A Subcutaneous Insulin Pharmacokinetic Model for Computer Simulation in a Diabetes Decision Support Role: Model Structure and Parameter Identification

Abstract

Objective

The goal of this study was to develop a unified physiological subcutaneous (SC) insulin absorption model for computer simulation in a clinical diabetes decision support role. The model must model the plasma insulin appearance of a wide range of current insulins, especially monomer insulin and insulin glargine, utilizing common chemical states and transport rates, where appropriate.

Methods

A compartmental model was developed with 13 patient-specific model parameters covering six diverse insulin types [rapid-acting, regular, neutral protamine Hagedorn (NPH), lente, ultralente, and glargine insulin]. Model parameters were identified using 37 sets of mean plasma insulin time-course data from an extensive literature review via nonlinear optimization methods.

Results

All fitted parameters have a coefficient of variation <100% (median 51.3%, 95th percentile 3.6-60.6%) and can be considered a posteriori identifiable.

Conclusion

A model is presented to describe SC injected insulin appearance in plasma in a diabetes decision support role. Clinically current insulin types (monomeric insulin, regular insulin, NPH, insulin, and glargine) and older insulin types (lente and ultralente) are included in a unified framework that accounts for nonlinear concentration and dose dependency. Future work requires clinical validation using published pharmacokinetic studies.

Introduction

For more than 80 years, administration of insulin via the subcutaneous (SC) route into the peripheral circulation has been the most widely used therapy in ambulatory diabetes. Since Binder,¹ the absorption kinetics of subcutaneously injected insulin has emerged as a mature research field.^2–4 Insulin absorption kinetics are complex and can be influenced in many complex ways. Insulin-independent factors include SC blood flow (which can be influenced by temperature,^2,5 exercise,^6,7 and even smoking^8,9), depth of injection,^10,11 and site of injection.⁴ There are also insulin-dependent factors, which include the species of insulin¹² and the concentration and volume of injected insulin.^1,4,10,13 As a result, SC insulin pharmacokinetics (PK) are variable enough to offer significant practical difficulties in providing consistent therapy for diabetes. Most studies report interindividual variation in key pharmacokinetic summary measures from 15¹⁴ to 107%,⁴ depending on insulin type.

Some diabetes decision support and control methods^15–18 rely on SC absorption models to deterministically predict plasma insulin appearance from SC injection, thus tracking on-board insulin from multiple doses over extended periods. Some would argue the clinical value to general diabetes management of such models given the inherent and significant intra- and interpatient variability in absorption.^14,19–23

Nevertheless, given the often limited data available for glycemic control in diabetes,^24,25 the qualitative estimation of the plasma insulin time course is not only useful but necessary for efficient dosing and reduced risk of hypoglycemia. Deterministic models of absorption can also form the basis of stochastic methods, as hinted at by Andreassen and colleagues¹⁸ with significant potential. These models have also proven useful for glycemic control simulation^15,18,26 and diabetes education.²⁷

Previous work in insulin PK modeling is voluminous. Kobayashi and associates²⁸ experimented with single-and dual-compartment models for regular insulin (RI) absorption. As most current regular insulin is regular human insulin, the term regular insulin in this study is taken to mean regular human insulin. Later, Kraegen et al.^29,30 developed a three-compartment model, which was refined with a minimal approach by Puckett et al.,³¹ modeling the long-acting ultralente insulin as an unlikely continuous flow in addition to RI. As the use of monomeric insulin (MI) types matured clinically, more models concentrated on its absorption kinetics. In this study and elsewhere, the term “monomeric insulin” is a convenient misnomer used for rapid-acting insulin analogues whose hexamers dissociate very rapidly into dimers/monomers in subcutaneous tissue, resulting in a monoexponential decay curve.³² With a similar three-compartment model, Shimoda et al.¹⁷ modeled both RI and MI absorption. More recently, Wilinska et al.³³ demonstrated a four-compartment model of MI absorption with fast/slow absorption channels and local insulin degradation.

All these models apply to prandial insulin only, which is limiting, as basal-acting insulin is required in ∼90% of all insulin-dependent diabetics not on insulin pump therapy (calculated based on estimated insulin pump use³⁴ and prevalence of diabetes³⁵ 2003 and 2005 statistics, U.S. figures only). The absorption dependency of RI on dose, concentration, and volume is also not modeled by any of these simpler compartmental models as noted earlier.

Noncompartmental approaches include Berger and Rodbard,²⁶ whose simulation model has been adopted by the AIDA decision support system.³⁶ A three-parameter logistic equation with linear dose dependency is used to describe the SC insulin plasma rate of appearance, while a two-compartment model describes plasma insulin kinetics and action. A wide range of older insulin types [e.g., RI, neutral protamine Hagedorn (NPH), lente, and ultralente] have been modeled and remain the only model for NPH-type insulins. While this model accounts for general dose dependency of absorption, it does not account for the underlying volume or concentration effects.

The model by Mosekilde et al.¹³ describes the absorption kinetics with coupled partial differential equations of detailed physicochemical properties of insulin. While accounting for dose, volume, and concentration dependency, solving the nonlinear-coupled differential equations it employs is computationally burdensome and only RI is modeled. This model was simplified by Trajanoski etal.³⁷ and subsequently extended by Tarin etal.³⁸ to insulin glargine, a long-acting basal insulin analogue, and is currently the only model for this newer insulin. While these noncompartmental models are more physiological than the compartmental models, the associated computational cost can be prohibitive, especially if the intended or potential end use is a realtime diabetes decision support system.

In summary, the complex, noncompartmental models better capture some published insulin absorption kinetics, while the computationally minimal compartmental models better suit simulation in a real-time diabetes decision support system. The more complicated kinetics of RI have not been modeled compartmentally nor have insulin glargine or even older intermediate and long-acting insulin types such as lente and ultralente. Most existing models also describe only one insulin type or a limited number with no commonality.¹⁷ Critical reviews of some of these models are available from Nucci and Cobelli.³⁹ Compartmental modeling methods are also well reviewed in Carson and Cobelli.⁴⁰

This article developed a physiological compartment model for a wide range of insulin types. Specifically, fast- and short-acting prandial insulins (MI and RI) and intermediate- and long-acting basal insulins (NPH, lente, ultralente, and insulin glargine) are modeled. The main principle was to more accurately capture the main dynamics of the absorption kinetics with a compartmental model using first-order kinetics. The secondary goal was to provide a computationally minimal, yet consistent, physiologically unified framework for all insulin types. The model also accounts for volume and concentration dependency on SC absorption of human insulins. The intended end use is as a simulation model for a real-time decision support system.

Methods

A diagram of the structure of the SC insulin absorption kinetic model is shown in Figure 1. The model equations are then listed, followed by a description of the individual sections of the model for each insulin type. After this description, a summary of the model is presented.

Figure 1.

Full structure of the overall SC insulin absorption kinetic model. The model is characterized by a common hexameric state compartment for RI, NPH, and lente insulins (x_h), whereas those for insulin glargine and ultralente (x_h,ulen and x_h,gla) are separate. A crystalline state compartment for NPH (c_NPH), lente (c_len), and ultralente (culen) insulins and a precipitate compartment for insulin glargine (p_gla) model these insulin-specific protraction mechanisms. All insulin types flow into common dimeric-monomeric state (x_dm), interstitium (x_i), and plasma (I) compartments.

Hexameric state: common to RI, NPH, and lente insulin types

$$\begin{array}{l}{\dot{x}}_h\left(t\right)=-\left(k_1+k_d\right)x_h\left(t\right)\\ +k_{crys,NPH}{c}_{NPH}\left(t\right)+k_{crys,len}{c}_{len}\left(t\right)\\ +u_{h,RH}\left(t\right)+u_{h,NPH}\left(t\right)+u_{h,len}\left(t\right)\end{array}$$ (1)

Dimeric/monomeric state: common to all insulin types

$$\begin{array}{l}{\dot{x}}_{dm}\left(t\right)=-\left(k_2+k_d\right)x_{dm}\left(t\right)+k_1{x}_h\left(t\right)\\ +k_{1,ulen}{x}_{h,ulen}\left(t\right)+k_{1,gla}{x}_{h,gla}\left(t\right)\\ +u_{mono}\left(t\right)+u_{m,RHt}\left(t\right)+u_{m,NPH}\left(t\right)++u_{m,len}\left(t\right)+u_{m,ulen}\left(t\right)+u_{m,gla}\left(t\right)\end{array}$$ (2)

NPH and lente insulin compartments

$${\dot{c}}_{NPH}\left(t\right)=-k_{crys,NPH}{c}_{NPH}\left(t\right)+u_{c,NPH}\left(t\right)$$ (3)

$${\dot{c}}_{len}\left(t\right)=-k_{crys,len}{c}_{len}\left(t\right)+u_{c,len}\left(t\right)$$ (4)

Ultralente insulin compartments

$${\dot{x}}_{h,ulen}\left(t\right)=-\left(k_{1,ulen}+k_d\right)x_{h,ulen}\left(t\right)+k_{crys,ulen}{c}_{ulen}\left(t\right)+u_{h,ulen}\left(t\right)$$ (5)

$${\dot{c}}_{ulen}\left(t\right)=-k_{crys,ulen}{c}_{ulen}\left(t\right)+u_{c,ulen}\left(t\right)$$ (6)

Insulin glargine compartments

$${\dot{x}}_{h,gla}\left(t\right)=-\left(k_{1,gla}+k_d\right)x_{h,gla}\left(t\right)+\min \left(k_{prep,gla}{p}_{gla}\left(t\right),r_{dis,\max }\right)+u_{h,gla}\left(t\right)$$ (7)

$${\dot{p}}_{gla}\left(t\right)=-\min \left(k_{prep,gla}+p_{gla}\left(t\right),r_{dis,\max }\right)+u_{p,gla}\left(t\right)$$ (8)

Mass input fractions and components

$$u_{total,NPH}\left(t\right)=u_{c,NPH}\left(t\right)+u_{h,NPH}\left(t\right)+u_{m,NPH}\left(t\right)$$ (9)

$$u_{total,len}\left(t\right)=u_{c,len}\left(t\right)+u_{h,len}\left(t\right)+u_{m,len}\left(t\right)$$ (10)

$$u_{total,ulen}\left(t\right)=u_{c,ulen}\left(t\right)+u_{h,ulen}\left(t\right)+u_{m,ulen}\left(t\right)$$ (11)

$$u_{total,gla}\left(t\right)=u_{p,gla}\left(t\right)+u_{h,gla}\left(t\right)+u_{m,gla}\left(t\right)$$ (12)

$$u_{total,RH}\left(t\right)=u_{h,RH}\left(t\right)+u_{m,RH}\left(t\right)$$ (13)

$$u_{total,mono}\left(t\right)=u_{mono}\left(t\right)$$ (14)

Mass input fractions and components

$$u_{c,NPH}\left(t\right)={\alpha }_{NPH}{u}_{total,NPH}\left(t\right)$$ (15)

$$u_{c,len}\left(t\right)={\alpha }_{len}{u}_{total,len}\left(t\right)$$ (16)

$$u_{c,ulen}\left(t\right)={\alpha }_{ulen}{u}_{total,ulen}\left(t\right)$$ (17)

$$u_{p,gla}\left(t\right)={\alpha }_{gla}{u}_{total,gla}\left(t\right)$$ (18)

$$u_{h,NPH}\left(t\right)+u_{m,NPH}\left(t\right)=\left(1-{\alpha }_{NPH}\right)u_{total,NPH}\left(t\right)$$ (19)

$$u_{h,len}\left(t\right)+u_{m,len}\left(t\right)=\left(1-{\alpha }_{len}\right)u_{total,len}\left(t\right)$$ (20)

$$u_{h,ulen}\left(t\right)+u_{m,ulen}\left(t\right)=\left(1-{\alpha }_{ulen}\right)u_{total,ulen}\left(t\right)$$ (21)

$$u_{h,gla}\left(t\right)+u_{m,gla}\left(t\right)=\left(1-{\alpha }_{gla}\right)u_{total,gla}\left(t\right)$$ (22)

where all variables in Equations 1–22 are defined:

xh(t)	Mass in the hexameric compartment (mU)
xh,ulen(t)	Mass in the ultralente hexameric compartment (mU)
xh,gla(t)	Mass in the glargine hexameric compartment (mU)
cNPH(t)	Mass in the NPH crystalline protamine compartment (mU)
clen(t)	Mass in the lente crystalline zinc compartment (mU)
culen(t)	Mass in the ultralente crystalline zinc compartment (mU)]
pgla(t)	Mass in the glargine precipitate compartment (mU)
xdm(t)	Mass in the dimer/monomer compartment (mU)
utotal,mono(t)	MI input (mU/min)
utotal,RH(t)	RI input (mU/min)
utotal,NPH(t)	NPH insulin input (mU/min)
utotal,len(t)	Lente insulin input (mU/min)
utotal,ulen(t)	Ultralente insulin input (mU/min)
utotal,gla(t)	Insulin glargine input (mU/min)
αNPH	Proportion of utotal,NPH(t) in protamine crystalline state at injection
αlen	Proportion of utotal,len(t) in zinc crystalline state at injection
αulen	Proportion of utotal,ulen(t) in zinc crystalline state at injection
αgla	Proportion of utotal,gla(t) precipitate insulin from at injection
uc,NPH(t)	NPH crystalline state insulin input (mU/min)
uc,len(t)	Lente crystalline state insulin input (mU/min)
uc,ulen(t)	Ultralente crystalline state insulin input (mU/min)
up,gla(t)	Glargine precipitate state insulin input (mU/min)
uh,NPH(t)	NPH hexamer state insulin input (mU/min)
uh,len(t)	Lente hexamer state insulin input (mU/min)
uh,ulen(t)	Ultralente hexamer state insulin input (mU/min)
uh,gla(t)	Glargine hexamer state insulin input (mU/min)
umono(t)	MI dimer/monomer state insulin input (mU/min)
um,RH(t)	RI dimer/monomer state insulin input (mU/min)
um,NPH(t)	NPH dimer/monomer state input (mU/min)
um,len(t)	Lente dimer/monomer state insulin input (mU/min)
um,ulen(t)	Ultralente dimer/monomer state insulin input (mU/min)
um,gla(t)	Glargine dimer/monomer state insulin input (mU/min)
kcrys,NPH	NPH protamine crystalline dissolution rate (min−1)
kcrys,len	Lente zinc crystalline dissolution rate (min−1)
kcrys,ulen	Ultralente zinc crystalline dissolution rate (min−1)
kprep,gla	Glargine precipitate dissolution rate (min−1)
Vinj	Insulin dose injection volume (ml or cm3)
n	Plasma insulin rate of clearance (min−1)
rdis,max	Maximum glargine precipitate dissolution rate (mU/min)
k1	Hexamer dissociation rate (min−1)
k1,ulen	Ultralente hexamer dissociation rate (min−1)
k1,gla	Glargine hexamer dissociation rate (min−1)
k2	Dimeric/monomeric insulin transport rate into interstitium (min−1)
k3	Interstitium insulin transport rate into plasma (min−1)
kd,i	Rate of loss from interstitium (min−1)
kd	Rate of diffusive loss from hexameric and dimeric/monomeric state compartments (min−1)

Regular Insulin Submodel Structure

The RI model [Equations 1 and 2] is based on insulin physicochemical properties.¹³ For soluble human insulins, it is generally accepted that the dynamic equilibrium of the hexameric, dimeric, and bound states characterizes absorption kinetics.^32,41–43 The equilibrium is concentration dependent and is destabilized by dilution and diffusion in the SC depot.^13,44 Qualitatively, the monomeric state has the highest absorption rate into plasma³² and becomes increasingly stable toward the end of the absorption process when insulin concentration at the site decreases.

For simplicity, the dimeric and monomeric states are lumped in x_dm(t) [see Equation 2]. Both states have higher relative absorption rates into plasma than the hexameric state, although the dimer is absorbed discernibly slower than the monomer.³² There is no provision for a reversible, bound state.¹³ However, for many common concentrations of insulin preparation, insulin binding has been shown to be negligible.³⁷ Reversible binding also becomes apparent only at low doses and concentrations.¹³ This result implies that the effect of insulin binding is relatively small, especially when large prandial injections are administered, and might be ignored for decision support.

Thus, the RI input [Equation 13] is assumed to consist of hexameric, x_h(t), and dimeric/monomeric, x_dm(t), states according to the equilibrium of Equation 33,¹³ but only at the t=0 injection [see Equation 24].

$$C_h=Q_D{C}_D^3$$ (23)

$$\frac{u_h\left(t=0\right)}{V_{inj}}=Q_D{\left(\frac{u_{dm}\left(t=0\right)}{V_{inj}}\right)}^3$$ (24)

where

Ch	Concentration of hexameric insulin [(liter/mU)2]
CD	Concentration of dimeric insulin [(liter/mU)2]
QD	Hexameric-dimeric equilibrium constant [(liter/mU)2]
Vinj	Insulin dose injection volume (ml or cm3)
um(t)	Dimer/monomer state insulin input (mU/min)
uh(t)	Hexamer state insulin input (mU/min)

This equation is an acknowledged simplification of the hexameric-dimeric state dynamic equilibrium, while still accounting for dose and concentration effects. Assuming a spherical depot, volume effect is modeled by a rate of diffusive loss, k_d, from both x_h(t) and x_dm(t) compartments in the SC depot [Equations 25 and 26].

$$k_d=\frac{3D}{r^2}$$ (25)

$$r={\left(\frac{3}{4}\frac{V_{inj}}{\pi }\right)}^{\frac{1}{3}}$$ (26)

where

kd	Rate of diffusive loss from hexameric and dimeric/monomeric state compartments (min−1)
D	Diffusion constant of hexameric and dimeric/monomeric insulin (cm2/min)
r	Radius of the SC depot (cm)
Vinj	Insulin dose injection volume (ml or cm3)

Both hexameric and dimeric/monomeric states are assumed to have the same diffusion constant, a further simplification of the absorption process that was also made by Mosekilde and associates.¹³

Neutral Protamine Hagedorn, Lente, and Ultralente Insulin Submodel Structures

The NPH, lente, and ultralente insulin submodel structures are similar to the RI model with an additional crystalline state compartment [Equations 1–6]. The crystalline state accounts for the protraction mechanism. Specifically, the formation of protamine [NPH, x_c,NPH(t)] or zinc crystals [lente, x_c,len(t) and ultralente, x_c,ulen(t)] delays the dissolution process^20,45 [Equations 3, 4 and 6]. These states then flow into the hexameric state after dissolution.

Compared to RI injection [Equation 13], a large proportion of the injected dose is crystalline [u_c,NPH(t), u_c,len(t), and u_c,ulen(t)], while the rest consists of hexameric [u_h,NPH(t), u_h,len(t), and u_h,ulen(t)] and dimeric/monomeric states [u_m,NPH(t), u_m,len(t), and u_m,ulen(t)] [Equations 9–11, (15)–(17), (19)–(21)]. Both the NPH and the lente insulin models incorporate the common hexameric state compartment as RI, x_h(t) [Equation 1], while a separate, slower hexameric state is introduced for ultralente insulin, x_h,ulen(t) [Equation 5].

Insulin Glargine Submodel Structure

The insulin glargine model structure has several key differences to the NPH, lente, and ultralente models [Equations 7 and 8]. The model structure consists of a precipitate compartment, p_gla(t) [Equation 8], similar in purpose to the crystalline state compartment for NPH and zinc-based insulins. Like the formation of crystals, the formation of an amorphous microprecipitate in neutral SC tissue is the primary protraction mechanism of insulin glargine, which has an acidic isoelectric point.⁴⁶ An empirical approximation is used to model the maximum dissolution rate, r_dis,max, of the precipitate [Equations 27 and 28] into a hexameric form unique to glargine, x_h,gla(t).

$$r_{dis,\max }\left(t\right)={\displaystyle \sum _{i=1}^N{r}_{dis,{\max }_i}}\left(H\left(t-t_{i-1}\right)-H\left(t-t_i\right)\right)$$ (27)

$$r_{dis,{\max }_i}=r_{dis,{\max }_i}\left(t_i<t<t_{i+1}\right)=\{\begin{array}{ll}\hfill 15\hfill & \text{if}\hspace{0.17em}{u}_{p,gla}\left(t_i\right)<30,000\hfill \\ 15\left(\frac{u_{p,gla}\left(t_i\right)}{30,000}\right)\hfill & \text{if}\hspace{0.17em}{u}_{p,gla}\left(t_i\right)\ge 30,000\hfill \end{array}$$ (28)

where H(t − t_i) is the Heaviside function defined as H(t − t_i) = 0 when t is less than t_i, and H(t − t_i) = 1 when t is greater than or equal to t_i.

Thus, r_dis,max is a function of dose size for doses >30 units and is a constant 15 mU/min for doses <30 units. This function has been selected based on the following model identification of the model parameters in this research. The glargine insulin hexamer is also strengthened to reduce dissociation,^46–48 resulting in greater stability in this state. This behavior is modeled using a separate hexameric state compartment x_h,gla(t) with a different hexameric dissociation (k_prep,gla) rate to the ones used for the intermediate (k_crys,ulen, k_crys,NPH) and zinc-based long-acting insulin (k_crys,ulen) [see Equation 7]. Thus, insulin glargine is injected in a mixture of precipitate [u_p,gla(t)], glargine hexameric [u_h,gla(t)], and dimeric/monomeric states [u_m,gla(t)] [Equations 12, 28, and 22]. Its rate of diffusive loss from hexameric and dimeric/monomeric states, k_d, remains the same as for RI.

Monomeric Insulin Submodel Structure

The MI model structure follows from the RI model structure in that the MI dose is assumed to be injected [u_mono(t)] entirely into the dimeric/monomeric [x_dm(t)] compartment as 100% dimers/monomers [Equation 14] and is immediately available for absorption into the interstitium after injection [Equation 2]. Thus, the absorption kinetics of MI injection is concentration independent, unlike RI,^49,50 and is effectively a three-pool model identical to the model of Shimoda and colleagues1⁷ [Equations 2, 29, and 30].

Interstitium and Plasma Insulin Model Structure

From the dimeric/monomeric state, insulin diffuses into interstitium, x_i(t) [Equation 29], and subsequently into plasma, I(t) [Equation 30]. Plasma insulin is represented by the widely accepted one-pool model^{17,28,30,31,51} in Equation 30.

$${\dot{x}}_i\left(t\right)=-\left(k_3+k_{d,i}\right)x_i\left(t\right)+k_2{x}_{dm}\left(t\right)$$ (29)

$$\dot{I}\left(t\right)=-nI\left(t\right)+k_3\left(\frac{x_i\left(t\right)}{V_i{m}_b}\right)$$ (30)

where

xi(t)	Mass in the interstitium compartment (mU)
I(t)	Plasma insulin concentration (mU/liter)
Vi	Insulin plasma distribution volume (liter/kg)
mb	Body mass (kg)
kd,i	Rate of loss from interstitium (min-1)

Summary of Model Structure

The total model structure consists of 10 compartments with 16 parameters for six SC injected insulin types. Each insulin submodel involves no more than 2 to 3 exclusive compartments and parameters, and each individual submodel is computationally modest as a result. The model structure is integrated in that all insulin types eventually emerge in the common, physiologically expected dimeric/monomeric state prior to transport into the interstitium and plasma [Equations 2, 29, and 30]. Except for the specific insulin glargine and ultralente insulin cases, the RI, NPH, and lente insulin types similarly share one hexameric state compartment with the same transport rates [Equation 1].

Increased NPH and lente duration of action is described by the formation of the crystalline state only [Equations 3 and 4]. Increased action duration of ultralente insulin and insulin glargine are modeled both by formation of the crystalline/precipitate state and by a more stable, slower dissociating hexamer [Equations 5–8]. The use of crystalline and precipitate compartments is physiological, but highly simplified compared to nonlinear noncompartmental approaches.^13,37,38

First-order transport rates are used except where in vivo knowledge is unpublished or unknown. In this case, empirical assumptions were made as in the case of the dose response of insulin glargine [Equations 27 and 28]. In this particular case, the empirical approximation is based on observations during model identification, as presented in this study.

While the absorption processes, i.e., dissolution, dissociation, and diffusion, are two way, only the net flow to plasma is modeled. This approach is another acknowledged simplification from other more complex models. It is similar to the recent compartment model of Clausen and colleagues⁵² for a biphasic protamine-retarded MI-MI preparation. This particular model also has a similar crystalline state compartment and models only net flows, but with no hexameric state due to the use of MI.

Model Parameter Identification

Certain model parameters are fixed to values in the literature and used as patient-independent population values (Table 1). All remaining parameters for all insulin types are patient specific and identified with nonlinear least squares (NLS) and unconstrained nonlinear optimisation methods. Data utilized are taken from 37 sets of plasma insulin time-course absorption curves (Table 2).

Table 1.

Fixed Mode Parameters to Literature

Parameter	Value	Reference
Vi	0.1421 (liter/kg)	15
n	0.16 (min-1)	51
kd,i	0.0029 (min-1)	17
D	0.9e-4 (cm2/min)	13
Qd	1.5e-12 [(liter/mU)2]	13

Table 2.

Published PK Studies Used for Model Parameter Identification

Insulin type	Dose	Concentration (U/ml)	Cohort studieda	Reference
MI	7.1 ± 1.3 U	100	T1 DM	57b,c,d
MI	0.12 U/kg	4	IDDM	17b,c,d
MI	10 U	100	T1 DM	58b,d
MI	0.05 U/kg	100	Normals	59d
RI (porcine)	0.15 U/kg (6–9 U)	40	IDDM/NIDDM	28b,c
RI	0.1 U/kg	—	Normals	60
RI (porcine)	10 U (rapid SC delivery)	3.3	Normals	29b
RI (porcine)	10 U (rapid SC delivery)	40	Normals	29b
RI	0.12 U/kg	4	IDDM	17b,c
RI	12 U	100	IDDM	61b
RI	10 U	—	Normals	2
RI	15 U	40	Normals	62b
RI	15 U	100	Normals	62b
RI	6 U	99	Normals	63
RI (porcine)	0.25 U/kg	—	Normals	4c,d (NPH study)
RI (porcine)	0.25 U/kg	—	Normals	4c,d (Lente study)
NPH	0.15 U/kg	100	Normals	64b
NPH	0.3 U/kg	100	T1 DM	23b
NPH	0.4 U/kg	—	Normals	21b (clamp 1)
NPH	0.4 U/kg	—	Normals	21b (clamp 2)
NPH	15 U	40	Normals	62b
NPH	15 U	100	Normals	62b
NPH	0.4 U/kg	40	Normals	65
NPH	14 U	95	Normals	63
NPH	0.25 U/kg	—	Normals	4c,d
NPH	0.4 U/kg	86.4	Normals	66c
Lente	0.25 U/kg	—	Normals	4c,d
Ultralente	0.4 U/kg	—	Normals	21b (clamp 1)
Ultralente	0.4 U/kg	—	Normals	21b (clamp 2)
Ultralente	0.3 U/kg	40	T1 DM	23b
Ultralente	0.3 U/kg	100.5	Normals	67b
Glargine	0.3 U/kg	100	T1 DM	23b,d
Glargine	0.15 U/kg	100	Normals	64b,d (15 μg/ml zinc)
Glargine	0.15 U/kg	100	Normals	64b,d (80 μg/ml zinc)
Glargine	0.4 U/kg	86.4	Normals	66c
Glargine	0.4 U/kg	—	Normals	21c (clamp 1)
Glargine	0.4 U/kg	—	Normals	21b (clamp 2)

^aT1 DM, type 1 diabetes mellitus; IDDM, insulin-dependent diabetes mellitus; NIDDM, noninsulin-dependent diabetes mellitus.

^bCorrected for endogenous glucose production, suppressed by protocol, or justified negligible by baseline C-peptide measurements (usually for NIDDM and normal cohorts only).

^cCorrected for insulin antibodies or justified negligible by measurement (usually for IDDM cohorts only).

^dCorrected for cross-reactivity of study insulin with insulin assay (usually insulin analogue or animal insulin studies only).

Data were collected via a literature review of relevant insulin PK studies searched in the MEDLINE and Science Citation Index Expanded (SCI-EXPANDED) databases. Only studies using direct measurement methods were considered⁵³ with data generally in the form of mean plasma insulin time-course measurements. These studies differ widely in cohort studied, methods, and protocol. However, data suffice for this study where the goal was to develop a mean PK simulation model for a diabetes decision support system. Rather than limiting the parameter identification to a specific cohort, experimental method, and/or protocol, a parameter fit across a broad range of studies is felt to be likelier to result in an averaged PK response suitable for clinical use over the similarly wide population encountered in the general diabetes control problem.

Several major factors may affect data used for parameter identification. First, insulin antibodies may not be quantified and/or cannot be presumed negligible. This problem is an issue more with IDDM study cohorts with an extended history of diabetes that have been or are being treated using older, highly immunogenic insulin types, e.g., porcine insulin. If the cohort is not naive to the specific insulin used, insulin antibodies can significantly affect the plasma insulin concentration measurement.

Insulin antibodies generally affect plasma insulin appearance proportionately, leading to inaccurate insulin distribution volume assumptions.²⁸ However, in a model-based diabetes decision support system where effective insulin sensitivity is optimized in real time,^51,54 a tradeoff between effective insulin sensitivity and insulin distribution volume can occur.^55,56 In this case, the shape of the plasma insulin curve is more critical within reasonable bounds than its exact magnitude.⁵⁵

Second, endogenous insulin production may not be corrected or be presumed negligible. This determination is typically performed using C-peptide measurements for noninsulin-dependent diabetes mellitus or normal study cohorts. In most cases, however, administered doses are sufficient to suppress endogenous insulin production, and its effect may largely be disregarded when using a large number of studies for an average identification.

Finally, in the case of insulin analogues, especially insulin glargine, or animal-based insulins, insufficient cross-reactivity with a nonspecific insulin assay may result in underestimated plasma insulin concentrations. With respect to these concerns, all studies are specifically identified in Table 2 where insulin antibodies and endogenous insulin production are accounted for. Similarly, if cross-reactivity with the insulin assay of the study insulin is sufficient. If concentration of the insulin preparation is not quoted, a typical concentration of 100 U/ml is assumed. If the insulin dose is quoted in units per kilogram, a body weight of 80 kg is assumed if no other information is provided. All of these assumptions are indicated where made and illustrate the lack of full data reporting that can often occur.

The model is built essentially by extensions from the MI model structure (Figure 1), which is identical to the one by Shimoda et al.¹⁷ and similar to the ones by Kraegen and Chisholm²⁹ and Puckett and Lightfoot.³¹ Equivalent parameter values from Shimoda and colleagues¹⁷ were thus able to be used as starting points for NLS optimization of k₂ and k₃ to MI studies data (Table 3). With fixed population values of k₂ and k₃ identified from MI data, a two-stage NLS optimization is used to optimize k₁, k₂, and k₃ using data from RI studies (Table 4). Note that the overall fitted k₂ and k₃ values for RI data (Table 4) are very close to the fitted k₂ and k₃ to MI data (Table 3) and that all fitted parameters display a low coefficient of variation (CV) as summarized in Table 9. Hence, the k₂ and k₃ parameters are consistent in describing these common physiological states during the SC insulin absorption process even among different insulin types.

Table 3.

Fitted k₂ and k₃ to Published MI PK Data

Parameter	Reference				Median	Mean (SD)	CV (%)
	57	17	58	59
k2 (min-1)	0.0106	0.0085	0.0119	0.0106	0.0106	0.0104 (0.0014)	14
k3 (min-1)	0.0473	0.0752	0.0355	0.0876	0.0613	0.0614 (0.0241)	39

Table 4.

Fitted k₁, k₂, and k₃ to Published RI PK Data

Parameter	Reference												Median	Mean (SD)	CV (%)
	28	60	29		17	61	2	62		63	4
			3.3 U/ml	40 U/ml				40 U/ml	100 U/ml		NPH study	Lente study
k1 (min−1)	0.0565	0.0217	0.0348	0.0264	0.0275	0.0266	0.0365	0.0229	0.0148	0.0870	0.0235	0.0233	0.0250	0.0331 (0.0200)	60
k2 (min−1)	0.0089	0.0148	0.0201	0.0205	0.0027	0.0088	0.0057	0.0086	0.0083	0.0067	0.0113	0.0104	0.0089	0.0106 (0.0054)	51
k3 (min−1)	0.0681	0.0619	0.0581	0.0621	0.0631	0.0615	0.0744	0.0613	0.0612	0.0839	0.0616	0.0614	0.0618	0.0649 (0.0073)	11

Table 9.

Summary of Parameter Identification to Published PK Data

Insulin type	Parameter (units)	Median	Mean (SD)	CV (%)
MI	k2 (min−1)	0.0106	0.0104 (0.0014)	14
MI	k3 (min−1)	0.0613	0.0614 (0.0241)	39
RI	k1 (min−1)	0.0250	0.0331 (0.0200)	60
RI	k2 (min−1)	0.0089	0.0106 (0.0054)	51
RI	k3 (min−1)	0.0618	0.0649 (0.0073)	11
NPH	kcrys,NPH (min−1)	0.0014	0.0016 (0.0011)	70
NPH	αNPH (unitless)	0.9432	0.9475 (0.0336)	4
Lente	kcrys,len (min−1)	0.0037	—	—
Lente	αlen (unitless)	0.9447	—	—
Ultralente	kcrys,ulen (min−1)	0.0013	0.0021 (0.0018)	84
Ultralente	k1,ulen (min−1)	0.0018	0.0023 (0.0014)	61
Ultralente	αlen (unitless)	1.0000	0.9724 (0.0552)	6
Glargine	kprep,gla (min−1)	0.0008	0.0011 (0.0006)	53
Glargine	k1,gla (min−1)	0.0084	0.0078 (0.0054)	69
Glargine	αgla (unitless)	0.9462	0.9192 (0.0738)	8
			Median (95th percentile)	51.3 (3.6–60.6)
			Range	3.6–80.4

Referring to Figure 1, two parameters each must be identified for the NPH (k_crys,NPH and α_len) and lente insulin (k_crys,len and α_NPH) data set, where fixed population k₁, k₂, and k₃ values from RI and MI data parameter identification are used. Likewise, three parameters must each be fitted for the ultralente (k_crys,ulen, k_1,ulen, and α_ulen) and insulin glargine (k_prep,gla, k_1,gla, and α_gla) data sets using the same fixed k₁, k₂, and k₃ population values.

Because of the larger number of data sets, parameters for the NPH, lente, ultralente, and insulin glargine submodels are fitted via unconstrained nonlinear optimization using a simplex search method for multiple variables, which is a quicker method. The objective function is the plasma insulin concentration sum squared error (SSE), defined in Equation 31 for the jth data set. As only mean insulin time-course data are used in this study, the variability of the mean value is neglected. Depending on the number of measurements in the calculation of the mean, the sum of the percentage measurement error would result in a wide error band around each mean value, which is of little value for model fit validation.

$$SS{E}_j={\displaystyle \sum _{i=1}^{N_j}{\left({\overline{I}}_{j,i}-I_j\left(t_{j,i}\right)\right)}^2}$$ (31)

where N_j is the number of plasma insulin data points in the jth data set, Ī_j,i is the ith plasma insulin concentration data point in the jth data set, and I_j(t_j,i) is the modeled plasma insulin concentration for the jth data set at t_j,i, the time at the ith plasma insulin concentration data point. There are 37 data sets in total for all insulin types (4 MI, 12 RI, 10 NPH insulin, 1 lente insulin, 4 ultralente insulin, and 6 insulin glargine).

Results for the NPH, lente, ultralente, and insulin glargine parameter identifications are shown in Tables 5–8. Note that even with fixed population values for k₁, k₂, and k₃, the CV of all fitted NPH, lente, ultralente, and insulin glargine model parameters was <100%. Specifically, a median CV of 57% and a 95th percentile of 3.6–69.1% were achieved (results not shown). Referring to Table 9, the median CV was 51.3% (95th percentile of 3.6-60.6%) across all parameters and insulin submodels. This precision of fitted parameters is adequate given the data, and the parameters can be considered a posteriori identifiable following the definition of Wilinska and colleagues.³³

Table 5.

Fitted k_crys,NPH and α_NPH to Published NPH PK Data

Parameter	Reference										Median	Mean (SD)	CV (%)
	64	23	21		62		65	63	4	66
			Clamp 1	Clamp 2	40 U/ml	100 U/ml
kcrys,NPH (min−1)	0.0018	0.0029	0.0017	0.0013	0.0015	0.0010	0.0038	0.0002	0.0004	0.0010	0.0014	0.0016 (0.0011)	70
αNPH (unitless)	0.9945	0.9471	0.9393	0.9501	0.9061	0.9018	1.0000	0.9386	0.9234	0.9737	0.9432	0.9475 (0.0336)	4

Table 8.

Fitted k_prep,gla to Published Glargine PK Data

Parameter	Reference						Median	Mean (SD)	CV (%)
	23	64		66	21
		80 μg/ml zinc	15 μg/ml zinc		Clamp 1	Clamp 2
kprep,gla (min−1)	0.0007	0.0019	0.0016	0.0005	0.0008	0.0008	0.0008	0.0011 (0.0006)	53
k1,gla (min−1)	0.0143	0.0018	0.0018	0.0062	0.0105	0.0124	0.0084	0.0078 (0.0054)	69
αgla (unitless)	0.9578	0.7694	0.9570	0.9426	0.9388	0.9498	0.9462	0.9192 (0.0738)	8

Table 6.

Fitted k_crys,len and α_len to Published Lente PK Data

Parameter	Reference
	4
kcrys,len (min−1)	0.0037
αlen (unitless)	0.9447

Table 7.

Fitted k_crys,ulen to Published Ultralente PK Data

Parameter	Reference				Median	Mean (SD)	CV (%)
	21		23	67
	Clamp 1	Clamp 2
kcrys,ulen (min−1)	0.0048	0.0012	0.0013	0.0012	0.0013	0.0021 (0.0018)	84
k1,ulen (min−1)	0.0015	0.0014	0.0044	0.0020	0.0018	0.0023 (0.0014)	61
αulen (unitless)	1.0000	1.0000	0.8897	1.0000	1.0000	0.9724 (0.0552)	6

Model Fit and Prediction Errors

In Table 10, model fit error (both absolute and absolute percentage errors) and model prediction errors using population parameters (both absolute and absolute percentage errors) are shown. Across all insulin types, median absolute model fit errors range from 0.62 to 3.20 mU/liter and median absolute percentage model fit errors range from 10.96% for glargine to 21.42% for lente, of which there was only one data set. This figure is hence unaffected by the averaging effect of multiple studies and may not be accurate. To calculate the model prediction errors, the plasma insulin concentration is generated using the model and the population parameters identified. As expected, the more stable insulin analogues, e.g., MI and glargine, have much lower model prediction errors (13.30 and 10.96%, respectively), whereas the more variable insulins, e.g., RI (18.35%) and NPH (16.25%), have considerably higher prediction errors. The more stable insulin analogues are hence better predicted using fixed population parameter values than less stable insulin types.

Table 10.

Model Fit and Model Prediction Errors

Insulin type	Error type (units)	Model fit [median (90% range)]	Model prediction using population parameters [median (90% range)]
MI	Absolute (mU/liter)	1.54 (0.84–3.25)	1.85 (0.78–3.69)
	Absolute percentage (%)	13.30 (5.18–32.45)	15.56 (5.69–41.00)
RI	Absolute (mU/liter)	2.65 (0.65–7.96)	6.49 (1.48–13.13)
	Absolute percentage (%)	18.35 (5.94–43.37)	38.95 (15.38–72.77)
NPH	Absolute (mU/liter)	1.10 (0.33–2.42)	1.57 (0.57–4.79)
	Absolute percentage (%)	16.25 (4.90–36.05)	22.70 (9.51–46.17)
Lente	Absolute (mU/liter)	3.20 (1.68–6.11)	3.20 (1.68–6.11)
	Absolute percentage (%)	21.42 (13.10–58.18)	21.42 (13.10–58.18)
Ultralente	Absolute (mU/liter)	0.80 (0.36–2.19)	1.81 (0.42–4.06)
	Absolute percentage (%)	14.32 (4.75–27.99)	31.05 (7.43–49.00)
Glargine	Absolute (mU/liter)	0.62 (0.26–0.93)	0.84 (0.38–1.31)
	Absolute percentage (%)	10.96 (5.61–19.85)	14.88 (8.43–28.68)

Conclusions

A simple, physiological compartmental model has been developed for computer simulation of SC injected insulin PKs for a diabetes decision support system. Most clinically current insulin types, including MI, RI, NPH, and insulin glargine, are modeled. The model accounts for concentration dependency of SC RI injection and models the dose dependency of insulin glargine absorption. In total, 13 patient-specific model parameters were fitted to 37 sets of plasma insulin mean time-course data over all insulin types from reported clinical studies. The remaining model parameters were assumed patient-independent constants and a priori identified from the literature. All fitted parameters have a coefficient of variation <100% (median 57%, 95th percentile 3.6-60.6%) and can be considered a posteriori identifiable. Hence, a model has been created based on known SC absorption kinetics and identified on a broad range of clinically reported studies. The precision in identified parameters is acceptable and the main clinically current insulin types have been modeled. Future validation of the model fit using clinical data is required.

Abbreviations

CV

coefficient of variation

MI

monomeric insulin

NLS

nonlinear least squares

NPH

neutral protamine Hagedorn

PK

pharmacokinetic

RI

regular insulin

SC

subcutaneous

SSE

sum squared error

Funding

The authors thank the Tertiary Education Commission Te Amorangi Matauranga Matua Bright Futures Top Achiever Doctoral Scholarship for financial support.