## GH-METHODS

Math-Physical Medicine

### NO. 152

Applying first-order perturbation theory of quantum mechanics to predict and build a postprandial plasma glucose waveform (GH-Method: math-physical medicine)

**Corresponding Author: Gerald C. Hsu, eclaireMD Foundation, USA.**

__Introduction__

In this paper, the author presents his techniques of applying first-order perturbation theory of quantum mechanics to predict and build a postprandial plasma glucose (PPG) waveform based on the “perturbation factor” of carbs/sugar intake amount. This is a part of his GH-Method: math-physical medicine research methodology.

__Methods__

Initially, he applied segmentation pattern analysis to analyze his 1,825 meals with 23,725 PPG Sensor data collected during a period of 5/5/2018 – 12/13/2019. Initially, his two segments were based on both “first factor” of meal’s carbs/sugar intake amounts and “second factor” of post-meal walking steps. His low-carb meals occupy about 2/3 of the total meals (1,209 meals with 8.5 grams per meal) and high-carb meals occupy about 1/3 of the total meals (615 meals with 27.1 grams per meal). A standard waveform (curve) contains 13 data points for each PPG curve and one input data for each 15-minute time segment. His post-meal walking steps are comparable (4,238 vs. 4,282 steps). Therefore, he decided to focus on the first factor of carbs/sugar intake amount only.

Next, he applied the first-order perturbation theory of quantum mechanics to continue and extend his glucose prediction research work. The perturbation equation is expressed in the following:

**A = ~ (***A*0 + ε **A*1 )

Where *A*0 would be the known solution to a simpler but solvable initial problem and *A*1 represents the first-order term which may be found interactively by some systematic procedure. For small *ε (epsilon), *this higher-order term in the series becomes successively smaller and derives to an approximate solution.

Since the second factor of post-meal walking steps are almost equal (4,238 vs. 4,282 steps) between the low-carb case and high-carb case, he will only focus on the first factor of carbs/sugar intake amount. The author conducted the two following perturbation analysis cases:

- Using a combination of weighted carbs/sugar amount, 14.6 grams, which is equal to (1/3 * high-carbs + 2/3 * low-carbs).
- Using an average carbs/sugar amount, 17.8 grams, which is equal to 1/2 * (high-carbs + low-carbs)

He will then be able to construct two new separate PPG waveforms (curves) between 0-minute throughout 180-minutes by applying the perturbation theory.

Finally, he used his collected data to calculate and construct a waveform with a gram of carb following very closely to the perturbed waveform with 17.8 grams of carb.

__Results__

Figures 1 and 2 display both the data table and waveforms chart of low-carb pattern vs. high-carb pattern. Although their opening glucoses at 0-minute (129 mg/dL vs. 131 mg/dL) and PPG curve shapes are quite similar (two “mountain” shapes with 58% correlation), their peak glucoses (140 mg/dL vs. 156 mg/dL) and closing glucoses at 180-minutes (127 mg/dL vs. 140 mg/dL) have different results. These differences have resulted from varying glucose decaying speeds after 60-minutes, which have deeper biomedical meanings, and are extremely critical to a patient’s risk probabilities of having diabetes complications. The significance of these differences from a segmentation analysis has already been discussed in his previous publications and presentations.

###### Figure 1: Data table of low-carbs and high-carbs PPG values

###### Figure 2: Graphic chart of low-carbs and high-carbs waveforms

Figures 3 and 4 illustrate both data table and two additional “perturbed waveforms” between low-carbs and high-carbs. It should be noted that the weighted combination PPG curve (14.6 g) almost completely matches with the original PPG curve generated with real data. However, the average carb PPG curve (17.8g) is a newly generated waveform by using the perturbed factor, carbs amount, which is similar to those two original curves, low-carbs and high-carbs, but are not the same if you examine them closely. Through application of perturbation theory concept of quantum mechanics, the author could generate a predicted PPG waveform entirely based on the selected “perturbation factor” of 17.8 grams of carbs/sugar intake amount. Of course, this perturbed waveform is only an approximated curve based on the first perturbation factor, carbs/sugar intake amount.

###### Figure 3: Data table of low-carbs, high-carbs, and two perturbed PPG values

###### Figure 4: Graphic chart of low-carbs, high-carbs, and two perturbed PPG waveforms

For clarity of waveform comparison, Figure 5 further demonstrates these two newly generated perturbed waveforms by using two slightly different perturbed carb values.

###### Figure 5: Waveforms comparison

In Figures 6 and 7, the author selected 84 meals with an averaged carb amount of 18g and constructed a new waveform between 0-180 minutes. This measured PPG waveform with 18g and 138.21 mg/dL is compared against the perturbed PPG waveform with 17.8g and 138.81 mg/dL to achieve a combined (adjusted) prediction accuracy of 98.4%.

The deviation amount of PPG is 0.5% and the deviation amount of carbs gram is 1.1% (in an opposite direction). Therefore, the combined deviation is 1.6%, which yields an accuracy rate of 98.6%.

###### Figure 6: Verification data comparison between measured PPG @ 18g and perturbed PPG @ 17.8g

###### Figure 7: Verification waveform comparison between measured PPG and perturbed PPG plus a 98.4% combined accuracy using perturbation prediction theory

__Conclusion__

Glucose variance is an extremely complex biochemical and biophysical phenomenon. After a diabetes patient measures and establishes two separate initial waveforms with one low-carb meal and another high-carb meal separately, we can then collect the patient’s PPG data and draw two separate PPG waveforms accordingly. As a result, we can predict the glucose behavior by using the perturbation theory of quantum mechanics to obtain an approximated PPG waveform according to this selected carbs/sugar intake. Of course, the same method can also be applied using the second per nation factor, post-meal waking steps. In this way, a patient will have the ability to predict his/her own PPG behavior before consuming a meal or initiate post-meal exercise.