Estimation of electrical conductivity of a layered spherical head model using electrical impedance tomography

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1 Estimation of electical conductivity of a layeed spheical head model using electical impedance tomogaphy M Fena ndez-coazza, N von-elleniede and C H Muavchik Laboatoio de Electo nica Industial, Contol e Instumentacio n (LEICI) Facultad de Ingenieı a, Univesidad Nacional de La Plata (UNLP), Buenos Aies, Agentina maianof.coazza@ing.unlp.edu.a Abstact. Electical Impedance Tomogaphy (EIT) is a non-invasive method that aims to ceate an electical conductivity map of a volume. In paticula, it can be applied to study the human head. The method consists on the injection of an unpeceptive and known cuent though two electodes attached to the scalp, and the measuement of the esulting electic potential distibution at an aay of sensos also placed on the scalp. In this wok, we popose a paametic estimation of the bain, scalp and skull conductivities using EIT ove an spheical model of the head. The fowad poblem involves the computation of the electic potential on the suface, fo given the conductivities and the injection electode positions, while the invese poblem consists on estimating the conductivities given the senso measuements. In this study, the analytical solution to the fowad poblem based on a thee laye spheical model is fist descibed. Then, some measuements ae simulated adding white noise to the solutions and the invese poblem is solved in ode to estimate the bain, skull and scalp conductivity elations. This is done with a least squaes appoach and the Nelde-Mead multidimensional unconstained nonlinea minimization method. 1. Intoduction Electical impedance tomogaphy (EIT) is a elatively new imaging method that has evolved ove the past 20 yeas. In the medical field it has a potential clinical value in diagnostics and monitoing of a numbe of disease conditions, fom the detection of beast cance [1], to monitoing bain function [2], and possibly stoke [3]. EIT consists on the injection of small electical cuents in diffeent points of the body and the measuement of the esulting electic potential distibution though a senso aay. It is safe, potable, inexpensive and fast [4]. At pesent, EIT does not have the spatial esolution of othe methods like Magnetic Resonance Imaging (MRI) o Compute Tomogaphy (CT), while its key advantage is the tempoal esolution, which is in the ode of milliseconds [3]. In neuoscience, EIT can be used fo at least two diffeent applications: to image conductivity changes o to measue absolute conductivities. The fome is based on measuing the conductivity changes poduced by blood flow and volume alteations duing evoked activity in the bain [4]. Also, a slight ( 1%) change of the impedance is poduced in the ceebal cotex duing neuonal depolaization [4]. But since the skull is moe esistive than the othe tissues of the head, the cuent that may each the bain is elatively small [4] (less than 15% of total

2 injected cuent in a simulation study [5]). This is why in ode to use EIT as a functional bain mapping tool, the shield effect of the skull to the injecting cuents should be estimated fist [6]. The second application is a technique fo measuing the electical conductivity of the head, useful to ceate an anatomical conductivity map. In this application, the EIT fowad poblem consists of computing the electic potential distibution geneated by a known injected cuent in a known head model with its coesponding conductivity map. The EIT invese poblem is the estimation of the electic conductivity map fom the electic potential measued at the electode positions. As mentioned, the spatial esolution that can be achieved fom EIT data alone is poo. We popose to paameteize the invese poblem by a few paametes being the electic conductivity of the tissues, assuming the spatial distibution of the tissues is known. In this study, this spatial distibution consists of thee concentic sphees. But next, ealistic geomety obtained fom some othe techniques such as MR o CT scalp will be used. The appoximate conductivity map will have then a vey good spatial esolution but with conductivity values that may diffe a little fom the eal ones since they ae assumed homogeneous in each tissue. The poposed methodology is expected to wok vey well to estimate the scalp and skull conductivities, and a combination with some othe techniques can complete the model. Diffusion Tenso Imaging (DTI) can be used to estimate the bain and ceebospinal fluid (CFS) conductivities [7]. The knowledge of the conductivity map is impotant in the study of electoencephalogaphy (EEG) poblems, in paticula fo the estimation of the position of neual souces. The EEG souce localization poblem aims to detemine the neual souces inside the bain that best explain the electical potentials measued on the suface of the scalp. The detemination of the souces is made though the use of mathematical models which descibe the head as an electical conducto. Then, the knowledge of the electical conductivity map of the head is impotant since it is known that the solution to the souce localization poblem is highly dependent on the values taken by the scalp, skull, and bain conductivities [8]. The scalp impedance plays a majo ole in head EIT, as it is engaged with cuents of the highest density coming fom the electodes, and theefoe it is the fist stage in which distibution of cuent into the est of the layes is detemined. Any inaccuacies at this stage will substantially affect the impedance eadings of the inne layes [9]. Seveal studies lead to a wide ange of values (see Hoesh gaphs and tables [9]) showing the impotance of in vivo and pe patient based infomation. Regading the skull, thee ae two kinds of matue bones: compact bone and spongy bone [9]. Many of the skull bones ae flat, consisting of two plates of compact bone (thickness of 1.7 to 4.3 mm) enclosing a naow laye of cancellous bone (thickness of mm) containing bone maow. Data published egading skull impedance also coves a wide ange of values. Dielectic popeties of the skull ae influenced by seveal factos: bone composition (skull layes-stuctue), non-unifom geomety, anisotopy and age (wate content is educed and bone maow is changing fom ed to yellow with age, which deceases the conductivity) [9]. In vivo and pe patient based infomation egading the skull impedance is needed fo building moe accuate models to use in EEG and functional EIT poblems. In the next section we descibe the adopted head model in detail. Then, we explain the mathematical backgound of the fowad poblem. Next, we state the invese poblem, and finally, the esults of seveal simulations fo diffeent numbe of electodes ae pesented and discussed. 2. Methods 2.1. Electical Model The head model consists of thee concentic spheical shells with the enclosed space between them epesenting the thee main tissues of the head: scalp, skull and bain (figue 1). Each laye

Electode positions based on the 10-10 system ove a spheical head. Black, black and shaded, and labeled cicles show the positions used fo 16, 32 and 64 configuations espectively.

3 is consideed as homogeneous and isotopic, i.e. conductivity is constant and with no pefeed diection. Figue 1. Electical model. σ1, σ2 and σ3 epesent homogeneous and isotopic conductivities of bain, skull and scalp espectively. +I and I epesent the positive and negative cuent injection points. Figue 2. Electode positions based on the system ove a spheical head. Black, black and shaded, and labeled cicles show the positions used fo 16, 32 and 64 configuations espectively. The electode placement used is based on the system fo 64, 32 and 16 sensos (figue 2). A pai of electodes is selected fo the cuent injection, while the othes ae dedicated to measue the esultant electic potential diffeence. One of the cuent electodes will supply a cuent of +I (cuent souce) and the othe a cuent I (cuent sink), both with the same absolute value (figue 1). EIT applications usually comply with the Intenational Electotechnical Commission (IEC) standad [10] which specify a patient auxiliay cuent limit of 100µA fom 0.1 Hz to 1 khz. This standad is based on limitation of the cuent to 10 % of the aveage theshold of sensation when cuent is applied to the skin using suface electodes. This coesponds to a widely cited study in which a mean peception cuent of 1.1 ma was obseved between DC and 200 Hz, fo hand holding of a coppe wie of d 3.3 mm in diamete [11]. This gives a theshold cuent density aound 2 to 15 A/m2. In anothe study ([12]), with abasion just unde a conventional cup electode (Ag/AgCl, 11 mm diamete), the theshold of sensation was µA and it became slightly unpleasant at 400µA. This gives a theshold cuent density simila to pevious study. Then, a cuent of 100µA is appopiate fo use in eal studies with patients. We use this value to poduce a ealistic simulation. Also, the model dimensions must be scaled to a human head. So, the oute shell adius is set to 10cm. The othe adii ae the ones that ae mostly used in spheical models [13]. Table 1 summaizes the inteface adii chosen fo the model. The emaining data to complete the model ae the isotopic conductivity values. These ae found in the liteatue with a high dispesion (table 2 shows some examples). Some of them ae measuements and some othes ae efeence values given in the studies. The values chosen fo the pesent wok ae shown in the last ow. The citeia was to select ealistic values, with two exact decimals fo bain:skull and bain:scalp conductivity atios in ode to ease the compehension of the esults.

4 Table 1. Model adii Inteface Label Radius [cm] Bain:Skull Skull:Scalp Scalp:Ai a b c Table 2. Conductivity values found in liteatue [S/m] Autho Bain Skull Scalp Zheng Xu et al [6] Gilad et al [12] Abascal et al [5] Adhazi et al [14] Gonc alves et al [8] Hoesh [9] Liston et al [15] Zlochive [16] Fena ndez-coazza (gey matte) (bone) (muscle) Fowad Poblem Assuming that the scalp, skull, and bain conductivities ae known, the poblem consists on estimating the electic potential on the outemost suface. This is an electomagnetism poblem with suface conditions and spheical geomety. Maxwell equations goven the physics of electomagnetism. In paticula, fo low fequencies, the quasi-static assumption can be used. This is common in many studies since it educes the complexity of the poblem [3]. The limitation is that the model is valid only fo fequencies below 10 khz, fo which the wave length is much geate than the head dimensions. Then, we can wite ~ E ~ 0, (1) ~ is the electic field. Fom (1), the electic field E ~ can be expessed as the gadient of whee E a scala potential Φ ~ Φ. ~ E (2) ~ [17]. It is known that the cuent density J~ is popotional to E ~ J~ σ E, Whee, σ is the electic conductivity of the medium. Applying the divegence to J~ and assuming that σ is constant, ~ J~ ~ (σ E) ~ σ( ~ E) ~ σ( ~ 2 Φ). (4) The cuent consevation law states that the divegence of the cuent density is zeo eveywhee in the sphee except in the injection points. With this appoximation, the electic potential Φ must satisfy the Laplace equation in each laye: ~ 2 Φ 0. (5)

5 Taking advantage of the poblem symmeties, fo each laye i, the solution of the potential Φ in evey point of the space (,θ,φ) can be expessed as a linea combination of the Legende Polynomials as shown in (6) [17] [18]. Φ(i) (, β, γ) X (+)(i) l (Al (+)(i) l 1 + Bl ).Pl (cosβ) X l0 Φ(i) (, β, γ) X ( )(i) l (Al ( )(i) l 1 + Bl ).Pl (cosγ) l0 (i) (i) (Al l + Bl l 1 ).(Pl (cosβ) Pl (cosγ)); i 1, 2, 3. (6) l1 (i) (i) Whee the constants Al, Bl depend on the bounday conditions, Pl (x) is the Legende Polynomial of ode l (whee P0 (x) 1), β(θ, φ) is the angle to the +I cuent souce and γ(θ, φ) to the I cuent sink. In laye 3, the potential is not exactly as (6) because the cuent injection points ae within this laye and thei effect must be included. Applying lineaity, the potential on this laye can be expessed as the potential in an unbounded medium (Φunb ) plus anothe with the expession of (6) (Φ ) that allows the total potential on this laye (Φtot ) to satisfy the bounday conditions. Then, Φtot Φunb + Φ, (7) whee, Φunb I 4πσ3 1 p 2 + c2 2c.cos(β) p c2 2c.cos(γ)!. (8) (1) Note that the constant Bl must be zeo in ode to have continuity at the oigin. So, fo each (1) (2) (2) ode l, thee emain five constants Al, Al, Bl, Al, Bl to be detemined. On the inteface between the layes, two conditions ae satisfied: the continuity of the nomal components of the cuent density and the continuity of the electic potential. They lead to five equations: σ1 Φ(1) Φ(2) σ2 σ3 σ2 Φ σ3 tot a a b Φtot Φ(2) 0 b (9) c Φ1 a Φ(2) Φ2 b Φtot a b Solving the equation system deived fom (9) the constants can be detemined. Then, expanding and egouping tems, the potential ove the outemost suface ( c) can be witten as:! X k K1.σ1/σ2 + K2.σ1/σ3 + K3.σ2/σ3 + K4 Φtot (c, θ, β). Φ + 1.(Pl (cosβ) Pl (cosγ)), c K5.σ1/σ2 + K6.σ1/σ3 + K7.σ2/σ3 + K8 l1 (10)

whee, k Φ (β, γ) K1 K2 K3 K4 K5 K6 K7 K8 I, 4πσ3 (11) 1 p 2 2 cos(β) p 1 2 2 cos(γ), (l + 1 + lb2l+1 )l(1 (a /b )2l+1 ), (1 b2l+1 )l(l + (l + 1)(a /b )2l+1 ), (1 b2l+1 )l(l + 1)(1 (a /b )2l+1 ), (l +

6 whee, k Φ (β, γ) K1 K2 K3 K4 K5 K6 K7 K8 I, 4πσ3 (11) 1 p 2 2 cos(β) p cos(γ), (l lb2l+1 )l(1 (a /b )2l+1 ), (1 b2l+1 )l(l + (l + 1)(a /b )2l+1 ), (1 b2l+1 )l(l + 1)(1 (a /b )2l+1 ), (l lb2l+1 )(l l(a /b )2l+1 ), l2 (l + 1)(1 (a /b )2l+1 )(1 b2l+1 ), 2l (l + (l + 1)(a /b ) )(l + (l + l)b2l+1 ), 2l+1 2l+1 (l + 1)l(1 (a /b ) )(l + (l + 1)b ), (l + 1)l((l + 1) + l(a /b )2l+1 )(1 b2l+1 ). (12) (13) (14) (15) (16) (17) (18) (19) (20) With a and b being the a and b adii elative to a sphee of adius 1, (i.e., a a/c and b b/c). Figue 3. Suface potential fo σ S/m, as seen fom behind. The positions and labels fo 64-electode configuation ae also displayed Figue 4. Suface potential fo σ S/m, as seen fom behind. The positions and labels fo 64-electode configuation ae also displayed Figues 3 and 4 show the esulting electic potential Φ ove the sphee suface fo two diffeent skull conductivity values. It can be seen that it changes with that paamete Invese Poblem Fo the invese poblem the conductivities ae assumed unknown, and the electic potential measued at the electodes is known. In ode to simulate the measuements, white gaussian noise is added to the solution of the fowad poblem with the pevious section fomulas (assuming the conductivities of table 2). This noise models the electonic noise of the amplifies and the skin/electode contact electochemical noise. We model it as Gaussian distibuted zeo mean and 1µA standad deviation.

7 We will estimate the conductivity atios instead of the conductivities themselves. This is convenient, fo example, in (10). But only two of them ae independent, so x1 σ1/σ2 and x2 σ1/σ3 ae chosen as the independent vaiables implying that σ2/σ3 x2/x1. Then, a least squaes appoach can be used to estimate the unknowns x1 and x2 that minimize the diffeence between each senso measuement and its coesponding fowad solution: (n 2) X min (Vi Φ (θi, βi, x1, x2 ))2, (21) x1,x2 i1 Whee, n is the numbe of electodes, Vi is the measued potential on the electode i and Φ (βi, γi, x1, x2 ) is the suface potential function obtained fom the fowad poblem (10). The uppe limit of the summation is (n 2) because the injection nodes ae excluded since thei potential is not measued. The optimization algoithm used is the Nelde-Mead multidimensional unconstained nonlinea minimization method [19]. Note that in the fomulation (21) thee is a constain (the unknowns must be positive), but as the stating point can be chosen elatively nea to the tue value, a faste unconstained method can be used. 3. Results The invese poblem (21) was solved fo 16, 32 and 64 electodes with 1µA standad deviation noise. It can be seen in figue 2 that thee is a symmety on the electode distibution between left an ight sides. This symmety does not exist between font and back because of the thee occipital electodes (Iz, P9 and P10) in the 64-electode configuation, and the Oz electode in the 32 and 16 configuations. So, all diffeent pais of electodes without an exact mioed configuation occuing on the left side wee consideed. Fo example, the poblem with the injection nodes being C4 and F4 leads to the exact same esults of the C3-F3 combination. Then, this last pai was excluded. The numbe of combinations used ae shown in table 3. Table 3. Injection node combinations Electodes Combinations Fo each combination, 100 diffeent noise ealizations wee simulated and the mean and vaiance of the x1 and x2 estimates wee computed. Figues 5 and 7 show, fo the 64 and 32 electode configuations, the mean of x1 as a function of the angle α between the injection electodes. The expected value is also displayed with a hoizontal line. Figues 6 and 8 show the same esults fo x2. Figue 9 shows, fo the 16-electode configuation, the mean and mean plus/minus vaiance of the x1 estimation fo diffeent combinations of injection electodes. A paticula injection pai has a vey high vaiance. This is easonable since it coesponds to the T7-T8 combination. In this case all measuement electodes ae close to the middle plane between these two positions, whee the electic potential is close to zeo. Figue 11 shows the esulting vaiance fo diffeent x1 simulations. Each coss epesents the mean vaiance fo each pai of injection electodes. The cental mak is the median, the edges

8 σ1/σ2 mean vesus α fo 64 Figue 5. electode configuation. σ1/σ3 mean vesus α fo 64 Figue 6. electode configuation. σ1/σ2 mean vesus α fo 32 Figue 7. electode configuation. σ1/σ3 mean vesus α fo 32 Figue 8. electode configuation. of the box ae the 25th and 75th pecentiles, and the whiskes extend to the most exteme data points not consideed outlies. Figue 12 shows the same esult fo x2. The combination T7-T8 was excluded fom both plots in the 16-electode configuation because, as explained befoe, it is a vey atypical case and would impact negatively in the plot eadability. Anothe simulation was done in ode to study the impact of the assumed noise vaiance. We know that the bain has nomal activity that can be detected with the EEG electodes. This activity means fo this application of EIT additional noise with peaks of aound 20µV. Although this noise might be disaffected using a pope technique (taking advantage of the fact that the fequency and phase of the injected cuent is well known), is impotant to know if such a technique effot is wothwhile. Then, the same simulation fo 64 electodes was an, but with a noise standad deviation of 10µV. Figue 10 compaes the 16 and 64 electode configuations with 1µV and 10µV of noise standad deviation espectively, whee the left box of the plot is the same than the ight one of figue Conclusions The fowad and invese poblems have been solved successfully fo 16, 32 and 64 electodes ove the thee shell spheical model and the expected conductivity values wee etieved. It is

9 Figue 9. σ1/σ2 mean and mean plus/minus vaiance fo 16-electode configuation vesus electode combinations Figue 11. Box plot of the vaiance fo estimations. σ1/σ2 Figue 10. Box plot of the vaiance fo σ1/σ2 estimations in 16 and 64 electode configuations with 1µV and 10µV standad deviation noise espectively. Figue 12. Box plot of the vaiance fo estimations. σ1/σ3 impotant to highlight that this was achieved with appopiate scales. Fo instance, the adius of the oute sphee was 10cm, close to a eal head size. Also, the injected cuent value was chosen with the same citeion as if it wee a eal expeiment. Futhemoe, the conductivity values wee close to eal values, and finally, a typical electonic noise standad deviation was chosen. Injection nodes with geate angle between them geneally achieve bette esults (less vaiance) but this is not a geneal ule, since the accuacy not only depends on that angle, but also on the suounding electodes fo measuement. Fo example, C3-C4 combination is bette than T7-Cz (less vaiance), although both have vey simila angle between the injection nodes. Anothe example is the atypical T7-T8 pai fo the 16 electode configuation, whee the angle α between the electodes is maximum and the vaiance of the conductivity estimation is the highest. In conclusion, the combinations with moe sensos suounding the injection electodes and a lage angle between them ae pefeed. On the othe side, the esults show an exponential-like dependence of the vaiance with the numbe of electodes. Of couse, moe electodes mean bette esults, but fom this study, and

10 also depending on the measuement pecision needed, a good choice on the sensos quantity could be pefomed. Moe impotant than the amount of electodes is the emoval of the bain activity fom the ecoded signals. A cap with 16 electodes and a system to emove the backgound bain activity is much bette than a 64 electodes system without it. The spheical model mainly seves as a validation tool fo numeical algoithms since the fowad poblem has analytical solution. With numeical methods like Bounday Element Method (BEM) o Finite Element Method (FEM), the algoithms can be independent of the geomety. The solutions fo the fowad and invese poblems with a eal shaped model using FEM ae in pogess. This model will use a standad shape model of the head and a standad aveaged conductivity distibution of the bain and CFS obtained via DTI. Then, it will be used to estimate the conductivities of skull and scalp. An advantage of using FEM instead of BEM is the possibility of include anisotopy in the model. It is known that skull has a tangential to adial anisotopy of aound 1:10 [13]. The same seems to happen in the scalp, but in this case the elation is lowe (1:1.5) [9] and it is not so elevant. So, the invese poblem will be extended to look fo fou unknowns: tangential skull, adial skull, tangential scalp and adial scalp conductivities. The development of an EIT device is also planned. Using the aveage head model o patient specific models (if possible) jointly with this device, eal estimations of scalp and skull conductivities will be pefomed. Since this infomation will be used to solve the EEG souce localization poblems, the eal impact of having patient specific conductivity maps will be studied. Acknowledgments This wok was suppoted by ANPCyT PICT , UNLP I127, CONICET and CICpBA. Refeences [1] Cheepenin V A, Kapov A Y, Kojenevsky A V, Konienko V N, Kultiasov Y S, Ochapkin M B, Tochanova O V and Meiste J D 2002 IEEE Tans. Med. Imaging 21(6) [2] Bagshaw A P, Liston A D, Bayfod R H, Tizzad A, Gibson A P, Tidswell A T, Spakes M K, Dehghani H, Binnie C D and Holde D S 2003 NeuoImage [3] Bayfod R 2006 Annu. Rev. Biomed. Eng [4] Holde D 2008 Automation Congess, WAC Wold pp 1 6 [5] Abascal J F, Aidge S R, Atkinson D, Shindmes R, Fabizi L, Lucia M D, Hoesh L, Bayfod R H and Holde D S 2007 Neuoimage 43(2) [6] Xu Z, He W, He C, Zhang Z and Liu Z 2008 Automation Congess, WAC Wold pp 1 5 [7] Tuch D S, Wedeen V J, Dale A M, Geoge J S and Belliveau J W 2001 Poceedings of the National Academy of Sciences of the United States of Ameica 98(20) [8] Gonc alves S I, de Munck J C, Vebunt J P A, Bijma F, Heethaa R M and da Silva F L 2003 IEEE Tans. Biomed. Eng. 50(6) [9] Hoesh L 2006 Ph.D. thesis Univesity College London [10] IEC Medical electical equipment - pat 1: geneal equiements fo basic safety and essential pefomance ed3.0 (Geneva: Intenational Electotechnical Commission) [11] Dalziel C F 1972 IEEE Spect. 9(2) [12] Gilad O, Hoesh L and Holde D S 2007 Med. Bio. Eng. Comput [13] Rush S and Discoll D 1968 Anasthesia and analgetica 47(6) [14] Ahadzi G, Gilad O, Hoesh L, Bayfod R and Holde D S 2004 ICEBI 04 - V Electical Impedance Tomogaphy [15] Liston A D, Bayfod R H, Tidswell A T and Holde D S Physiol. Meas [16] Zlochive S 2005 Ph.D. thesis Tel Aviv Univesity [17] Jackson J D 1975 Classical Electodynamics Second Edition (New Yok: John Wiley & Sons) [18] Fank E 1952 Jounal of applied physics 23(11) [19] Lagaias J C, Reeds J A, Wight M H and Wight P E 1998 SIAM Jounal of Optimization 9(1)

Contact impedance of grounded and capacitive electrodes

Contact impedance of grounded and capacitive electrodes Abstact Contact impedance of gounded and capacitive electodes Andeas Hödt Institut fü Geophysik und extateestische Physik, TU Baunschweig The contact impedance of electodes detemines how much cuent can