory system	dL/kgBWa	0.5	[1, 29]
$V_{{SM}_N}^{\mathit{I}}$	Normalized volume of distribution of insulin in skeletal muscle	dL/kgBWa	1.2	[1, 29, 30]
$V_{{SM}_N}^{\mathit{tiss}}$	Normalized volume of tissue in skeletal muscle domain	mL/kgBWa	540	[18]
${\text{r}}_{{CS}_N}^I$	Normalized clearance rate of insulin in circulatory system	1/min	0.32	[31–33]b
${\text{r}}_{{SM}_N}^I$	Normalized clearance rate of insulin in skeletal muscle	1/min	0.02	[31–33]b
Qarest	Tissue perfusion rate in absorbing (SC) tissue at rest	mLb/mLtiss/min	0.028	[18]
Qdrest	Tissue perfusion rate in delivering (SM) tissue at rest	mLb/mLtiss/min	0.038	[18]
h	Hematocrit percentage in blood	1	0.4	[29]
λa	Slope relating Q in absorbing (SC) tissue to exercise	mLb/mLtiss/min	0.071	[5, 50–52]
λd	Slope relating Q in delivering (SM) tissue to exercise	mLb/mLtiss/min	1.1	[19]
$P{S}_{a_{rest}}^I$	Permeability surface area to insulin in absorbing (SC) capillaries	mLb/mLtiss/min	0.005	c
$P{S}_{d_{rest}}^I$	Permeability surface area to insulin in delivering (SM) capillaries	mLb/mLtiss/min	0.005	[27, 28]
Ra	Capillary recruitment factor in absorbing (SC) tissue	1	0.40	[46, 47]d
Rd	Capillary recruitment factor in delivering (SM) tissue	1	1.46	[24, 26, 49]d
γ	Saturation coefficient for PSI	1	10	[24, 26, 49]d

Superscripts and subscripts indicate the context of the parameter. I indicates an insulin-specific parameter. tiss indicates volume of tissue. Subscript CS or SM indicate the domain. Subscript a or d indicate ‘absorbing’ capillaries of the CS domain or ‘delivering’ capillaries of the SM domain. Subscript N indicates a normalized parameter

^aMultiplied by BW prior to being used in model. See demographics for BW

^bDerived from total clearance rate

^cAssumed to be the same as $P{S}_{d_{rest}}^I$

^dDerived by curve fitting to MBV literature data in SC and SM tissues

Here, we briefly describe how parameters and inputs were obtained. Refer to the Supplementary Material for a more complete explanation.

The estimated exercise intensity E is defined as a fraction of $\text{V}{0}_{2_{max}}$

$$E=\frac{\text{V}{0}_2}{\text{V}{0}_{2_{\max }}}.$$ (14)

E = 0 signifies that a subject is at rest and E = 1 is at maximal exercise. E is a continuous input to the insulin kinetics model. We consider E to be the exercise signal above rest. Q and PS depend directly on E, simulating the effects of capillary recruitment. In our human trials, exercise is done at 50% $\text{V}{0}_{2_{max}}$ and so E is set to 0.5 during exercise, and 0 otherwise.

We use data taken from independent studies to quantify the tissue perfusion rate Q [18, 19]. Q is linearly related to E with slopes λ_a = 0:071 and λ_d = 1.1 (mL_b/mL_tiss/min). Q is different in the SC and SM tissues because during exercise Q in SC tissue increases only twofold compared to 20-fold in SM tissue [18, 19]. The percent hematocrit h is assumed 40%.

PS is a measure of how permeable a capillary bed is to a certain solute. We assume that baseline PS in the two tissues at rest are equal $P{S}_{d_{rest}}^I=P{S}_{a_{rest}}^I$, as there is a dearth of data to suggest otherwise. To describe recruitment we define the second term in Eq. 9 as a new function f_d that describes recruitment

$$f_d=R_{d}tanh(\gamma E).$$ (15)

f_d can be thought of as the proportional increase in the number of perfused capillaries in the delivering (SM) tissue. A similar expression is used for the SC tissue, replacing R_d with R_a. R_a and R_d are the recruitment factors that describe the proportional increase in capillary surface area induced by exercise. γ is a free parameter that serves to modulate how quickly the function reaches its plateau value. Values for these three parameters do not exist in the literature and we estimate them using a commonly used proxy for capillary surface area, the microvascular blood volume (MBV), as measured in several studies [24–26]. We assume that the proportional change in MBV induced by exercise is equivalent to a change in f_d. We plot this proportional change at various exercise levels in Fig. 3. The ratios show considerable variation, but generally describe a saturable process. The solid curve is Eq. 15 with best-fit parameters R_d = 1.46 and γ = 10 (R² = 0.55). Because quality of fit is relatively insensitive to γ, we assign γ = 10. We recognize that because the fit is insensitive to γ, that the expression in Eq. 15 is technically non-identifiable from the available data. However, the expression concisely describes the concept of a finite reservoir of capillaries that can be recruited as a function of E, and thus is suitable for our purpose. Data on capillary recruitment in SC tissue during exercise is extremely limited and so we set R_a = 0.4, which is the recruitment factor in the SC following a meal. The baseline $P{S}_{d_{rest}}^I$ is 0.005 (mL_b/mL_tiss/min) as measured by [27, 28].

Fig. 3.

The curve describes the concept of a finite reservoir of capillaries that can be recruited as a function of exercise intensity. The black dots are derived from the experimental results of various studies on capillary recruitment [24, 48, 49]. Recruitment f_d (Eq. 15) is defined to be zero at rest and is fit to data to find R_d and γ in Eq. 15. f_d rapidly increases to a plateau value of R_d = 1.46, indicating a 146% increase in capillary density during exercise

The volume of distribution we use for the CS domain is equal to the volume of the blood plasma, which normalized by the body weight (BW) of an average man is $V_{C{S}_N}^I=0.5\left(\text{dL}/{\text{kg}}_{BW}\right)$ [1, 29]. The volume of distribution that we use in the SM domain is roughly equal to the volume of the ISF, which normalized by BW is $V_{S{M}_N}^I=1.2\left(\text{dL}/{\text{kg}}_{BW}\right)$ [1, 29, 30]. There may be a small redistribution of fluids during exercise that may decrease the volume of plasma by 10–15% [22], but this change is small and we treat the volume of distribution as constant and independent of exercise.

V^tiss is fundamentally different from the volume of distribution; it is the volume of tissue in the compartment. This parameter only shows up in the SM compartment (Eq. 3) because we must know the total volume flow rate $\left(Q_d^I\cdot V_{SM}^{tiss}\right)$ in order to calculate the mass flux of insulin between compartments. The volume of tissue in the SM compartment is $V_{SM}^{\mathit{\text{tiss}}}=0.54\left(\text{L}/{\text{kg}}_{BW}\right)$ [18], i.e. 54% of body weight at an assumed 1 (kg/L) density.

Metabolic clearance of insulin occurs in both the CS domain and the SM domain. We estimate each domain’s contribution to insulin clearance by setting total clearance ${\text{r}}_{\mathit{\text{tot}}}^I$ (estimated to be on average 17·BW (mL/min) [31–33]) to be equal to the sum of the clearance in each compartment. We estimate ${\text{r}}_{SM}^I=200(\text{mL}/\text{min})$ and ${\text{r}}_{CS}^I=1370(\text{mL}/\text{min})$. This corresponds to volume-normalized clearance rates of ${\text{r}}_{S{M}_N}^I=0.02(1/\text{min})$ and ${\text{r}}_{C{S}_N}^I=0.32(1/\text{min})$.

Human subjects data

The data used in this study comes from two separate research studies on T1D human subjects and is the secondary use of existing data. The main difference between the two cohorts is that one examined T1D patients under resting conditions (T1DR), and the other examined T1D patients under exercising conditions (T1DE). The protocols are illustrated in Fig. 4. Data from T1DR was previously published in [34] where diurnal variations in insulin sensitivity were examined, however that study did not focus on modeling insulin absorption and delivery. The T1DE data is previously unpublished. All data has been provided by the Mayo Clinic through their own internal human subjects research (HSR) data protection protocols. For brevity, only the relevant protocol information will be discussed here. Detailed protocol information on T1DR can be found in [34]. The meal and exercise protocol for T1DE is the same as that of a similar study on healthy patients [35] (the patients in the present study were T1D). Important dosing and demographic information is summarized in Table 2.

Fig. 4.

Timeline for the mixed-meal tolerance test (MMTT) T1D resting protocol (T1DR, top) and the T1D exercising protocol (T1DE, bottom). At time 0 the subjects in T1DR ingested a 50 (g) CHO mixed-meal and the subjects in T1DE ingested 75 (g). The insulin bolus was infused through an insulin pump at time 0, with bolus size calculated according to the subject’s normal insulin-to-carb ratio. The T1DR group rested throughout the entire study period. The T1DE group exercised at 50% $\text{V}{0}_{2_{max}}$ for four 15 min exercise periods, starting at minute 120

Table 2.

Demographics of T1D subjects

Variable	Resting group (T1DR)	Exercising group (T1DE)
n	17	12
Age (years)	39.7 ± 14.3a	43.5 ± 12.9a
Sex (M/F)	9M, 8F	7M, 5F
Weight (Kg)	74.4 ± 5.3a	85.9 ± 21.9a
HbA1C (%)	7.11 ± 0.6a	7.5 ± 0.6a
Average bolus size (U)	4.1	6.7
V02max	Not measured	23.8 ± 5.4a

^a± Standard deviation

Cohort of subjects with type 1 diabetes at rest (T1DR)

T1DR consisted of 19 subjects. All meals were provided by the clinical research unit (CRU) metabolic kitchen promptly at 0700 h. Subjects received weighed meals, with each comprising 33% of the total estimated calorie intake based on Harris Benedict calorie requirements, including a low level of physical activity, with ∼50 g of carbohydrate in each meal. The meal consisted of Jell-O with dextrose, eggs (scrambled or omelet), and ham slices. At meal time, an insulin dose that corresponded to the patient’s normal insulin-carb ratio was infused through an insulin pump.

All subjects were on an insulin pump. Thirteen subjects were taking insulin aspart, whereas the remaining six were taking insulin lispro. Insulin pump data was downloaded and used to quantify the insulin infusion rate $\mathcal{U}$(t). Insulin levels were measured by a two-site immunoenzymatic assay performed on the DxI automated immunoassay system (Beckman Coulter, Inc., Chaska, MN). Of the 19 patients in the T1DR group, 2 patients were removed due to missing data, leaving 17 for analysis.

Cohort of subjects with type 1 diabetes with exercise (T1DE)

T1DE consisted of 14 patients. Subjects did not engage in vigorous physical activities for 72 h before screen and study visits. In the screening visit participants performed a graded exercise test on a treadmill to determine $\text{V}{0}_{2_{max}}$ according to guidelines (American College of Sports Medicine Guidelines for Exercise Testing and Prescription, 7th Edition) and ensure stable cardiac status. Expired gases were collected and analyzed using indirect calorimetry. $\text{V}{0}_{2_{max}}$ was determined when at least two of the following three criteria were met: (1) participant too tired to continue exercise, (2) respiratory exchange ratio exceeded 1.0; or (3) a plateau was reached in oxygen consumption with increasing workload. The purpose of this test was to use individual $\text{V}{0}_{2_{max}}$ data to determine workload during the moderate-intensity $\text{V}{0}_{2_{max}}$ protocol during the study day.

For the study visit all subjects spent 40 h in the CRU. On day 1 subjects were admitted to the CRU at 1600. They were then provided a standard 10 kcal/kg meal (55% carbohydrate, 15% protein, and 30% fat) consumed between 1700 and 1730 h. No additional food was provided until the next morning. On day 2 at 0600 h, an intravenous cannula was inserted retrogradely into a hand vein for periodic blood draws. The hand was placed in a heated (55C) Plexiglas box to enable drawing of arterialized-venous blood for glucose and hormone analyses. At 0700 h a mixed-meal study was performed.

A mixed meal containing 75 g of glucose was ingested at time 0. At meal time, an insulin dose that corresponded to the patient’s normal insulin-carb ratio was infused through an insulin pump. At 120 min following the first bite, subjects stepped on a treadmill to exercise at moderate-intensity activity (50% $\text{V}{0}_{2_{max}}$): i.e., four bouts of walking at 3–4 miles/h for 15 min with rest periods of 5 min between each walking bout: total duration 75 min. The workload during physical activity was continuously monitored by measurements of V0₂ during exercise to maintain target 50% $\text{V}{0}_{2_{max}}$ exercise intensity. Following the last blood draw the hand vein cannula was removed. Lunch at 1300 h and dinner at 1900 h were provided, each meal contributing 33% of daily estimated caloric intake. The patient was discharged the next morning at 0800 h.

Of the original 14 patients in the T1DE group, 2 patients were deemed outliers because they exhibited extremely rapid absorption rates compared to the rest of the group. They were subsequently removed from the analysis.

Simulation, input, and data considerations

Equations 1–9 are solved using MATLAB ode15, a built in explicit ODE solver. The time step was 0.1 (s). The initial conditions were found by solving the steady-state resting Eqs. 11–13. t_o = 420 min was the initial time of the simulation which corresponded to about 0000 h (midnight) when the subjects were sleeping and had approximately steady-state and basal insulin levels.

The averaged insulin pump infusion rate $\mathcal{U}$(t) and exercise intensity E(t) were used as model inputs (top frame of Fig. 5) and the measured plasma insulin concentration I_data was used for comparison to model predictions. Slight bumps in $\mathcal{U}$(t) prior to time 0 occurred in the averaged input because some patients administered small correction boluses prior to the meal test. The initial input $\mathcal{U}$(t_o) was set to the average infusion rate at t_o. For the T1DR cohort E = 0 throughout. For the T1DE cohort E = 0.5 during exercise periods (120–135, 140–155, 160–175, 180–195 min) and E = 0 elsewhere. A small insignificant time delay with a time-constant of 2 min was given to the E signal to account for lag of physiological changes after exercise begins and ends.

Fig. 5.

Simulated insulin concentration for a resting (left) and exercising (right) average subject. Inputs $\mathcal{U}$(t) and E(t) are shown in the top frames. Predicted (I) and measured (I_data) plasma insulin concentrations are shown in the bottom frame with error bars representing the standard-deviation of the group. Skeletal muscle interstitial fluid concentration (S) is shown as a dotted line. 15 min exercise bouts with 5 min breaks are shown as shaded regions. The predicted increases in I and S during exercise are a result of the enhanced absorption rate $k_a^l$ and delivery rate $k_d^I$ associated with capillary recruitment. The predicted increases in insulin concentration during exercise may have significant implications on glucose dynamics, potentially leading to hypoglycemia during exercise

Results

Model comparison with resting patient cohort (T1DR)

Using the average measured input $\mathcal{U}$(t) from the resting cohort (T1DR), the insulin kinetics model was simulated for an average resting patient to produce predictions of I and S. Figure 5 (left) plots the predictions along with the measured insulin concentration I_data. Overall, predictions reasonably agree with data, indicating that the important physical phenomena underlying the model were captured, however there are some notable discrepancies. The predicted insulin reaches a peak of 28 (μU/mL), a slight underestimation of the measured peak plasma insulin concentration 32 (μU/mL). The peak insulin occurs only 20 min after injection, compared to 45 min for the measured data. These discrepancies occur because of underestimated parameter values and because the model does not include the insulin hexameric dissociation step in the SC. This will be further explained in the discussion section.

The concentration in the SM domain, S, slowly rises because of the slow diffusion of insulin into the SM tissue. It eventually reaches a maximum of 13 (μU/mL), up from the baseline concentration of 6 (μU/mL). S was not measured in the current study and hence we cannot verify the predictions for S, however they are in reasonable agreement with lymph insulin data taken by Bergman after an intravenous insulin infusion [36].

Model comparison with exercising patient cohort (T1DE)

Next, using the average measured input $\mathcal{U}$(t) and E(t) from the exercising cohort (T1DE), the insulin kinetics model was simulated for an average exercising patient to produce predictions of I and S. Fig. 5 (right) shows the comparison of simulation to measured data.

Overall, I in the exercise simulation reasonably follows I_data, however again there are notable discrepancies. In the resting period I reaches a peak of 30 (μU/mL), a slight underestimate of the measured peak of 33 (μU/mL). Again, the predicted peak occurs much earlier than the measured peak because the model does not include insulin dissociation in SC tissue.

Once exercise begins, I rapidly increases by approximately 30% from 25 (μU/mL) to 32 (μU/mL). This occurs as a result of the 45% increase in the insulin absorption rate $k_a^I$ brought on by capillary recruitment in the SC tissue (see Fig. 2). I_data shows a similar approximately 30% increase from 22 to 29 (μU/mL).

A sawtooth pattern in the predictions emerges as a result of 15 min on, 5 min off, exercise bouts. During the 5 min rest we predict capillary de-recruitment, which reduces the absorption rate and thus lowers insulin concentration in plasma. I_data shows a similar pattern. After exercise ends, the predictions closely follow the data up until the cessation of the study at minute 360.

S slowly rises after the bolus because of the slow diffusion of insulin into the SM tissue. Then S reaches a local maximum near minute 60 at a concentration of 13 (μU/mL). It maintains this level until exercise commences at minute 120. At this point, the rate of insulin delivery $k_d^I$ increases by 150% because of capillary recruitment and as a result, S begins a rapid rise, peaking at a concentration of 21 (μU/mL), a 60% increase, until the end of the exercise period.

Predicted insulin kinetics

To illuminate the effect that capillary recruitment has on insulin delivery to SM tissue, we have calculated estimates of two often-reported kinetic measures. The first is the extraction fraction of insulin (ef^I) (%) which measures the percentage arteriovenous change in insulin concentration after blood passes through SM tissue. The second is the insulin uptake by SM capillaries ($\dot{\mathcal{S}}$_d) (μU/min).

The extraction fraction of insulin (ef^I) is defined as

$$e{f}^I=\frac{I_{arterial}-I_{venous}}{I_{arterial}}\cdot 100.$$ (16)

ef^I is a measure of how effectively insulin in the CS is delivered to the SM tissue. A high ef^I means that a large amount of insulin is removed from the CS on each pass through the SM tissue. ef^I can be directly measured using arteriovenous catheterization, however in the present study ef^I is not measured. Instead, we will use the developed model to express ef^I as a derived quantity. ef^I can be written in terms of I and S by approximating that I_arterial = I and $I_{venous}={b_d|}_e$ (the calculated exit concentration of the capillary bed). Using the solution for b found in the Supplementary Material and substituting S for c we can estimate ${b_d|}_e$ and then substitute into Eq. 16 to get

$$e{f}^I=\frac{\left(1-e^{\frac{-P{S}_d^I}{Q_d^I}}\right)(I-S)}{I}\cdot 100.$$ (17)

where $P{S}_d^I$ is the permeability surface area and $Q_d^I$ is tissue perfusion rate in the SM. A linear error analysis (not shown) reveals that the approximation of I_arterial = I leads to about a 10% error in ef^I, which is acceptable for our purposes of exploring how exercise generally affects insulin kinetics.

The other quantity, insulin uptake ($\dot{\mathcal{S}}$_d) can also be estimated from the model. We do this by multiplying the total flow rate to the SM tissue $V_{SM}^{tiss}{Q}_d^I$ by the predicted arteriovenous difference (numerator of Eq. 17) to get

$${\dot{\mathcal{S}}}_d=Q_d^I{V}_{SM}^{\mathit{\text{tiss}}}\left(1-e^{\frac{-{PS}_d^I}{Q_d^I}}\right)(I-S).$$ (18)

ef^I and $\dot{\mathcal{S}}$_d are plotted versus time in Fig. 6 for both the resting (T1DR, blue line) and the exercising (T1DE, orange line) groups. The exercise periods are shown in the shaded regions, but do not apply to the resting group.

Fig. 6.

(left) Calculated insulin extraction fraction ef^I from the resting (T1DR, blue line) and exercising (T1DE, orange line) simulations. (right) Corresponding insulin uptake $\dot{\mathcal{S}}$_d. The exercise periods are shown in the shaded regions, but do not apply to the resting group. Initial resting ef^I is 10%. Upon meal ingestion ef^I temporarily increases because of an increase in I from the injected insulin. During the exercise periods ef^I drops as a result of the increase in $Q_d^l$, which cuts the transit time by a factor of 10. Still, even with the significant drop in ef^I during exercise, the magnitude of $\dot{\mathcal{S}}$_d still increases because of increased surface area. This large increase in $\dot{\mathcal{S}}$_d is responsible for the increase in S shown in Fig. 5 (right)

Baseline ef^I is about 10%, which is of the same order but slightly lower than the 15%−20% measured in [2, 37]. Upon meal ingestion in both groups ef^I increases temporarily because of an increase in I − S. Upon the commencement of exercise at minute 120 in the T1DE group there is a precipitous drop in ef^I. This is because of a tenfold increase in the tissue perfusion rate $Q_d^I$, which cuts the transit time by a factor of 10, reducing the arteriovenous difference. However, the drop in ef^I during exercise does not completely characterize insulin delivery because it does not account for the perfusion rate. ef^I drops by roughly a factor of 5, but $Q_d^I$ increases by a factor of 10, and hence the insulin delivery rate $\dot{\mathcal{S}}$_d increases 2–3 fold, as shown in Fig. 6 (right). This tremendous increase in insulin delivery, from 3000 (μU/min) to nearly 9000 (μU/min) is responsible for the increase in S shown in Fig. 5.

Investigative case: glucose uptake and the timing of exercise

Many with T1D report that it is difficult to avoid hypoglycemia when exercising shortly after a meal [38]. The main motivation for the development of the present model is to understand how exercise-induced changes to insulin kinetics affect glucose uptake (GU). To investigate this we simulate a subject with T1D undergoing an exercise bout with varying amounts of IOB.

We make a rough estimate of insulin-dependent GU in the periphery by simulating a subject under a basal glucose clamp with a 45 min exercise bout starting at different times with respect to a 6 [U] insulin bolus administered at time 0. There are certainly other mechanisms of GU during exercise, such as insulin-independent GU [39], but quantifying these are beyond the scope of this investigation. Future work will investigate the effects of exercise on glucose uptake under more general conditions. Following Castillo and Bergman [1, 2] the rate of glucose uptake (RGU) (mg/min) in the SM is roughly proportional to insulin concentration in the SM, S, times the insulin sensitivity ${\text{r}}_{SM}^{GI}$

$$RGU={\text{r}}_{SM}^{GI}S.$$ (19)

${\text{r}}_{SM}^{GI}$ was estimated to be roughly 10 (mg/min per μU/mL) at basal glucose concentration [2]. To quantify GU in a given time period, we integrate Eq. 19 over the time frame t₁ to t₂

$$GU={\text{r}}_{SM}^{GI}{\int }_{t_1}^{t_2}Sdt.$$ (20)

We have assumed constant ${\text{r}}_{SM}^{GI}$. To calculate GU, we simulate S from – 400 to 360 min, while specifying a 45 min period of exercise at E = 0.5. Five cases are simulated, each with a different exercise time frame: from minute – 300 to 255, 30 to 75, 90 to 135, 150 to 195, and 210 to 255. Results of these five simulations are plotted in Fig. 7 (left) with solid lines overlayed. For comparison a resting case was also simulated (black dotted line). GU is proportional to the S area under the curve (AUC) for each test case. The AUC of S for the five exercising cases are shown as shaded colored regions in Fig. 7 (left). The AUC of the resting case is shown as a darker region below the dotted line. The corresponding GU, calculated with Eq. 20, is plotted in Fig. 7 (right).

Fig. 7.

Investigative Case The timing of exercise with respect to the bolus at time 0 has a significant impact on peak insulin S and glucose uptake (GU). (left) Five test cases with moderate (E = 0:5) 45 min exercise periods were simulated (colored solid lines overlayed). For comparison a resting case was simulated (black dotted line). As more time is put between the bolus and the exercise bout, the increase in S is diminished because there is less IOB (IOB is not shown). (right) Peripheral GU was estimated with Eq. 20 from the shaded AUC regions. Clearly, the sooner exercise occurs after the bolus at time 0, the more significant the rise in S and the more profound the increase in GU

The model predictions suggest that the timing of the exercise bout has a significant effect on max S. The earlier the exercise after the bolus the more IOB (IOB is not shown but decreases with time following a bolus) and thus the more significant the effect of capillary recruitment on S. This leads to an increase in GU; exercise 30 min after insulin injection suggests an increase in GU from 5.4 (g) for the period at rest, to 9 (g) for the same period with exercise, a 70% increase in GU. In comparison, for the case with only basal IOB that started at minute – 300, GU increased from 2.5 to 3.7 (g), only a 50% increase.

Discussion

Enhanced insulin absorption and delivery rates during exercise may have significant implications on the rate of glucose uptake and can potentially cause hypoglycemia. This is one reason why exercise is considered a major hurdle to closed loop control in T1D. A more complete understanding of how insulin kinetics are altered during exercise can potentially inform insulin dosing decisions prior to exercise.

The objective of this work is to develop a physiology-based model of insulin kinetics to understand why insulin has been observed to absorb from SC tissue and deliver to SM tissue more rapidly during exercise [3–7, 40]. We hypothesize that these enhanced rates of transport are due to capillary recruitment, which increases surface area available for insulin transport.

Model development

Starting with first principles, we developed a model of insulin absorption/delivery that included the effects of capillary recruitment on solute delivery, as first described by Renkin in 1966 [41]. To simulate exercise and capillary recruitment we allow the surface area and tissue perfusion rate to increase. We included this microscale transport model (Eq. 10) into a broader macroscale insulin kinetics model.

When developing the model, our goal was to keep the model as physiologically-based as possible, so that predictions could be understood in the context of real physical phenomena such as increased blood flow or increased surface area. All the parameters in the model were taken directly from the literature or extracted from experiments found in the literature. Though this approach did not use best-fit parameters to describe the current data, it afforded us the chance to evaluate whether the physiological concepts that were included in the model can describe the data in general.

Model comparison with data

To test the model we compared plasma insulin predictions I to plasma insulin measurements I_data in Fig. 5. In both the resting and exercising groups, the model reasonably agrees with the data, indicating that the most important underlying physics are being captured. However, there are two distinct model discrepancies. First, the model slightly underestimates the peak concentration of I. This is likely due to nonoptimal parameter values, as population averaged values (Table 1) were used for the simulation. Second, predictions did not exhibit the same time-lag seen in the measurements. This is because insulin hexameric dissociation was not modeled. Each of these two discrepancies are important to consider in future iterations of the model.

When exercise occurs at minute 120, both predictions and measurements rapidly increase by 30%. This is a result of the 45% increase in $k_a^I$ (see Fig. 2)—primarily due to the 40% increase in surface area. An alternative but unmodeled explanation is that I_data increases because $V_{CS}^I$ decreases during exercise due to the volume-contraction effect, estimated to be 10% [22]. However, it is doubtful that a 10% volume contraction could cause the observed 30% increase in I_data. Hence, it is our interpretation that capillary recruitment causes enhanced absorption and is the primary mechanism responsible for the observed increase in I_data.

Insulin absorption from subcutaneous depot

Transport of large molecules with low permeability such as insulin is typically limited by capillary surface area [17]. This is contrary to the commonly held notion that it is the increase in blood flow that is responsible for enhanced insulin absorption [10–12, 42]. Surely, there is a correlation between blood flow and absorption rate, however, according to our transport model (see Fig. 2), the associated surface area increase is what predominantly causes the enhanced absorption. We have estimated that the capillary surface area in SC tissue increases by 40% during exercise. This increase in $P{S}_a^I$, along with a modest increase in $Q_a^I$, causes $k_a^I$ to increase by 45% during exercise. Previous studies of the exercise-induced increase in $k_a^I$ have ranged from 49 to 100% for insulin injected into the thigh [3–6], and from 0–20% for insulin injected into the abdomen [4, 6]. In the present study insulin was infused into the abdomen with an insulin pump.

Why do we predict a 45% increase in $k_a^I$, while the literature only indicates a 0–20% increase? Most studies on insulin absorption during exercise were carried out in the 1980s [4, 6] using regular human insulin—not the rapid acting insulins that are typically used today. Modern rapid acting insulins dissociate much faster than human insulin [20] and the rate-limiting step of absorption is likely penetration of the endothelial walls. In older insulins the rate limiting step was more skewed toward hexameric dissociation, and thus older insulins may have been less sensitive to changes in surface area brought on by capillary recruitment. To test this theory, new studies should be designed to estimate the effects of exercise on the absorption of newer rapid acting insulins through injection into the abdominal region.

It is also worth mentioning that the present hypothesis that capillary recruitment increases the insulin absorption rate may explain the observed increase in insulin absorption rate when administered via new delivery devices. Insulin administered into the abdominal region by a jet injector [43] or a localized heating pad [42] has been shown to absorb significantly faster than with a typical injection. If capillary recruitment occurs because of tissue trauma from the jet or heating from the pad then the enhanced absorption rate could be explained by the present hypothesis.

Insulin delivery to skeletal muscle

Insulin delivery to skeletal muscle has been shown to be the rate-limiting factor to glucose uptake because insulin concentration in SM, S, is directly proportional to peripheral glucose uptake [1, 2, 36]. For this reason we included the SM as a domain in our model.

The model allows for $k_d^I$ to increase by 150% during exercise and as a result S is predicted to increase by 60% (Fig. 5). S was not measured in the present study so we cannot verify these model predictions. However, others have measured an increase of 10–25% during exercise, albeit under different conditions [7, 40]. To understand why S is predicted to increase dramatically, we calculate two standardized measures of insulin delivery: the insulin extraction fraction ef^I (Eq. 17) and the insulin uptake $\dot{\mathcal{S}}$_d (Eq. 18).

Baseline ef^I is estimated to be 10% (Fig. 6 (left)). This is lower than the 15–20% measured in [2, 37]. This difference is likely because of an underestimation of the $P{S}_d^I$ used in the model. During exercise ef^I drops by roughly 80% because $Q_d^I$ increases tenfold, reducing the transit time and therefore the arteriovenous difference I – S. No data exists for ef^I during exercise and thus this drop cannot be verified.

We also calculate insulin uptake by SM capillaries $\dot{\mathcal{S}}$_d and it is plotted in Fig. 6 (right). Baseline $\dot{\mathcal{S}}$_d is estimated to be 1190 (μU/mL) or 18 (fmol/min/100mL_tiss) when normalizing by $V_{SM}^{tiss}$ and converting to mol. This is remarkably close to the 15 (fmol/min/100mL_tiss) measured by Eggleston [37], and gives some confidence that the model accurately predicts insulin delivery, at least in baseline resting conditions.

During exercise $\dot{\mathcal{S}}$_d increases twofold, from 3000 to 9000 (μU/mL). Direct measurements in the literature show an increase from 0 to 200% [8, 44]. The predictions match the top end of 200%, however, because of the rapidly changing dynamics it is difficult to compare directly with measurements, which are typically taken at steady-state. Because of the difficulty of measuring $\dot{\mathcal{S}}$_d during exercise, the mechanisms governing the phenomenon remain an open question—the model merely provides a plausible explanation based on capillary recruitment.

Modeling limitations

When deriving the insulin kinetics model we have tried to balance model parsimony with accuracy. To do this we neglected certain phenomena. The most significant assumption was the decision not to include the insulin hexameric dissociation step. This was done for two reasons. First, dissociation is a more rapid process than penetration of insulin through the capillary wall [20], and thus is not the rate-limiting step in insulin absorption. Second, the physics of dissociation is a non-linear process that if included would greatly increase the complexity of the model, without adding much value to fulfilling the goal of this work.

Another important but complex phenomenon that was not included is capillary bed heterogeneity. Renkin [17] showed that heterogeneity may explain why the capillary transport model he derived (and we derived independently as Eq. 10) is inaccurate for predicting solute delivery. Capillary recruitment may not just increase surface area, but also redistribute flow to longer capillaries, enhancing transport [17]. To include heterogeneity would involve defining a distribution of capillary lengths for inclusion in the capillary bed description. This may bring the model closer to ‘truth’ but the introduction of uncertainty through unknown distribution parameters may negate any gains made by including heterogeneity.

Some have also suggested that insulin transport may be a saturable process [37]. However, given the lack of understanding and data to quantify this phenomenon, there is not a clear path to include this in a model of insulin kinetics. Last, lymphatic insulin absorption was not included because it is believed to be of secondary importance [21].

Parameter selection and uncertainty

To keep the model rooted in physiology, we chose parameters in the literature that reflected real physical properties of the system, such as tissue perfusion rate and permeability surface area. For many of these properties, little is known about parameter variability and uncertainty, and hence not enough information is available to adjust the parameters using a Bayesian approach with least-squares curve fitting. This resulted in predictions that did not ‘optimally’ follow the data. To address this, more information on parameter uncertainty and variability needs to be gathered.

The permeability surface area of insulin PS^I is difficult to measure and highly uncertain. This is because PS^I cannot be directly measured with current techniques, and thus its value is based on assuming a homogeneous capillary bed model and measured related quantities [27]. In other words, PS^I is a model-dependent parameter. New methods need to be developed to better measure PS^I, especially during exercise.

Another parameter that warrants further discussion is r^I. Typically r^I is measured at steady-state during rest and assuming that clearance occurs from a single compartment. This is known to be incorrect, as insulin has been shown to be cleared directly from both the CS and the SM [1]. To account for this, we derived a value for ${\text{r}}_{SM}^I$ that was based on the assumption that insulin clearance is linear in each compartment. Also, little is known about how clearance changes during exercise. To obtain a higher fidelity model, more clearance measurements need to be made during both rest and exercise.

Last, MBV has been used sparingly to characterize capillary recruitment in SC tissue [45–47] (MBV is more often used to characterize recruitment in SM tissue). In fact, we found no studies that examined changes in SC MBV during exercise and we had no choice but to use measurements obtained during a resting postprandial period in developing the recruitment model for the absorbing tissue $P{S}_a^I$. This is undoubtedly a very important parameter in our model, and its uncertainty is a major drawback of our approach. More experimental studies need to be done to characterize capillary recruitment in SC tissue during exercise.

Model insights

Those with T1D often report difficulty avoiding hypoglycemia when exercising shortly after a meal [38]. The investigative case shown in Fig. 7 examined how the timing of exercise following a bolus affects GU. The investigation showed that exercise affects insulin kinetics and thus GU most significantly when performed shortly after a bolus, when IOB is greatest. This increase in GU demonstrates how more rapid insulin absorption and delivery can lead to hypoglycemia. The investigation did not take insulin-independent GU mechanisms into account or the effects on endogenous glucose production during exercise. Future work will address both of these.

The model also provides an interesting insight about the natural limits of insulin absorption from subcutaneous tissue. The transport model (Eq. 10) shows that for typical values of PS and Q, insulin absorption is a surface area dependent phenomenon. If we further assume, as we did when deriving the model, that absorption is driven by pure diffusion, then the rate $P{S}_a^I$ may represent a hard upper-limit on the absorption rate from the SC. This is because even if hexameric dissociation occurs instantaneously upon injection, the permeability of the capillary walls would still provide resistance to transport. It can be argued that the relative importance of the time-lag of hexameric dissociation is small compared to the time it takes for insulin to penetrate the capillary walls. This means that there may be little room for the design of faster absorbing insulins, at least through the SC injection route. This thought is in agreement with a recent article on insulin dissociation by Gast et al. [20], where it was concluded that to expedite insulin absorption, newer formulations must focus on improving insulin transport through capillary walls, rather than just quickening the dissociation step.

The present insulin kinetics model also helps to explain how lack of capillary recruitment is a possible cause of insulin resistance. If capillary recruitment is attenuated, as it has been shown to be in those with type II diabetes [26], then the model suggests that insulin is more slowly delivered to SM. This effectively causes insulin resistance because GU in the peripheral tissue would correspondingly be reduced.

Future directions

The model gives reasonable predictions for an average patient, providing confidence that the most important physical mechanisms were modeled. As a result of the reasonable agreement, we were able to draw some conclusions about the underlying physiology of insulin absorption, which was the ultimate objective of this work. However, the model in its current form cannot be implemented directly in an artificial pancreas control system because we have not yet included patient variability. In addition, future work will need to evaluate whether neglecting hexameric dissociation has a significant effect on the current model predictions in the context of artificial pancreas dosing algorithms.

Insight about the variability of insulin absorption can be gained from characterizing the variability of tissue vascularization in patient populations. The incorporation of the variability of capillary beds into the model could potentially provide useful information about patient variability in insulin absorption.

In the future, we would also need to validate the model on various levels of exercise intensity E. In this study we only evaluated the model for E = 0.5. The model may also benefit from splitting the SM domain into ‘resting’ and ‘exercising’ muscle compartments to simulate different types of exercise. Also, an add-on to the current model that relates heart rate and other signals to exercise intensity remains a very import part of future work.

Finally, to understand how exercise-induced changes in insulin kinetics will affect glucose dynamics, it is necessary to include the present model in a broader model of glucose dynamics. This will make the relationship between exercise, insulin kinetics, and glucose dynamics much more evident and is the primary focus of future work.

Conclusion

The insulin kinetics model that we developed is, to the authors’ knowledge, the first insulin kinetics model that has been explicitly designed to include the effects of exercise. This was done by modeling the effects of exercise-induced capillary recruitment on two processes: insulin absorption from the subcutaneous injection site and insulin delivery from the blood to skeletal muscle. The model provides understanding on how these two rates change during exercise and how they are critically important because they determine insulin concentration, and thus directly affect rates of glucose uptake, potentially causing hypoglycemia. We have compared predictions from the insulin kinetics model to data from two cohorts of T1D subjects, one resting and one exercising. The results compare favorably, and also agree with direct measurements of insulin absorption and delivery rates during exercise from literature. In the future, the model can be incorporated into a broader model of glucose dynamics, which can provide a greater understanding of how to handle exercise in those with type 1 diabetes.