**6. Medical physiology: A non-dimensional diabetes index with respect to Oral-Glucose-Tolerance testing**

Fig. 16. OGTT response-Curve: A=1.3hr-1 (i.e., higher damping.coefficient value) for the normal subject for the diabetic patient A=0.808 hr-1, i.e., the damping coefficient is smaller [7, 8].

Oral Glucose Tolerance Test (OGTT) is a standard procedure for diagnosing diabetes and risk of becoming diabetic. However, the test data is assessed empirically. So, in order to make this procedure more reliable, we have carried out a biomedical engineering analysis of the OGTT data, to show how to distinguish diabetes subjects and those at risk of becoming diabetic. For Oral-glucose Tolerance Test simulation (entailing digestive & blood-pool chambers), the

differential equation is as follows [7,8]:

$$\begin{aligned} \, &y'' + 2Ay' + \alpha\_n^2 y = \text{G}\, \delta(t); \quad \text{y in gms/liter, G in gm/liter/hr} \\ \, ∨\_{\text{\tiny $y'' + \lambda T\_d$ }} \, &y' + \lambda T\_d \, y' + \lambda y = \text{G}\, \delta(t) \end{aligned} \tag{1}$$

wherein ( ) 1/2 ω λ *<sup>n</sup>* = is the natural oscillation-frequency of the system, *A* is the attenuation or damping constant of the system, ω = is the (angular) frequency of damped oscillation of the system, λ*= 2A/Td*=, <sup>2</sup> vω*<sup>n</sup>* with (λy) representing the proportional-control term of blood-glucose concentration (y).

(λTdy' )is the derivative feedback control term with derivative-time of *Td G t* δ( ) represents the injected glucose bolus.

The input to this system is taken to be the impulse function due to the orally ingested glucose bolus [G], while the output of the model is the blood-glucose concentration response y(t). For an impulse glucose-input, a normal patient's blood-glucose concentration data is depicted in **Figure 16** by open circles. Based on the nature of this data, we can simulate it by means of the solution of the Oral-glucose regulatory (second-order system) model, as an under-damped glucose-concentration response curve, given by:

$$
\mathbf{y}\left(\mathbf{t}\right) = \left(\mathbf{G} / \alpha\right) \mathbf{e}^{-\mathbf{A}t} \text{sinot},
\tag{2}
$$

$$
\alpha = \left(\alpha\_n^2 - A^2\right)^{\bigvee}\_2
$$

wherein A is the attenuation constant, is the damped frequency of the system, thenatural frequency of the system = ωn, and λ = 2A/Td.

The model parameters λ and Td are obtained by matching eqn.(1) to the monitored glucose concentration y(t) data (represented by the open circles). The computed values of parameters are: λ = 2.6 hr-2, Td = 1.08 hr. This computed response is represented in Figure 1 by the bottom curve, fitting the open-circles clinical data.

ParametricIdentification (sample calculation for Normal Test Subject No.5)

$$\begin{aligned} \text{y } (1/2) &= (\text{G } / \,\rho\text{)e}^{-\text{A}/2} \sin \theta / \,\text{2} = 0.34 \text{ g } / \,\text{L} \\ \text{y } (1) &= (\text{G } / \,\rho\text{)e}^{-\text{A}} \sin \phi = 0.24 \text{ g } / \,\text{L} \\ \text{y } (2) &= (\text{G } / \,\rho\text{)e}^{-2\text{A}} \sin 2\phi = -0.09 \text{ g } / \,\text{L} \end{aligned}$$

Using trignometry relations, we get

$$\begin{aligned} \text{A = } &0.8287 \text{ hr}^{-1} \\ \text{A = } &\alpha\_{\text{n}} = \text{A}^2 + \alpha^2 = \text{(0.82875)}^2 + \text{(2.0146)}^2 = 4.7455 \text{ hr}^{-2} \\ \text{T}\_{\text{d}} = 2 \text{A / } &\lambda = 0.3492 \text{ hr} \end{aligned}$$

Upon substituting the above values of λ and Td, the value of the third parameter,

$$\text{G = } 1.2262 \text{ g (l)}^{-1} \text{hr}^{-1}$$

For a diabetic subject, the blood-glucose concentration data is depicted by closed circles in Fig 16. For the model to simulate this data, we adopt the solution of model eqn(17), as an over-damped response function:

$$\mathbf{y}(\mathbf{t}) = \left(\mathbf{G} / \,\alpha\right) \mathbf{e}^{-\text{At}} \text{sinhot} \tag{3}$$

The solution (y = (G/ ω) e-Atsinhωt) is made to match the clinical data depicted by closed circles, and the values of λ and Td are computed to be 0.27 hr-2 & 6.08 hr, respectively. The top curve in **Figure 16** represents the blood-glucose response curve for this potentially diabetic subject. The values of Td, λ and A for both normal and diabetic patients are indicated in the figure, to provide a measure of difference in the parameter values.

It was found from these calculations that not all of the normal test subjects' clinical data could be simulated as under-damped response. Similarly, not all the diabetic test subjects' clinical data corresponded to over-damped response.

However it was found that the clinical data of these test subjects (both normal and diabetic) could indeed be fitted by means of a critically-damped glucose-response solution of the governing equation.

$$\mathbf{y}(\mathbf{t}) \, \mathbf{ = G} \, \mathbf{t} \, \mathbf{e}^{\tau^{\mathrm{At}}} \tag{4}$$

for which, ω = 0, ωn2 = A2 = λ , and Td = 2A/λ = 2 Clinically-based Diagnosis:

The blood glucose 'normal' values, used for the clinical studies, were: Fasting: 70 to 115 mg/dl, At 30th min.: less than 200 mg/dl,

At 1st hour: less than 200 mg/dl, At 2nd hour: less than 140 mg/dl,

Modeling-based Diagnosis:

24 Biomedical Science, Engineering and Technology

The input to this system is taken to be the impulse function due to the orally ingested glucose bolus [G], while the output of the model is the blood-glucose concentration response y(t). For an impulse glucose-input, a normal patient's blood-glucose concentration data is depicted in **Figure 16** by open circles. Based on the nature of this data, we can simulate it by means of the solution of the Oral-glucose regulatory (second-order system) model, as an

wherein A is the attenuation constant, is the damped frequency of the system, thenatural

ω = ω*<sup>n</sup>* − *A*

( ) <sup>2</sup> <sup>1</sup> <sup>2</sup> <sup>2</sup>

The model parameters λ and Td are obtained by matching eqn.(1) to the monitored glucose concentration y(t) data (represented by the open circles). The computed values of parameters are: λ = 2.6 hr-2, Td = 1.08 hr. This computed response is represented in Figure 1

– A/2

= =

 ω

( )( )

Upon substituting the above values of λ and Td, the value of the third parameter,

λ = ω = A +ω = 0.82875 + 2.0146 = 4.7455 hr ,

( ) <sup>1</sup> <sup>1</sup> G 1.2262 g l hr <sup>−</sup> <sup>−</sup> <sup>=</sup>

For a diabetic subject, the blood-glucose concentration data is depicted by closed circles in Fig 16. For the model to simulate this data, we adopt the solution of model eqn(17), as an

> ( ) At y t (G / ) e sinh t ω

The solution (y = (G/ ω) e-Atsinhωt) is made to match the clinical data depicted by closed circles, and the values of λ and Td are computed to be 0.27 hr-2 & 6.08 hr, respectively. The

<sup>−</sup> = ω (3)

2 22 2 2 – 2 <sup>n</sup>

 ω

y 1/2 (G / )e sin /2 0.34 g /L

 ω

– A – 2A

= = = = −

ω

y 1 (G / )e sin 0.24 g /L y 2 (G / )e sin 2 0.09 g /L δ

( ) -At y t = (G /ω)e sinωt, (2)

( ) represents

)is the derivative feedback control term with derivative-time of *Td G t*

under-damped glucose-concentration response curve, given by:

frequency of the system = ωn, and λ = 2A/Td.

Using trignometry relations, we get

d

over-damped response function:

by the bottom curve, fitting the open-circles clinical data.

( ) ( ) ( )

–1

A = 0.8287 hr

T = 2A / λ = 0.3492 hr

ParametricIdentification (sample calculation for Normal Test Subject No.5)

ω

ω

(λTdy'

the injected glucose bolus.

The test subjects have been classified into four categories:

Normal-test subjects based on under-damped model-response;

Normal test-subjects based on critically-damped model-response, at risk of becoming diabetic;

Diabetic test-subjects based on critically-damped model-response, being border-line diabetic;

Diabetic test-subjects based on over-damped model response;

### **Non-Dimensional Number for Diagnosis of diabetes:**

We decided to develop a unique diabetes index number (DIN) to facilitate differential diagnosis of normal and diabetic states as well as diagnose supposedly normal but high (diabetic) risk patients and diabetic patients in early stages of the disorder [8].

$$\text{DIN} = \frac{y(\text{max})}{G} \times A \times \frac{T\_d}{T(\text{max})} \tag{5}$$

wherein,

y(max) = maximum blood glucose value in gram/liter

G = glucose dose administered to the system in gram/liter hour

A= attenuation constant in 1/hour

Td= derivative-time (α+δ) in hours

T(max) = the time at which y(max) is attained in hour

This non-dimensional number DIN consists of the model parameters (A & Td) or (A & ωn) or (λ & Td). The DIN values for all four categories were computed from equation (5). A distribution plot of the DIN is plotted in fig 17, wherein the DIN is classified into sections with 0.2 increments (for all the four categories of subjects) and the number of subjects which fall into these sections (frequency) is determined.

In the distribution plot (shown in Fig 17), the DIN values 0-0.2 is designated as range 1, the DIN 0.2-0.4 is range 2, 0.4-0.6 is range 3, and so on up to DIN 2.2-2.4, which is range 12.

As can be seen from figure 17, normal (i.e., non-diabetic) subjects with no risk of becoming diabetic, will have DIN value less than 0.4, or be in the 1 – 2 range. Distinctly diabetic subjects will have DIN value greater than 1.2, or be in the 7 – 12 range categories.

Supposedly, clinically-identified normal subjects who have DIN values between 0.4 and 1.0, or are in the 3 – 5 range, are at risk of becoming diabetic.

On the other hand, clinically-identified diabetic subjects with DIN value between 0.4 – 1.2, or in the 3 – 6 range category are border-line diabetics, who can become normal (with diet control and treatment).

Fig. 17. DIN distribution plot of all the four categories subjects [8]. In the figure, diabetic (critically damped) category of subjects are designated as border-line diabetic; normal (critically damped) category of subjects are designated as normal subjects at risk of becoming diabetic.
