Vol 16, #2 - A Multidimensional Optimization Process For Enhancing Electroacupuncture Efficiency

Vol. 16, #2

A Multidimensional Optimization Process For
Enhancing Electroacupuncture Efficiency
Bin Chen, PhD
Thomas Yee, MD
Mehlika Ayla Kiser
Bingmei Fu, PhD

ABSTRACT
Most electroacupuncture devices function by adjusting variables such as current strength, frequency, stimulating pattern, and duration. To achieve a therapeutic result, repeated stimuli must be applied to at least 10 acupuncture points each time. Four variables on an individual point would produce numerous possible combinations and the therapy would be too time-consuming. The goal of our study was to improve acupuncture efficiency by identifying the optimal combinations of these variables. To do this, we first quantified the stimulated feeling as a comfort level and measured this comfort level as a function of current strength, frequency, stimulating pattern, and duration for each acupuncture point. Experiments were conducted on 3 subjects. Least-squares multidimensional curve fitting method was applied to these experimental data to determine the closed-form of this function (comfort level k = function k (current, frequency, pattern, duration); k = 1, 2, 3 ...n; n is the total number of the acupuncture points, n $10). Multivariable optimization technique (Quasi-Newton method) was then used to discover the value of current strength, frequency, stimulating pattern, and duration that gave the best comfort levels in terms of a comprehensive effect for n points.
KEY WORDS
Acupuncture, Efficiency, Comfort Level, Least-Squares, Quasi-Newton, Electrostimulation

INTRODUCTION
Acupuncture began among the Chinese 5,000 years ago.¹ Ancient diagrams depicting human and animal bodies labeled with acupuncture points are evidence of this method of healing. Though acupuncture has been practiced for thousands of years in Asia, the mystery surrounding this method has only recently begun to fade in Western societies and acceptance is gradually taking root.

Acupuncture did not enter modern Western consciousness until the 1970s.² For the past 30 years, Western physicians have been studying and incorporating acupuncture into Western medical culture, successfully complementing modern medicine with the ancient Chinese techniques.³

According to traditional Chinese medicine (TCM), a form of bodily energy is generated in internal organs and systems. This energy combines with the breath and circulates throughout the body, forming paths or "meridians."² In China, this energy force is deemed the "life force" and is termed "Qi" or bioenergy.⁴ The meridians form a complex multilevel network that connects the various areas of the body, from the surface to the internal organs. All of the various meridian systems work together to ensure the flow and distribution of Qi throughout the body, thus controlling all bodily functions.²

Chinese medical theory states that illness is attributed to Qi in the meridian systems. Treatment is applied to acupuncture points, which are located throughout the body. Twelve regular meridians, also called major trunks, run deep within the tissues and organs of the body, surfacing these points on the skin.4 Originally, 365 such points existed, corresponding to the days of the year. The number identified by proponents during the past 2,000 years has increased gradually to about 2,000.^2,5
Some practitioners place needles at or near the site of disease, whereas others select points on the basis of symptoms. In traditional acupuncture, a combination of both selection methods is widely used.

Since the 1950s, the electrical properties of acupuncture meridians have been extensively studied by many researchers.⁶ Nakatani⁷ found that many points on the skin have high electrical potential. He called the lines connecting these points the "ryodoraku," which means "good conduction lines." Most of these characteristic points and lines were confirmed to be acupuncture points and meridians of TCM.^8,9 Reichmanis et al10 found that acupuncture points have a larger capacitance than the neighboring skin. Chen^11,12 also found that the meridian system exhibited special electrical properties (current conduction and wave propagation). Furthermore, a model has been proposed by Shang¹³ suggesting that acupuncture points are organizing centers. At the macroscopic level, they are singular points (e.g., sinks, sources) in the electromagnetic field.

Electroacupuncture (EA) is based on these theories and experiments and was introduced in the 1950s in connection with the development of acupuncture anesthesia in China.14 During acupuncture anesthesia, it is necessary to stimulate the acupuncture points for a long period. Mechanical stimulation by hands can be difficult; electrical current stimulation was thus first introduced and further developed in the subsequent years not only for anesthesia, but also for the daily practice of therapeutic acupuncture. Electroacupuncture is now considered a routine method of stimulation. Though perhaps somewhat less effective than traditional acupuncture, EA does have several advantages over the manual techniques. The main advantage is that of ease. Traditional acupuncture should be practiced only by an experienced physician. An EA device is portable and can be used at home, school, in the office, and while traveling. Because of the extra stimulation provided by the electrical current, the treatment of a point with EA is performed within seconds rather than minutes.¹⁵ Electroacupuncture devices can also decrease the risk of infection while in traditional acupuncture, the needles, if not properly sterilized, can spread infectious diseases.

Most EA devices function by adjusting variables such as current strength (c), frequency (f), stimulating pattern (p), and duration (d). To achieve a therapeutic result, stimuli need to be applied to at least 10 acupuncture points in a patient simultaneously. Four variables on an individual point of more than 10 points would result in numerous possible combinations and make the therapy too time-consuming. For example, Figure 1 shows the 14 acupuncture points for headaches.¹⁶The aim of our study was to improve EA efficiency by searching the best combinations of these variables. To do this, we first quantified the stimulated feeling as a comfort level and measured this level as a function of "c, f, p, d" for each acupuncture point. Least-squares multidimensional curve fitting method was applied to these experimental data to determine the closed-form of this function (comfortk = f k (c, f, p, d), k = 1, 2, 3, ...n, n is the total number of the acupuncture points, n$10). Multivariable optimization technique (Quasi-Newton method) was then used to find the value of c, f, p, d that produced the best comfort levels in terms of a comprehensive effect for n points.

MATHEMATICAL MODELING
Least-Squares Multidimensional Regression Method
We now considered the problem of fitting a function to a relatively large amount of data that contain errors. It would not be sensible or computationally possible to fit a very high-order polynomial to a large amount of experimental data as may be done for interpolation. What is required is a function that stabilizes the fluctuations in the data due to errors and reveals any underlying trend. We therefore adjusted the coefficients of a chosen function to provide a "best fit" according to some criterion. For example, the criterion may be to minimize the maximum error, the sum of the modulus of the errors, or the sum of the squares of the errors between the chosen function and the actual data points. The last criterion, also called "least-squares method," is the most widely used criterion.

We have m sets of observations of y and of p separate explanatory variables, x₁, x₂ ... x_p. The variables x₁, x₂ ... x_p are generally independent, which are measured with negligible error or even controlled in an experiment, and y is a single dependent variable, uncontrolled and containing random measurement errors. For the ith observation, the relationship between the equation and the data can be expressed generally as follows:

(1)

where yi = measured value of the dependent variable, i = number of measurements, f ( x_1ix_2i ... x_pi; a₀ a₁a₂... a_q) = a function of the independent variables x₁, x₂ ... x_p and of the parameters a₀, a₁ ... a_q, and the ´i = a random error. Initially, we simply assume that these random errors are identically distributed with a zero mean and a common unknown variance, s2, and that they are independent of each other. This implies that

(2)

represents the fitted function value at x1i, x2i ... xpi.

So, the difference between the observed and the fitted value for the ith observation

(3)

is called the residual and is an estimate of the corresponding ´i. Our criterion for choosing a0, a1 ... aq is that they should minimize the sum of the squares of the residuals, which is often called the sum of squares of the errors (SSE). Thus,

(4)

To perform the necessary calculation efficiently, we rewrite the model in a matrix form as follows. We define a as a (q+1)x1 vector such that

(5)

Similarly, we define e, y, u to be m31 vectors thus:

(6)

(7)

(8)

From Equations 2 and 3, the residuals may be written as: e = y – Xa, where X is a variable matrix, each of whose component corresponds to a constant in a and the SSE. From Equation 4, is then

(9)

Differentiating the SSE with respect to a, we get a vector of partial derivatives, as follows:

(10)

Equating this derivative to zero we have

(11) Xa = y

These equations could be solved directly to obtain the coefficients a, which minimize SSE.

Figure 1. Fourteen acupuncture point locations
for the treatment of headache

Multidimensional Regression
The current strength that a person can bear is roughly up to 2 mA.13 Many EA devices actually do not mark the real current values, but rank the values from 0 to 9, with 0 representing no current and 9 representing 2 mA current. The duration is normally from 5 to 30 seconds. In most cases, 2 frequency choices exist: f1 = 50 Hz and f2 = 100 Hz, and 3 pattern choices: double burst (p1), train-of-four (p2), and random burst (p3). In our model, we assume that the stimulated feeling can be quantified as a comfort level, and we can measure this comfort level as a function of c, f, p, d for each acupuncture point. Since we only have 2 frequency choices and 3 pattern choices, there are 6 combinations of f, p: f1p1, f1p2, f1p3, f2p1, f2p2, and f2p3, for each c and d at each acupuncture point, where 6 independent functions of c and d can be obtained as:

(12)

where i = number of measurements for each combination of f and p at an individual point, j = 6 combinations of f, p, and k = number of acupuncture points. Least-squares regression leads to a fitting function Comjk (c,d) at each point for each combination of f and p. Finally, the comprehensive effect of n points for one combination of f, p on a patient's body is:

(13)

Figure 2 shows the experimental data (scattered stars) for point

BL 9 in Figure 1 by using f = f1 (50 Hz) and p = p1 (double burst). To fit the data shown in Figure 2, we assume that c and d contribute equally to the comfort level, and a 3rd order polynomial is applied to represent the fitting function,

(14)

In Equation 14, aj0k, aj1k, aj2k, aj3k, aj4k, aj5k, and aj6k are constants to be determined based on the experimental data using the least-squares regression method described. Correspondingly, in our acupuncture model,

(15)

(16)

(17)

We can apply Equation 11 to calculate all the constants in Equation 14 and eventually, the comfort level function (smooth surface in Figure 2) can be determined for each combination of f and p at an individual point.

Figure 2. Experimental data (*) and the fitting surface by using the multidimensional least-squares regression method for point BL 9 in Figure 1 under f1 = 50 Hz and p1 = double burst. The unit for duration is second.

Quasi-Newton Optimization Method
Although a wide spectrum of methods exist for optimization, methods can be broadly categorized in terms of the derivative information that is, or is not, used. One is "search method," the other is "gradient method." Gradient methods are generally more efficient when the function to be optimized is continuous in its 1st derivative, such as in our case. In these methods, the most favored are the Quasi-Newton methods, which are based on Newton's methods.

In the unconstrained optimization problem, we seek a local maximum of a real-valued function, f(x), where x is a vector of p variables. That is, we seek a vector, x*, such that f (x*) Ž f(x) for all x close to x*.

Global optimization algorithms try to find an x* that maximize f over all possible vectors x. This is a much harder problem to solve. We do not discuss it here because at present, no efficient algorithm is known for performing this task. For many applications, local maxima are sufficient, particularly when the user can rely on his/her own experience to find the global optima out of local ones.

Newton's method gives rise to a wide and important class of algorithms that require computation of the gradient vector

(18)

And the Hessian matrix (H),

(19)

Newton's method forms a quadratic model of the objective function around the current iterate xk. The model function is defined by

(20)

At the optima,

(21)

If | H |> 0 and d²f/dx^{2 > 0}, then f has a local minimum

If | H |> 0 and d²f/dx^{2 < 0}, then f has a local maximum

If | H |< 0 then f has a saddle point.

Thus, combining Equations 18 and 21, at the optima,

(22)

If H is nonsingular, the iterating formula for finding optima is:

(23)

Quasi-Newton methods seek to estimate the direct path to the optimum in a manner similar to Newton's methods. They can be used when the Hessian matrix is difficult or time-consuming to evaluate. Notice that the Hessian matrix in Equation 19 is composed of the 2nd derivatives of f that vary from step-to-step. Quasi-Newton methods attempt to avoid these difficulties by approximating H with another matrix, using only 1st partial derivatives of f. A large number of Hessian updating methods have been developed. Generally, the formula of Broyden,17 Fletcher,18 Goldfard,19 and Shanno20 is thought to be the most effective for use in a general-purpose method. The formula is

(24)

where S_k =x_k+1–x_k, qk = Df(x_k+1) – Df(x_k). As a starting point, H0 can be set to any symmetric positive definite matrix; for example, the identity matrix I. In our study, we used the algorithm in MATLAB 6.0 optimization toolbox to perform this process.

EXPERIMENTAL PROTOCOLS
Equipment
The device used for electrical stimulation is peripheral nerve stimulator (Model MS-III, MiniStim). The stimulator has 3 operational variables: frequency (f), stimulating pattern (p), and current strength (c). The frequency has 2 setting options: f1 = 50 Hz and f2 = 100 Hz. Three patterns are: p1 = double burst, p2 = train-of-four, and p3 = random burst. The scale of the current strength is set from 0 to 9, representing the current from 0 to 2 mA. The stimulator has output electrodes in the form of 2 stainless steel nodes, 1 positive and 1 negative, so that the generated current will flow from the positive node to the negative node.

Each node has a diameter of approximately 0.06 in (1.6 mm) and a length of 0.44 in (11.1 mm). The nodes have a distance of 0.2 in (4.8 mm) between them. Acupuncture points have been measured to be approximately 0.08 in2 (2 mm2), so the distance between the nodes is more than enough to cover the area and stimulate the points.¹⁵

^{Point BL 9 in Figure 1 (for subject 1)}

^{Figure 3. Fitting surfaces for the comfort level as a function of current c and duration d for point BL 9: (a) f1 = 50 Hz, p1 = double burst, (b) f1 = 50 Hz, p2 = train-of-four burst, (c) f1 = 50 Hz, p3 = random burst, (d) f2 = 100 Hz, p1 = double burst, (e) f2 = 100 Hz, p2 = train-of-four burst, (f) f2 = 100 Hz, p3 = random burst. The unit for duration is second. "•" indicates the optimal point.}

Comfort Level Definition
There is ambiguity and subjectivity in defining the comfort level. In our study, comfort was defined as a recognizable feeling of stimulation that is free from pain. The "optimal" comfort level we wanted to find is the most desired feeling for the patient when using the EA device. The "uncomfortable" level is determined by a recognized feeling of stimulation that is painful but bearable. Finally, "intolerable" signifies that the recognizable feeling of stimulation has a degree of pain that cannot be endured beyond a second or so. The numerical value for each level was assigned specifically for our optimization purposes. The most desired state (ideal maximum comfortable level without feeling pain) was assigned the highest numerical value, 10. The level without recognized feeling, or with intolerable feeling, was assigned the value of 0. Comfort levels between no feeling, intolerable feeling, and most desired feeling are assigned numerical values greater than 0 but less than 10, somewhat subjectively.

SUBJECTS
Three subjects were treated in our experiment. Because we tested the comfort level of subjects during EA application, the subjects could be in any state of health. Subject 1 was a 70-year-old man with Parkinson disease, high blood pressure, and headache; subject 2 was a 57-year-old woman with frequent headaches; and subject 3 was a healthy 23-year-old man with occasional headaches.

Acupuncture Points for Testing
Five of the 14 points for headache were chosen for testing to provide several different acupuncture points on various regions of the body. The 5 points were BL 9, BL 56, GV 19, LI 4, and ST 36, which are located on the neck, foot, head, leg, and hand (Figure 1).

Data Acquisition and Analysis
Figure 2 shows the example for data acquisition and analysis using point BL 9. Frequency and stimulating pattern in Figure 2 were f1 (50 Hz) and p1 (double burst), respectively. We adjusted the current strength (c) starting from 0 (0 mA), gradually increasing to 8 (~1.8 mA) by 5-6 increments. The level 1.8 mA was the maximum current strength the subject could bear in our experiments. For each current strength, we measured the comfort levels for the duration of 5, 10, 15, 20, 25, and 30 seconds. Least-squares regression method was applied to the measured data for the comfort level and a fitting function Comf1p1 (c, d) was obtained. This function is a 3rd-order polynomial as expressed in Equation 14. The corresponding fitting surface is shown in Figure 2.

At the same point, we repeated the same procedures for 5 other combinations of frequency and pattern: f1p2, f1p3, f2p1, f2p2 and f2p3. Five more fitting surfaces for the point BL 9 of subject 1 are shown in Figure 3, in addition to the one for f1p1 shown in Figure 2. Measurements were repeated on 4 other acupuncture points; results are shown in Figure 4(a). Results for subjects 2 and 3 are shown in Figures 4 (b,c).

RESULTS
For each acupuncture point k, we can get 6 fitting functions for 6 combinations of f and p: Com ^j_k (c,d) , j = 1,2...6, and k = 1,2...n. n is the total number of points for a treatment. We use the 5 acupuncture points for headache, BL 9, BL 56, GV 19, LI 4, and ST 36, as an example to demonstrate our model results for several subjects. Figure 3 shows the 6 fitting functions at point BL 9 in Figure 1 for subject 1. These 6 plots in Figure 3 are similar in shape, which indicates 2 parameters for stimulation, frequency, and pattern may not be that sensitive in the EA treatment for individual points. In fact, for all 6 combinations of frequency and pattern, the maximum comfort levels ranging from 5.9 to 8.3 were achieved when the current strength was ~1-2 (0.2-0.4 mA) and duration was ~5 to 12 seconds.

Figure 4a, 4b, and 4c. Fitting functions for the comfort level at 5 points for headache treatment under f1p1: BL 9, BL 56, GV 19, LI 4, and ST 36. The unit for duration is second. See the text for a description of each subject.

4a

4b

4c

Another observation from Figure 3 is that the higher the frequency is applied, the higher the maximum comfort level can be achieved. However, the comfort levels do not increase remarkably with increasing frequency. When frequency is doubled, from 50 to 100 Hz, the percentage increases in the maximum comfort levels are 19%, 17%, and 5% for p1, p2, and p3. Therefore, in this way, we can stimulate an individual point by adjusting other parameters to achieve the therapeutic effect without changing frequency and pattern.

In contrast, the same frequency and pattern may induce distinctly different effects on different acupuncture points. Figure 4(a) shows the fitting functions at 5 points in Figure 1: BL 9, BL 56, GV 19, LI 4, and ST 36 for the same f1p1 for subject 1. These plots have various appearances. While the most desired level is achieved for point BL 9 when c ~= 2, d ~= 5 seconds, it is achieved when c ~= 3.2 and d ~= 25 seconds for point BL 56. Figures 4(b,c) and are for subject 2 and 3, respectively. They have similar appearances as those for subject 1 for the same acupuncture point regardless of ages and health conditions.

If we have enough instruments, we can apply different parameter values on each point to achieve the best treatment effect. To decrease the cost and make the treatment easier and more convenient, for each combination of frequency and pattern, we can define the comprehensive effect function at these n points as

The comprehensive effect function of our 5 testing points (n = 5) under f1p1 for subject 1 is shown in Figure 5(a). From this function, we can discover the maximum value for the comfort level, ~7, when the current strength ~ 2 (~ 0.44 mA) and duration ~ 5 seconds by using Quasi-Newton's method. Comfort level × 7 is the local optimum when c × 2, d × 5 seconds f = f1 (50 Hz), and p = p1 (double burst) for the comprehensive effect. After obtaining the comprehensive effect functions for other 5 combinations of frequency and pattern, we can compare the 6 local optima for the 6 combinations of frequency and pattern to find the global optimum for subject 1. Figures 5(b,c) show the comprehensive effect function of those 5 testing points under f1p1 for subjects 2 and 3, respectively.

Comparing Figures 5(a,b,c) for different subjects with different
ages and health conditions, we can see that their maximum comfort levels happen almost at the same stimulating conditions when the current = 2 and the duration = 5 seconds. This suggests that we can apply the same stimuli to different subjects to achieve the same effect, regardless of age, sex, and health conditions. As shown in Figure 5, the younger and healthier subject (3) achieved the highest comfort level among all 3 subjects. Subjects 1 and 3 in general had higher comfort levels than did the woman subject (2). This suggests that males of younger ages are more sensitive to EA treatment than females and older patients.

So far, by using this optimization process on measured data from the subjects, we can determine the best combination of 4 parameters, c, d, p, f, for the treatment of each individual patient. When the subject presents again with the same illness, we can use this set of values for the treatment. In addition, the same treatment parameters can be applied to different patients with the same illness regardless of their sex, age, and general health conditions. In this way, the treatment efficiency can be improved.

Figure 5. Comprehensive effect function of the 5 points for headache for 3 subjects under f1p1. The unit for duration is second. "•" indicates the optimal point

(a) Subject 1

(b) Subject 2

(c) Subject 3

DISCUSSION
In this study, we proposed a multivariable optimization method aimed at enhancing the efficiency of EA treatment. Experiments were designed to determine the best combination of parameters in EA on multiacupuncture points that would yield the most desired therapeutic effect. The results are bound to several uncertainties. For example, we obtained the data from the subjective perception of the EA stimulation which change under different health conditions. In addition, this study only tested the short-term effects. The next step would be to test the long-term effects of the application of EA.

CONCLUSION
The aim of EA research is to find the combination of parameter settings or to search for an improved design of an EA device that is efficient and can bring better health benefits. To obtain more comprehensive and conclusive results regarding the effects on different age and sex groups, further studies involving more subjects are needed.

ACKNOWLEDGEMENTS
We would like to thank George Ladkany, Vitality Skrinpken, and Jason Gough for their help in processing the experimental data.

FUNDING/SUPPORT
This work was supported by grants from Hypermedical Inc./ PrimaryCare Inc., University of Nevada, Las Vegas Applied Research Initiatives (ARI), and National Science Foundation Grant for Research experiences for undergraduate (REU).

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AUTHORS' INFORMATION
Bin Chen is a post-doctoral researcher in the College of Pharmacy at Ohio State University and has a PhD in Mechanical Engineering.
Bin Chen, PhD
E-mail: chen.1148@osu.edu
Dr Thomas Yee is a Surgeon in the Department of Surgery at the University of Nevada, School of Medicine.
Thomas Yee, MD
E-mail: r1888@aol.com
Mehlika Ayla Kiser is an undergraduate student in the Department of Mechanical Engineering at the University of Nevada, Las Vegas, Nevada.

Bingmei Fu is Associate Professor, Department of Biomedical Engin-
eering, at The City University of New York in New York, NY.
Bingmei Fu, PhD*
Asso Professor, Dept of Biomedical Engineering
City University of New York
138th at Convent Ave
New York, NY 10031
Phone: 212-650-7531 • Fax: 212-650-6727
E-mail: fu@engr.ccny.cuny.edu

*Correspondence and reprint requests

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