A study for the measurement of muscle electrical properties using a multi-contact needle

Bachelor’s degree thesis A study for the measurement of muscle electrical properties using a multi-contact needle Albert Cheto Barrera Advisors: Benjam´ın S´anchez Terrones (BIDMC - HMS) Ramon Brag´os Bardia (UPC) In partial fulfillment of the requirements for the: Bachelor’s degree in Mathematics Bachelor’s degree in Industrial Technology Engineering May 2020 ii Abstract A study for the measurement of muscle electrical properties using a multi-contact needle by Albert Cheto Barrera Neuromuscular disorders are a broad range of conditions affecting the health of the nerves and muscles. Disease-induced alterations affecting muscle composition and structure alter the ability of the muscle to conduct electrical current and store charge within, as quantified by the electrical properties of the muscle. Measuring the electrical properties of human muscle can thus provide actionable information for assessment of patients with neuromuscular diseases. In this project, we propose different methods to obtain the electrical properties of muscle including their directional dependence, a concept also known as anisotropy. The data is obtained performing a single needle insertion with a novel cross-type structure. This document introduces the methods to the reader in increasing order of complexity, including experimental work for validation purposes. Here, we develop a combined engineering and mathematical approach leveraging the knowledge obtained in the Bachelor’s degree in Mathematics and the Bachelor’s degree in Industrial Technology Engineering. Keywords: Electrical properties, anisotropy, skeletal muscle, needle. Mathematics Subject Classification American Mathematical Society code: [92C50] iii Resumen Estudio para la medici´on de propiedades el´ectricas musculares utilizando una aguja multicontacto por Albert Cheto Barrera Los trastornos neuromusculares son una gran variedad de afecciones del estado de salud de nervios y m´usculos. Las alteraciones inducidas por la enfermedad que afectan la composici´on y estructura del m´usculo, alteran la capacidad de este para conducir corriente el´ectrica y almacenar carga en su interior, cuantificado en las propiedades el´ectricas del m´usculo. De este modo, la medici´on de las propiedades el´ectricas puede proporcionar informaci´on ´util para la evaluaci´on de pacientes con enfermedades neuromusculares. En este proyecto, se proponen diferentes m´etodos para obtener las propiedades el´ectricas del m´usculo incluyendo su dependencia direccional, concepto tambi´en conocido como anisotrop´ıa. Las mediciones se realizan con la ´unica inserci´on de una novedosa aguja con secci´on en forma de cruz ’+’. Este documento presenta al lector los m´etodos en orden creciente de complejidad, incluyendo trabajo experimental con finalidad validadora. El trabajo desarrolla un enfoque tanto ingenieril como matem´atico para hacer uso de los conocimientos obtenidos en el Grado de Matem´aticas y en el Grado en Ingenier´ıa en Tecnolog´ıas Industriales. Palabras clave: Propiedades el´ectricas, anisotrop´ıa, m´usculo esquel´etico, aguja. C´odigo de clasificaci´on de la American Mathematical Society: [92C50] iv Resum Estudi per mesurar propietats el`ectriques musculars emprant una agulla multicontacte per Albert Cheto Barrera Els trastorns neuromusculars s´on una gran varietat d’afeccions de l’estat de salut de nervis i m´usculs. Les alteracions indu¨ıdes per la malaltia que afecten la composici´o i estructura del m´uscul, alteren la capacitat d’aquest de conduir corrent el`ectrica i d’emmagatzemar c`arrega al seu interior, quantificat a les propietats el`ectriques del m´uscul. D’aquest mode, la mesura de les propietats el`ectriques pot proporcionar informaci´o valuosa per l’avaluaci´o de pacients amb malalties neuromusculars. En aquest projecte, es proposen diferents m`etodes per a obtenir les propietats el`ectriques del m´uscul incloent la seva depend`encia direccional, concepte altrament conegut com a anisotropia. Les mesures es realitzen amb una ´unica inserci´o d’una agulla novedosa d’estructura en forma de creu . Aquest document presenta al lector els diferents m`etodes en ordre creixent de complexitat, incloent treball experimental amb finalitat validadora. El treball desenvolupa un enfocament tant enginyeril com matem`atic per a posar de manifest els coneixements obtinguts en el Grau en Matem`atiques i el Grau en Enginyeria en Tecnologies Industrials. Paraules clau: Propietats el`ectriques, anisotropia, m´uscul esquel`etic, agulla. Codi de classificaci´o de l’American Mathematical Society: [92C50] v Acknowledgements First of all, a special thanks to Dr. Benjam´ın S´anchez for being my supervisor and closely guiding my research during all the project. It has been more than an academic supervisor. I would also like to thank Dr. Seward Rutkove, Principal Investigator at the Neurology Department at Beth Israel Deaconess Medical Center, for giving me this great opportunity of getting to know the research world from the inside. Thank you also to CFIS and Fundaci´o Privada Cellex for the financial support during the last years, facilitating me to focus on the academic work. My gratitude also goes to Dr. Ramon Brag´os for being my co-advisor of the thesis. Last but not least, I would like to thank my family and friends for all the support along this experience. Moltes gr`acies. Sabadell, May 2, 2020. vi Contents List of figures ix 1 Introduction 1 1.1 Neuromuscular disorders ............................ 1 1.2 Motivation and objectives ........................... 1 1.3 Document overview ............................... 2 2 Preliminary concepts 3 2.1 Electrical properties .............................. 3 2.2 Anisotropy .................................... 4 2.3 Electric potential ................................ 5 3 Cross-type ’+’ needle 7 3.1 Needle structure ................................. 7 3.2 Electrode position and coordinate system ................... 9 3.3 Apparent distance ................................ 12 3.4 Impedance configurations ............................ 13 3.5 Inverse problem ................................. 21 3.5.1 Method 1 ................................ 21 3.5.2 Method 2 ................................ 23 4 T-type needle 25 4.1 Needle structure ................................. 25 4.2 Electrode position and coordinate system ................... 26 4.3 Apparent distance ................................ 27 vii 4.4 Impedance configurations ............................ 28 4.5 Inverse problem ................................. 30 4.5.1 Method 1 ................................ 31 4.5.2 Method 2 ................................ 31 4.5.3 Method 3 ................................ 31 5 Experimental part 34 5.1 Needles design and fabrication ......................... 34 5.2 Experiments ................................... 35 5.2.1 Devices ................................. 36 5.2.2 Methodology .............................. 37 5.3 Results ...................................... 38 6 Conclusions and future work 41 Bibliography 43 viii List of Figures 3.1 (A) Schematic representation of the cross-type needle (B) Needle’s geometry (not to scale) (C) Needle’s top view. ................... 9 3.2 (A) Insertion of the needle to the skeletal muscle. (B) Enlarged insertion across the skin and subcutaneous fat. (C) Schematic drawing representing a model abstraction of an arbitrary geometrical arrangement of muscle domain Ω and needle to illustrate the spherical coordinate system used. . . 9 3.3 Eight different distances between two same face electrodes. Face [ES] in default position shown. ............................. 14 3.4 Configurations 1 to 19. Distances for V++,V−−,V+−and V−+shown in blue, red, yellow and black, respectively. ................... 15 4.1 (A) Schematic representation of the ’T’-type needle (B) Needle’s geometry front view (C) Needle’s geometry rear view (D) Needle’s top view. Default position for φ=ϕ= 0. ............................. 26 4.2 Notation of the [N] face electrodes from the ’T’-type needle. ........ 26 4.3 Distances for the [N] face of the ’T’-type needle. ............... 27 4.4 Configuration 20(2) for [N] face. Distances for V++,V−−,V+−and V−+ shown in blue, red, yellow and black, respectively. ............. 29 5.1 ’T’-type needle CAD design. .......................... 35 5.2 PCB designs used to facilitate the electrodes selection. ........... 36 5.3 (A) ’+’ cross-type needle, [WS] and [ES] faces (B) ’T’-type needle, [WS] and [ES] faces (C) ’T’-type needle, [N] face (D) ’T’-type needle. ...... 36 ix 5.4 Conductimeter (left) and impedance (right) apparatus used in the experiments. ...................................... 37 5.5 Relative error of the experimental conductivty σexp while changing the frequency. .................................... 38 5.6 Experimental conductivity σexp vs. theoretical conductivity σtfor a specific frequency in logarithmic scale shown in blue. (Left) f = 9.3kHz and (Right) f = 90kHz. Reference line σexp =σtshown in black. ............. 39 5.7 Plots of the conductivity value (top row) and the relative error of the conductivty (bottom row) while changing the frequency. Experimental, corrected and theoretical values shown in blue, orange and black dashed, respectively. From left to right, σt={0.04,0.17,0.38,1.39}......... 40 7 Chapter 3 Cross-type ’+’ needle In this chapter, we describe an approach to measure the impedance of skeletal muscle by using a multi-face cross ’+’ section needle. First, we define the needle structure and a coordinate system for it. Then, we derive analytical expressions for impedance using an infinite medium model. After that, we present a way to obtain the anisotropy ratio and the electrical properties of the muscle. For both this chapter and the following, we will assume that we insert the needle into the skeletal muscle across the skin and subcutaneous fat. We also assume the muscle to be an infinite domain Ω ⊆R3. The face of the needle makes the domain semi-infinite. We will start considering the forward problem, which consists in obtaining the impedance value knowing all the other variables, i.e., knowing the the resistivity and reactivity in the longitudinal and transverse directions, orientation of the needle and anisotropy ratio we are able to obtain the impedance. Once the forward problem is set out, we will propose different methods for the inverse problem, which consists in obtaining the anisotropy ratio, needle orientation, resistivity and reactivity of the muscle from impedance measurements. 3.1 Needle structure We propose a cross-type ’+’ needle as shown in figure 3.1A. Having multiple electrodes distributed in different faces allows to perform more than one measurement with a single 3.1. Needle structure 8 needle insertion unlike other studies [11]. To perform an impedance measurement, we need four electrodes being positive and negative current and voltage electrodes. For this reason, we choose that each of the faces is formed by 4×2 electrodes distributed in 4 rows and 2 columns as you can see in figure 3.1B. We will denote each electrode by εik, where i∈ {1,2,3,4}and k∈ {1,2}are the row and column index of the electrodes, respectively. By convention, we define iincreasing towards the needle tip whereas kdecreasing towards the needle’s major axis. When we refer to the electrodes’ roles, we denote them by εI+, εI−,εV+and εV−. The subscripts I+,I−,V+and V−denote the positive and negative current and voltage roles. The distance a(m) is defined between the outermost electrodes and the nearest electrodes, i.e., ε1kto ε2kand ε3kto ε4k, while the distance b(m) is defined between the inner electrodes ε2kto ε3k. The vertical distance from both the needle tip to the nearest electrodes ε1kand from the needle’s upper edge to the nearest electrodes ε4kare a0(m) and c0(m), respectively. The distance between εi1and εi2is s(m). Finally, both distances from εi1to the needle’s major axis and from εi2to the lateral edge is c(m). We then assume that the four arms of the needle define four semi-infinite subdomains in figure 3.1C, namely Ω1={x > 0, y > 0, z ∈R}, Ω2={x < 0, y > 0, z ∈R}, Ω3={x < 0, y < 0, z ∈R}and Ω4={x > 0, y < 0, z ∈R}. In other words, the needle arms’ thickness is considered negligible. We also consider that the electrodes are point-like, i.e., their diameter is negligible. The needle is formed by eight faces as you can see in figure 3.1C. We use the subscript ·[F1F2]to denote each face, where F1denotes the face’s cardinal point and F2denotes the face’s orientation, i.e., F1, F2∈ {N,S,W,E}, so that when F1∈ {N,S}then F2∈ {E,W} and when F1∈ {E,W}then F2∈ {N,S}. The impedance measurements are performed by inserting the needle through the skin and subcutaneous fat tissues into the skeletal muscle as in figure 3.2A. An enlarged representation is shown in figure 3.2B. The skeletal muscle is considered to be infinite and is modeled as the domain Ω ⊂R3, with Ω = Ω0 1∪Ω0 2∪Ω0 3∪Ω0 4and where Ω0 {1,··· ,4}denotes Ω{1,··· ,4}with the reference system shown in figure 3.2C. 3.2. Electrode position and coordinate system 9 BA C y x z [EN] [ES] [WN] [WS] [NE][NW] [SE][SW] W1 W2 W3 W4 e41 ε41 e42 e42 e31 ε31 e32 e32 e21 ε21 e22 e22 e11 ε11 e12 e12 [ES][WS] a b a a0 c0 c cs e41 e42 e31 e32 e21 e22 e11 e12 [SE] [NE] z y x Figure 3.1: (A) Schematic representation of the cross-type needle (B) Needle’s geometry (not to scale) (C) Needle’s top view. Figure 3.2: (A) Insertion of the needle to the skeletal muscle. (B) Enlarged insertion across the skin and subcutaneous fat. (C) Schematic drawing representing a model abstraction of an arbitrary geometrical arrangement of muscle domain Ω and needle to illustrate the spherical coordinate system used. 3.2 Electrode position and coordinate system To describe a general position of the electrodes in R3, we must first define a general position of the needle. The skeletal muscle fibers are oriented along the longitudinal direction that we define as the z-axis. The x, y-axes are defined perpendicular to the fibers direction, i.e., along the transverse direction, to form the canonical basis in R3. We note that the skeletal muscle presents an axial symmetry with the muscle fiber as the axis. We also note that the muscle fiber is seen equal in both directions of the z-axis as 3.2. Electrode position and coordinate system 10 it is modelled as an infinite cylinder. Therefore, we define the x, y-axis for the needle’s major axis to be located in the first quadrant of the yz plane to avoid ambiguity in the definition. We will use spherical coordinates to define the position of the needle’s major axis. We will need and additional angle to describe the needle rotation with respect to its major axis. The origin of coordinates defined by convention is located in the needle’s tip as in figure 3.1B. The angles used to define the needle’s major axis are ϕfor the polar angle and θfor the azimutal. The restrictions regarding the symmetry presented above allow us to define the angles as ϕ∈[0, π/2) and θ= 0, where ϕis defined as the angle between the needle’s major axis and the muscle fibers and θis the angle of the needle’s projection to the xy plane and the y-axis. The additional angle to describe the needle’s rotation with respect to its major axis is φand its range is φ∈[0,2π). As a convention, when θ=ϕ=φ= 0, then the needle’s major axis is aligned with respect to the z-axis and the [ES] face is the yz plane. Finally, we define when ϕ=π/2 then φ∈[0, π). This last definition is needed to avoid a given expression to be expressed in two different ways. For the particular case of ϕ=π/2, it is not clear with respect to which direction of the fiber is the angle defined. Restricting the values of φ, this indeterminacy is solved. We note that the {y, z}-plane is not well defined when ϕ= 0 because the needle’s major axis and the fibers only define a line and not a plane. We can assume that this situation never happens as it would mean inserting the needle parallel to the fibers. Henceforth, we will restrict ourselves to face [ES]. If that is not the case, subscript ·[F1F2]will be used to denote the referred face. Following the definitions of the needle structure, the position Pik := (xik, yik, zik)∈R3of electrode εik by default, i.e., ϕ=φ= θ= 0, is P11 = (0, c, 2a+b+a0),P12 = (0, c +s, 2a+b+a0), P21 = (0, c, a +b+a0),P22 = (0, c +s, a +b+a0), P31 = (0, c, a +a0),P32 = (0, c +s, a +a0), P41 = (0, c, a0),P42 = (0, c +s, a0), as we see in figure 3.1B. For notational simplicity, we denote the electrode position by 3.2. Electrode position and coordinate system 11 (0, yik, zik) := Pik for i= 1,2,3,4 and k= 1,2. To describe the general position of an electrode we use three dimensional rotation matrices. The rotation matrices along x-, y-, and zaxes with an angle δusing right-hand rule are Rx(δ) :=      1 0 0 0 cos δ−sin δ 0 sin δcos δ      , Ry(δ) :=      cos δ0 sin δ 0 1 0 −sin δ0 cos δ      and Rz(δ) :=      cos δ−sin δ0 sin δcos δ0 0 0 1      . The rotated position of an electrode Qik ∈R3is defined as Qik := Rx(−ϕ)Rz(φ)Pik, that is Qik =     1 0 0 0 cos(−ϕ)−sin(−ϕ) 0 sin(−ϕ) cos(−ϕ)           cos φ−sin φ0 sin φcos φ0 0 0 1           0 yik zik      =     1 0 0 0 cos ϕsin ϕ 0−sin ϕcos ϕ           −yik sin φ yik cos φ zik      =     −yik sin φ yik cos φcos ϕ+zik sin ϕ −yik cos φsin ϕ+zik cos ϕ      . Therefore, for i= 1,2,3,4 and k= 1,2, Qik =     −yik sin φ yik cos φcos ϕ+zik sin ϕ −yik cos φsin ϕ+zik cos ϕ      ,(3.1) where yi1=s1,yi2=s2,z1k= 2a+b+a0,z2k=a+b+a0,z3k=a+a0, and z4k=a0. 3.3. Apparent distance 12 If we want to describe the position of an electrode in another face it is equivalent to consider the electrode to be in the [ES] face changing the angle φfor φ+tπ/2 with t∈Z. For example, Qik[SE] =     yik(−sin(φ−π/2) cos θ−cos(φ−π/2) cos ϕsin θ)−zik sin ϕsin θ yik(−sin(φ−π/2) sin θ+ cos(φ−π/2) cos ϕcos θ) + zik sin ϕcos θ −yik cos(φ−π/2) sin ϕ+zik cos ϕ      , where yi1=s1,yi2=s2,z1k= 2a+b+a0,z2k=a+b+a0,z3k=a+a0, and z4k=a0. 3.3 Apparent distance In the previous chapter we have seen the formula of the voltage generated by a point-like current electrode in equation 2.3. In this formula, we have the apparent distance in the denominator. We will now rewrite this apparent distance in a more convenient way using the rotated positions of the electrodes. Let’s consider electrodes with rotated positions Qij and Qkl. We define the distance dα(Qij,Qkl) as dα(Qik,Qnl) := p(Qik(x)−Qnl(x))2+ (Qik(y)−Qnl(y))2+α2(Qik(z)−Qnl(z))2, (3.2) where Qik(c) denotes the ccomponent of electrode’s εik position. This distance can be seen as the k·k2between the apparent vector position of two points, where the apparent position of point (x0, y0, z0) is (x0, y0, αz0). We note that this definition is the same as in 2.2 with a different notation. Using the rotated positions in (3.1) we can rewrite this distance as dα(Qik,Qnl) = p(yik −ynl)2A+ (zik −znl)2B+ (yik −ynl)(zik −znl)C, (3.3) where A:= sin2φ+ cos2φ(cos2ϕ+α2sin2ϕ), B:= sin2ϕ+α2cos2ϕ, C:= 2(1 −α2) cos φsin ϕcos ϕ= (1 −α2) cos φsin 2ϕ, (3.4) 3.4. Impedance configurations 13 are variables that depend on the needle orientation and the anisotropy ratio, i.e., A=A(φ, ϕ, α2), B=B(φ, ϕ, α2) and C=C(φ, ϕ, α2). Given two electrodes from the same face and considering the face’s dimensions (figure 3.1B) there are eight different distances between electrodes. These different distances are defined as d1,...,8and are presented in figure 3.3. d1:= dα(Qik,Q(i+1)k) = a√B, i ∈ {1,3}, k ∈ {1,2}, d2:= dα(Qik,Q(i+2)k)=(a+b)√B, i ∈ {1,2}, k ∈ {1,2}, d3:= dα(Q1k,Q4k) = (2a+b)√B, k ∈ {1,2}, d4:= dα(Qi1,Qi2) = s√A, i ∈ {1,2,3,4}, d5:= dα(Qik,Q(i+1)n) = √s2A+a2B+saC, i ∈ {1,3},{k, n}={1,2}, d6:= dα(Q2k,Q3n) = √s2A+b2B+sbC, {k, n}={1,2}, d7:= dα(Qik,Q(i+2)n) = ps2A+ (a+b)2B+s(a+b)C, i ∈ {1,2},{k, n}={1,2}, d8:= dα(Q1k,Q4n) = ps2A+ (2a+b)2B+s(2a+b)C, {k, n}={1,2}. (3.5) 3.4 Impedance configurations To perform an impedance measurement using the four-electrode technique we use positive and negative current and voltage electrodes denoted by εI+,εI−,εV+and εV−, respectively. Their positions are denoted by QI+,QI−,QV+and QV−. Considering the apparent distance we rewrite the electrical potential (2.3) as Vpm =καI Kdα(QVp,QIm),(3.6) where Kis the domain factor, p, m ∈ {+,−} and Vpm is the electric potential measured at εVpwhen current Igenerated at εIm. Hereafter, we consider K= 2πas we assume that the electrodes of the needle are located in a plane that makes the domain semi-infinite. We note that in this equation, the apparent distance is presented as the distance of two electrode roles instead of two specific electrodes. That is because to define Vpm we need to define the distance between the position of electrode roles. Later, for a given configuration of electrodes, i.e., knowing which electrode has each of the roles, we will be able to present 3.4. Impedance configurations 14 d1 z y x d2 z y x d3 z y x d4 z y x d5 z y x d6 z y x d7 z y x d8 z y x Figure 3.3: Eight different distances between two same face electrodes. Face [ES] in default position shown. this apparent distance between the position of two electrodes. As defined in equation 2.4, the impedance is Z=V++ −V+−−V−++V−− I. If we choose four electrodes on the same face for an impedance measurement, i.e., εI+,εI−,εV+and εV−, there are 8 4·4! = 1680 possibilities. Most of the configurations give redundant information so we restrict ourselves to 19 configurations (see figure 3.4). Configurations 1 to 16 in figure 3.4 verify {ε11, ε12} ∈ εI+,{ε41, ε42} ∈ εI−,{ε21, ε22} ∈ εV+ and {ε31, ε32} ∈ εV−. Configurations 17 to 19 are rectangular configurations. We can rewrite the impedance (2.4) using the voltage equation (3.6) as Z=κα K1 dα(QV+,QI+)−1 dα(QV+,QI−)−1 dα(QV−,QI+)+1 dα(QV−,QI−).(3.7) Note that from this equation it is easier to see that two configurations that have the sets {dα(QV+,QI+), dα(QV−,QI−)}and {dα(QV+,QI−), dα(QV−,QI+)}equal give the same impedance equation. 3.4. Impedance configurations 15 1 z y x 2 z y x 3 z y x 4 z y x 5 z y x 6 z y x 7 z y x 8 z y x 9 z y x 10 z y x 11 z y x 12 z y x 13 z y x 14 z y x 15 z y x 16 z y x 17 z y x 18 z y x 19 z y x Figure 3.4: Configurations 1 to 19. Distances for V++,V−−,V+−and V−+shown in blue, red, yellow and black, respectively. Below, we derive the impedance equations for each of the configurations. The 19 configurations are valid for each of the eight faces of the needle. 3.4.1 Configuration 1 For configuration 1 we have elelectrodes ε11, ε21, ε31, ε41 with the following roles QI+= Q11,QV+=Q21,QV−=Q31,QI−=Q41. E.g. QI+=Q11 means that electrode ε11 is acting as the positive current electrode. Using equations (3.5) and (3.6) we have V++ =καI K 1 dα(Q21,Q11)=καI K 1 d1 , V+−=καI K 1 dα(Q21,Q41)=καI K 1 d2 , V−+=καI K 1 dα(Q31,Q11)=καI K 1 d2 , 3.4. Impedance configurations 16 V−− =καI K 1 dα(Q31,Q41)=καI K 1 d1 . Using this voltage expressions in (3.7) we obtain Z1=2κα K1 d1−1 d2, where Zcdenotes the impedance Zin configuration c,c∈ {1, ..., 19}. 3.4.2 Configuration 2 For configuration 2 we have elelectrodes ε11, ε21, ε31, ε42 with the following roles QI+= Q11,QV+=Q21,QV−=Q31,QI−=Q42. Using equations (3.5) and (3.6) we have V++ =καI K 1 dα(Q21,Q11)=καI K 1 d1 , V+−=καI K 1 dα(Q21,Q42)=καI K 1 d7 , V−+=καI K 1 dα(Q31,Q11)=καI K 1 d2 , V−− =καI K 1 dα(Q31,Q42)=καI K 1 d5 . Using this voltage expressions in (3.7) we obtain Z2=κα K1 d1−1 d7−1 d2 +1 d5. 3.4.3 Configuration 3 For configuration 3 we have elelectrodes ε11, ε21, ε32, ε41 with the following roles QI+= Q11,QV+=Q21,QV−=Q32,QI−=Q41. The impedance equation for configuration 3 is the same than in configuration 2 because the sets {dα(QV+,QI+), dα(QV−,QI−)}and {dα(QV+,QI−), dα(QV−,QI+)}are the same in both configurations. Therefore, Z3=Z2. 3.5. Inverse problem 23 and α2and its definitions, we can obtain ρL,ρT,τLand τTas                ρL= Re(καα), ρT= Re(κα/α), τL= Im(καα), τT= Im(κα/α), where Re(z) and Im(z) represent the real and imaginary part of the complex number z, respectively. For this method, the needle orientation is already known prior to the insertion. Therefore, a solve for the needle orientation is not needed. 3.5.2 Method 2 We assume that the needle major axis is perpendicular to the muscle fibers direction (ϕ:= π/2). With this assumption in (3.4) we have A= sin2φ+α2cos2φ,B= 1 and C= 0. We notice that applying the same assumptions in face [SE] we can obtain A[SE] = sin2(φ−π/2) + α2cos2(φ−π/2) = cos2φ+α2sin2φ, which implies A+A[SE] = 1 + α2. Following the same rationale as in Method 1, we take equations Z1and Zwwhere w∈ {17,18}from Table 3.1 to obtain καand A κα=Z1Ka(a+b) 2b, A =2καw s(ZwKw + 2κα)2 . If we repeat both measurements Z1and Zwin face [SE], i.e., Z1[SE] and Zw[SE], we can obtain κα[SE] =Z1[SE]Ka(a+b) 2b, A[SE] =2κα[SE]w s(Zw[SE]Kw + 2κα[SE])2 . We note that by definition καdoesn’t depend on the needle orientation. Therefore, the following equality holds κα=κα[SE]. Next, the anisotropy ratio can be solved as α2=A+A[SE] −1. 3.5. Inverse problem 24 The values of καand α2are used to obtain ρL,ρT,τLand τTanalogously to Method 1. In this method, Acan be rewritten in a way that makes clear its range and evinces the numerical problems when solving for the needle orientation. A= sin2φ+α2cos2φ =1−cos 2φ 2+α21 + cos 2φ 2 =1 2(1 + α2) + cos 2φ 2(α2−1). As cos 2φ∈[−1,1] =⇒A∈[α2,1]. And from this expression we can solve for φas φ=1 2arccos 2A−1−α2 α2−1. Note that when α2≈1, a small error obtaining Atraduces to an enlarged error in φ. The next natural method is to make no assumptions in the angle positioning of the needle. Ideally, the inverse problem for this method could be solved with this same needle. However, the interaction of the variables in the impedance equations (Table 3.1) make it inviable. We must use the geometry of the needle cleverly to simplify the equations. For this reason, we come up with the following needle, which is a variation of the present needle and makes viable a solve for the method with no angle assumptions. 25 Chapter 4 T-type needle This chapter is analogous to Chapter 3with a different needle. In this case, we talk about a multi-face ’T’-type needle. The modified needle will allow us to solve a more general problem than the ones solved with the previous needle. This problem consists in obtaining the anisotropy ratio and the impedivity of the muscle making no assumptions in the needle orientation with respect to the muscle fibers. 4.1 Needle structure The ’T’-type needle we propose is shown in figure 4.1. We consider the muscle fibers to be oriented along the z-axis, i.e., we define x,y,z-axes the same as in Chapter 3. The faces [SE], [ES], [SW] and [WS] are identical to the ones in the previous needle. Face [N] is formed by 32 electrodes distributed in two concentric circumferences of 16 electrodes each and radii r1and r2. The distance from the circumference center to the needle tip is b0. To name these electrodes we use a double subscript εs1s2. The first one, s1∈ {1,2} represents the circumference where the electrode is located (1 for the inner circumference and 2 for the outer). The second one, s2∈ {01,02, ..., 16}is a two digit number starting from the rightmost electrode and increasing counterclockwise along the circumference. The naming of these electrodes is represented in figure 4.2. We define the value of angle γequal to 0 when the electrode has s2= 01 and increasing counterclockwise as seen in figure 4.1C. The angle γfor electrode εs1s2is γ=2π 16 (s2−1). 4.2. Electrode position and coordinate system 26 41 ε41 4242 31 ε31 3232 21 ε21 2222 11 ε11 1212 [ES][WS] a b a a0 c s 41 42 31 32 21 22 11 12 [SE] z y x e e e e e e e e e e e ee ee e e ee c c0 [N] [SW] r1 r2 b0z y x W1 W2 W4 y x z[ES] [N] [WS] [SE][SW] g e e A B C D Figure 4.1: (A) Schematic representation of the ’T’-type needle (B) Needle’s geometry front view (C) Needle’s geometry rear view (D) Needle’s top view. Default position for φ=ϕ= 0. [N] ε201 ε202 ε203 ε204 ε205 ε206 ε207 ε208 ε209 ε210 ε211 ε212 ε213 ε214 ε215 ε216 ε101 ε102 ε103 ε104 ε105 ε106 ε107 ε108 ε109 ε110 ε111 ε112 ε113 ε114 ε115 ε116 z yx Figure 4.2: Notation of the [N] face electrodes from the ’T’-type needle. 4.2 Electrode position and coordinate system We can use the same rotated electrode’s position as in the cross-type needle. The only difference is the default position for electrodes in [N] face, which is Ps1s2:= (0, rs1cos γ, b0+rs1sin γ), 4.3. Apparent distance 27 where s1∈ {1,2},s2∈ {01,02, ..., 16},γ= (s2−1)π 8and the functions sine and cosine are expressed in radians. Defining (0, ys1s2, zs1s2) := Ps1s2we have the same as for the cross-type needle Qs1s2=     −ys1s2sin φ ys1s2cos φcos ϕ+zs1s2sin ϕ −ys1s2cos φsin ϕ+zs1s2cos ϕ      .(4.1) 4.3 Apparent distance The apparent distance is the same as in (3.3). As well as having the eight different distances as in (3.5) we have two additional distances in face [N] if we only consider distance between radially electrodes, i.e., electrodes with γ=ct. or γ=ct. +πThese distances are d9:= dα(Qil,Qnl)=(r2−r1)pA+B+C, d10 := dα(Qil,Qnk) = (r2+r1)pA+B+C, (4.2) and they are shown in figure 4.3, where A=Acos2γ,B=Bsin2γ,C=Csin γcos γ, l∈ {1, ..., 16},k=l±8 and {i, n}={1,2}. The expression l±8 denotes l+ 8 (mod 16) given that l+ 8 ≡l−8 (mod 16). Therefore, if l+ 8 >16 we take k=l−8, otherwise we take k=l+ 8. Hereafter, t±8 will also denote t+ 8 (mod 16). [N] d9 z yx [N] d10 z yx Figure 4.3: Distances for the [N] face of the ’T’-type needle. 4.4. Impedance configurations 28 4.4 Impedance configurations Configurations 1 to 19 presented in Table 3.1 are also valid in faces [ES], [SE], [WS] and [SE] of the ’T’-type needle. In face [N] we consider the configuration made from four aligned electrodes as in figure 4.4. We denote this configuration Z20. For this configuration, we have V++ =καI K 1 dα(Q1l,Q2l)=καI K 1 d9 , V+−=καI K 1 dα(Q1l,Q2(l±8))=καI K 1 d10 , V−+=καI K 1 dα(Q1(l±8),Q2l)=καI K 1 d10 , V−− =καI K 1 dα(Q1(l±8),Q2(l±8))=καI K 1 d9 . where l∈ {01,02, ..., 16}. Using this voltage expressions in (3.7) we obtain Z20 =2κα K1 d9−1 d10 . Applying the distances definition we rewrite Z20 as Z20 =2κα K 2r1 r2 2−r2 1 1 pA+B+C. This impedance equation is valid ∀γ∈[0,2π). However, the given electrode distribution forces us to restrict ourselves in multiples of π/8. For this reason, to distinguish the different configurations 20 we can find in [N] face we define Z20(g):= Z20|γ=(g−1) π 8,(4.3) where g∈ {1,2, ..., 8}. Example: For g= 2, the configuration 20(2) has the electrodes εI+=ε202,εV+=ε102, εV−=ε109 and εI−=ε209. Its representation is shown in 4.4. 4.4. Impedance configurations 29 [N] γ z yx Figure 4.4: Configuration 20(2) for [N] face. Distances for V++,V−−,V+−and V−+ shown in blue, red, yellow and black, respectively. We claim that A+B+C≤1. If we prove it and show that there exists an angle γ∗such that A+B+C= 1 the impedance equation will be simplified. The need to simplify the impedance equation is motivated by the need to find an inverse problem method independent of the needle orientation. Lemma 4.1. There exists an angle γ∗∈[0,2π)such that A+B+C= 1. In fact, this angle is the one that verifies tan γ∗ tan ϕ= cos φ. Proof. ∃: We note that A+B+C= 1 ⇐⇒ the segment verifies z=ct.. That is because by definition of dα,z=ct. ⇐⇒ d(Qik,Qnl) = dα(Qik,Qnl), where d(·,·) is the euclidean distance between two electrodes. The [N] face is a plane in R3and it is immediate that there is at least one line in [N] verifying z=ct. which concludes the existence proof. Characterization of γ∗: We start noting that if ϕ=π/2 and φ=π/2 the [N] face forms a plane z=ct. so the equation is verified ∀γ∗∈[0,2π) because tan γ∗ tan π/2=tanγ∗ ∞= 0 = cos π/2 ∀γ∗∈[0,2π). Otherwise, we take the third component of points Qs1s2and Qs1s2from (4.1) where s1, s1∈ {1,2}and s2∈ {s2−8, s2, s2+ 8} ·If s2=s2=⇒ |Qs1s2(z)−Qs1s2(z)|= (rs1−rs1)(cos γcos φsin ϕ−sin γcos ϕ). Using the hypothesis tan γ∗ tan ϕ= cos φwe have |Qs1s2(z)−Qs1s2(z)|= (rs1−rs1)(cos γtan γ tan ϕsin ϕ− sin γcos ϕ) = 0 so we have that if gamma verifies the equation, the segment is in a line 4.5. Inverse problem 30 z=ct. ·If s2=s2±8 =⇒ |Qs1s2(z)−Qs1s2(z)|= (rs1−rs1)(cos(γ+π) cos φsin ϕ−sin(γ+ π) cos ϕ= (rs1−rs1)(−cos γcos φsin ϕ+ sin γcos ϕ) which is proved analogous to the previous case. This proves that tan γ∗ tan ϕ= cos φ=⇒z=ct.. To prove the reciprocal, we must prove z=ct. =⇒tan γ∗ tan ϕ= cos φ. Using the hypothesis, z=ct. =⇒ |Qs1s2(z)−Qs1s2(z)|= 0 =⇒(cos γcos φsin ϕ− sin γcos ϕ) = 0 =⇒tan γ∗ tan ϕ= cos φwhich concludes the proof. Corollary 4.1.1. If γ=γ∗=⇒Z20 =2κα K 2r1 r2 2−r2 1. Proof. Immediate from the four electrodes being aligned in configuration 20 and having the angle such that A+B+C= 1. We define Z20∗as the impedance equation obtained in the previous Corollary. Lemma 4.2. A+B+C≤1. Proof. Let d(·,·) be the euclidean distance between two electrodes. Firstly, we have that the euclidean distance between electrodes ε101 and ε102 in face [N] is d(ε101, ε102) = r2−r1. Secondly, we have that the apparent distance between electrodes ε101 and ε102 in face [N] is dα(ε101, ε102)=(r2−r1)pA+B+C. By definition, α2≤1 =⇒d(ε101, ε102)≥ dα(ε101, ε102). Therefore, substituting each distance by its value we have A+B+C≤1. Corollary 4.2.1. Z20∗≤Z20. Proof. Immediate from the previous Lemma and the definitions of Z20 and Z20∗. 4.5 Inverse problem Below we present different methods to solve the inverse problem, which consists in obtaining the anisotropy ratio, the impedivity and the needle orientation from impedance measurements. From the impedivity value, it is straightforward to obtain the resistivity and reactivity in both longitudinal and transverse directions. 4.5. Inverse problem 31 4.5.1 Method 1 Same as Method 1 in section 3.5.1 4.5.2 Method 2 Same as Method 2 in section 3.5.2 4.5.3 Method 3 For this method, we make no assumptions about the needle orientation with respect to the fibers. As opposed to the cross-type needle the ’T’-type needle structure enables to solve this method. The essential part of this method is to find γ∗to simplify the equation Z20∗(Corollary 4.1.1) to only one unknown, the impedivity κα. In Corollary 4.2.1 we note that finding the minimum value of Z20 ∀γ∈[0,2π) suffices to determine the value of Z20∗. Representing the values of Z20 in R2in polar coordinates with angle γand radius Z20 we obtain an ellipse [11]. The value of Z20∗corresponds to the minor axis of this ellipse. We start taking equations Z20(g),g∈ {1,2, ..., 8}. Then, we find the ellipse that best approximates the values of Z20(g). This ellipse has center in the origin so its equation can be written as C(x, y) : Ex2+Fxy +Gy2= 1, where E,Fand Gare the coefficients of the ellipse. This coefficients are found using minimum least squares. We define M∈ M8×3(R) and ~ 1∈ M8×1(R) as M:=         Z2 20(1) cos2(0) Z2 20(1) cos(0) sin(0) Z2 20(1) sin2(0) Z2 20(2) cos2(π 8)Z2 20(2) cos(π 8) sin(π 8)Z2 20(2) sin2(π 8) . . .. . .. . . Z2 20(8) cos2(7π 8)Z2 20(8) cos(7π 8) sin(7π 8)Z2 20(8) sin2(7π 8)         ,~ 1 :=         1 1 . . . 1         , 4.5. Inverse problem 32 so we can solve using minimum least squares as follows      E F G      = (MMT)−1MT~ 1. The associated matrix of the ellipse is M2:=   E F/2 F/2G . There exists another basis in which C(x, y) can be rewritten as ˜ C(˜x, ˜y) : ˜x2 ˜ F2+˜y2 ˜ G2= 1, where ˜ Fand ˜ Gare the axes of the ellipse and 1 ˜ F2and 1 ˜ G2are the eigenvalues of M2. Solving for the eigenvalues of M2we have 1 ˜ F2,1 ˜ G2=E+G±p(E−G)2+F2 2. Therefore, we have Y= min{˜ F, ˜ G}=s2 E+G+p(E−G)2+F2. We realize that Yrepresents the minimum value of Z20 when computed ∀γ∈[0.2π). On account of that and Corollary 4.2.1 we have Z20∗=Y. Taking the equation presented in Corollary 4.1.1 we can isolate καas follows κα=Z20∗K(r2 2−r2 1) 4r1 .(4.4) Taking the impedance equations from face [ES] of configurations 1, 18 and 19 from Table 3.1 and substituting the distance values we have Z1=κα K 2b a(a+b) 1 √B,(4.5a) Z18 =2κα K (a+b)√B−(s2−s1)√A (a+b)(s2−s1)√AB !,(4.5b) Z19 =2κα K (2a+b)√B−(s2−s1)√A (2a+b)(s2−s1)√AB !.(4.5c) From (4.4) we know the value of κα. As a result of that, we can isolate B from (4.5a) B=κα2b Ka(a+b)Z12 . Dividing equations (4.5b) and (4.5c) we get the ratio between Aand Bas follows A=(a+b)(2a+b)(Z18 −Z19) s(Z18(a+b)−Z19(2a+b))2 ·B. 5.3. Results 39 corrective factors to reduce it. Figure 5.6: Experimental conductivity σexp vs. theoretical conductivity σtfor a specific frequency in logarithmic scale shown in blue. (Left) f = 9.3kHz and (Right) f = 90kHz. Reference line σexp =σtshown in black. The corrective factors are the result of performing a linear regression with ln(σexp) and ln(σt) as dependent and independent variables, respectively, for all frequencies. The relation between the theoretical and experimental conductivity can be expressed as ln(σexp) = c1ln(σt) + c2(5.1) where c1:= c1(w) and c2:= c2(w) are the corrective factors that depend on the frequency. For a better understanding of their values, if the blue points in figure 5.6 were perfectly aligned over the orange line, the values would be c1= 1 and c2= 0 for all frequencies. After calculating them for each frequency, we obtain          min(c1) = 1.016 c1= 1.021 max(c1) = 1.048          min(c2) = −0.166 c2=−0.149 max(c2) = −0.127 where cdenotes the mean of the elements in c. These values support that the behaviour of the measurements is what was expected (c1≈1) but we have an offset (c26= 0). For this reason, we repeat the measurements with a new set of saline solutions. We aim to use the corrective factors calculated from the first set of measurements in the second set to confirm that it reduces the relative error. With c1and c2calculated, the corrected conductivity (σcorr) can be expressed as σcorr = (e−c2σexp)1/c1, 5.3. Results 40 obtained from solving for σtin equation (5.1). Below in figure 5.7 we present data obtained from the second set of saline solutions. We also include the corrected values and a reference line representing the theoretical conductivity. Figure 5.7: Plots of the conductivity value (top row) and the relative error of the conductivty (bottom row) while changing the frequency. Experimental, corrected and theoretical values shown in blue, orange and black dashed, respectively. From left to right, σt={0.04,0.17,0.38,1.39}. Figure 5.7 shows a significant improvement towards error reduction. As expected, using the corrective factors from the first set of saline solutions in the second set, the error gets reduced significantly to approximately 5%. 41 Chapter 6 Conclusions and future work Motivated by the need of reducing the number of needle insertions of previous research [12], this thesis has been able to present a method to measure the electrical properties of human skeletal muscle with a unique needle insertion. The thesis covers from the theoretical study to the design, fabrication and some experimental work. We can conclude that the goal of developing a method to measure electrical properties and their directional dependence has been achieved. Limitations in previous research included the need to perfectly align the electrodes in a specific direction. We have found a novel method presented with the ’T’-type needle which is independent of the needle orientation and overcomes previous limitations. Therefore, there is no need to align the needle in any particular direction. A consequence of having explicit impedance expressions is that you can foresee the data measured if you consider the electrical properties known. This work is going to be presented in a paper currently in preparation. Future work needs to be done to validate the different methods presented. Research should focus in doing numerical simulations of the needle to validate the different methods presented as well as to be a reference for the experimental data. Once the methods are validated, experiments could be repeated in an anisotropic material. Prior to any experiment in biological tissue, it would be interesting to try the needle in an anisotropic material without being invasive in biological tissues. Because of the needle’s size, an anisotropic material would have to be build. We propose a combination of agarose and saline solutions 42 to replicate the muscle fibers. 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