The subject is a mid-sixty healthy male of 180 lbs with 5'10" frame, leading a productive professional life. He has been diagnosed with type 2 diabetes for more than 30 years. Initially, he was on diet regimen for nearly twenty years and then was instructed by his physician to dispense 5 mg glucotrol once every morning. He experienced frequent acute hypoglycemia that led him to discuss a possible self-managed regimen with his family physician.
Lunch was chosen as the test meal for having sufficient time to take post-prandial measurements. The test meals were 15 sets of lunches that consisted either (1) 10 to 12 oz of steamed rice, stir-fried vegetables with 4 oz canned tuna (or steamed cod), or (2) 10 to 12 oz spaghetti with 6 medium sized meat balls (from Sam's family package). Five sets of data each were collected from: (i) without taking hypoglycemic pills before test meals; (ii) 1/4 size of 5 mg glyburide pills were dispensed pre-prandially right before the meal and (iii) 1/4 size of 5 mg glyburide pills were dispensed pre-prandially an hour before the test meals. One pre- and 8 to 12 post-prandial blood glucose measurements were taken at 30-minute intervals starting at the beginning of a meal (meal is usually consumed in 15 minutes): (i) for 6 hours, (ii) for 5 hours, and (iii) for 4 hours. In addition, for case (iii) two reference measurements were taken with one right before dispensing the pill and one an hour after completion of the 8 post prandial measurements, i.e., at hour 5, for a total of 11 readings.
The purpose of the first set of measurements was to establish the baseline for this diabetic subject: the recovery period of post-prandial blood glucose excursion without medication. The second and the third sets of the trials were designed to quantitatively measure the hypoglycemic drug effects and the most optimal time frame to administer the pills. Raw data were averaged and the corresponding standard deviations were also calculated for 5 replicates at given times. The averaged data were then used for modeling analysis.
Model formulation
The post-prandial blood glucose excursion can be considered as a hormone regulated resilient system. The food intake is treated as a bolus injection of glucose, and thus the impulse force f(t); effects of exercises and hypoglycemic medication are lumped as the damping factor, β. The differential equation of such an oscillatory system, that is used to describe post-prandial blood glucose excursions, can be found in many physics texts:
where x represents blood glucose level over the baseline at time t, ω0 is the system natural frequency [12]. The pre-prandial blood glucose levels are generally fluctuating with relatively insignificant magnitudes thus can be approximated as a flat level. If the impulse force f(t) takes the form of the Dirac delta function, F δ(t-0) with F being a food intake dependent parameter, the solution of Eq. (1) is
where
is the frequency of the system. Equation (2) is a three parameter model: F, ω and β. Implications of these three parameters not only could reveal distinctive characteristics between diabetic and non-diabetic individuals but also provide guidelines to adjust one's lifestyle.
Parametric estimation
For a given blood glucose excursion, data was taken every 30 minute interval from the time a meal was initially consumed, from which the excursion peak (MR), x
max
, and the corresponding time τ to reach MR can both be estimated. Setting dx/dt = 0 in Eq. (2), the time τ can be expressed as:
Substituting Eq. (3) into Eq.(2), we have
The area under an excursion curve, AUC, can also be obtained:
where T = 2π/ω is the period of oscillation. The reason for setting the upper integral limit to T/2 is because the damping factor β effectively depresses the glucose excursion levels x near zero for t >T/2, i.e., it ripples about pre-prandial level. The time T/2 is therefore defined as the recovery period (RP). For type 2 diabetic patients who are not in a properly structured regimen, the recovery periods are often longer than 5 hours, by which time the next meal arrives and induces another blood glucose upswing.
Equations (3) – (5) can be used to estimate the three parameters, F, ω and β, from the measurable quantities of τ, x
max
, and AUC. The procedure is briefly described below:
-
1.
Assign T as twice the roughly estimated recovery period in hours, which can be obtained from the raw data and thus ω = 2π/T.
-
2.
The damping factor β can be estimated from Eq. (3):
, and thus
.
-
3.
The estimation of food intake-dependent impulse force F can be obtained from Eq. (4):
.
-
4.
Fine tune these three parameters by using MATLAB function fminsearch to minimize [AUCdata - AUC(F, β, ω)]2, where AUCdata is calculated from the averaged data points by the trapezoidal rule and AUC(F, β, ω) is calculated from Eq. (5).
-
5.
These three parameters can further be fine-tuned by fminsearch (sum of squared errors between the averaged data points and the model predicted values).
Two MATLAB user defined functions: GlucoseModel (for No pill and Pill at meal) and GlucoseModel1 (for Pill one hour prior) to estimate these model parameters and calculating the relevant diabetic characteristic measures: τ, x
max
, AUC are listed in the Additional files 1 and 2, respectively.