Solving for Complex Permittivity of Biomedical Tissue ...

2017 International Symposium on Multimedia and Communication Technology August 23 – 25, 2017, Classic Kameo Hotel, Ayutthaya, Thailand

Solving for Complex Permittivity of Biomedical Tissue from Open-Ended Probe Measurement Artit Rittiplang*,1 and Pattarapong Phasukkit*,2 * Faculty of Engineering, King Mongkut’s Institute of Technology Ladkrabang, Thailand 1 [email protected] and [email protected] Abstract— This paper describes basic mechanisms to solve a complex permittivity of a nonlinear equation from the openended probe measurement in biomedical tissue. It’s basic conception to solve and study the higher order nonlinear equations in biomedical engineering fields. The mechanisms are proposed the step-by-step as follows. Firstly, the equation is represented as complex function. Then, Newton’s method is used to calculate the complex permittivity. The calculation, concerned parameters of the equation are obtained from an experiment of a reliable reference.

II.

The open-ended probe is a simply non-destructive measurement and fast process based on the “S11” reflected wave of interface as shown in the Fig 1. However, this measurement is used for the Non-magnetic lossy media.

Keywords: complex permittivity, nonlinear equation, Newton method, and numerical method, and biomedical tissue

I.

Vector Network Analyzer

INTRODUCTION

Nowadays, an open-ended probe based on the “S11” reflected wave has been widely applied to measure a complex permittivity (εc) of a material under test (MUT). Because the non-destructive and non-invasive are required analyzing the properties of biological tissue, chemical material, etc. In medical engineering fields, the εc value is very importance to diagnose some abnormalities, such as breast cancer diagnostic. Also, that value can be used to design the thermo-therapeutic applicators which the electromagnetic energy is converted into thermal in the pathological tissue [1]. In chemical substances, the dielectric substrates are important in microwaves application to design the microwave drying of textile or microwave heating of liquids [1]. Some papers have commonly presented a suitable “openended” probe type and better technics in suitable applications for measuring the accurate εc value [1]–[5]. However, the basic step of εc calculation is not clear because it’s computed by a ready-made program or calculator website. Thereby, this paper proposed the complex function and Newton’s method to solve the accurate εc value of a nonlinear equation. For basic conception it brings to study and solve the higher order nonlinear equations in biomedical and other applications. In the section II, we propose the mechanism to form and evaluate the εc value of the nonlinear equation from an open-ended probe measurement and the process of the calculation is explained as a flowchart. In the section III, the “εc” calculations of the proposed method are illustrated to compare with a reliable referenced paper. Finally, the results are a conclusion in the section IV.

MECHANISMS

Coaxial Line

MUT 𝜀𝑐 = 𝜀𝑐′ − 𝑗𝜀𝑐′′

Probe

Fig 1. The principle of reflection method with material under test

The vector network analyzer operates the broadband frequency range from 300 kHz to 3 GHz. The SMA connector is applied the probe and the panel of the probe is removed as shown in the Fig 2.

Fig 2. (a) Photograph of the SMA probe, the copy right [2].

The equivalent circuit of the Fig. 1 is also presented as shown below 𝑌0

𝑌

𝐺0 𝜀𝑐5

2

𝐶0 𝜀𝑐

Fig 3. The equivalent circuit of the Probe with a MUT

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The simple admittance equation (Y) of the probe can be represented as the equation (1) [1]-[3].

Two nonlinear equations are formed to implicit function 5

𝑌 = 𝑗𝜔𝐶0 𝜀𝑐 +

𝐺0 𝜀𝑐5 2

(1)

Where, the C0 and G0 are the capacitance and conductance, respectively. They are dependent on a MUT. As well as the εc value is unknown permittivity of a MUT. The admittance is related to the measured reflection coefficient (S11). 1 − 𝑆11 𝑌 = 𝑌0 ( ) 1 + 𝑆11

(2)

The coaxial line is Y0 = 1/(50 Ω) = 0.02 S. Details for solving the εc value: 1. The C0 and G0 values are computed by testing a material with a known εc value. After, the computed C0 and G0 values are substituted to the equation (1). 2. The equation (1) much set the two equations for solving the real and imaginary parts of permittivity. Mechanisms are explained as shown below The permittivity (εc = εc' - jεc'') is formed to a complex exponential function 𝜀𝑐 = 𝑟𝑒 𝑗(𝜃+2𝜋𝑘) ; 𝑘 = 0,1,2,3, …

𝑓1 (𝑟, 𝜃) = 𝐺0 𝑟 2 cos (

5

𝑓2 (𝑟, 𝜃) = 𝐺0 𝑟 2 sin (

𝑌 = 𝑗𝜔𝐶0 (𝑟𝑒 𝑗(𝜃+2𝜋𝑘) ) + 𝐺0

5

5

(𝜃+2𝜋𝑘)

[ (4)

𝜕𝑓2 (𝑟, 𝜃) 𝜕𝑓2 (𝑟, 𝜃) ∆𝑟 + ∆𝜃 𝜕𝑟 𝜕𝜃

(13)

(𝑛+1)

− 𝑓1

(𝑛)

𝑓2

(𝑛+1)

− 𝑓2

(𝑛)

𝜕𝑓1 (𝑟, 𝜃) 𝜕𝜃 ] [ ∆𝑟 ] 𝜕𝑓2 (𝑟, 𝜃) ∆𝜃 𝜕𝜃

(14)

5𝜃 + 10𝜋𝑘 ) − 𝜔𝐶0 𝑟sin(𝜃 + 2𝜋𝑘)] 2 5 5𝜃 + 10𝜋𝑘 +𝑗 [𝐺0 𝑟 2 sin ( ) + 𝜔𝐶0 𝑟cos(𝜃 + 2𝜋𝑘)] (7) 2 5

𝜕𝑓1 (𝑟 (𝑛) , 𝜃 (𝑛) ) 𝜕𝑟 = 𝜕𝑓2 (𝑟 (𝑛) , 𝜃 (𝑛) ) [ 𝜕𝑟

𝜕𝑓1 (𝑟 (𝑛) , 𝜃 (𝑛) ) −𝑓 𝜕𝑟 [ 1 (𝑛) ] = 𝜕𝑓2 (𝑟 (𝑛) , 𝜃 (𝑛) ) −𝑓2 [ 𝜕𝑟 (𝑛)

(8)

5𝜃 + 10𝜋𝑘 ) + 𝜔𝐶0 𝑟cos(𝜃 + 2𝜋𝑘) 2

(9)

𝜕𝑓1 (𝑟 (𝑛) , 𝜃 (𝑛) ) (𝑛+1) (𝑛) 𝜕𝜃 [ 𝑟 (𝑛+1) − 𝑟 (𝑛) ] (𝑛) (𝑛) −𝜃 𝜕𝑓2 (𝑟 , 𝜃 ) 𝜃 ] 𝜕𝜃 (15)

𝜕𝑓1 (𝑟 (𝑛) , 𝜃 (𝑛) ) (𝑛+1) (𝑛) 𝜕𝜃 [ 𝑟 (𝑛+1) − 𝑟 (𝑛) ] (𝑛) (𝑛) −𝜃 𝜕𝑓2 (𝑟 , 𝜃 ) 𝜃 ] 𝜕𝜃 (16)

From (16) the equation is reformed for calculating r (n+1) and  (n+1) values as shown below (𝑛+1) (𝑛) 𝑓1 (𝑟 (𝑛) , 𝜃 (𝑛) ) [ 𝑟 (𝑛+1) ] = [ 𝑟 (𝑛) ] − 𝑱(𝑛)−1 [ ] 𝜃 𝜃 𝑓2 (𝑟 (𝑛) , 𝜃 (𝑛) )

Hence, real (G) and imaginary (B) parts are shown as + 10𝜋𝑘 ) − 𝜔𝐶0 𝑟sin(𝜃 + 2𝜋𝑘) 2

]

f1(n+1) and f2(n+1) values are set to zero, as well-known Newton’s method or Newton-Raphson method.

𝐺 + 𝑗𝐵 = [𝐺0 𝑟 2 cos (

5

∆𝑓2 (𝑟, 𝜃) ≈

𝑓1

(5)

The above equation is rewritten the two parts

𝐵 = 𝐺0 𝑟 2 sin (

(12)

Numerical method is

𝐺 + 𝑗𝐵 = 𝑗𝜔𝐶0 𝑟[cos(𝜃 + 2𝜋𝑘) + 𝑗sin(𝜃 + 2𝜋𝑘)] 5 5𝜃 + 10𝜋𝑘 5𝜃 + 10𝜋𝑘 +𝐺0 𝑟 2 [cos ( ) + 𝑗sin ( )] (6) 2 2

𝐺=

𝜕𝑓1 (𝑟, 𝜃) 𝜕𝑓1 (𝑟, 𝜃) ∆𝑟 + ∆𝜃 𝜕𝑟 𝜕𝜃

𝜕𝑓1 (𝑟, 𝜃) ∆𝑓1 (𝑟, 𝜃) [ ] = [ 𝜕𝑟 𝜕𝑓2 (𝑟, 𝜃) ∆𝑓2 (𝑟, 𝜃) 𝜕𝑟

(3)

Euler’s formula is applied the equation (5) as

5 5𝜃 𝐺0 𝑟 2 cos (

∆𝑓1 (𝑟, 𝜃) ≈

From the equation (12) and (13) are formed as matrix system.

And Y = G + jB, where G is conductance and B is susceptance. 𝐺 + 𝑗𝐵 = 𝑗𝜔𝐶0 𝑟𝑒 𝑗(𝜃+2𝜋𝑘) + 𝐺0 𝑟 2 𝑒 𝑗2

5𝜃 + 10𝜋𝑘 ) + 𝜔𝐶0 𝑟cos(𝜃 + 2𝜋𝑘) − 𝐵 2 (11)

Partial differential can be used to approximate the equations (10) and (11).

The equation (3) is substituted into the equation (1) as 5 (𝑟𝑒 𝑗(𝜃+2𝜋𝑘) )2

5𝜃 + 10𝜋𝑘 ) − 𝜔𝐶0 𝑟sin(𝜃 + 2𝜋𝑘) − 𝐺 2 (10)

Where J(n) is Jacobian Matrix of position “n”

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(17)


𝜕𝑓1 (𝑟 (𝑛) , 𝜃 (𝑛) ) 𝜕𝑟 𝑱(𝑛) = 𝜕𝑓2 (𝑟 (𝑛) , 𝜃 (𝑛) ) [ 𝜕𝑟 Where

𝜕𝑓1 (𝑟 (𝑛) , 𝜃 (𝑛) ) 𝜕𝜃 𝜕𝑓2 (𝑟 (𝑛) , 𝜃 (𝑛) ) ] 𝜕𝜃

III. (18)

3 𝜕𝑓1 (𝑟, 𝜃) 5 5𝜃 + 10𝜋𝑘 = 𝐺0 𝑟 2 cos ( ) − 𝜔𝐶0 sin(𝜃 + 2𝜋𝑘) 𝜕𝑟 2 2 (19) 5 𝜕𝑓1 (𝑟, 𝜃) 5 5𝜃 + 10𝜋𝑘 = − 𝐺0 𝑟 2 sin ( ) 𝜕𝜃 2 2 −𝜔𝐶0 𝑟cos(𝜃 + 2𝜋𝑘)

The measured Y admittance is 1.71*10-3 + j*2.72*10-2 at the frequency operation of 931MHz, the calculated C0 and G0 are 5.96*10-14 and 8.94*10-9, respectively. These parameters are referenced from the experiments [2] under a 25ºC temperature. From the equation (24) the initial value is εc(0) = 78.0177 - j4.9048 that is formed as radial (r) and phase angle () below. (0)

𝜀𝑐 (20)

And the r (0) and  (0) initial values can be approximated by (0)

𝑌 = 𝑗𝜔𝐶0 𝜀𝑐 (0)

𝜀𝑐

=

𝑌 = 𝑟 (0) ∠𝜃 (0) 𝑗𝜔𝐶0

(25)

(0) (1) 𝑓1 (𝑟 (0) , 𝜃 (0) ) [ 𝑟 (1) ] = [ 𝑟 (0) ] − 𝑱(0)−1 [ ] 𝜃 𝜃 𝑓2 (𝑟 (0) , 𝜃 (0) )

(26)

Substituting these values, we have (1) 78.1717 [ 𝑟 (1) ] = [ ] −0.0628 𝜃 307.5 2861 4.7708𝑒 − 04 ] (27) −[ ][ −36.6 3.9 −7.5505𝑒 − 05 (1) 78.2410 [ 𝑟 (1) ] = [ ] −0.0450 𝜃

(23) (24)

(0) 78.1717 = [ 𝑟 (0) ] = [ ] −0.0628 𝜃

At n = 0, From the equation (17) the function input f1(0), f2(0), and Jacobian matrix J(0) are computed.

3 𝜕𝑓2 (𝑟, 𝜃) 5 5𝜃 + 10𝜋𝑘 = 𝐺0 𝑟 2 sin ( ) + 𝜔𝐶0 cos(𝜃 + 2𝜋𝑘) 𝜕𝑟 2 2 (21) 5 5𝜃 + 10𝜋𝑘 𝜕𝑓2 (𝑟, 𝜃) 5 = 𝐺0 𝑟 2 cos ( ) − 𝜔𝐶0 𝑟sin(𝜃 + 2𝜋𝑘) 2 2 𝜕𝜃 (22)

CALCULATION

(28)

First approximation of the permittivity (1)

𝜀𝑐

Simple process is shown as

= 𝑟 (1) ∠𝜃 (1) = 78.1618 − 𝑗3.5218

(29)

We repeat the whole process to find an even better approximation. Explaining n= 1, 2, 3, 4 in the below table with error and operation time.

The admittance equation is set to the two equations by complex exponential function

n 1 2 3 4

Obtained the Y, C0, G0, frequency f. Then, the initial value εc(0) is calculated by (23)

TABLE I NUMBER OF REPETITION εc(n) Error % 78.1618 - j3.5218 7.1578 78.173... - j3.5245… 1.2178 78.173… - j3.5245… 5.4990e-06 78.173…- j3.5245... 6.938e-16

Tims (s) 0.075 0.091 0.1 0.12

Basic Newton-Raphson Method algorithm From the Table I, this algorithm code is computed using the MATLAB program with the Core 2 Duo CPU of 2.2 GHz. The above table, we suggest the number of repetition at n = 2, the εc value is

Complex permittivity εc Fig 4. Mechanisms Flowchart.

(2)

𝜀𝑐

As stated above, we propose mechanisms to solve the complex permittivity of the nonlinear equation. In the next section, this method is illustrated with the parameters in the experiment of reference paper.

= 78.17 − 𝑗3.52

(30)

And the reference paper is 𝜀𝑐 = 78.1 − 𝑗3.53

(31)

Next, we test the calculation at three temperatures to compare to the calculation in [2] reference.

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2017 International Symposium on Multimedia and Communication Technology August 23 – 25, 2017, Classic Kameo Hotel, Ayutthaya, Thailand IEEE Trans. Microwave Theory and Techniques, Vol. 56, No. 10, October 2008. [5] M, Abbas and B, Hamdoun, “Measurement of Complex Permittivity of Adhesive Materials using a short Open-Ended Coaxial Line Probe,” Journal of Microwaves and Optoelectronics, Vol. 3, No 4, October 2004.

TABLE II CALCULATE THE PERMITTIVITY COMPARE TO [2] AT 931 MHZ Tº 15ºC

Re[Y] *10-3 2.22

Im[Y] *10-2 2.78

C0 *10-14 5.83

G0 *10-9 9.59

25ºC

1.71

2.72

5.96

8.94

29ºC

1.54

2.70

6.01

8.53

εc [2]

εc

81.5j5.94 78.1j3.53 76.7j4.19

81.76j4.82 78.17j3.52 76.92j3.12

From the Table II at 15ºC and 29ºC temperatures, the proposed mechanism gives the accurate εc values than the reference paper. Furthermore, the εc values of the proposed mechanism are very close the values of ready-made program and calculator website. IV. CONCLUSIONS Some papers have presented the better technics in suitable applications for measuring the accurate εc value. However, the basic step of εc calculation is not clear because it’s computed by a ready-made program or calculator website. Thereby, we proposed the basic mechanism to solve the accurate εc value of a nonlinear equation. For basic conception it brings to study and solve the higher order nonlinear equations in biomedical and other applications. Furthermore, the εc values of the proposed mechanism are very close the values of ready-made program and calculator websites. ACKNOWLEDGMENT The authors would like to thank CAPT. Anusorn Yungkumyart at Royal Thai Navy for supporting the electrical engineering tools. Also, the authors would like to thank the Biomedical Tissue Ablation Research Center, Biomedical Engineering, Department of Electronics, Faculty of Engineering at King Mongkut’s Institute of Technology Ladkrabang, Thailand, for supporting the MATLAB program. REFERENCES [1] R. ZAJÍČEK, L. OPPL, J. VRBA, “Broadband Measurement of Complex Permittivity Using Reflection Method and Coaxial Probes,” Radio Engineering, Vol.17, No. 1, April 2008. [2] S. Seewattanapon, P. Akkaraekthalin, “ A Broadband Complex Permittivity Probe Using Stepped Coaxial Line,” Journal of Electromagnetic Analysis and Applications, 2011, 3, 312-318 doi:10.4236/jemaa.2011.38050 Published Online August 2011. [3] D. BCrubC, F. M. Ghannouchi,, and P. Savard, “A Comparative Study of Four Open-Ended Coaxial Probe Models for Permittivity Measurements of Lossy Dielectric/Biological Materials at Microwave Frequencies,” IEEE Trans. Microwave Theory and Techniques, Vol 44, No 10, October 1996. [4] Bilal Filali, François Boone, Jamal Rhazi, and Gérard Ballivy, “Design and Calibration of a Large Open-Ended Coaxial Probe for the Measurement of the Dielectric Properties of Concrete,”

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