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Abstract

Objective:

Accurate, safe glycemic management requires reliable delivery of insulin doses. Insulin can be delivered subcutaneously for action over a longer period of time. Needle-free jet injectors provide subcutaneous (SC) delivery without requiring needle use, but the volume of insulin absorbed varies due to losses associated with the delivery method. This study employs model-based methods to determine the expected proportion of active insulin present from a needle-free SC dose.

Methods:

Insulin, C-peptide, and glucose assay data from a frequently sampled insulin-modified oral glucose tolerance test trial with 2U SC insulin delivery, paired with a well-validated metabolic model, predict metabolic outcomes for N = 7 healthy adults. Subject-specific nonlinear hepatic clearance profiles are modeled over time using third-order basis splines with knots located at assay times. Hepatic clearance profiles are constrained within a physiological rate of change, and relative to plasma glucose profiles. Insulin loss proportions yielding optimal insulin predictions are then identified, quantifying delivery losses.

Results:

Optimal parameter identification suggests losses of up to 22% of the nominal 2U SC dose. The degree of loss varies between subjects and between trials on the same subject. Insulin fit accuracy improves where loss greater than 5% is identified, relative to where delivery loss is not modeled.

Conclusions:

Modeling shows needle-free SC jet injection of a nominal dose of insulin does not necessarily provide metabolic action equivalent to total dose, and this availability significantly varies between trials. By quantifying and accounting for variability of jet injection insulin doses, better glycemic management outcomes using SC jet injection may be achieved.

Keywords

jet injection insulin kinetics hepatic clearance subcutaneous insulin identification OGTT insulin modified identification error

Introduction

Insulin jet injectors are an alternative insulin delivery method for individuals requiring glycemic management, such as people with diabetes. A jet injector uses a high-pressure stream of liquid to penetrate the skin and deliver insulin subcutaneously, as opposed to standard syringe delivery. This delivery method aimed to reduce delivery pain associated with reduced rates of patient self-injection, and provide an acceptable alternative for individuals with a phobia of needles. In addition, this method achieves total insulin action equivalent to needle injection with significantly faster onset and reduction in glucose levels.¹

Jet injectors are most discussed in literature from the late 1980s to the 1990s,^2-6 but have not become a common method for insulin administration.⁷ Where usage of jet injectors has been examined in literature, a number of issues have been reported by users, such as prolonged pain at injection site relative to syringe delivery, bleeding or bruising at injection site, and incomplete delivery due to leaking of injected liquid.^3,8,9 In addition, jet injectors require a higher degree of training and more cost than alternatives.⁵ As such, jet injectors have not been regularly recommended for regular insulin administration.¹⁰ In addition, uptake of modern insulin pump therapy may have supplanted jet injection as the preferred alternative to needle injection.¹¹

Variance in delivery profile is a notable issue for achieving accurate and precise glycemic control (GC). Studies where incomplete jet injection delivery is quantified provide grouped statistics,^4,6,9,12 but no methodology to reliably assess proportion of insulin absorbed on an individual basis. Furthermore, studies of jet injectors do not take account for potential inter-subject losses relative to needle injection.^13,14 Jet injection still presents potential advantages for glycemic management, including higher insulin absorption and faster reduction in glucose levels,^2,4,6 so long as potential delivery loss is accounted for.

Validated mathematical models describe dynamics of a subject’s insulin delivery and resultant glycemic state.^15-18 Where dynamics can vary, identification of certain model parameters allows a model to explain variation between cohorts, subjects, or individual trials as a difference in parameter values.^19,20 To assess the degree of subcutaneous (SC) jet injection loss in a trial, a model can be expanded with a parameter representing loss of delivered insulin. Identification of this loss parameter using measured assay data allows assessment of subject-specific or trial-specific loss, and more completely explains subsequent insulin kinetics. This process will help quantify a range of expected losses from SC jet injection delivery, and inform future jet injection development and treatment.

Method

Subjects and Data

A SC insulin-modified oral glucose tolerance test trial was performed at the University of Canterbury. N = 9 tests from seven subjects were analyzed. Table 1 provides the subject demographics for this trial, where suffixes “a” and “b” denote different tests performed on the same subject over two weeks apart. Subjects were fasted for 12 hours prior to the start of the trial. For each trial, one plasma sample was collected at basal state, followed by eight post-stimulatory samples over a 120-minute trial period. The sampling and delivery protocol is shown in Figure 1. At t = 0 min, subjects consumed 35 g of glucose dissolved in 200 ml of water. At t = 15 minutes, a nominal 2.0 units of fast-acting insulin in 0.4 ml was delivered using a jet injector device developed by the Auckland Bioengineering Institute’s Bioinstrumentation Laboratory (University of Auckland, Auckland) into the subject’s thigh. Collected samples were assayed for plasma insulin and C-peptide concentrations using the Roche Cobas e411 analyzer (Roche Diagnostics, Basel). C-peptide measurements estimate pancreatic insulin secretion rates during trial.²¹

Table 1.

Subject Demographics for the OGTT Trial.

Subject identifier	Age (years)	Sex
1a	56	Male
1b	56	Male
2a	26	Female
2b	26	Female
3	30	Male
4	24	Female
5	21	Male
6	22	Male
7	26	Male

Suffixes “a” and “b” denote different tests performed on the same subject over two weeks apart.

Abbreviation: OGTT, oral glucose tolerance test.

Figure 1.

Trial protocol and timing.

Model

Subcutaneous delivery of insulin is modeled by a two-compartment model showing transfer of insulin from the SC injection site $I_{S C}$ to the local interstitium $Q_{L o c a l}$ .^22,23 The parameters of this model are defined in Table 2. The SC delivery model is expanded with the additional parameter $L_{e x}$ to account for any loss of exogenous insulin during jet injection, effectively equivalent to the absolute bioavailability of jet-injected insulin. $L_{e x} = 0$ indicates none of the nominal dose entered plasma, and $L_{e x} = 1$ indicates the entire nominal dose entered plasma:

{\dot{I}}_{S C} = L_{e x} I_{b o l u s} - k_{s 2} I_{S C}

(1)

{\dot{Q}}_{L o c a l} = k_{s 2} I_{S C} - (k_{s 3} + k_{d i}) Q_{L o c a l}

(2)

Table 2.

Parameter Descriptions and Values for the Subcutaneous Delivery Model.

Parameter		Value	Unit	Notes
$L_{e x}$	Proportion of insulin bolus delivered after losses, absolute bioavailability	(Identified)	—	Trial-specific
$I_{b o l u s}$	Nominal amount of insulin injected	2.0	U	Trial-specific
$k_{s 2}$	Transfer rate from subcutaneous injection site to local interstitium	0.0104	min⁻¹	Wong et al24
$k_{s 3}$	Transfer rate from local interstitium to plasma	0.06	min⁻¹	Wong et al24
$k_{d i}$	Rate of interstitial degradation	0.006	min⁻¹	Wong et al24

Glucose-insulin kinetics in plasma are modeled over time with a clinically validated three-compartment GC model,¹⁵ whose parameters are defined in Table 3. A diagram of this model and the SC model is provided in Figure 2. The compartments of interest, plasma insulin $I (mU / L)$ and interstitial insulin $Q (mU / L)$ , are defined as follows:

\dot{I} = \frac{U_{e x}}{V_{I}} + (1 - x_{L}) \frac{U_{e n}}{V_{I}} - n_{K} I - n_{L} \frac{I}{1 + α_{I} I} - \frac{n_{I}}{V_{I}} (I - Q)

(3)

where for SC delivery $U_{e x} = k_{s 3} Q_{L o c a l}$ , and $Q$ is defined as follows:

\dot{Q} = \frac{n_{I}}{V_{Q}} (I - Q) - n_{C} Q

(4)

Table 3.

Parameter Descriptions and Values for the Glycemic Control Model.

Term		Value	Unit	Notes
$U_{e x}$	Exogenous (injected) insulin	$k_{s 3} Q_{L o c a l}$	mU·L⁻¹	For subcutaneous delivery, from Equation (2)
$V_{I}$	Plasma insulin volume of distribution	4.0	L	Lin et al15
$x_{L}$	First-pass hepatic insulin clearance (fractional)	(Identified)	—	See the “Parameter identification” section
$U_{e n}$	Endogenous (pancreatic) insulin secretion	(Calculated from C-peptides, time-varying)	mU·min⁻¹	Van Cauter et al²¹
$n_{K}$	Renal insulin clearance	(Calculated)	min⁻¹	Van Cauter et al,²¹ Lotz et al²⁵
$n_{L}$	Hepatic insulin clearance	(Identified, time varying)	min⁻¹	See the “n_L spline approach” section
$α_{I}$	Saturation parameter of hepatic liver clearance rate	$0.0017$	L·mU⁻¹	Lin et al,¹⁵ Lin et al²⁶
$n_{I}$	Trans-endothelial insulin diffusion	(Calculated)	L·min⁻¹	Van Cauter et al,²¹ Lotz et al²⁵
$V_{Q}$	Interstitial insulin volume of distribution	$1.2 \times V_{I}$	L	Van Cauter et al,²¹ Krebs et al²⁷
$n_{C}$	Peripheral degradation of insulin	$0.5 \times n_{I} / V_{Q}$	min⁻¹	Krebs et al,²⁷ Lotz et al²⁸

Figure 2.

Diagram of SC and GC model compartments. The introduced delivery loss factor, $L_{e x}$ , is highlighted. Abbreviations: GC, glycemic control; SC, subcutaneous delivery.

Parameter Identification

Introducing additional identified parameters to a model increases model complexity and contributes to theoretical and practical identification issues.^19,29,30 Simultaneous identification can also involve trade-off between parameter values, introducing further error.^19,31 Thus, consideration must be given to the order and method by which parameters are identified. As such, when introducing a new parameter which interacts with others, some parameters must be fixed to a population or cohort constant value, or chosen parametrically per subject without mathematical identification. This approach limits possible parameter trade-off and identification issues, and can yield more accurate identification. Analysis of parameter trade-off between $n_{L}$ and $x_{L}$ 31 shows optimal values of $x_{L}$ vary less than $n_{L}$ . Thus, to support the addition and identification of $L_{e x}$ , $x_{L}$ was fixed to $x_{L} = 0.6$ , the mean optimum across all subjects. A subsequent sensitivity analysis will quantify the robustness of this choice.

$n_{L}$ spline approach

A subject’s rate of hepatic insulin clearance $n_{L}$ varies over time inversely to plasma glucose concentration.^32-34 To model this time-varying action, $n_{L} (t)$ is modeled as a weighted sum of M basis spline (or B-spline) functions, $Φ_{i} (t)$ . The resulting $n_{L}$ profile is a sum of spline functions, each multiplied by an identified weighting, ${\hat{n}}_{i}$ . This approach captures patient-specific variability in relation to glucose concentration without requiring a predefined population-based shape function which may not fit all cases.

n_{L} (t) = \sum_{i = 1}^{M} {\hat{n}}_{i} \cdot Φ_{i, k} (t)

(5)

A single $k th$ order B-spline function $Φ_{i, k} (t)$ intersects $k$ adjacent splines on one side at defined times $T_{i}$ (“knots”). Thus, M kth order splines require M+k+1 knot locations. At all points of time, the sum of all basis splines is 1. B-splines are defined over time for zeroth order $(k = 0)$ , and recursively for higher orders $(k > 0)$ by interpolating lower order splines:

Φ_{i, 0} (t) = {\begin{matrix} 1 \\ 0 \end{matrix} \begin{matrix} T_{i} < t < T_{i + 1} \\ otherwise \end{matrix}

(6)

\begin{array}{l} Φ_{i, k} (t) = \frac{t - T_{i}}{T_{i + k} - T_{i}} Φ_{i, k - 1} (t) + \frac{T_{i + k + 1} - t}{T_{i + k + 1} - T_{i + 1}} \\ Φ_{i + 1, k - 1} (t) for k > 0 \end{array}

(7)

To ensure clearance profiles reflect measured dynamics and do not overfit the data, a subject with N insulin measurements uses knots located at these measurement times to model $n_{L} (t)$ .³⁵ Splines of order $k = 3$ , an odd number, are selected to place the peak of each spline function near a single knot, rather than between two. Because $k th$ order splines interpolate lower-order splines $k$ knots away, $k$ additional knots are placed at each end to ensure basis splines sum to 1 over the entire measurement period before weighting. Therefore, $(N + 2 k)$ total knots are required to ensure N splines comprise $n_{L} (t)$ over the measurement period.

The spline definition in Equation (5) is applied to the compartment equation in Equation (3), where $U_{e x} = k_{s 3} Q_{L o c a l}$ for SC injection. For $k$ = 3 (third-order splines) and N = 9 measurement times, the equation yields 12 splines per $n_{L} (t)$ profile:

\dot{I} = \frac{k_{s 3} Q_{L o c a l}}{V_{I}} + (1 - x_{L}) \frac{U_{e n}}{V_{I}} - n_{K} I - \underset{n_{L} (t)}{\underset{︸}{\sum_{i = 1}^{12} {\hat{n}}_{i} \cdot Φ_{i} (t)}} \cdot \frac{I}{1 + α_{I} I} - \frac{n_{I}}{V_{I}} (I - Q)

(8)

For this definition, $L_{e x}$ is assumed to be 1.0. This allows the identification process to fit overall clearance dynamics over all time, under the standard assumption the entire dose is delivered. Subsequent identification of $L_{e x}$ (See the section “Identification of $L_{e x}$ ”) determines where this assumption is invalid and loss is present in the trial due to error in insulin profile post-bolus delivery.

Spline weightings ${\hat{n}}_{i}$ are identified by the following process³⁶:

Interpolate N insulin assay measurements to get a minute-wise $I$ profile.

Analytically solve Equation (4) to get a minute-wise estimate of $Q$ .

Substitute $I$ and $Q$ into Equation (8), set $L_{e x} = 1.0$ , and calculate resultant $k_{s 3} Q_{L o c a l}$ term via Equations (1) and (2).

Numerically integrate Equation (8) between pairs of measurement times $T_{i}$ and $T_{i + 1}$ , yielding $N - 1$ integral equations.

Using linear least squares regression solve for optimal weightings ${\hat{n}}_{i}$ which minimize total error all integral equations. This regression was performed using the lsqlin function in MATLAB (MathWorks, Natick, Massachusetts).

To ensure the identified $n_{L} (t)$ profile is physiologically realistic,^25,37-39 the regression problem is constrained. Physiological analyses have shown $n_{L} (t)$ exhibits an inverse relationship to plasma glucose dose and concentration.33,34 Thus, the solver constrains the direction of change in ${\hat{n}}_{i}$ between adjacent time steps to be opposite to the change in measured glucose values (G_i) at those time steps:

({\hat{n}}_{i + 1} - {\hat{n}}_{i}) = - k_{i} (G_{i + 1} - G_{i}) for some positive constant k_{i}

(9)

In addition, the rate of change of ${\hat{n}}_{i}$ over time is limited to 0.001 min⁻² as per observed dynamics32 to prevent rapid nonphysiological fluctuations and thus minimize overfitting:

\frac{| {\hat{n}}_{i + 1} - {\hat{n}}_{i} |}{T_{i + 1} - T_{i}} < 0.001 \min^{- 2}

(10)

Mathematically, these constraints are applied at all knot locations, $i$ . Overall, the resulting identified $n_{L} (t)$ profile tries to optimally equate the quantities of insulin entering and leaving the compartment within physiological reason defined by the constraints and model dynamics.

Identification of $L_{e x}$

With the overall dynamics of the $I$ compartment accounted for by identifying $n_{L} (t)$ , an optimal $L_{e x}$ can be identified. Changing $L_{e x}$ alters the total appearance of exogenous insulin to the compartment post-delivery and thus changes the shape of the insulin fit after t = 15 minutes. Therefore, a high-resolution line search of $L_{e x}$ values is performed to find the value further minimizing insulin fit error. This line search starts at $L_{e x} = 1.0$ and reduces in increments of 0.01. While this two-step identification is straightforward, it ensures the identified $L_{e x}$ value is not affected by identification issues by first accounting for major insulin clearance dynamics in the compartment, then identifying loss to further optimize fit to the measured insulin data.

Analyses

The primary analysis is determining the range of optimal loss factor values present for this cohort. Comparing the accuracy of the modeled $I (t)$ profile both with and without identifying $L_{e x}$ will validate the parameter’s inclusion in the model. Furthermore, optimal $L_{e x}$ values will demonstrate the range of potential jet injection losses in a clinical setting. Results are compared by assuming $L_{e x}$ = 1.0 and there is no loss, where the impact is assessed by change in identified model error to the measured insulin data.

The second analysis involves the magnitude of general hepatic clearance $n_{L} (t)$ over time and relative to plasma glucose levels. Validation is based on comparison with expected trends and the level of variability, the latter of which indicates the potential error introduced by a constant clearance value in this analysis, and similar previous studies. The values and variability obtained provide a range and variability for physiological expectations and are compared with expected physiological values.

Finally, the sensitivity of identified $L_{e x}$ values to changes in fixed parameter values is analyzed. Fixing $x_{L} = 0.6$ for this cohort is justified by prior studies showing its lower sensitivity versus general hepatic clearance $n_{L}$ .³¹ To analyze the impact of this choice, for each subject, the constant $x_{L}$ value was iterated across the physiological range^25,37-39 from $x_{L} = 0.5 to 0.9$ , in intervals of 0.05. At each iteration, $n_{L} (t)$ was identified per the process in the section “n_L spline approach,” and $L_{e x}$ identified with a subsequent line search. Results are compared with the values found using $x_{L} = 0.6$ to assess the sensitivity of this choice.

Results

A selection of $n_{L} (t)$ profiles with constituent splines, alongside resulting insulin profiles, are provided in Figure 3. These plots demonstrate the placement of knots at insulin measurement times, as well as the weightings of splines which sum to give $n_{L} (t)$ . Plots for all subjects are provided in the Appendix.

Figure 3.

Selection of four $n_{L} (t)$ profiles and resultant insulin fits. The weighted splines comprising $n_{L} (t)$ over the measurement time are shown beneath each $n_{L} (t)$ plot. Basis splines positioned outside the measurement range are not shown.

Table 4 presents the $L_{e x}$ values identified for each patient and the resulting fit error, compared with the case where the nominal dose is assumed to completely enter plasma and insulin loss is not modeled. Where a trial’s optimal $L_{e x}$ differs from 1.0, the fit accuracy is significantly improved. Trials with little change in fit error have an optimal $L_{e x} 1.0$ . Results show optimal $L_{e x}$ differs from 1.0 by more than 5% for three of the nine trials, and for two subjects with multiple trials, $L_{e x}$ varied between trials. This latter result indicates $L_{e x}$ is capturing delivery-specific jet injector loss, not a more general physiological effect.

Table 4.

Insulin Fit RMS Errors for Cases.

Subject	(a) No modeled exogenous loss $L_{e x} = 1.0$		(b) Including identified exogenous loss factor $L_{e x} 1.0$
Subject	$L_{e x}$	RMS error of insulin fit (mU/L)	$L_{e x}$	RMS error of insulin fit (mU/L)	Change in RMS error (mU/L) (%)
1a	1.00	3.44	0.78	3.16	−0.28	(−8.0)
1b		1.95	0.90	1.84	−0.11	(−5.4)
2a		1.02	1.00	1.02	0.00	(0.0)
2b		2.50	0.84	2.35	−0.15	(−6.1)
3		3.27	0.97	3.26	−0.01	(−0.2)
4		4.36	0.90	4.29	−0.07	(−1.6)
5		1.36	1.00	1.36	0.00	(0.0)
6		4.01	0.95	3.90	−0.11	(−2.6)
7		3.87	0.95	3.84	−0.03	(−0.9)

(a) Exogenous loss factor is not modeled (ie, $L_{e x} = 1.0$ ), and (b) exogenous loss is modeled by identifying $L_{e x}$ .

Abbreviation: RMS, root mean square.

Where insulin fit error is high relative to other subjects, such as for subjects 1a, 3, and 4, this is consistently due to modeled insulin not rising fast enough to meet measured data points during first-phase secretion $(t = 10 - 30 minutes)$ . Because identified $L_{e x}$ is optimal to measured data over all time and only impacts post-bolus appearance, there is no way to improve modeling of these peaks by modification of $L_{e x}$ alone. However, higher peaks would be explained by greater endogenous secretion rate $U_{e n}$ , which has been shown to vary up to 15% from modeled values, primarily at first-phase secretion.21,40 Figure 4 compares insulin fit accuracy and identified optimal $L_{e x}$ with and without 15% scaling on first-phase endogenous secretion, occurring 10 to 30 minutes after glucose bolus delivery at $t = 0 minute$ .

Figure 4.

Insulin profiles. (a) As originally modeled with no modification to $U_{e n}$ . (b) With +15% scaling on the first-phase endogenous secretion response.40 With scaling, insulin fit accuracy is improved while identified loss factor $L_{e x}$ changes less than 6%.

The sensitivity of $L_{e x}$ to choice of $x_{L}$ value is analyzed in Table 5. For each subject, the mean identified $L_{e x}$ for fixed $x_{L}$ values in the physiological range (0.5-0.9) is compared with that identified for $x_{L} = 0.6$ . These results show the identified $L_{e x}$ for $x_{L} = 0.6$ is within one standard deviation of the mean over a physiological $x_{L}$ range for seven of the nine trials, and within two standard deviations for all nine trials, justifying this selection and further validating the low sensitivity of this parameter value.31 Where mean $L_{e x}$ is unphysiologically greater than 1.0, most clearance values in the assessed range are too high to be accurate for the data. High clearance causes $L_{e x}$ to increase above 1.0, raising $I$ compartment input to balance increased output. These unphysiological results further validate the choice of $x_{L} = 0.6$ , in the lower end of the physiological range.

Table 5.

Sensitivity Analysis of $L_{e x}$ With Respect to a Choice of Fixed First-Pass Hepatic Clearance Value $x_{L}$ .

Subject	Mean of identified $L_{e x}$	Standard deviation of identified $L_{e x}$	$L_{e x}$ at $x_{L} = 0.6$
1a	0.76	0.01	0.78
1b	0.87	0.06	0.90
2a	1.00	0.01	1.00
2b	0.87	0.04	0.84
3	0.96	0.02	0.97
4	0.89	0.02	0.90
5	1.00	0.01	1.00
6	0.97	0.02	0.95
7	0.97	0.03	0.95

The mean identified $L_{e x}$ value is provided for fixed $x_{L}$ values in the range 0.5 to 0.9. This mean is compared with the identified value at the chosen value of $x_{L} = 0.6$ .

Discussion

Outcomes

Modeling accuracy is significantly improved when accounting for potential SC delivery losses, as observed in Table 4. Where a subject’s optimal loss factor is $L_{e x} < 0.95$ , a significant loss proportion, there is a significant decrease in insulin fit error. The decrease is greater than this 5% threshold for four of the nine trials, where optimal $L_{e x}$ ranges from 0.78 to 0.90 in these cases. Previous analyses have determined the bioavailability of jet-injected insulin to be approximately 80%,⁴¹ in agreement with this study’s findings. Where optimal fitted $L_{e x} 1.0$ , the insulin fit error does not change significantly, as expected. This result validates the loss factor $L_{e x}$ in this identification approach is not highly associated with other variables, as it only grows in value to provide a better fit in cases where this dynamic can play such a role.

Multiple factors may contribute to this apparent loss, including pooling at injection sites42 and incomplete ejection from the jet injector or incomplete injection into the skin. These factors are difficult or impossible to measure directly. Thus, identifying this loss on a per-subject basis from clinical assays and modeling post-test are a useful result when considering this type of new technology where losses are a potential likelihood. While inaccurate model parameters could cause apparent loss, these models have been well validated in clinical trials^28,43,44 and for assessing insulin kinetics⁴⁵ and its parameter values validated with research of human physiology.^21,24,27,37 Thus, the loss factor, $L_{e x}$ , and methods for its identification presented appear sound.

The analysis in Figure 4 shows significant insulin fit errors are not due to identification of $L_{e x}$ . Variation in modeled endogenous secretion explains modeling error around first-phase insulin secretion, while identified optimal $L_{e x}$ does not vary greatly. The increased insulin input over all time is accounted for with parameter trade-off, where clearance $n_{L} (t)$ increases to preserve overall model fit. The optimal loss factor $L_{e x}$ depends only on the post-bolus profile, which is relatively unchanged due to this trade-off. Thus, model accuracy can be improved by considering variation of endogenous secretion without compromising the loss results identified by this methodology.

The sensitivity analysis in Table 5 shows for any physiologically realistic choice of $x_{L}$ , the optimal loss factor $L_{e x}$ remains broadly consistent. At the selected value of $x_{L} = 0.6$ , every trial’s optimal identified $L_{e x}$ value is within one standard deviation of the mean $L_{e x}$ value across physiological values of $x_{L}$ .

Therefore, for the purposes of identifying a loss factor which best explains the appearance of exogenous insulin, it is justified to choose a fixed value of $x_{L} = 0.6$ for this cohort of healthy subjects. Any similar study on other cohorts should also consider a sensitivity analysis to confirm this value and choice hold for that study and/or cohort.

Overall, these results demonstrate a significant degree of loss may be present for some subjects during SC jet injection delivery of insulin. Expanding the modeling approach to capture this effect allows indirect measurement of SC delivery loss, and improves the accuracy of modeled insulin profiles.

Limitations

Due to the number of variable parameters in the model, the value of $x_{L}$ was fixed. Parameter trade-off inherent to this or any similar model means subsequent identification of $n_{L} (t)$ is influenced by the selected value of $x_{L}$ .^19,29-31 However, the sensitivity analysis in Table 5 shows the optimal $L_{e x}$ is invariant over the physiological range of $x_{L}$ values. The flexibility of modeling $n_{L} (t)$ using physiologically realistic spline functions allows this parameter to vary, so overall dynamics present in the data are preserved in the model for any physiological choice of $x_{L}$ . $L_{e x}$ can then be altered to best-fit exogenous dynamics independently of the numerical values of $x_{L}$ or $n_{L}$ Ultimately, a fixed $x_{L}$ value is appropriate for the purposes of analyzing exogenous loss provided the chosen $x_{L}$ and identified $n_{L}$ are physiologically reasonable. The decision to fix $x_{L} = 0.6$ specifically was justified for this cohort by sensitivity analysis in Table 5, where $L_{e x}$ was shown not to vary significantly across the physiological range of $x_{L}$ . While this approach is valid for this cohort, the appropriate value of $L_{e x}$ may vary across cohorts of different demographics.^25,31,38 For this approach to be robust for a general population, this analysis should be expanded to further and larger cohorts, demonstrating a fixed $x_{L}$ is appropriate in general for this analysis.

Subcutaneous absorption rate parameters $k_{s 2}$ and $k_{s 3}$ were fixed at physiological values for standard needle injection as listed in Table 2. However, clinical trials have demonstrated faster absorption from jet injection relative to needle injection,¹ likely due to increased surface area of dispersion pool.⁴⁶ Due to variation in delivery completeness, as demonstrated in this study, we would expect these rates to vary between subjects and thus require individual identification. However, our model exhibits trade-off between these absorption rates, loss factor $L_{e x}$ , and output rate $n_{L}$ , similar to that described above for $x_{L}$ . This three-parameter trade-off, therefore, requires either $L_{e x}$ or the SC absorption rates to be fixed for reliable identification, as simultaneously identifying all three parameters is infeasible. Thus, in our method, any SC absorption variability is captured by the identified $L_{e x}$ . This consideration further highlights the importance of identifying SC injection loss on a per-subject basis.

While the number of subjects examined in this trial could be seen as a limitation, Table 4 shows every subject experienced improved insulin fit accuracy where any significant loss over 5% was identified. Given delivery loss is a known factor in this type of exogenous delivery, accounting for it in the model can only improve fit accuracy. If no loss is present, the optimal loss factor will be near 1.0, as demonstrated by subjects 2a and 5 in Table 4. As such, further subjects would demonstrate the same improvements in accuracy where any insulin loss is present, as seen for subjects 1a, 1b, and 4. Thus, this cohort is sufficient to demonstrate modeling potential loss allows for a significant improvement in fit accuracy. However, the results showing 33% of subjects had a significant loss over 5% of the delivered amount might not hold in a larger study or with greater use of the device.

Due to safety limitations on this specific trial and the healthy cohort subjects used, the quantity of insulin delivered was limited to 2U to avoid hypoglycemia. A cohort with type 1 or severe type 2 diabetes, of more relevance to jet injection therapy, would be treated with a larger dose, whether fixed for each subject or relative to subject weight. However, the dynamic insulin sensitivity and secretion test protocol on which our methods are based was designed with a low-quantity, fixed dose,²⁸ and exhibits no difference to identification between fixed or weight-dependent doses.³⁷ This protocol is proven to accurately identify insulin kinetics and sensitivity;³⁶ therefore, a 2U dose is appropriate for demonstrating the identification protocol used in this study.

Table 4 quantifies the proportion of nominal insulin dose which enters plasma. However, the above data and analysis do not consider the potential of constant insulin losses, for example, a fixed volume lost to surface pooling It is possible the degree of loss is dependent on dosage volume and amount,⁴⁶ where these were constant and relatively low in this study. These results alone cannot conclude what level of consistent or proportional exogenous insulin loss is expected. Rather, the method presented identifies a trial-specific proportion of loss, and provides a method for identifying this value when it is significantly different from 1.0 (no loss). Future research should analyze this loss effect with different volumes or amounts of insulin using the methods presented in this study.

Implications

The methodology described above has been shown to determine trial-specific magnitude of exogenous loss factors in a clinical setting when using jet injection delivery or any similar delivery method which might have significant losses. Furthermore, the methodology can quantify apparent losses of unknown quantity, such as pooling or leakage at an injection site. These results demonstrate loss is variable between subjects, and between trials on one subject, where this is not considered in some analyses of jet injectors.^13,14 Understanding the degree and variability of jet injection delivery can provide significant insight into subject-specific control outcomes, as well as a means to evaluate new technologies with typical, traditional metabolic tests. Thus, for accurate assessment and glycemic management of an individual, delivery-specific loss must be considered, and accounted for where dose accuracy is a priority.

Conclusion

Modeling insulin loss in SC jet injection delivery has enabled quantification of loss on a per-subject basis and demonstrated its intra-subject variability. Identified losses are in accordance with clinical assessments of bioavailability of jet-injected insulin. For all subjects with loss greater than 5% of the nominal dose, the accuracy of insulin prediction improved when loss was included in the model. Quantification of jet injection loss and variability without novel measurement techniques will support further development of jet injectors and other promising insulin therapy technologies.

Footnotes

Appendix

Subject 1a	Subject 1b
Subject 2a	Subject 2b
Subject 3	Subject 4
Subject 5	Subject 6
Subject 7

Abbreviations

SC, subcutaneous; OGTT, oral glucose tolerance test; GC, glycemic control; BMI, body mass index; DISST, dynamic insulin sensitivity and secretion test.

ORCID iDs

Alexander D. McHugh

J. Geoffrey Chase

Jennifer L. Knopp

Tony Zhou

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors are supported by New Zealand Science for Technological Innovation National Science Challenge [#CRS-53-2019] and MedTech Core [TEC #370571].

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Determining Losses in Jet Injection Subcutaneous Insulin Delivery: A Model-Based Approach

Abstract

Objective:

Methods:

Results:

Conclusions:

Keywords

Introduction

Method

Subjects and Data

Model

Parameter Identification

n L spline approach

Identification of L e x

Analyses

Results

Discussion

Outcomes

Limitations

Implications

Conclusion

Footnotes

Appendix

Abbreviations

ORCID iDs

Declaration of Conflicting Interests

Funding

References

$n_{L}$ spline approach

Identification of $L_{e x}$