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**|**J Diabetes Sci Technol**|**v.3(5); 2009 September**|**PMC2769898

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Article sections

- Abstract
- Motivation and Context
- Background
- Simplified Model Development
- Model-Based Controller Development
- Time-Activity Profiles and Insulin-on-Board Computations
- Simulation Model and Protocol
- Feedback Control of Glucose Concentration
- Conclusions
- References

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J Diabetes Sci Technol. 2009 September; 3(5): 1005–1013.

Published online 2009 September. doi: 10.1177/193229680900300503

PMCID: PMC2769898

B. Wayne Bequette, Ph.D.

Department of Chemical and Biological Engineering, Rensselaer Polytechnic Institute, Troy, New York

Correspondence to: B. Wayne Bequette, Ph.D., Department of Chemical and Biological Engineering, Rensselaer Polytechnic Institute, Ricketts Building, Room 129, 110 8th St., Troy, NY 12180-3590; email address ude.ipr@etteuqeb

Copyright © 2009 Diabetes Technology Society

This article has been cited by other articles in PMC.

Most current insulin pumps include an insulin-on-board (IOB) feature to help subjects avoid problems associated with “insulin stacking.” In addition, many control algorithms proposed for a closed-loop artificial pancreas make use of IOB to reduce the probability of hypoglycemic events that often occur due to the integral action of the controller. The IOB curves are generated from the pharmacodynamic (time-activity profiles) actions of subcutaneous insulin, which are obtained from glycemic clamp studies.

Glycemic clamp algorithms are reviewed and *in silico* studies are performed to analyze the effect of glucose meter bias and noise on glycemic control and the manipulated glucose infusion rates. The glucose infusion rates are used to obtain insulin time-activity profiles, which are then used to generate IOB curves.

A model-based, three-step-ahead controller is shown to be equivalent to a proportional-integral control algorithm with time-delay compensation. A systematic glucose meter bias of +6 mg/dl results in a decrease in the glucose area under the curve of 3% but no change in the IOB profiles.

Based on these preliminary simulation studies, a substantial amount of glucose meter bias and noise during a glycemic clamp can be tolerated with little net effect on the IOB curves. It is suggested that handheld glucose meters can therefore be used in clamp studies if the measurements are filtered (averaged) before processing by the control algorithm. Clinical studies are needed to confirm these preliminary results.

Many control algorithms proposed for a closed-loop artificial pancreas make use of insulin-on-board (IOB) information that accounts for the insulin time-action profile (subcutaneous insulin pharmacodynamics).^{1,2} Indeed, most insulin pumps also have an IOB feature to prevent the effect of “insulin stacking,” where previously infused insulin still has an effect on future glucose values.^{3,4} Subcutaneous insulin pharmacodynamic models are usually based on glycemic clamp studies, where intravenous glucose is delivered to maintain constant blood glucose concentration in response to an insulin bolus.

One of the goals of this article is to provide a concise review of glycemic clamp protocols (control algorithms) and to show via simulation studies the effect of glucose measurement delay and uncertainty on glycemic control. Another goal is to analyze the effect of glycemic clamp variability, possibly due to the use of handheld glucose meters, on IOB calculations.

Euglycemic glucose clamps are the standard method for determining insulin pharmacokinetics and pharmaco-dynamic behavior.^{5} A bolus of insulin is usually given subcutaneously (0.2 U/kg is typical), and glucose is infused intravenously to maintain a constant glucose concentration. The rate of glucose infusion is then used as a measure of insulin action; glucose infusion is usually scaled by the subject's weight and presented with units of mg/kg/min.

The glucose clamp technique was originally developed based on steady-state intravenous insulin infusion and adjustment of the glucose infusion rate to maintain a constant glucose concentration. The components required include a method to take blood glucose samples, a blood glucose analyzer, and a glucose infusion pump. DeFronzo and colleagues^{6} give a thorough overview of the glucose clamp technique, including a method to correct the glucose infusion rate to compensate for changes in glucose concentration (“space correction”) and urinary loss. They also provide an algorithm for the real-time adjustment of the glucose infusion rate to maintain constant blood glucose concentrations. Insel and associates^{7} use clamp studies to estimate parameters for a three-compartment model of insulin and glucose dynamics.

Clemens and coworkers^{8} present the glucose clamping algorithm used by the Biostator, which was originally developed as a bedside or clinical closed-loop artificial pancreas.^{9–11} The Biostator uses an automated glucose sampling and analysis device to provide glucose concentrations every minute and to adjust the glucose infusion rates every minute. The glucose clamping (control) algorithm appears to be linear, whereas the Biostator artificial pancreas algorithm (where insulin infusion is manipulated) is nonlinear.

Ponchner and colleagues^{12} compare the performance of the Biostator glucose clamping algorithm with the manual algorithm of Reference 6 and conclude that the manual algorithm is simpler and yields better performance. The Biostator algorithm that they present, however, appears similar to the nonlinear algorithm used to infuse insulin in the artificial pancreas version of the Biostator. It is fairly clear that Ponchner and colleagues^{12} were not comfortable with the frequent calibration and tuning involved with the Biostator and felt that the manual method was clearer to use. Based on my reading of the Biostator literature, I must agree with them. In my view, a *properly designed and implemented* closed-loop glucose infusion algorithm should always have better performance than the manual method.

Palazzo and Viti^{13} simulate the behavior of proportional-integral (PI) controllers applied to the two-state Bergman minimal model;^{14} they present tuning parameters as a function of Bergman model parameters. Campostano and associates^{15} implement the proposed Palazzo and Viti tuning parameters based on estimates of the patient's sensitivity. Four clinical subjects were studied using a euglycemic clamp, while hyperglycemic clamps were applied to 14 subjects.

The glucose clamping algorithms and studies by DeFronzo and coworkers,^{6} Clemens and colleagues,^{8} and Ponchner and associates^{12} are based on steady-state insulin sensitivity analysis. That is, intravenous insulin is infused at a constant rate, and intravenous glucose infusion is used to control glucose concentration to a desired setpoint. Certainly, by the end of the experiment, the glucose concentration is held relatively constant. The glucose clamps used in subcutaneous insulin bolus pharmacokinetic–pharmacodynamic (PKPD) studies, however, are inherently dynamic in nature. Manual clamps, in particular, can result in significant perturbations of the glucose concentration from desired values (setpoints), and the glucose infusion rates should be corrected to estimate the actual rate of glucose metabolized. Few PKPD articles provide significant detail on the mathematical algorithms used to correct the glucose infusion rate. Most of the subcutaneous insulin pharmacodynamic studies published by Heinemann and colleagues^{16–19} are based on the use of the Biostator, where presumably the glucose is controlled tightly enough that no glucose infusion rate compensation (“space correction”) is necessary. Most other authors appear to use a manual clamping technique, where it is expected that glucose concentration will vary significantly.

Furler and coworkers^{20} present what they call a proportional-derivative controller since the difference between two successive glucose values is used in the algorithm. The resulting controller, however, is a PI controller, as is shown in this article.

Glucose clamp protocols usually use a laboratory-quality glucose measurement device, such as the YSI (Yellow Springs Instrument) glucose oxidase analyzer. Cohen and colleagues^{21} conclude that the FreeStyle Mini glucose meter is accurate enough for glucose clamp studies, based on clamp procedures conducted on seven volunteers with type 2 diabetes. Hompesch and Rave,^{22} however, argue that the systematic bias of 6% would result in substantial underestimation or overestimation of glucose requirements; in addition, the bias would hamper the comparison of the absolute amount of glucose infused between different studies.

In this article, a simple model of glucose dynamics is used to develop a model-based glycemic clamp protocol (control algorithm). Then I examine the effect of glucose measurement uncertainty (bias and noise) on glucose control during euglycemic clamps and on the resulting time-action profiles and IOB curves; these studies are based on the use of a simulation model, which is of much higher complexity than the model used for controller design, in keeping with the consensus of a meeting summarized by Steil and Reifman.^{23}

A simple model relating glucose infusion to glucose concentration is

$${V}_{G}\frac{dG}{dt}={G}_{IR}-{G}_{up}$$

(1)

where *V _{G}* represents the volume of the glucose compartment per unit weight of the subject (liters/kg),

When discretized, **Equation (1)** is

$${V}_{G}\frac{\left({G}_{k+1}-{G}_{k}\right)}{\Delta t}={G}_{IR,k}-{G}_{up,k}$$

(2)

where subscript *k* represents the beginning of the *k*th sample interval. This can also be written as

$${G}_{k+1}={G}_{k}+\frac{\Delta t}{{V}_{G}}\left({G}_{IR,k}-{G}_{up,k}\right)$$

(3)

which represents an exact integration of **Equation (1)** under the assumption that the glucose infusion and uptake rates are constant over the sample time. Now consider a desire to reach the glucose setpoint in *N* sample times. The discretization of **Equation (1)** for *N* sample times in the future is

$${V}_{G}\frac{{G}_{k+N}-{G}_{k}}{N\Delta t}={G}_{IR,k}-{G}_{up,k}$$

(4)

which, again, is exact under the limiting assumption that the glucose infusion and uptake rates are constant over the *N* sample times into the future.

Here we use the simple model of glucose dynamics as a basis for controller design. While there are many different possible model-based controller design techniques, we develop one with a goal of reaching the setpoint at step *k* + *N* (that is, *N* sample times in the future) by making a change in the glucose infusion rate and holding it constant over the *N* sample times. Substituting *G _{sp}* for

$${G}_{IR,k}={\widehat{G}}_{up,k}+{V}_{G}\frac{{G}_{sp}-{G}_{k}}{N\Delta t}$$

(5)

where ^{∧} is the estimate of the uptake rate since it is not known. The uptake rate at the previous step can be estimated from the most recent glucose measurements (at steps *k* and *k* − 1),

$${\widehat{G}}_{up,k-1}={G}_{IR,k-1}+{V}_{G}\frac{\left({G}_{k-1}-{G}_{k}\right)}{\Delta t}$$

(6)

if it is assumed that the uptake rate remains constant over the next *N* time steps. Substituting **Equation (6)** into **Equation (5)**, we find

$${G}_{IR,k}={G}_{IR,k-1}+{V}_{G}\frac{\left({G}_{sp}-{G}_{k}\right)}{N\Delta t}+{V}_{G}\frac{\left({G}_{k-1}-{G}_{k}\right)}{\Delta t}$$

(7)

Recognizing that the difference between the desired setpoint and the glucose measurement is the error,

$${G}_{k-1}-{G}_{k}=\left({G}_{sp}-{G}_{k}\right)-\left({G}_{sp}-{G}_{k-1}\right)={e}_{k}-{e}_{k-1}$$

(8)

we can rewrite **Equation (7)** as

$${G}_{IR,k}={G}_{IR,k-1}+\frac{{V}_{G}}{N\Delta t}{e}_{k}+\frac{{V}_{G}}{\Delta t}\left({e}_{k}-{e}_{k-1}\right)$$

(9)

It should be noted that the discrete velocity form of a PI controller^{24} is

$${u}_{k}={u}_{k-1}+\left(\frac{{k}_{c}\Delta t}{{\tau}_{I}}\right){e}_{k}+{k}_{c}\left({e}_{k}-{e}_{k-1}\right)$$

(10)

where *u _{k}* is the insulin infusion rate at time

$${k}_{c}=\frac{{V}_{G}}{\Delta t},{\tau}_{I}=N\Delta t$$

(11)

The glucose compartment volume is typically in the range of *V _{G}* = 1 to 2 dl/kg. If

There is typically a delay of 2–4 min between the glucose sample and the availability of the glucose measurement. When the glucose sample time is 5 min, it is important to compensate for this delay. The glucose infusion rates used in **Equations (7)** and **(9)** are the average over the sample interval, thus, to be more precise, we write **Equation (7)** as

$${G}_{IR,k}^{ave}={G}_{IR,k-1}^{ave}+{V}_{G}\frac{\left({G}_{sp}-{G}_{k}\right)}{N\Delta t}+{V}_{G}\frac{\left({G}_{k-1}-{G}_{k}\right)}{\Delta t}$$

(12)

and we recognize that the average infusion rate over the interval from *k* − 1 to *k* is

$${G}_{IR,k-1}^{ave}=\frac{{t}_{d}}{\Delta t}{G}_{IR,k-2}+\frac{\left(\Delta t-{t}_{d}\right)}{\Delta t}{G}_{IR,k-1}$$

(13)

where *t _{d}* is the sample delay (the amount of time it takes before the glucose measurement at sample

$${G}_{IR,k}=\frac{\Delta t}{\Delta t-{t}_{d}}{G}_{IR,k}^{ave}+\frac{{t}_{d}}{\Delta t-{t}_{d}}{G}_{IR,k-1}$$

(14)

so the new infusion rate, written in terms of the previous infusion rates, is

$$\begin{array}{l}{G}_{IR,k}=\frac{\left(\Delta t-{t}_{d}\right){G}_{IR,k-1}+{t}_{d}{G}_{IR,k-2}}{\Delta t}+\\ \text{\hspace{1em}\hspace{1em}\hspace{1em}}{V}_{G}\frac{\left({G}_{sp}-{G}_{k}\right)}{N\Delta t}+{V}_{G}\frac{\left({G}_{k-1}-{G}_{k}\right)}{\Delta t}\end{array}$$

(15)

which is virtually the algorithm proposed by Furler and colleagues^{20} written as

$$\begin{array}{l}{G}_{IR,k}=\frac{\left(\Delta t-{t}_{d}\right){G}_{IR,k-1}+{t}_{d}{G}_{IR,k-2}}{\Delta t}+\\ \text{\hspace{1em}\hspace{1em}\hspace{1em}}{V}_{G}\frac{\left({G}_{sp}-{G}_{k}\right)}{{k}_{p}\Delta t}+{V}_{G}\frac{\left({G}_{k-1}-{G}_{k}\right)}{{k}_{d}\Delta t}\end{array}$$

(16)

where clearly the *k _{p}* term is the

When there is substantial glucose measurement noise, it is important to include a “filter” to reduce the effect of the noise on the control computation. A simple averaging filter has the form

$${G}_{k}^{f}=a{G}_{k-1}^{f}+\left(1-a\right){G}_{k}$$

(17)

where the superscript *f* indicates the filtered glucose value. This filtered value can be substituted into the previous equations to reduce the effect of sensor noise. If a continuous-time filter time constant, τ_{f}, is used, it can be implemented in discrete form as

$$a=\mathrm{exp}\text{\hspace{0.17em}}\left(-\Delta t/{\tau}_{f}\right)$$

(18)

Note that a smaller value of *a* (smaller value of τ_{f}) results in less filtering.

The rate that glucose is metabolized is a function of the glucose infusion rate, the rate of glucose lost through the urine, and the amount that enters/leaves and raises/lowers the glucose concentration in the glucose compartment.^{6} In the studies that follow, we neglect the amount lost to the urine since a typical rate is less than 0.2 mg/kg/min, as reported by DeFronzo and associates.^{6}

The average glucose uptake during the *k*th time interval is computed by

$${\widehat{G}}_{up,k-1}^{ave}={G}_{IR,k-1}^{ave}+{V}_{G}\frac{\left({G}_{k-1}-{G}_{k}\right)}{\Delta t}$$

(19)

where the rightmost term represents the “space correction”^{6} to account for the portion of glucose infused that increases the glucose concentration. For a retrospective analysis of the glucose infusion rates to calculate IOB (duration of action) plots, the glucose infusion rates are averaged over 10 min intervals.

The fractional insulin activity remaining (IOB) is calculated by

$$\begin{array}{l}IOB\left(t\right)=1-\frac{{\displaystyle \int {\widehat{G}}_{upR}+\left(t\right)dt}}{{G}_{up,total}},\\ \text{where\hspace{1em}}{G}_{up,total}=AUC={\displaystyle {\int}_{t=0}^{t={t}_{final}}{\widehat{G}}_{up}\left(t\right)dt}\end{array}$$

(20)

and is typically plotted over a 6–8 h time period.

The simulation model for insulin glucose dynamics used in our *in silico* studies is based on the compartmental model by Hovorka coworkers,^{25} with a revised subcutaneous insulin kinetic model by Wilinska and associates;^{26} there are a total of nine differential equations in this model. We have reduced the published insulin sensitivities by 50% to better reflect match the basal insulin and carbohydrate-to-insulin ratios of typical adults.^{27} In this simulation protocol, we start with the subject at a constant steady-state glucose concentration of 100 mg/dl, based on a constant basal infusion of insulin. The subject is then given a subcutaneous insulin bolus of 0.2 U/kg, and glucose is manipulated to maintain glucose concentration at the desired setpoint value of 100 mg/dl.

We first compare the results of two different clamps. The first is based on a continuous adjustment of glucose infusion rates using continuous, noise-free glucose sensor values. The second is based on sampling glucose at 5 min intervals, with a 2 min time delay to report the glucose values to the controller; glucose measurement uncertainty and resolution are neglected. Although the discrete controller does not maintain as tight regulation of glucose as the continuous controller, as shown in **Figure 1**, the resulting glucose infusion rates are nearly identical, except for the first 10 min, as shown in **Figure 2**.

In the second study, we assume measurement noise with a mean of 0 and a standard deviation of 1 mg/dl, with integer reporting of glucose values. Further, a glucose infusion pump threshold resolution of 0.1 mg/kg/min is used. The actual glucose values (uncorrupted by noise and delay) are shown in **Figure 3**. The noise and finite resolution of the sensor and pump lead to much more active glucose infusion rates, as shown in **Figure 4**.

The previous simulations were pertinent to the use of a YSI glucose measurement, which we assumed to have no bias, and a measurement standard deviation of 1 mg/dl (1% in the range of our measurements). Now we consider the use of blood glucose meters, which can have significant uncertainty, including bias. The study results of Cohen and colleagues^{21} and the related critique by Hompesch and Rave^{22} indicate that a glucose meter may have a bias of +6 mg/dl (6% in our range of measurements), with a standard deviation of 5 mg/dl (5% in our range of measurements).

In the next set of simulations, we study the effect of a systematic glucose meter bias of ±6 mg/dl. The meter bias clearly results in an “offset” for the actual glucose concentrations, as shown in **Figure 5**, since the controller is regulating the glucose concentration to 100 mg/dl based on the glucose meter readings. The corresponding glucose uptake rates (based on the glucose infusion rates and the “space correction”) are shown in **Figure 6**. As suggested by Hompesch and Rave,^{22} the biased values do result in slightly different areas under the glucose uptake cures. For the unbiased meter the area under the curve (AUC) = 1.915 g; for the +6 mg/dl bias,the AUC = 1.859 g; and for the −6 mg/dl bias, the AUC = 1.970 g. Thus a meter bias of 6 mg/dl (6%) results in AUC estimates that are in error by 3%. There is no effect of glucose meter bias on the IOB curves, however.

Effect of glucose meter bias (±6 mg/dl) on actual blood glucose concentrations during a euglycemic clamp. Since the controller is regulating glucose to a set point of 100 mg/dl based on the glucose meter readings, +6 and −6 mg/dl biases **...**

The previous subsection illustrated the effect of glucose measurement bias. Here we study the effect of a standard deviation of 5 mg/dl in the glucose meter values, in addition to a bias of +6 mg/dl. The large glucose meter noise (with bias) causes the actual glucose to vary substantially from the setpoint (100 mg/dl), as shown in **Figure 7**. The corresponding glucose infusion rates are shown in **Figure 8**, where the simple measurement filter (*a* = 0.5) reduces the variability in the glucose infusion rates. The time-activity profiles are compared in **Figure 9**; even with the “space correction” for glucose variability and the averaging over 10 min periods, there is substantial variation in the profiles due to sensor noise. On the other hand, this “noise” is smoothed by the process of integration used in the IOB computations, as shown in **Figure 10**.

Comparison of glucose concentrations for the following controllers: discrete (with no noise) and discrete with bias and substantial noise (filtered and unfiltered measurements). The actual glucose concentrations are plotted.

Comparison of glucose infusion rates (mg/kg/min) for the following controllers: discrete (with no noise) and discrete with bias and substantial noise (control algorithm based on filtered and unfiltered measurements).

Comparison of time-action profiles (glucose uptake rate, mg/kg/min) for continuous, discrete (no noise), and discrete with bias (+6 mg/dl) and substantial noise (standard deviation = 5 mg/dl; filtered and unfiltered).

Comparison of IOB profiles for the results in **Figure 9**.

Mudaliar and associates^{28} use a manual clamp to compare insulin aspart and regular human insulin using three injection sites (abdomen, deltoid, and thigh) on 20 healthy male subjects with average characteristics of 31.1 years of age and a body mass index of 23.6 kg/m^{2}.The time-action profiles from their abdomen studies using aspart are compared with the *in silico* noisy (but filtered)data in **Figure 11**. Naturally, the averaging effect over 20 subjects makes the glucose uptake rate curves much smoother than a single simulated subject with high measurement noise. The IOB curves are very similar, however, as shown in **Figure 12**, since the integration in **Equation (20)** naturally smoothes the data.

Comparison of time-action profiles (glucose uptake rate, mg/kg/min) for continuous, discrete with bias (+6 mg/dl) and substantial noise (standard deviation = 5 mg/dl; filtered), and the curve published by Mudaliar and colleagues^{28} (scaled by 75 kg).

Comparison of IOB curves for the time-action profiles in **Figure 11**.

It is important to note that not all insulin pumps use IOB curves that have the characteristic shown in **Figures 10** and **1212**. The Deltec and Insulet pumps, for example, use linear approximations to these curves. In addition, all pumps have a “duration of insulin action” parameters to allow the user to adjust the rate of decrease in insulin activity. A review of these basic pump features is provided by Zisser and coworkers.^{4}

A simple model-based glucose controller has the form of a PI controller with time-delay compensation. Even relatively noisy glucose meters with bias can result in reasonable glucose control and glucose infusion rates as long as the noisy measurements are filtered (averaged) before using them in the control algorithm. While biased measurements do have a minor effect on the area under the glucose uptake curves, there is no effect on the IOB curves that are used in smart insulin pumps and many proposed closed-loop artificial pancreas control algorithms.

- AUC
- area under the curve
- IOB
- insulin on board
- PI
- proportional integral
- PKPD
- pharmacokinetic–pharmacodynamic

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