Bioelectrode Models with Fractional Calculus

Transcription

1 INTERNATIONAL JOURNAL OF INTELLIGENT CONTROL AND SYSTES VOL. 17, NO. 1, ARCH 1, 7-13 Bioelectrode odels with Fractional Calculus Emmanuel A. GONZALEZ, L ubomír DORČÁK, and Celso B. CO Abstract The introduction of fractional calculus has spurred a lot of possibilities in biomedical engineering and bioengineering research and development, especially in the modeling, design, and analysis of bioelectrodes. With the advent of memristors and fractional calculus, it is seen that these concepts would have several advantages and applications in such fields. In this short note, we present various bioelectrode equivalent circuit models including their impulse and frequency responses. The equivalent models presented utilize the concepts of memristors and fractional calculus. Index Terms Fractional calculus, fractor, memristor, bioelectrode. I. INTRODUCTION Recording and stimulating bioelectrodes is a very important part in the field of bioengineering as it deals with the optimal design of such components for electrode/tissue interface applications. The design of such components would, however, start with the development of a mathematical model and possibly an equivalent circuit model for the electrical impedance of the interface. A lot of efforts have also been directed toward the development of accurate impedance models for electrophysiological recording and electrical simulation [11], [13], [16], [18]. Such bioelectrodes can be modeled using passive electric components, i.e. resistors, inductors, and capacitors. However, most of time, models only lie with resistors and capacitors due to practical reasons. The resistance in the model of an electrode usually represents the intrinsic resistance of the electrolyte, while the capacitance is associated with the interface. By making appropriate assumptions for the values of the resistance and capacitance, the impedance then determines the response of the electrode in time and frequency domain in which, most electrodes especially designed for electrocardiogram ECG) systems, cell suspensions, and whole tissues are of the first-order [1], [1], [], [19]. Although there is a great amount of literature involving the use of integer-order dynamic models for bioelectrodes, there are still some cases in which the agreement between E. A. Gonzalez corresponding author) is with the Department of Computer Technology, College of Computer Studies, De La Salle University anila, 41 Taft Ave., alate anila 14, Philippines, and with the School of Electrical Engineering, Electronics and Communications Engineering, and Computer Engineering, apua Institute of Technology, uralla St., Intramuros anila, Philippines. He is also with Jardine Schindler Elevator Corporation, 8/F Pacific Star Bldg., Sen. Gil Puyat Ave. cor. akati Ave. akati City, Philippines emm.gonzalez@delasalle.ph) L. Dorčák is with the Department of Applied Informatics & Process Control as well as a B.E.R.G. faculty at the Technical University of Kosice, B. Nemcovej 3, 4 Kosice, Slovak Republic lubomir.dorcak@tuke.sk) C. B. Co is with the Department of Electronics, Computer, and Communications Engineering, School of Social Science and Engineering, Ateneo De anila University, Katipunan Ave., Quezon City, Philippines celso.co@gmail.com) the theoretical model of the bioelectrode s impedance and the observed impedance seems to be questionnable especially if the angular frequency of characterization follows a noninteger-order power [14]. In other words, the mathematical model describing the bioelectrode may not follow an integerorder dynamics, consequently leading to fractional calculus that is able to describe fractional dynamics dynamics of system having non-integer-order representations. II. FRACTIONAL CALCULUS Fractional calculus is a topic that has been dealt for more than three centuries up to this date, and researches in the theories and applications of this field are still rapidly growing. The term fractional calculus is actually a misnomer since the actual designation is integration and differentiation of arbitrary order, which is rather approriate in the current setting. It started in 1695 when L Hospital raised a question to Leibniz as to the meaning of d n y/dx n if n = 1/, that was answered:...this is an apparent paradox from which, one day, useful consequences will be drawn... d 1/ x will be equal to x dx : x... The derivative of abitrary real order α has the notation adt α f t), where a and t are the limits of operation, and α is the order of differentiaton or integration. For fractional differentiation, it is assumed that the order α is positive., i.e. α >. If α <, then the operation becomes a fractional integration having the general notation a Dt α f t). Fractional differentiation and integration come in many definitions as they are studied and developed throughout the years. One of the most-common definition is the Grünwald- Letnikov GL) differintegration that is defined by n ) GL a Dt α f t) = lim h ;nh=ta hα 1) r α f t rh), r r= 1) which represents the derivative of order m if α = m and the m-fold integral if α = m. The values of a and t are the limits of operation. Definition 1) is derived from the wellknown definition of the nth-order derivative f n) t) = lim n h n n 1) r n r r= ) f t rh), ) ) n where n is the order of differentiation and is the usual r binomial coefficient notation. In the particular case of 1), the bionmial coefficient results in the expression ) α Γ α + 1) = r Γ r + 1) Γ α r + 1). 3)

2 8 INTERNATIONAL JOURNAL OF INTELLIGENT CONTROL AND SYSTES, VOL.17, NO. 1, ARCH 1 Another definition of fractional differintegration is the Riemann-Liouville RL) definition given by RL a D α t f t) = d n 1 Γ n α) dt n ˆ t a f τ) αn+1 dτ, 4) t τ) for n 1 < α < n. However, especially in the frequency domain, both the RL and GL definitions lead to initial conditions containing limit values that may not be physically interpretable. In spite of having some fractional calculus intial value problems that are solved mathematically through RL and GL definitions, their solutions are practically useless as there may be no physical interpretations for such types of initial conditions. This problem is solved by. Caputo by allowing the formulation of initial conditions for intial-value problems for fractional-order differential equations in a form involving only integer-order limit values. Caputo s approach is defined by C a Dt α 1 f t) = Γ n α) ˆ t a f n) τ) dτ αn+1, 5) t τ) for n 1 < α n. Looking into Caputo s definition, the Laplace transform of the Caputo fractional derivative of order n 1 < α n is L { C a Dt α f t) } n1 = s α F s) s αk1 f k) ). 6) k= If initial conditions are not considered, especially in impedance studies, then the Laplace-transform of 5) would simply result to L { C a D α t f t) } = s α F s). 7) A thorough discussion of fractional calculus and fractional differential equations can be found in [17]. III. THE FRACTOR A fractor [8], [15] is defined as either a capacitive or inductive element that is following fractional-order properties. Fractors were most probably formally introduced by G. E. Carlson and C. A. Halijak, which is also presented by S. Westerlund and L. Ekstram, who obviously are unaware about Carlson and Halijak s work [1], [], [3], [4], [1]. Fractors are considered to be a special case of memristors a theoretical concept introduced by L. O. Chua in 1971 [5], [6], [7], which was then discovered by S. Williams and his research group of scientists in HP Labs. Fractors can be deduced as memristive devices, having properties equivalent to real electrical elements with fractional order mathematical models generally described as Z s) = Ks k, 8) where K R + is a constant representing a physical element that be a resistor, capacitor, inductor, or a generic memristor, and k R is the order of the memristive device. If k 1, ), then the memristive device is considered as partially-capacitive. On the other hand, if k, 1), then the memristive device is considered as partially-inductive. Having k = simply means that the memristive device is purely resistive, and having k equal to 1 and 1 would make the device purely inductive, and purely capacitive, respectively. One thing to be noted is that fractors can physically represent components having fractional-order properties, and may be the actual physical representation of some bioelectrodes previously tested. IV. FRACTIONAL BIOELECTRODE ODELS AND PROPERTIES In this section, we present a class of bioelectrode models that can be used for bioelectrode benchmarking purposes. The models presented in this section are of the fractional-order type that can be physically implemented using memristors, as well as resistors and capacitors. Throughout this section, the following notations will be used: R to denote the internal resistance value, C to denote the internal capacitance value, L to denote the internal inductance value, to denote the memristance value. The order of value of α in most cases positive, is expected to be non-integer in value. A. Fractor F) Class The simplest class of a bioelectrode model that is derived from fractional calculus has the impedance of the form Z F s) = s α 9) where > and α / Z. However, one should be careful in generalizing the order α. In particular cases where α = 1,,, this class would just be equivalent to a pure differentiator of order 1,,, and magnitude responses will result in a slope of db per decade db/dec) per pure differentiator. For example, if α = 1, then the model will similarly result in a pure inductor where = L being the equivalent value of inductance. For a particular case where < α < 1, the model in 9) will result in a partially-inductive element. On the other hand, having α = 1,, would make 9) equivalent to a pure integrator of order 1,,, thus having magnitude responses equivalent to -db/dec per pure integrator, and = 1/C, where C being the equivalent value of capacitance. The magnitude spectrum of this class is Z F jω) = jω) α = ω α 1) which will have a slope of αdb/dec in the entire frequency spectrum. If α = 1/, then the model will be a semidifferentiator having a slope of 1dB/dec for the entire frequency spectrum. This can be graphically proven by plotting the magnitude spectrum 1) in a logarithmic frequency scale and a magnitude scale with db as the unit of measure see Fig.1). The impulse repsonse of 9) is determined to be h F t) = tα+1) Γ α) 11) in the restriction that α is not an integer. In 11), the term Γ ) is known as the Euler s gamma function.

3 GONZALEZ et al: Bioelectrode odels with Fractional Calculus Figure 1. agnitude and phase responses of a semi-differentiator, i.e., Z s) = s 1/. The slope of the magnitude is 1dB/dec. Figure 3. agnitude and phase responses of the SFR class with transfer functions Z 1 s) = s and Z s) = s Impulse Response for Class F Systems B. Series Fractor-Resistor SFR) Class The series combination of a resistor and a fractor has the impedance form of Z SF R+ s) = R + s α, 1) Amplitude, h 6 4 alpha = 1.5 alpha =.5 alpha =.5 alpha = Time, t Figure. Impulse response 11) with α = {.5,.5, 1., 1.5}. Looking further into the impulse response 11) and letting = 1 without loss of generality, one can see that the initial value of the impulse response would depend on the value of order α since the term Γ α) dictates if the entire response is positive of negative. If k 1 < α < k for k = 1, 3, 5,, then the impulse response becomes negative. Otherwise, if k =, 4, 6,, then the impulse response becomes positive. However, in the case where the model is capacitive, i.e., 1 < α <, the impulse response becomes positive. When α 1, the model becomes unstable since lim t h t), where is any vector norm. The impulse response 11) with = 1 and for different values of α are shown in Fig.. where R, >, for α > s.t. α / Z +, and Z SF R s) = Rsα + s α, 13) for α < s.t. α / Z. This class is treated separately for positive and negative values of α. 1) α > : The magnitude response of 1) is given by Z SF R+ jω) = R + jω) α = R + ω α + Rω α cos 4n + 1) απ, 14) for n =, 1,,. The Bode plots for this class are depicted in Fig 3. As the value of frequency increases from its cut-off, i.e., ω >> ω C, the magnitude response increases by a slope of αdb/dec. However, when frequency decreases from its cut-off, i.e., ω << ω C, the magnitude reponse tends to be a constant with a slope of approximately db/dec. The impulse response of 1) is determined to be h SF R+ t) = Rδ t) + tα+1) Γ α), 15) where δ t) is the well-known Dirac-delta or impulse function. Furthermore, it is seen in 15) that the value of resistance doesn t affect much the impulse response as it only provides a spike at t =. When R =, the impulse response results in something similar to a simple fractor, i.e., F class. When k 1 < α < k for k = 1, 3, 5,, the impulse response becomes negative. Furthermore, for k =, 4, 6,, the response becomes positive.

4 1 INTERNATIONAL JOURNAL OF INTELLIGENT CONTROL AND SYSTES, VOL. 17, NO. 1, ARCH Figure 4. agnitude and phase responses of the SFR class with transfer functions Z 1 s) = s and Z s) = s Figure 5. agnitude and phase responses of the PFR class with transfer functions Z 1 s) = s.3 / s ) and Z s) = s.8 / s ). ) α < : The magnitude response of 13) is given by Z SF R jω) = = 1 ω α R jω) α + jω) α + R ω α + Rω α cos 4n + 1) απ, 16) for n =, 1,,. The magnitude and phase responses for this type are depicted in Fig 4. It can be seen in Fig. 4 that the magnitude reponse tends to be constant as the value of ω increases from its cut-off. However, below cut-off, the magnitude changes with frequency. The impulse response of 13) is determined by h SF R t) = Rδ t) + t α 1). 17) Γ α ) Similarly to the previous type where α > s.t. α / Z +, the impulse response also has a spike generated by the resistance value R at t =. When 1 < α <, the impulse response becomes positive. However, when α 1, the model becomes unstable, having a phenomenon similar to that of the F class. C. Parallel Fractor-Resistor PFR) Class The parallel combination of a resistor and a fractor has the impedance form of when α > s.t. α / Z +, and Z P F R+ s) = Rsα s α + R, 18) Z P F R s) = s α +, 19) R when α < s.t. α / Z. In both cases,, R >. However, each type must be considered separately. 1) α > : The magnitude spectrum of this type is R jω) α Z P F R+ jω) = jω) α + R Rω α = ω α + R ωα cos 4n+1)απ + ), ) R for n =, 1,,. With high values of ω >> ω C, where it is assumed that ω C is the cut-off frequency, the magnitude tends to be constant with a magnitude of R. On the other hand, at lower frequencies, ω << ω C, the magnitude tends to increase with frequency. This would then result in a magnitude response that is similar to a high-pass filter, except that the slope at lower frequencies is not necessarily a multiple of db/dec. The Bode plots of ) for the case where, R = 1 and α = {.3,.8} are shown in Fig. 5. The impulse response of 18) is h P F R+ t) = R R α,α R, t ) = R k= ) R k t αk)1, Γ αk) 1) where R q,v a, t) is known as the Generalized R function see page 3 of [9]) defined by a k t k+1)q1v R q,v a, t) = ) Γ q k + 1) v) k= having the Laplace transform of s v / s q a). ) α < : The magnitude spectrum of this type is Z P F R+ jω) = jω) α + R = ω α + R ωα cos 4n+1)απ + ), 3) R for n =, 1,,. Unlike in the previous type where α >, this type is similar to a low-pass filter. When the operating

5 GONZALEZ et al: Bioelectrode odels with Fractional Calculus Figure 6. agnitude and phase responses of the PFR class with transfer function Z 1 s) = 1/ s ) and Z s) = 1/ s ). Figure 7. agnitude and phase responses of the PFC class with transfer function Z 1 s) = s.5 / s ) and Z s) = s.1 / s ). frequency is much greater than its cut-off, i.e., ω >> ω C, its magtniude decreases with frequency. However, when the operating frequency is much less than its cut-off, i.e., ω << ω C, its magnitude remains at constant equal to R. The Bode plots of 3) with, R = 1 and α = {.3,.8} are shown in Fig. 6. The impulse response of 19) is h P F R t) = F α R, t ) = k= ) k R t αk+1)1, Γ α k + 1)) 4) where F q a, t) is known as the Rabotnov-Hartley function [9] defined by F q a, t) = k= a k t k+1)q1 Γ q k + 1)) having the Laplace transform of 1/ s q a). D. Parallel Fractor-Capacitor PFC) Class 5) This model depicts an intrinsic capacitor connected in parallel with a fractor. The impedance of the PFC class is defined by Z P F C+ s) = 1 s α C s α+1 + 1, 6) where, C > and α > s.t. α / Z +, Z P F C,1 s) = where 1 < α < and α + β = 1, and Z P F C, s) = where α < 1 s.t. α / Z and α γ = 1. 1/C s ), α s β + 1 7) 1/C s ), α s γ + 1 8) 1) α > : The magnitude response of 6) is defined by ZP F C)+ jω) 1 jω) α = C jω) α = 1 C ω α ω α+1) + ) ω α+1 cos 4n+1)α+1)π + ). 1 9) The shape of the magnitude response of such type would depend on the value of α. As α approaches, the response becomes similar to a low-pass filter. However, as α increases, the shape becomes similar to that of a narrow band-pass filter with the left and right slopes having different values. Fig. 7 shows that Bode plots of such type with α = {.5,.1}, while Fig. 8 show the Bode plots of such type with α = {.8, 1.}. Without loss of generality, it is assumed that = 1 and C = 1. The impulse response of 6) is h P F C+ t) = 1 C R α+1,α = 1 C k= ), t ) k t kα+1) 1 1 Γ k α + 1) + 1), 3) where R q,v a, t) is known as the Generalized R function discussed in the previous section. ) 1 < α < : The magnitude response of 7) is defined by Z P F C,1 jω) = 1 1 C ) jω) α jω) β + 1 = 1 1 C ω ω α β + ) ωβ cos 4n+1)βπ + ), 31) 1 where β = 1 α. When α approaches, the magnitude response becomes similar to that of a low-pass filter. However,

6 1 INTERNATIONAL JOURNAL OF INTELLIGENT CONTROL AND SYSTES, VOL. 17, NO.1, ARCH Figure 8. agnitude and phase responses of the PFC class with transfer function Z 1 s) = s.8 / s ) and Z s) = s 1. / s. + 1 ). Figure 1. agnitude and phase responses of the PFC class with transfer function Z 1 s) = 1/s 1.1 s ) and Z s) = 1/s 6.1 s ) Figure 9. agnitude and phase responses of the PFC class with transfer function Z 1 s) = 1/s.1 s ) and Z s) = 1/s.9 s ). 3) α < 1: The magnitude response of 8) is defined by 1 1 Z P RC, jω) = C jω) α jω) γ ) + 1 = 1 1 C ω ω α γ + ) ωγ cos 4n+1)γπ + ), 33) 1 where γ = α 1. As α approaches, the slope of the magnitude response increases negatively. The slope of the magnitude response before and after the cut-off frequency, however, are different. This can be seen in Fig. 1 for α = {1.1, 6.1} with = 1 and C = 1 without loss of generality. The impulse response of 8) is h P F C, t) = 1 C R γ,1/ α = 1 C k= 1 1 ), t ) k t γk+1)11/ α Γ γ k + 1) 1/ α ). 34) when α approaches 1, the magnitude response becomes similar to that of an integrator. This observation is depicted in Fig. 9, where without loss of generality, = 1, C = 1 and α = {.1,.9}. The impulse response of 7) is E. Addition of a Delay Factor In addition to the models above, if a dynamic impulse response of the actual bioelectrode in measure exhibits a delay compared to the fractional model, then a delay factor into impedance can be added, thus resulting in the model Z s) = Z X s) e τs, 35) h P F C,1 t) = 1 C R β,1/ α 1 ), t = 1 ) 1 k t βk+1)11/ α.3) C Γ β k + 1) 1/ α ) k= where τ > is the time delay in seconds, and Z X s) is the impedance model discussed in the previous subsections. It is also important to note that the addition of this delay factor does not change the magnitude response of the model. Phase reponse, however, is modified by a factor of ωτ, which also corresponds to an impulse response h t τ).

7 GONZALEZ et al: Bioelectrode odels with Fractional Calculus 13 V. CONCLUSION In this short note, we present the use the fractional calculus in the mathematical modeling of bioelectrodes with the combination of the traditional passive elements and memristive devices, i.e., fractors. odels presented in this paper include impedances represented by 1) a single fractor, ) series combination of a fractor and a resistor, 3) parallel combination of a fractor and a resistor, and 4) parallel combination of a fractor and a capacitor. The impulse and frequency responses of the impedances of these models are also presented and discussed in detail. Finally, we also show indirectly that fractional calculus is being ubiquitous and its use in modeling and simulation of electric circuits and their equivalent bioelectrode models must also be considered. [17] I. Podlubny, Fractional Differential Equations, in athematics in Science and Engineering, vol. 198, San Diego, California: Academic Press, [18] J. P. Reilly, Applied Bioelectricity, From Electrical Stimulation to Electropathology. New York: Springer, [19] H. P. Schwan, Electrode polarization impedance and measurements in bioogical materials, Ann. NY Acad. Sci., vol. 148, pp , [] H. P. Schwan, Electrical properties of tissue and cell suspensions, Advances in Biological and edical Physics, vol. 5. New York: Academic Press, [1] S. Westerlund and L. Ekstram, Capacitor theory, IEEE Trans. Dielec. Elec. Insul., vol. 1, no. 5, pp , Oct ACKNOWLEDGEENT This work is supported under a research grant by Jardine Schindler Elevator Corporation, 8/F Pacific Star Building., Sen. Gil Puyat Ave. cor. akati Ave., akati City, Philippines, and under grant VEGA 1/44/8 from the Slovak Grant Agency of Science. REFERENCES [1] G. E. Carlson, Investigation of fractional capacitor approximations by means of regular Newton processes, Kansas State Univ. Bull., vol. 48, no. 1, [] G. E. Carlson and C. A. Halijak, Simulation of fractional derivative operator s and the fractional operator 1/ s, Kansas State Univ. Bull., vol. 47, no. 7, [3] G. E. Carlson and C. A. Halijak, Approximation of fixed impedances, IRE Trans. Circ. Theor., vol. CT-9, no. 3, pp. 3-33, 196. [4] G. E. Carlson and C. A. Halijak, Approximation of fractional capacitors 1/s) 1/n by a regular Newton process, IRE Trans. Circ. Theor., vol. CT-11, no., pp. 1-13, [5] L. O. Chua, emristor the missing circuit element, IEEE Trans. Circ. Theor., vol. CT-18, no. 5, pp. 5-53, Sep [6] L. O. Chua and S.. Kang, emristive devices and systems, Proc. IEEE, vol. 64, no., pp. 9-3, Feb [7] L. O. Chua and C.-W. Tseng, A memristive circuit model for p-n junction diodes, Int. J. Circ. Theor. App., vol., no. 4, pp , Dec [8] C. Coopmans, I. Petráš, and Y. Chen, Analog fractional-order generalized memristive devices, Proc. ASE 9 Int. Des. Eng. Tech. Conf. & Comp. Inf. Eng. Conf. IDETC/CIE 9), San Diego, California, Aug. 3 - Sep., 9. [9] S. Das, Functional Fractional Calculus for System Identification and Controls, New York: Springer, 8. [1] A.. Dymond, Characteristics of the metal-tissue interface of stimulation electrodes, IEEE Trans. Biomed. Eng., vol. 3, pp. 74-8, [11] D. C. Grahame, athematical theory of the faradic admittance, J. Electrochem. Soc., vol. 99, pp. 37C-375C, 195. [1] W. Greatbatch and W.. Chardack, yocardial and endocardial electrodes for chronic implanation, Ann. NY Acad. Sci., vol. 148, pp , [13] R.. Gulrajani, Bioelectricity and Biomagnetism. New York: Wiley, [14] R. L. agin, Fractional Calculus in Bioengineering, Connecticut: Begal House, Inc., 6. [15] S. ukhopadhyay, C. Coopmans, and Y. Chen, Purely analog fractional order PI control using discrete fractional capacitors fractors): synthesis and experiments, Proc. ASE 9 Int. Des. Eng. Tech. Conf. & Comp. Inf. Eng. Conf. IDETC/CIE 9), San Diego, California, Aug. 3 - Sep., 9. [16] R. Plonsey, Bioelectric Phenomena. New York: cgraw-hill, 1969.