3.1. Measurement and Analysis of the Glucose Level
The dynamics of the morning BGL during the acupuncture treatment of Patient 1 is shown in
Figure 1.
Figure 1a shows the glucose level during the first 12 acupuncture treatments after which the patient contracted COVID-19, interrupting the acupuncture for 3 weeks. After that break, eight additional treatments were performed, and the glucose level measured for that period is shown in
Figure 1b. Both curves show clear oscillations, with the peaks determined by the time of applied acupuncture treatment, so the following text is focused on their description and analysis.
The time of each acupuncture treatment is indicated by vertical pink dashed lines in
Figure 1 and numerated by pink numbers above each dashed line. Thus, the treatments were performed twice a week. Usually, two treatments were applied with two days separation, followed by a five day acupuncture-free period, as described previously.
The BGL (black curve in
Figure 1a) has an initial value of 7.2 mmol/L before the acupuncture. This significantly lowers after each treatment, followed by an increase back to the high value in the acupuncture-free period, causing its oscillatory behavior, as visible from
Figure 1a. The minima of the BGL oscillations are indicated by blue arrows and numerated by blue numbers. Thus, each minimum (i.e., the blue number) can be easily related to the corresponding number of the applied treatment (pink number). It is clearly visible that each treatment causes a significant lowering of the BGL (indicated by green arrows), however, the lowering effect does not appear immediately after the acupuncture, but is shifted for 2–3 days. Thus, each treatment and its corresponding minimum, having the same order numbers, are always shifted for 2–3 days.
The lowering effect is especially strong after the 2 closer treatments (2 day separation). Therefore, we can observe the strong lowering effect in the graph after the 2 closer treatments (1 and 2, 3 and 4, 5 and 6 …) with the 2 minima corresponding to the 2 treatments. After each strong lowering effect, the glucose level raises back to the initial high level (7.2 mmol/L) during the longer acupuncture free period (5 days) for the first 9 treatments. The increase is emphasized by red arrows in
Figure 1. Finally, an overall lowering effect, indicated by violet arrows, is observed after the 10th treatment. In that case the glucose level does not increase back to the initial level of 7.2 mmol/L, but starts to lower as well. The maximal increase of about 6.2 mmol/L was observed after the 12th treatment. After that, Patient 1 contracted COVID-19, so the acupuncture and measurements were interrupted for three weeks.
Three important features can be resolved from the measured BGL shown in
Figure 1a and they are used for the predictive modelling of the BGL during acupuncture (next section):
- (i)
Lowering effect. The glucose level significantly lowers after each single acupuncture treatment (green arrows); the reduction is especially strong after 2 treatments that are closer together in time (2 days separation). A delay in the response of the body of about 2–3 days is evident.
- (ii)
Rising effect. The glucose level rises back to the initial high value during the 5 day acupuncture-free period for the first 9 acupuncture treatments (red arrows).
- (iii)
Overall lowering effect. An overall lowering effect, i.e., BGL normalization, was observed after the 10th treatment, so the glucose level dropped below 6.2 mmol/L after the 12 treatments (violet dashed-line arrow).
Figure 1b shows the BGL measurements after the illness. First, we notice that the glucose level increased to the initial level of about 7.2 mmol/L during disease. However, the same pattern of glucose level behavior can be observed as in
Figure 1a. More precisely, the measurements again show the characteristic lowering and rising effects after each treatment, but the measurements are much noisier than those from
Figure 1a. We attribute the noise directly to SARS-CoV2 and its effect on glucose metabolism [
31,
32,
33]. Another important finding is that the overall lowering effect was achieved after only four treatments, and the glucose level of about 5.7 mmol/L generally remained constant after the acupuncture treatment is stopped. The same (average) level is kept for one year after the treatment, and is shown in
Figure 1c. The average (mean) value is 5.76 mmol/L, with a standard deviation of 0.54 mmol/L, which is almost the normal BGL value. This is achieved only by acupuncture and a healthy lifestyle.
The presented BGL measurements are compared with those taken from Control 1, who has a similar disease history and similar nutrition and exercise habits. They are shown in
Figure 1d. The BGL measurements are taken after the initial rapid reduction from 9 mmol/L caused by the drugs. The mean value is 6.11 mmol/L and standard deviation 0.31 mmol/L. This BGL curve is similar to the measurements shown in
Figure 1c, taken for Patient 1 one year after the acupuncture. However, it shows significantly different properties than the BGL measured during the acupuncture treatment, shown in
Figure 1a,b. The first difference refers to the BGL fluctuations which are more rapid and random. No correlation with the day of treatment is visible as for the case of acupuncture treatment (lowering and rising effects). The overall lowering happened immediately after the therapy was started and since then the BGL fluctuate slightly around the value of about 6 mmol/L.
Thus, both treatment methods produced similar final results, lowering the BGL from about 9 mmol/L to near 6 mmol/L for both patients (Patient 1 and Control 1). The main differences are: (i) the targeted BGL value was achieved only by acupuncture treatment for Patient 1 (no drugs included); (ii) the normalization of BGL was much faster for the metformin-based therapy (Control 1), but constant therapy is necessary, which is not the case for Patient 1; and (iii) the oscillations related to the day of treatment are absent for the drug-based therapy (Control 1), which shows only standard BGL fluctuations.
The same procedure is applied to the BGL levels of Patient 2, and Control 2. The measurements are shown in
Figure 2. The main differences in the experimental conditions with respect to
Figure 1, are: (i) the frequency of the acupuncture therapy changed slightly for Patient 2. It started with 1 therapy per week (6- and 8-day separation), and continued with 2 per week (2- and 5-day separation); (ii) Patient 2 and Control 2 do not have as healthy a lifestyle as Patient 1 and Control 1, including their adherence to diet and exercise; and (iii) Patient 2 has been undergoing a metformin-based therapy in addition to the acupuncture, while Patient 1 received only acupuncture. Due to the larger separation of treatments the overall time for 12 treatments was longer for Patient 2 (67 days) than for Patient 1 (43 days).
The BGL measurements of Patient 2 during the acupuncture treatments before and after the break are shown in
Figure 2a and
Figure 2b, respectively. First, we can note that the lowering, rising and overall lowering effects are also clearly visible for Patient 2 (indicated by green, red and violet arrows respectively) as was the case for Patient 1. The effect again follows the frequency of the acupuncture, although it is different than for Patient 1 and changes during the treatment. The effect of the overall lowering also begins after the 10 therapies, as it does for Patient 1 (
Figure 2a), and the BGL increased to the elevated value during the break. The overall lowering was also achieved after the four therapies after the break (
Figure 2b) as is the case in
Figure 1b. Finally, the reduced BGL value was also present at the check performed one year after the treatment as visible in
Figure 2c. The mean value of 5.95 mmol/L and the standard deviation of 0.27 mmol/L show that the values are kept close to normal. Thus, all of the main effects of the acupuncture treatments are the same for both patients, though they have different backgrounds and different acupuncture frequencies.
The main difference in the results is that the measurements of Patient 2 are noisier, which we can attribute to the non-healthy lifestyle affecting the BGL variation, as will be explained in
Section 4. The measurements for Control 2, with similar habits and undergoing only drug therapy, are shown in
Figure 2d. In this case, the mean BGL value is 8.2 mmol/L with a standard deviation of 0.63 mmol/L. Patient 2 had a similar mean BGL value before the acupuncture treatment, when undergoing only a metformin-based therapy as Control 2.
A detailed analysis of the BGL measurements from
Figure 1 and
Figure 2 was performed to enable better visibility and understanding of the observed effects. It includes a determination of the maximal, minimal and averaged (main) values of the measured BGL during each week of the treatments. These are shown in
Figure 3. The measurements from
Figure 1 and
Figure 2 are indicated by light gray lines.
The maximal BGL values, shown by the red line and symbols in
Figure 3a, show the maximal values of BGL for Patient 1 during each of the 1 week periods (two treatments). It has an approximately constant value of 7.2 mmol/L, until the 10th treatment. This fact shows that the BGL rises back to its initial (high value), after each reduction by acupuncture in that period. After the 10th treatment, the maximal value dropped to the lower value of about 6.9 mmol/L, showing the beginning of the overall lowering effect. The average BGL value, denoted by the green line and symbols, shows that the average BGL lowers with each new acupuncture treatment, as early as the second treatment. This drop is larger after the 10th treatment when the overall lowering is observed. The minimal values of the BGL are shown by the blue line and symbol. This curve shows a similar lowering effect as the average value curve, and drops in the normal BGL range after the 10th treatment.
The same properties, but with a more obvious BGL drop for all values (maximal, average and minimal) are visible in
Figure 3b. This shows the properties of the eight treatments performed for Patient 1 after contracting COVID-19 (measurements from
Figure 1b). The maximal values start to lower after the four treatments (16th treatment) as well as the average and minimal values. All values are below the diabetes BGL limit (6.9 mmol/L). The values stabilized after the 18th treatment and stayed nearly constant even one year after the acupuncture, as is visible in
Figure 3c. The minimal and most of the average values are in the regime of normal BGL, while the maximal values are very slightly above the normal value. The values for Control 1, shown in
Figure 3d, are also nearly constant and slightly above the normal values.
The analogue analysis for Patient 2 and Control 2 are shown in
Figure 3e–h. They all show the same main properties, including overall reduction (
Figure 3e,f), achieving an almost normal BGL value which was kept for at least 1 year after the treatments (
Figure 3g). The main difference is the smaller BGL variability of the achieved BGL for Patient 2, which is probably the consequence of the metformin therapy. Another difference is the much higher BGL values for Control 2 (
Figure 3h), which are all still in diabetes range. We believe this fact is the consequence of a nonhealthy lifestyle and probably to some extent to differences in their age.
In summary, each acupuncture treatment lowers the blood glucose level, and the lowering effect is maximal approximately 2–3 days after the acupuncture. However, the glucose level rises back to the initial high value in the acupuncture-free period. These oscillations in the BGL curve occur until some critical treatment, after which it starts to lower as well, so an overall lowering is visible. The critical treatment, i.e., the overall lowering, is achieved after the 10th treatment in the first acupuncture cycle, and after the 4th treatment in the second acupuncture cycle. The drug-based therapy causes fast overall lowering, but no oscillations related to the time of treatment are visible. Another relevant feature is that all of the important effects of the acupuncture are the same for both treated patients, although they are very different in their lifestyle and the way in which they are taking drugs for diabetes. This eliminated many alternative possible reasons beyond acupuncture that might cause BGL normalization.
3.2. Modelling of the Glucose Levels
Based on the presented measurements, a model is proposed that mathematically describes the BGL over time during the acupuncture treatment. The function that describes the BGL dependence on time (
t) is denoted by
Gl (
t), and the total number of applied acupuncture treatments is denoted by
NA.
Gl (
t) has a constant value equal to the elevated BGL named
Ghigh, before the first treatment (
i = 0), where
i is the number of the already applied acupuncture treatments. For details, please see the Equations (A1) and (A2) given in
Appendix A.
Gl (
t) assumes the existence of three main effects, as was observed in the experiment presented in the previous section. Thus, it actually consists of three sub-functions (named
F1–
F3), each of which describe one of the observed effects. Their main features are shown in
Figure 4 and include:
1. Establishment of the normal glucose level by a single acupuncture treatment—lowering effect. This is described by the function
F1 (
t) (for details see
Appendix A, Equations (A3)–(A6)), which defines the lowering of the glucose level over time from the elevated level
Ghigh towards the normal level
Gnorm. The function is illustrated in
Figure 4a by the green line. The time of the acupuncture is denoted by
tAP, and the time when the effect of the acupuncture starts to be measurable is
ta. Thus, a delay in the body’s response to the acupuncture exists as was observed in the experiment, and that time is denoted by
tdelay.
2. Raising the glucose level from the normal value back to the elevated value
Ghigh after a single acupuncture treatment—rising effect. This function, named by
F2 (
t) is plotted by the red line in
Figure 4a, and its details are given in
Appendix A, Equations (A7)–(A10). The same delay time of the response (
tdelay) is assumed as that for the
F1.
The sum of these two effects is illustrated by the blue dash-dot line in
Figure 4a, and corresponds to the oscillations of the glucose level observed in the experiment after each acupuncture treatment (
Figure 1). The functions
F1 and
F2 are described by parameters
Cdown and
Cup, respectively, which determine the slope of lowering and rising. Simulations of the functions
F1 and
F2 using different values of the parameters
Cdown and
Cup are shown in
Figure 4b–d. Clearly, the increase of the parameters
Cdown and
Cup causes the slower lowering and rising effects, respectively. These actually describe the level of the body response to the treatment. We additionally assume that the constants
Cdown and
Cup in functions
F1 and
F2 depend linearly on the number of the already applied treatments. This effect is described by the two constants
a1 and
a2 (
Appendix A, Equations (A6) and (A10)). Thus, their values associated with BGL lowering and rising may change during the multiple acupuncture treatments.
The simulation of the BGL in a multiple acupuncture treatment using only functions
F1 and
F2 is shown in
Figure 4e. Although the functions clearly describe the oscillations visible in the experiment, the glucose level always rises back to its initial value
Ghigh after the acupuncture treatment. The overall lowering effect that is observed experimentally cannot be described using those functions alone. Therefore, a third function
F3 is needed to obtain the complete description, as it is described in the following text.
3. Lowering the value of
Ghigh over time towards a normal level
Gnorm—overall lowering effect. This function describes the lowering of the maximal elevated sugar level after several acupuncture treatments. It was observed experimentally after the 10 and 4 acupuncture treatments in
Figure 1a,b and
Figure 2a,b, respectively. This actually represents the long-term effect of the acupuncture treatment. It is named
F3 (
t) and is shown in
Figure 4f. This function is needed in addition to
F1, because
F2 always raises the glucose level back to the
Ghigh value after the treatment is stopped.
F3 is described by parameters
Cslow and
a3, defining the strength of the lowering effect. Formula and all details of
F3 are given in the
Appendix A Equations (A11)–(A13). In the simulation shown in
Figure 4f, this function has no effect until the eighth acupuncture treatment (indicated by violet vertical dashed line). After that, the function slowly lowers the
Ghigh value to some equilibrium level
Geq at which it again stays constant. In an ideal case,
Geq is equal to the
Gnorm; however, this may be different depending on the specific case. This function has similar properties to fracture healing vs. time after the fracture [
31]. The healing shows a nearly constant value several days after the fracture, then starts rising toward the completely healed bone, and then again has a constant value. The final function
Gl (
t) used for the simulations and for the fitting of the experimental data consists of all three of the functions
F1–
F3. The formula for this final function for the simulation of BGL, obtained after
NA treatments is given in
Appendix A, Equations (A14) and (A15).
3.3. Simulations and the Predictions of the Glucose Levels
The proposed model is used to simulate and predict the effects of acupuncture on BGL normalization. The simulations of the BGL during acupuncture, obtained using the model (Equation (A15) given in the
Appendix A), and their main properties are shown in
Figure 5. The simulations are performed for different times between two acupuncture treatments (acupuncture periods), assuming the same initial, high value of BGL
Ghigh = 7.2 mmol/L.
Figure 4a shows the simulated glucose levels as a function of time for three separations (periods) between the treatments,
T1 = 7 days,
T2 = 3 days and
T3 = 1 day, and for one two-period separation (
T4a = 2 days,
T4b = 5 days). The last case means that the acupuncture was performed on the 2nd and 7th day each week.
The first period (
T1) is longer than the time needed for the glucose level to rise back to the initial
Ghigh value, while the other two periods (
T2 and
T3) are shorter than that time. Therefore, the glucose level rises back to the initial value between two acupuncture treatments for the
T1, while it stays lower for the
T2 and
T3. The fourth simulation in
Figure 5a is calculated for two different separations between the treatments, as was the case in the experiment (shown in
Figure 1), so it shows mixed behavior as will be explained.
All simulations show regular oscillatory behavior with periodic lowering and rising effects. However, the shape of the simulated curves depends strongly on the time between the two treatments. The simulation with a long time between the two treatments (
T1) shows deep oscillations because the BGL rises back to the initial high value between the two treatments, which is not the case for the shorter period (
T2 and
T3). The unequally spaced treatments (
T4) show the characteristically deep double-minima that occurs after the two closer treatments (separated by 2 days), followed by a rising of the BGL back to the initial high value after the longer acupuncture-free period (separated by 5 days), as was observed in the experiment shown in
Figure 1.
All simulations assume a delay in the body’s response
tdelay = 3 days. The other parameters describing body response (
Cdown = 0.7,
Cslow = 14.0), and their dependence on the number of already applied treatments (
a1 =
a2 = 0), and strength of the lowering effect (
a3 = 1) are taken to be the same for all four simulations. The function
F3 is assumed to lower
Ghigh after the eighth (
iD = 8) acupuncture treatment for all cases. This is clearly visible in the simulations from
Figure 5a. The glucose level rises completely to its initial
Ghigh value for the period
T1. For the other two cases the next acupuncture occurs before the glucose levels raise back to the initial value, so the maximal glucose levels are smaller than the initial
Ghigh value. The function
F3 lowers their values as well after the eighth treatment, but its effect is not so obvious as for the
T1.
The main properties of the different periods are more visible in
Figure 5b, where the maximal, average and minimal values of the simulated glucose levels are shown. The simulated values are shown by the light blue line. The averaging and the determination of the maxima and minima are performed over the time of two acupuncture periods (14 days for
T1, 6 days for
T2, 2 days for
T3 and 7 days for
T4).
For the largest separation between the acupuncture (
T1), the maximal value is constant and equal to
Ghigh before the characteristic 8th treatment (indicated by the vertical black dashed line in
Figure 5b), and lowers after it. Minimal and average values lower slowly before the 8th treatment and more rapidly after it, due to the effect of
F3. For the other two equal periods (
T2 and
T1) the lowering effect is visible from the beginning of the treatment and increases with the decrease of the period between the treatments. This is expected because the time between the treatments is shorter than the time needed for the glucose level to rise to the
Ghigh value, so each new treatment starts lowering from the smaller BGL maximal value. Therefore, the average value of the BGL is smaller for the shorter periods. The effect of
F3 is not so obvious as for the case of
T1, but the function
F3 is still needed to fully describe the BGL normalization.
From this consideration it seems that the most efficient period of acupuncture treatments is the one that is shorter than the time in which the BGL raises back to the initial high value. That time is characteristic of the body’s response, and it differs for different people. Of course, each person reacts differently, so the real numbers of parameters used for the simulation should be introduced to the model to obtain a simulation for each particular case. These numbers of parameters can be estimated after several acupuncture treatments.
At the end we conclude that the simulations clearly describe all of the main experimentally observed features of the BGL during the multiple acupuncture treatments. First, the lowering and rising effects are clearly visible and their values depend on the time between the two treatments. If that time is short (such as a 2 day separation in the experiment) the glucose level stays low, while it raises back to the high value during the longer (5 day) acupuncture-free time. The delay in the response of the body is included in the model as was observed experimentally. Finally, the overall lowering effect which lowers the maximal values of the blood sugar towards the normal value is taken into account.
3.4. Analysis of the Measured Glucose Using the Proposed Model
In this section we apply the above given model to the analysis of the measured BGL shown in
Figure 1 (Patient 1), and to those measured under similar conditions for Patient 2. The experimentally measured data are fitted to the model and the best-fit parameters are determined. The experimental data and the best-fit simulated BGL curves for both patients are shown in
Figure 6. The parameters of the model are determined to best fit the experimental data, using the function
Gl (
t), given in
Appendix A, Equation (A15). The results of the fitting for Patient 1 and Patient 2 are shown in
Figure 6a and 6b, respectively. The experimental data are shown by the green line and symbols while the fit is shown by the light-blue full line. First, we note that the acupuncture periods (the time between the treatments) for Patients 1 and 2 are slightly different for the first several treatments. Patient 1 has two close treatments (2 day separation) followed by a 5 day acupuncture-free period. The separations for Patient 2 are longer (6 and 8 days). This strongly affects the shape of the BGL curves, as well as the fitted curves. In the first case the curve shows deep double-peaks (similar to the simulation for
T4 in
Figure 4), while the oscillations for Patient 2 are more regular due to the similar and longer times between the treatments (similar to
T1 in
Figure 5).
From the comparison of the experimental and model curves, we can see that the model curve well describes the experimental data. All of the main features (BGL lowering, rising and overall lowering) are visible in the fitted curves. They also well follow the shape of the oscillations caused by different times between the treatments. Patient 1 was very disciplined in the application of diet and exercise proposed for diabetics, so the measured BGL level is practically noise-free, especially before the break. Therefore, the experimental and the modelled curves are very similar. So, the BGL is mainly affected by the acupuncture, and not by improper eating or body activity. However, the measurements of Patient 2, and the measurements of Patient 1 after the break caused by COVID-19 contains additional ‘noise’ that originates from the non-ideal diet and exercise application, and from the recovery of the illness in the case of Patient 1. Of course, even in ideal conditions, the BGL oscillates slightly due to the normal body functions. Therefore, the data for Patient 2 are much noisier, causing larger differences between the experimental and the modelled curves. However, all of the main effects mentioned above are clearly visible for both patients.
The main properties (maxima, average, minima) of the experimental and simulated BGL curves for Patients 1 and 2, as well as the parameters obtained by the fits are given in
Figure 7. The maximal, minimal and average values for Patients 1 and 2 are compared in
Figure 7a and b, respectively. It follows that the main properties of the simulated curves are very similar to the experimental curves, especially the maximal and average values, for both patients. The minima of the simulated curves are slightly higher than the experimental, probably due to the limitations of the functions used for the description. We intentionally use simple functions that can be easily used by non-experts. In addition, it is important to note that the maximal and minimal values are strongly affected by the random fluctuations of the experimentally measured BGL, while they are not included in the model. Therefore, for example, the last maximal value of the experimental curve deviates from the modelled one in
Figure 7b. Interestingly, for both patients the overall lowering effects starts after the 10th treatment, and becomes obvious after the 14th treatment what will be discussed later in the text.
The parameters of the fits that describe the simulations, and their dependencies on the number of treatments are shown in
Figure 7c,d for Patients 1 and 2 respectively. Four parameters are fitted:
Ghigh,
Cup,
Cdown and
tdelay. The value of the
Ghigh (shown by the blue line and symbols) is constant until the 10th and 14th treatments in the first and second sets of the measurements, respectively, as was experimentally observed for both patients. It then lowers toward the normal BGL. The constant
Cdown (black line and symbols) has smaller values than the
Cup (red line and symbols), also for both patients, showing that the glucose level drops faster towards the normal level after each single treatment than it rises back to the high value. Another important feature of these two constants is that
Cdown increases with the number of treatments, while
Cup decreases, also for both patients. This shows that the glucose level lowering effect becomes faster with each new treatment, while its rising back to the high value becomes slower. Finally, we have determined that the time of the body response to the acupuncture is between 2 and 3 days for the first set of the acupuncture treatments. This shows similar values at the beginning of the second set of treatments, but drops toward a value of 1 after 8th treatment of the second set.
The main difference between the response of the two patients is the value of the constant C
up. However, this constant shows the same type of behavior as just described, with Patient 1 having significantly lower values (about two-times lower). Its influence on the BGL lowering and rising is shown in
Figure 7e, where the effects of a single treatment of acupuncture on BGL are compared for Patients 1 and 2. Patient 1 shows significantly faster rising of the BGL after the acupuncture before and after the break, (i.e., the smaller values of the constants) than Patient 2. This effect may be attributed to the metformin, which is taken only by Patient 2, as will be discussed later.
In summary, the results of the BGL modelling show that both patients have very similar types of reaction to acupuncture. They both show similar effects of BGL lowering and rising, similar response time to acupuncture, and practically the same overall lowering of BGL after the 10th and 14th treatments. The main difference is the level of the body’s response which is slightly stronger for Patient 2, and the time of the BGL rising to the elevated (high) value, which is also longer for Patient 2.