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The application of FEM technique to electromagnetic models for magnetic neural stimulation Le Pocher, Herve 1993

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The Application of FEM Techniqueto Electromagnetic Models forMagnetic Neural StimulationByHerve Le PocherB.Sc., (Biology) The University of Ottawa, 1984B.A.Sc., (E.E.) The University of Ottawa, 1989A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE FACULTY OF GRADUATE STUDIESDepartment of Electrical EngineeringWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAOctober 1993© Herve Le Pocher, 1993In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)Department of  fiec.."-et'ca g en-prieept,The University of British ColumbiaVancouver, CanadaDate ^DE-6 (2/88)AbstractSeveral aspects of magnetic neural stimulation are investigated by applying an advanced finite-elementanalysis. The objective is to demonstrate that the finite element method is well suited for realistic bio-electromagnetic modelling. The focus of our research is to explore four factors which affect induced electricfields: 1) magnetic coil diameter, 2) anisotropic tissue conductivity, 3) inhomogeneous tissue conductivityand 4) finite tissue volumes. All results are analyzed in terms of the second space derivative of thepotential along the nerve fiber path 82 V/82 called the "activation function".If two magnetic coils have the same outer diameter than the one with the larger inner diameter willinduce the largest electric field and activation function. This result has direct clinical implications formodern commercial magnetic coil designs which are characterized by small inner diameters. In addition,a magnetic coil was modelled over a semi-infinite planar tissue and a finite cylindrical tissue. The electricfield induced in the finite volume was 25% smaller than in the planar volume; however, the activationfunction magnitude remained essentially constant.The fields generated by two magnetic coil designs were then applied to arms modelled with isotropicand anisotropic conductivities. The results show that the electric field is 13% lower and that the activationfunction is 24% lower in anisotropic tissue relative to the values seen in the isotropic tissue. These fieldreductions may be attributable to changes in the distribution of surface charges. The isotropic modelresults for the arm were compared to previous studies where fields were modelled within a planar isotropictissue. This analysis indicates that a larger activation function can be induced in the arm due to rapidspatial changes in the electric field magnitude, even though a larger electric field magnitude is present inthe planar tissue. Conversely, if the magnetic coil is too large surface charges in the arm will significantlydecrease both the activation function and the electric field.A magnetic coil was modelled above the gyrus of the primary motor cortex. The gyrus conductivity wasmodelled in three ways: 1) homogeneous, 2) inhomogeneous-isotropic and 3) inhomogeneous-anisotropic.Only minor differences in the results could be attributed the anisotropic conductivity parameters. Severalfeatures of the modelled electric fields within the inhomogeneous tissue were different from those foundin the homogeneous model: 1) the electric fields were not all parallel to the planar air/tissue interface,2) there was a significant increase in the electric field above the gyrus and within the crest of the gyrus,3) decreases in the current density were found within the gyrus and 4) large electric field gradients wereobserved at conductivity interfaces. A typical pyramidal fiber path was then defined within the gyrus andits activation function was calculated. We found two distinct peaks in the activation function, one relatedto a conductivity interface and the second related to a bend along the fiber path. The magnitude andspatial span of both these peaks are unlike any previously reported for magnetic stimulation. Either ofthese two peaks may account for both the direct and indirect pyramidal stimulation reported in clinicalinvestigations. These model results indicate the importance of several tissue characteristics which havenot been investigated in previous studies, and demonstrate how these characteristics interact with eachother and their relationship to effective nerve stimulation.iiContents Abstract ^List of Figures  iiiList of Tables  vAcknowledgments ^ viChapter 1^Introduction ^  1^1.1^The Nerve Cell and Its Stimulation^ 2^1.2 Magnetic Stimulation^ 5^1.3^Measurements and Models of Nerve Activity^ 71.4 The Activation Function 8Chapter 2^A Review of Magnetic Coil Analysis ^  12^2.1^Previously Used Modelling Methods 122.2 Results from MC Modelling 15Chapter 3^Theory and Methods in FEM Modelling ^  19^3.1^Variational Solution^ 203.2 Finite Element Method 233.3^FEM Method for Electromagnetics^ 253.4 Generating a FEM Model^ 28Chapter 4^Validation of FEM Results  344.1^Introduction^ 344.2 Initial Validation 354.3^Validation Using Published Results^ 364.4 Discussion^ 39Chapter 5^Modelling Commercial MC Designs  425.1^Introduction 425.2 FEM Modelling^ 435.3^FEM Model Results 445.4 Discussion 46Chapter 6^Modelling Anisotropic Tissue in the Arm ^  486.1^Introduction^ 486.2 FEM Model of Square MC Over the Arm 506.3^FEM Model of Figure-8 MC Over the Arm^ 596.4 Discussion^ 67Chapter 7^Modelling Inhomogeneous Tissue in the Brain  70^7.1^Introduction 707.2 The Central Nervous System^ 717.3^FEM Model^ 777.4 Homogeneous Model Results 847.5^Isotropic Model Results^ 877.6 Anisotropic Model Results 937.7^Stimulus Fields for a Pyramidal Fiber^ 997.8 Discussion^ 101Chapter 8^Discussion and Conclusion ^  1078.1^Clinical Data 1078.2 Summary^ 112Bibliography  117Appendix A^Acronyms  122List of FiguresFigure 1.1^Nerve Stimulation Apparatus ^  1Figure 1.2 Clinical Magnetic Stimulation  2Figure 1.3^Elements of Nerve Cell ^  3Figure 1.4 Equivalence Between Electrode and Magnetic Stimulation ^ 4Figure 1.5^Electrode Stimulation of Long Straight Axon ^  5Figure 1.6 Cable Model for Extracellular Stimulation  9Figure 1.7^Clinical Magnetic Stimulation Conditions ^  10Figure 2.1 Magnetic and Electrostatic Contributions to E-field ^  13Figure 2.2^Typical Magnetic Coil Designs ^  15Figure 3.1 Functionals and Path Variations  20Figure 3.2^Electrostatic Example ^  21Figure 3.3 Basic FEM Model Components ^  23Figure 3.4^Generation of a Finite Element Mesh  30Figure 3.5 FEM Model Symmetry ^  31Figure 4.1^MC Orientations and Coordinate System ^  34Figure 4.2 Axial Magnetic Flux Density for the N1H and N1V MCs ^ 36Figure 4.3^Horizontal N1H MC Results ^  37Figure 4.4 Vertical N1V MC Results  38Figure 4.5^Validation of N250H MC Results ^  39Figure 5.1 Vertical MC Over Finite Tissue  43Figure 5.2^Induced [-fields Along the X and Y Axes ^  44Figure 5.3 Induced Activation Functions ^  45Figure 6.1^Structured Membrane Geometry in Muscle Tissue ^  48Figure 6.2 Model Geometry ^  50Figure 6.3^Four Views for Square-MC Results Presentation ^  50Figure 6.4 Gray Scale for Isocontour Plots ^  51Figure 6.5^Fields Along Axon ^  54Figure 6.6 Results for Square-MC Over Arm [Angled View] ^  55Figure 6.7^Results for Square-MC Over Arm [End View]  56Figure 6.8 Results for Square-MC Over Arm [Lateral View]^  57Figure 6.9^Results for Square-MC Over Arm [Medial View]  58Figure 6.10 Model Geometry ^  59Figure 6.11^Four Views for Figure-8 MC Results Presentation^  60Figure 6.12 Fields Along Axon ^  62Figure 6.13^Results for Figure-8 MC Over Arm [Angled View] ^  63Figure 6.14 Results for Figure-8 MC Over Arm [End View]  64ivFigure 6.15^Results for Figure-8 MC Over Arm [Lateral View] ^  65Figure 6.16 Results for Figure-8 MC Over Arm [Medial View]  66Figure 7.1^The Brain ^  72Figure 7.2 Coronal Section of Cerebral Hemisphere and Motor Cortex Sequence Map ^ 73Figure 7.3^Cytoarchitecture of Cerebral Cortex ^  74Figure 7.4 Coordinate System for Cortical Conductivity ^  75Figure 7.5^Subcortical Fiber Tracts ^  76Figure 7.6 Figure-8 MC for Cortical Stimulation ^  78Figure 7.7^Views of Modelled Precentral Gyrus and MC^  79Figure 7.8 Four Views of the FEM Mesh for Precentral Gyrus and MC ^ 82Figure 7.9^FEM Mesh for Head ^  83Figure 7.10 Angled View of Field Magnitudes in Head^  84Figure 7.11^Anterior and Posterior Views of Field Magnitudes in Precentral Gyrus ^ 85Figure 7.12 Angled and Medial Views of Field Magnitudes in Precentral Gyrus ^ 86Figure 7.13^Angled View of Field Magnitudes in Head^  88Figure 7.14 Angled View of Field Magnitudes in Precentral Gyrus ^  89Figure 7.15^Medial View of Field Magnitudes in Precentral Gyrus  90Figure 7.17 Posterior View of Field Magnitudes in Precentral Gyrus ^ 91Figure 7.16^Anterior View of Field Magnitudes in Precentral Gyrus  92Figure 7.18 Angled View of Field Magnitudes in Head^  94Figure 7.19^Angled View of Field Magnitudes in Precentral Gyrus ^  95Figure 7.20 Medial View of Field Magnitudes in Precentral Gyrus  96Figure 7.21^Anterior View of Field Magnitudes in Precentral Gyrus^  97Figure 7.22 Posterior View of Field Magnitudes in Precentral Gyrus  98Figure 7.23^E-fields Along a Typical Pyramidal Fiber ^  100Figure 7.24 Modes of Brain Stimulation ^  105Figure 8.1^Threshold Stimuli and Electrode Distance^  109VList of Tables Table 4.1Table 5.2Table 5.3Table 6.4Table 6.5Table 7.6Table 8.7Published MC Modelling Results 41Commercial Magnetic Coil Designs 43MC Dimensions and Model Results 46Admittance Ratio for Monkey Muscle Tissue 49Review of Modelled AF Functions 68Cortical Conductivities 75Threshold Levels for Modelled AF Functions 110AcknowledgmentsI wish to thank Dr. C.A. Laszlo and Dr. W.D. Dunford for their supervision and suggestions throughoutthe course of this research. In addition, the discussions with Dr. P.C. Vaughan on the physiologic aspectsof this thesis were much appreciated.I am also thankful for the use of the computing facilities offered by Dr. S. Cherukupalli at PowertechLabs, the extensive technical assistance provided by Chuck Figer at the MacNeil Schwendler Corporationand the effective management of our Department's computing facilities by Dave Gagne.I would also like to thank the British Columbia Science Council for funding this project and the NaturalScience and Engineering Research Council of Canada for providing me with a Postgraduate Scholarship.Finally, I thank my wife, Theresia, for her encouragement and steadfastness.viivoltagesource •^•electrodesElectric Stimulator— I cmMagnetic StimulatorChapter 1IntroductionIt is possible to stimulate the human brain through the intact scalp and skull in conscious subjects. Itis also possible to stimulate the human peripheral nervous system, in particular the motor nerves whichactivate skeletal muscles. The stimulus may be administered by the cutaneous application of a voltagesource (electrodes) or by the non-contact application of an intense burst of a time varying magnetic fieldfrom a current-loop called a magnetic coil (MC). It is assumed that both types of stimuli (electric andmagnetic) function by generating electric fields (E-fields) along the length of the target nerve(s). Clinicalnerve stimulation a) serves as a diagnostic tool, b) is used to investigate nerve function and c) is used toactivate nerves and muscles which have lost their function due to disease or injury.Modern clinical electrode stimulation has been in practice since the 1950's [1]. A low-impedance,pulsed voltage source is used (Fig. 1.1). The electrodes are put onto the skin where the target nerveis located. The patient's response to the stimulus is gauged by noting muscle contraction strength or bymeasuring the electric field generated by the target nerve or its associated muscle(s).Clinical use of magnetic stimulation has only become partially accepted within the last decade [2-4].It differs from electrode stimulation in that no direct skin contact is necessary. A high intensity pulsedmagnetic field is used to induce electric fields within the patient. A typical modern magnetic stimulator issimple in concept, and may consist of four major components: a high voltage discharge capacitor (1000V,200 uF), a power supply, a multi-turn palm-sized stimulator coil (20 turns, 20pH), and switches (rectifiers)to recharge the capacitor and to discharge of the capacitor through the coil (Fig. 1.1).Magnetic stimulation is currently being promoted as a painless method of performing routine nerveconduction studies of the peripheral and more importantly, the central nervous system. The two mostcommonly investigated locations for magnetic stimulation are the head and wrist (Fig. 1.2). The MC isplaced next to or on the body part containing the target nerve. The orientation of the MC with respectto the skin surface and nerve location are important factors for effective stimulation. Several popular MCorientations are shown in Fig. 1.2.Figure 1.1 Nerve Stimulation Apparatus1(A) Medial nerve stimulation using tangential MC orientation over wrist.(B) Medial nerve stimulation using orthogonal MC orientation over wrist.(C) Brain stimulation using tangential MC orientation over vertex of head.(D) Brain stimulation using lateral-sagittal MC orientation at side of head.Figure 1.2 Clinical Magnetic StimulationThe clinical merit of magnetic stimulation as a standard diagnostic tool is presently being debated.Some of its weaker points are lack of focusing ability [5-8], submaximal stimulation [6, 5, 9-11], andreproducibility [5, 12]. Reasonable success has been achieved in several clinical domains: the assessmentof central conduction times in multiple sclerosis, Parkinsonism and other degenerative disorders [13, 14,6], evaluating patients with peripheral facial nerve disorders such as Bell's palsy [15, 16], monitoring theintegrity of the corticospinal tract during spinal cord surgery and other interventions [17-19], localizationof lesions of the central nervous system and peripheral nervous system [10], studies in motor systemreorganization in quadriplegics [5], and functional mapping of the human cortex [20].1.1 The Nerve Cell and Its StimulationThe measurement and interpretation of clinical results obtained from nerve stimulation require anunderstanding of the structures and mechanisms involved in the initiation and transmission of nerveimpulses. The nervous system is that part of the anatomy which deals with the acquisition, transferand processing of nerve impulses (information). This network of nerve fibers is divided into two parts: a)the central nervous system (CNS) which includes the brain and spinal cord and b) the peripheral nervoussystem (PNS) which consists of bundles of sensory and motor axons called nerves, radiating from thebrain and spinal cord in a consistent fashion. The neuron is the anatomical unit of the nervous system.These cells are specialized for the reception and transmission of excitatory or inhibitory stimuli and areconsequently responsible for most functional characteristics of the nervous system. A neuron generallyconsists of : 1) a cell soma which contains the nucleus and varies in length from 5 to 135 urn, 2) an axonwhich can vary in length from less than a millimeter to several meters, 3) an axon hillock where APs areinitiated and conduct away from the cell soma and toward the synaptic knobs at the end of the axon, 4)many dendrites which are typically short, branching processes which conduct impulses towards the cellbody from up to 5000 neural sources (i.e. the synaptic knobs of other neurons), (Fig. 1.3). Axons with adiameter greater than 2 prIl are enveloped in a fatty myelin sheath which insulates the axon and increasesthe speed of impulse conduction along its length. Connections among neurons are called synapses. Atypical synapse can occur between a synaptic knob and: a dendrite, a cell body or an other axon. At the2A. cell somaB. axonC. synaptic knobD. dendriteE. axon hillocksynapse the two cells do not make physical contact but are about 20 nm apart allowing chemical diffusionto act as a rapid, target specific mechanism of information transfer. The information transfer consists ofeither promoting or inhibiting the initiation of a single AP. Neurons can be classified into three groups: a)efferent neurons conduct impulses from the CNS to the PNS (eg: motor neurons), b) afferent neuronsconduct impulses from the PNS to the CNS (eg: sensory neurons), and c) intemeurons form a network ofinterconnecting neurons between efferent and afferent neurons. The PNS consists mainly of afferent andefferent neurons while the CNS consists largely of interneurons. Although all parts of the neuron cell areinvolved in nerve impulse transmission the axon is our main focus of study for nerve modelling.Clinical stimulation of nerves is dependent on the E-field distribution along the length of the targetnerve and is not dependent on whether these E-fields are sourced by an electrode or induced by a pulsedmagnetic field (Section 1.4). In general, clinical test results using magnetic stimulation on the CNS andthe PNS are very similar to those found using regular electrode stimulation. One important characteristicwhich separates these two stimulation methods is that magnetic stimulation cannot deliver a monopolar(cathodal or anodal) stimulus since these two poles are always relatively closely spaced when using aMC, (Fig. 1.4 and 1.7). Thus magnetic stimulation is best compared to bipolar rather than monopolarelectrode stimulation.The initiation of a nerve impulse is associated with a local depolarization across the nerve membrane.At rest the cytoplasmic potential is about 65mV below the extracellular potential. The initiation andpropagation of an AP depends on the potential across the cell membrane, and the consequent non-passiveion-channel activities which ensue. Electrode and magnetic stimulation work by reducing the extracellularpotential towards that found in the cytoplasm, i.e. transmembrane depolarization. For example, an APmight be initiated if the extracellular potential changed from OV to —10mV thus depolarizing the membranefrom its resting state of —65mV to —55mV. In contrast, changes in the intracellular potential are muchsmaller than changes seen in the extracellular potential and do not account for any significant portionof the total transmembrane depolarization. A highly localized extracellular potential gradient may not beeffective for stimulation because of the nerve's tendency to dissipate the stimulus by a process calledelectro-diffusion and by the leakage of ions through the membrane. It is the absolute difference betweenthe extracellular potentials on the surface of the same neural element within distances of a length constant(almost two internodal lengths or 4mm) which are important [21].Figure 1.3 Elements of Nerve Cell3A spatial analogy can be made for the axonalstimulation sites associated with bipolar electrodeand magnetic stimulation. The MC plane is tan-gential to the tissue surface containing the targetnerve. For maximal stimulation the MC is posi-tioned off center relative to the axon. Note theMC current polarity is opposite to the magneti-cally induced current and that magnetic stimula-tion is associated with a virtual cathode and an-ode.Bipolar ElectrodesDepolarizedRegion1-lyperpolarizedRegionCurrent Polarityin SalineTo simplify the discussion we investigate the effect of the cathode and anode separately. Figure1.5 depicts the currents, charge distribution, and transmembrane polarization associated with cathodaland anodal stimulation of long axons where end-effects can be neglected. The superimposition of theseanodal and cathodal results is equivalent to bipolar stimulation and is nearly equivalent to the membranepolarization profile associated with a MC.Both the cathode and the anode are capable of depolarizing the nerve, but in different degrees.Threshold cathodal stimulation requires three to eight times less current than does anodal stimulation [21];this is due to the greater depolarization generated by a cathode for a given electrode current level. Forany outward current there must be an inward current elsewhere. Inward current locally hyperpolarizes themembrane and conversely the associated outward current depolarizes another region.A fiber is depolarized directly below a cathode and hyperpolarized on either side (anodal surround).The inward current is spread out over a larger area so the magnitude of hyperpolarization is less thanthe magnitude of the depolarization. An AP generated at the depolarized site can propagate through ahyperpolarized site if the hyperpolarization is not too large. Conversely, the anodal stimulus depolarizesthe nerve on either side of the anode and strongly hyperpolarizes the fiber at the center.Once the magnitude of the depolarized region(s) reaches threshold an AP is generated and beginsto propagate in both directions. If the cathodal current is two to three times above the threshold levelthen AP propagation can be inhibited by the surrounding hyperpolarized regions; this effect is calledanodal block [22]. Anodal block is not present when using an anode since the depolarized regions arenot surrounded by hyperpolarized regions. Similarly when using bipolar electrodes or an MC at levelsseveral times above threshold an AP initiated at the (virtual) cathode may propagate in only one direction,away from the anode [21].Figure 1.4 Equivalence Between Electrode and Magnetic Stimulation4AnodeCathode^ +1 1 b\P-■■^A.A)/Th^1\4---■------11++^ ++ 0^N-----------Axon.+++++   ^4 0^_i_i_*------A.0,7---------'++ -4— DepolarizationX—Distance MOn xon Length./VHyperpolarizationDepolarizationOV Distance Alon^on LenHyperpolarizationSpatial profile of transmembrane polarization associated with cathodal and anodal stimulation of peripheralnerve. Relative resting potential is at 0 volts, the actual transmembrane resting potential is —65mV.Figure 1.5 Electrode Stimulation of Long Straight Axon1.2 Magnetic StimulationThe previous section indicates how electrode and magnetic neural stimulation have many similaritiessince they are both based on generating potentials across the nerve membrane (depolarization). Herewe discuss the progression in the knowledge of magnetic stimulation and some of its distinguishingcharacteristics with respect to electrode stimulation. The excitatory effect of magnetic fields on the bodyhas been studied since the turn of the century [23] when it was noted that weak fields as low as 10mTcaused flashes of light (magnetophosphenes) in the ultra-sensitive retina. In 1965 magnetic stimulation ofsuperficial peripheral nerves was reported [24, 25]. In 1970 muscle tissues were directly stimulated andthe estimated threshold was 1.2 V/cm, twelve times greater than that of nerve tissue [26].Between 1970 and 1986 several investigators explored the possibility of focusing the magnetic fieldusing high permeability materials. Rather than using a large MC, as in Fig. 1.2, these studies stimulatedexcised nerves using the lumen or gap of small magnetic cores [27-32]. A recent study [33] suggeststhat their results were due to electric (capacitive coupling) rather than magnetic (inductive coupling) nervestimulation. The modern era of clinical magnetic nerve stimulation began in 1985, when Barker [2-4]stimulated deep peripheral nerves and, more importantly, the human brain.A basic feature of clinical cutaneous electrode stimulation is that the entire stimulus current flow mustboth enter and exit the patient through the skin, where numerous pain-sensing nerves are located. Inaddition, underlying brain tissues are in effect insulated by high resistance bone (skull). Electrode currentsthus follow the least resistance path along the surface of the scalp (skin tissue) towards the ground5electrode, [34]. Thus a large 2kV [35]pulse (transcranial and transcutaneous) voltage is required to supplya clinically useful current level to the deeper regions of the brain [36]. These factors combined explainwhy electrode stimulation in general, and cortical stimulation in particular, is accompanied by a significantamount of pain. In contrast, magnetic stimulation induces current distributions which have no poles butsimply form closed current loops. Unlike electrode stimuli, these magnetically induced eddy currents arenot very focused spatially or oriented normal to the surface of the skin. These factors leads to the greatestpractical advantage offered by magnetic stimulation: lower patient discomfort.Magnetic stimulation offers the clinical advantage of being a non-contact and a non-invasive technique.Because living tissue is transparent to low frequency magnetic fields, magnetic stimulation is technicallysuperior to electrode stimulation in its ability to reach deep lying nerves without requiring high currentdensities at the skin surface. Magnetic stimulation is not significantly attenuated by the lossy (conductive)characteristic of living tissue, in fact attenuation of the magnetically induced stimulus is more a functionof (free space) distance and of the general tissue geometry (shape and size). Magnetic stimulation stillinduces maximal current densities at the skin surface but these currents are tangent rather than normal tothe skin surface; additionally these cutaneous current densities are not as great as in electrode stimulation,and they experience a lower attenuation at deeper tissue depths.The frequency bandwidth of a pulsed MC-current consists of long wavelengths, A=30km --* f=10kHz.This is several orders of magnitude larger than the MC dimensions or the distance between it and thetarget nerve. For this reason our study is concerned with the near zone rather than the far zone of theMC's electromagnetic field. In other words, the electric and magnetic fields associated with our study arereactive fields rather than travelling waves. Reactive fields are found within low frequency inductors, ornear an antenna. These types of fields are rapidly attenuated at short distances from the source anddecrease at rates of i/r2 or 1/r3 . In contrast, the typical attenuation seen for travellingwaves follows the inverse distance relationship 1/r. The electric and magnetic components of a travellingwave are coupled, i.e. in phase, whereas in reactive fields these two components are uncoupled, i.e.out of phase by 90 degrees. This uncoupled characteristic may be one reason why the term "magnetic"stimulation is used rather than the less common term "electromagnetic" stimulation [37-39]. In our MCstudy these field components are "reacting" to the inductive reactance of the magnetic coil.Many factors will influence the success of an applied stimulus as well as determining which neuralstructures are excited: the location, orientation and shape of the MC, the intensity, risetime and durationof the stimulus [40], the intrinsic excitability of the neural elements and their interconnections (synapses).Many clinical papers have studied the optimal conditions for neural stimulation. The parameters mostinvestigated are MC orientation over tissue, MC design, and "facilitation" (see page 106) of corticalresponses via voluntary muscle contraction. The optimal condition(s) should exhibit one or more ofthe following reproducible characteristics at the target muscle: 1) a supramaximal compound motor unitaction potential (CMAP) response level, 2) a minimal threshold level, 3) a focused stimulation to avoidcoactivation of nearby nerves and 4) for brain stimulation, either preferential direct or indirect stimulationof the corticospinal tract.61.3 Measurements and Models of Nerve ActivitySince early Roman times it has been observed that electricity, however cognized, could activatesomatic responses (e.g. electric eel placed on head). In 1791 Luigi Galvani further elucidated thischaracteristic by electrically stimulating severed limbs (nerve and muscle) [1]. In 1896 d'Arsonval describedthe excitation of the retina by magnetic fields [23]. From 1791 to 1950 related nerve studies experimentallydetermined and refined such classical characteristics as the effect of the electrode current (magnitude,pulse length, pulse rate, polarity) and the electrode position relative to nodes of Ranvier (myelinated nerve).In order to comprehend and predict the process of clinical magnetic stimulation it is necessary toassemble conceptual representations (models) of the governing factors. Two quantitative models arecentral to the investigation of magnetic stimulation: 1) the active cable model of the nerve fiber and 2) thetechniques which solve for the magnetic and electric fields within various tissues. In contrast to the longhistory of empirical knowledge, these modelling tools have been available for a relatively short time.The foundation of modern nerve modelling was laid in 1952 when Hodgkin and Huxley proposedthe first comprehensive and quantitative description of the multi-state dynamics of the transmembraneionic conductances and their interaction with the passive electrical properties of the axon [41-43]. Thebasic simplification (experimental and theoretical) which allowed them to solve such a complex problemwas to place the stimulus electrode internally (intracellularly) along the entire length of the axon (spaceclamping). This allowed them to assume that the solution exterior to the nerve was isopotential andeliminated complexities associated with wave propagation and ionic electro-diffusion.In 1976 the first model addressing the issue of extracellular electrode stimulation was published [44].A simple analytic equation was used to describe the E-field produced by one or more electrodes in aninfinite conductor and not in direct contact with the nerve. In 1986 this model was further developed byintroducing the activation function (AF) which defines the component of the extracellular E-field which isresponsible for nerve activation [45, 46].The solution for the extracellular E-field distribution induced by a MC has only received seriousattention since 1989. The methods used are all variations of the semi-analytic method which reducesthe MC to many small current elements and analytically solves for each of their vector magnetic potentialfields (A-field). The net E-field is solved by numerically combining the contributions from each coil element.Several studies limit their scope to cases where the MC plane is parallel to the tissue surface [47, 36,48, 49]. More advanced models account for surface charges induced when the MC plane is not parallelto the tissue surface. Several studies have compared various MC designs in terms of induced E-fielddistributions [48, 50] and their AF functions [51, 52, 47, 53]. There is also interest in how surface chargesat the boundary of small tissue volumes affect the induced fields, [36, 54, 53]. Results using thesesemi-analytic methods generally agree with the scarce experimental data available [55-57].These electromagnetic models rely, to varying degrees, on solution methods which are only validunder circumscribed conditions, but offer reductions in the computer time required, [51, 52]. The limitationsimposed by this approach requires that problems to be reasonably simple both in geometry and in material7characteristics. Thus, the finite volume conductors which are analyzed have simple geometric shapes,[58, 50], with a homogeneous and isotropic conductivity. These and other simplifications are neitherrequired nor are of great advantage when using a comprehensive finite element method (FEM) whosesolution sequence is directly founded upon Maxwell's equations. With FEM analysis the problem space isdivided into much smaller elements. A linearized but general form of the Maxwell equations is applied toeach element, and a correct solution is obtained by minimizing an energy functional associated with theentire problem space. The major draw back of this method is that models with many finite elements requiresignificant computer time and memory. Although the FEM method has not been used for solving MC fields,it has appeared in related studies concerning electrode stimulation, and magnetoencephalography [59-61].1.4 The Activation FunctionWe have described nerve stimulation in physiological and topological terms. We now review amore analytic interpretation of the mechanisms involved in electrode and magnetic nerve stimulation.Once a nerve's transmembrane potential reaches a threshold value further depolarization becomes apredetermined and active process resulting in a traveling action potential (AP). This means the neuron iscapable of information transfer. During the regular biological activity of a system, this threshold potentialis initiated by chemical means. In contrast, clinical (human intervention) nerve stimulation is initiated byexposing the target nerve to an E-field. This externally applied stimulus must have sufficient amplitudeand duration to cause current (mostly capacitive) to flow through the cellular membrane and induce atransmembrane depolarization. Threshold dynamics are very complex due to the nonlinear nature of theactive process of depolarization, as described by the Hodgkin-Huxley equations [41-43]. Modelling studiesoften simplify this situation by only considering the passive, linear, subthreshold characteristics associatedwith the nerve's membrane structure.Hodgkin and Huxley stimulated squid giant axon in-vitro using a long thin intracellular cathode. Theirexperimental method lead to a model which assumed that the extracellular space was isopotential while theintracellular potential was increased (depolarized). This is a reasonable assumption since the extracellularsaline was reasonably bulky and conductive and because the largest field it would be exposed to wasthat of an AP which is a disturbance mostly limited to intracellular space. They determined the dynamicsof the ion-channels, and combined this with passive nerve characteristics to predict a propagating APdescribed by the cable equation82v„ 1 = ^av„ Tinas2 Ri  `-'m at ± Rm(1)where: Rm (Q cm) is the membrane resistance times unit length, Ri (Q/cm) intracellular resistance per unitlength, Cm (F/cm) is the membrane capacitance per unit length, and Vn is the transmembrane potential.The Hodgkin-Huxley analysis is not directly applicable to regular clinical conditions where the stimulusis applied on the epidermis, i.e. extracellularly. Extracellular stimulation is dependent on the generation of alarge extracellular E-field. The transmembrane depolarization which causes APs in extracellular stimulation8• Vr is the transmembrane resting potential• ye is the intracellular potential• Vi is the intracellular potential• x is a differential length of axon and thesubscript n refers to the nodeExtracellular SpaceVe,nt ^p.Ve,n-1 Ve,n+14.XIntracellular Spaceis largely due to the extracellular potential becoming more negative while changes in intracellular potentialare negligible [21]. In 1978 McNeal first modelled extracellular stimulation where the electrodes were notin direct contact with the axon [44]. Rattay in 1986 defined the "activation function" which describes thecritical E-field component required for nerve stimulation [45].Using Fig. 1.6 Rattay developed an expression analogous to equation 1 but for extracellular nervestimulation,{ Vn_i — 214, + Vn-Fi  1 + { Ve,n-1 — 2Ve,n + Ve,n-Fi  } fcm dlin ± Vn 1 (2)dt^Rrn 1 .Ax2Ri^ Ax2RiThe net transmembrane depolarization is given by: Vn = Vi,n — V,,, + V,-, Vr is the transmembraneresting potential, (ye, Vi) are the extra- and intracellular potentials respectively, Ax is a differential lengthof axon and the subscript n refers to the node. Rattay defined the AF function by isolating the extracellularpotential terms in equation 2, taking the limit for Ax ---+ 0 and calling it the AF function,ASAx■—.0S(x, t) = ( Ve ' n-1 — 2Ve,n + Ve,n+1)AX2S(X, t) = —192 Ve = AF .ax2(3)One intermediate step in determining the AF function is by taking the first derivative of the external potentialalong the length of the nerve axis, i.e. the E-field component tangent to the axon path. We refer to thisfunction as the electric function (EF): EF = —8V/Oz. To emphasize the role and importance of theAF function, equation 2 is put in a form similar to the cable equation 1,A2e _ A2.92vn +Toy. ±vn82x^82x^ot (4)\ I RmA =^and T = CmRmRiIt must be stressed here that equation 4 only models the passive aspects of nerve structure and that acomplete model of AP initiation requires the nonlinear Hodgkin-Huxley equations which are not explicitlyincluded in the above analysis.Figure 1.6 Cable Model for Extracellular Stimulation9Magnetic CoilAirMC CurrentTissue0 coDepolarizedNerve LengthNote that the AF function associated with electrodes have spatial profiles identical to those depictedin Fig. 1.5. A typical AF function induced by a vertical magnetic coil is described in Fig. 1.7. In essence,the AF function is the second derivative of the external potential along the length of the nerve axis and isresponsible for all nerve activation where end-effects are not a complicating issue, e.g. peripheral nervestimulation conditions. There is no known simple expression or function which relates the E-field to nervestimulation of short axons where end-effects must be considered. It is presumed that a straight nerve in auniform E-field will be stimulated at one of its ends. The associated AF function has a magnitude of zeroand thus cannot account for nerve stimulation. Consequently we define the EF function in an attempt toquantify the effectiveness of a stimulus field where end-effects dominate. An analysis of the EF and AFfunctions within tissues is required for the development of magnetic coil designs and a further understandingof the clinical results obtained from magnetic stimulation of the peripheral and central nervous systems.A MC is placed over a tissue volume containinga long nerve axon. The MC plane is verticalto the tissue surface and parallel to the axonpath. That portion of the nerve length wherethe AF function is positive (shaded region)will be depolarized. There is also an adjacenthyperpolarized region where the AF function isnegative.Figure 1.7 Clinical Magnetic Stimulation Conditions10The remainder of this thesis is developed and presented in the following fashion:0 In Chapter 2 we review the methods and results from previously published MC modelling studies. Weexplore the motivation for electromagnetic models, their limitations, assumptions and the major findingsreported by investigators.0 In Chapter 3 we present the motivation and method for our electromagnetic models. The theory offinite element analysis is discussed in terms of Maxwell's equations, energy minimization, solutionapproximations, and the practical components of a finite element model.0 Chapter 4 is concerned with the validation of our general FEM modelling procedure. Our model resultsare compared with analytic solutions, experimental data, and finally with previously published modelresults.0 In Chapter 5 we investigate the effect of inner and outer MC diameters on the induced E-field. Therole of tissue volume is also explored and our field results are then analyzed in terms of the EF andAF functions.0 In Chapter 6 we present the first model for magnetic stimulation which accounts for anisotropic conduc-tivity in tissue. The arm is modelled both as an isotropic and an anisotropic tissue with homogeneouscharacteristics. Two MC designs are modelled and the EF and AF functions are analyzed.0 In Chapter 7 we present the first modelled E-field distribution for magnetic stimulation which accounts forinhomogeneous conductivity in tissue. We model the gyrus of the primary motor cortex within a planartissue volume and a horizontal figure-8 MC positioned above the gyrus. The effect of anisotropic andinhomogeneous conductivity on the induced E-field is reported and two novel AF functions are analyzed.0 A discussion and summary of the results is presented in Chapter 8.11Chapter 2A Review of Magnetic Coil AnalysisThe earliest studies of electrode stimulation were characterized by clinical/experimental observationsof the effect(s) of various test conditions on nerve stimulation [23, 62-67]. The elucidation of active ion-channel behavior motivated many studies [45, 46, 68] in which the E-field distribution of an electrodewas modelled. The modelled E-field results were then used as an input parameter to the Hodgkin-Huxleynerve model [41-43].A review of the magnetic stimulation literature shows a similar development. Early studies, from 1959to 1989 [27, 5], presented clinical observations. The use of analytic techniques was both slow to start andslow to develop. From 1965 to 1989 analytic (engineering) studies concerning magnetic stimulators weremainly confined to experimental measurement of the nerve response to various MC-current characteristics(magnitude, duration, polarity, rise time and current damping) and their relation to nerve responses [25,34, 40, 69, 70].Not until 1989 [47, 52] did MC modelling of E-fields, induced within tissue, receive serious attention.This late date is probably attributed to the recent clinical work by Barker in 1985 which ushered theintroduction, if not general acceptance, of magnetic stimulation in the clinical community [2-4]. A secondfactor which may have delayed E-field modelling is the availability and capability of computers necessary tosolve electromagnetic problems. A knowledge of the induced extracellular E-field is a basic requirement forthe further development of MC designs and to the formulation of models explaining the diverse observationsreported in the clinical literature.2.1 Previously Used Modelling Methods2.1.1 General MethodThe methods previously used to model induced E-fields are all variations of a method which reducesthe MC to many small current elements and analytically solves for each of their E-fields. The net E-field issolved by numerically combining the contributions from each coil element. Published MC E-field modellingstudies have focused on variations of fast but simple solutions which are only valid for a restricted set oftest conditions. The total E-field (ET) is treated as the sum of the E-field induced by the vector magneticpotential (EA) and the electrostatic E-field (E0) generated by surface charges (if applicable). EA is equatedwith the time derivative of the vector magnetic potential (A) and E0 with the gradient of the electrostaticpotential (0):ET = EA EQdA=^— v0 •dt(5)12An example of these two field components are presented in Fig. 2.1. A general relationship [71] definesthe vector magnetic potential induced at a point (r) by a closed current-loop consisting of differential lengths(vectors) dL each carrying a current (I) at point (r) as:A(r) = V i ir arl .^ (6)The electrostatic E-field at a point (r) generated by a charged surface (S') with a surface charge densityof ps(r) can be defined [71] by the general expression:Eq(r)f=^Mr') dS' r —r'J 4ircolr — r'12 Ir — el •These equations assume a homogeneous, isotropic tissue such that the E-field ET is independent oftissue conductivity [47, 50].2.1.2 Examples, Variations and LimitationsTofts [72] and Durand [53] used conventional techniques to resolve the E-fields for a vertical MC overa planar semi-infinite tissue referred to hence forth as a planar tissue. If the MC is assumed to be circularthen the vector magnetic potential only has one component (A0 in cylindrical coordinates) and equation(6) can be expressed as a sum of elliptic integrals:1A9 = poNI ( a) 2 [K(k) (1_ _k2) _ E(k)]irk^p)^2k2—4ap(a -I- p)2 - I - z2 '(8)where the measurement point is defined by the distance (z) below the plane of the MC and the distance(p) from the MC axis, K(k) and E(k) are complete elliptic integrals of the first and second kind respectively[73-75] and (a) is the MC radius. Thus, for a circular MC Ao forms concentric circles about the MC axis.r vector defining measurement pointr' vector defining source pointlr-r'l distance between measurement and source pointsdEo(r)^differential electrostatic fielddA(r)^differential vector magnetic potentialdL'^differential current length carrying current IdS' differential charged surfaceSee text for details. dL'Figure 2.1 Magnetic and Electrostatic Contributions to E-field(7)S13If the MC plane is not parallel to the tissue surface then the air/tissue interface will be charged. Theelectrostatic field normal to a sheet of charge with a surface charge density of (Ps) is given by EQ„ L-_-P3/2c0. At the air/tissue interface the normal component of EQ must be equal in magnitude and oppositein direction to the normal component of EA. Thus, the charge density is given by ps :--- 2E0 x dAnIdt,where An is the normal component of the vector magnetic potential. Substitution of this charge densityterm into equation (7) allows both E0 and ET to be defined in terms of A and A. The surface integral inequation (7) is then evaluated along the planar air/tissue interface. Note that this surface integral requiresa significant amount of time to calculate but that ET need be only calculated at desired locations, usuallyalong some planar surface (rather than the entire tissue volume). Finally, this method is limited to circularMCs since elliptic integrals were used to determine the A vector field. Using the specific situation describedabove, five published variations differ in the following ways:I. Grandori also uses elliptic integrals but only considers MCs parallel to planar tissue surface,i.e. EC) =0 [36].II. Cohen [48] and Roth [49] do not use elliptic integrals and solve equation 6 directly. Thisallows them flexibility in modelling non-circular MC designs using groups of short lineelements. The E-fields are approximated by summing the contributions from each currentsegment. They only consider coils parallel to a planar tissue surface.III. Mouchawar also solves equation 6 directly [76]. Surface charges were solved on allsix sides of a finite tissue volume, shaped in a cube. The model solution combinedLaplace's equation, C720 = 0 with equation 7, subject to the air/tissue boundary condition,&Han = n • EA . Laplace's equation requires that the tissue be linear, isotropic and thatit contain no free charge (i.e. homogeneous conductivity).IV. Roth solves for the E-field within a nonterminating cylinder (arm) and a sphere (head) [54,50] using similar methods as Mouchawar.V.^EsseIle derives a novel method which is computationally very efficient but is limited to casesinvolving a planar tissue volume [51].All the modelling methods described above depend on the following assumptions:I. Quasi-Static Analysis: The air/tissue boundary condition^&Han = n • EA^suggests that there isno current normal to the boundary surface; however, such a current is required to generate the surfacecharge density ps. The time constant which describes the exponential decay of this current is given byT = c0€7./a- and is small (3.54 ns) for standard tissue characteristics: ir =80, cr=0.2 S/m. Thus for astepped pulse of a magnetically induced E-field (EA), 37% of the surface charge density Ps is generatedwithin less than 4 ns. The highest significant frequency component from a typical commercial MCdesign is about 10kHz [36, 77] consequently, it is assumed that the charge redistribution is effectivelyinstantaneous, i.e. quasi-static conditions.II.Skin Depth: Ohmic losses within a conducting material will exponentially attenuate a uniform planewave as described by the skin depth (6) expression: 6 = liVirfpa- . For a standard magnetic14stimulator ( f=104Hz, it=po, and o-=0.2 S/m) the skin depth for the E-field is greater than 11 meters.Although magnetic stimulation deals with non-planar near fields this expression does indicate, to a firstapproximation, that attenuation (to 37% of peak) of the induced currents due to ohmic losses shouldnot be significant during clinical applications where the nerve is only 5 to 15 mm from the MC.III. Secondary Fields: The alternating magnetic fields generated by the induced currents are ignored sincethey are much smaller than those produced by the MC current [54].Guidi presents the only published MC study which uses a FEM method of analysis [78]. The methodused was adapted from a finite-difference time-domain analysis developed for RE absorption in humantissue [79]. This FEM method is dependent on different simplifying assumptions and only preliminaryresults were presented (Section 2.2.2).2.2 Results from MC ModellingInvestigators have calculated magnetically induced E-fields using various test conditions in an effortto understand the factors underlying the published clinical data and to improve MC designs. There arethree main characteristics by which it is possible to categorize any given study on MC E-field modelling:1) what MC designs are considered, 2) what MC orientations are modelled, and 3) what type of tissuevolume is modelled.Several MC designs that have been modelled are illustrated in Fig. 2.2. The spiral MC has long beenthe standard design for commercial manufacturers and, consequently, for clinical studies. The loop MCdesign is examined in nearly every modelling study. The interest in powerful but spatially confined fieldgradients (focused magnetic stimulation) has motivated the modelling of other more exotic MC designssuch the figure-8 MC design [48, 36, 76, 50, 58, 51], the tear drop coil [48, 50], the square loop [51] andthe quadruple loop [36, 51].During actual clinical stimulation at the wrist and head many MC positions have been investigated.The majority of these MC positions are poorly defined in the literature because of the spatial complexityof the body limbs and their relationship to the MC location and orientation. For electromagnetic modellingpurposes this diversity of MC positions is reduced to two basic orientations referred to here as the horizontaland vertical MC orientations. These two MC orientations are illustrated in Figure 4.1. A vertical MC hasits plane oriented normal to the tissue surface while a horizontal MC's plane is tangential to the tissueFigure 2.2 Typical Magnetic Coil Designs15surface. Cases involving horizontal MCs over a planar tissue are simpler to model since no electrostaticfields are generated at the air/tissue interface [48, 80, 26, 54, 58, 47, 36, 49]. If either of these conditionsare not met then surface charges must be accounted for in the model [53, 76, 50, 51, 72, 54, 81, 80, 58].The study of induced currents from MCs is a relatively novel line of research. Most clinical magneticstimulation is performed on relatively small tissue volumes yet only one modelling study has solved forthe E-fields within the arm [54] while three studies have modelled the head [36, 50, 58]. In contrastmost published results deal with planar tissue volumes [48, 26, 54, 51, 72, 47] or with cube shaped finitevolumes [80, 53, 76]. Below we present results and conclusions from previously published MC modellingstudies. Note that relatively few studies [55, 77, 72, 47, 53, 81] have validated their modelled results byusing either experimental or other independent modelled results.2.2.1 MC Designs and OrientationsScaling the coil size and all spatial dimensions by the same factor while holding the time varyingMC current constant results in the same level of induced E-field, [76, 36, 51]. Investigators have takeadvantage of this fact to present normalized plots of induced E-fields at several relative distances fromthe MC's plane [36, 51]. It follows from this scaling principle that smaller MCs will provide more focusedE-fields [50] and that deeper stimulus penetration is possible with larger MCs [36, 54, 76].The theoretical comparison of the relative efficacy of one MC design over another has been a tenuousaffair at best. Early studies simply describe the E-field distribution of various MCs [48, 50] withoutpresenting some type of metric by which to evaluate its anticipated clinical effectiveness. Some of theirgeneral conclusions are: a loop MC induces a more focalized E-field than a spiral MC [48], and a verticalMC induces a weaker but more focalized E-field than a horizontal MC [53, 50, 55, 72, 54, 76].Recently, investigators have further refined these findings by determining the AF function, [51, 52,47, 53]. In these studies it is implied that the modelled peak value of the AF function is a direct measureof expected clinical effectiveness. The effectiveness of various MC designs are ranked by comparing themagnitude and focality of the induced E-field and/or the AF function. The more focal E-field and AF functiongenerated by non-circular MC designs have been emphasized, e.g. square, figure-8, teardrop, quadruple-loop [51, 52, 48, 50]. However, if the MC dimensions and the MC current derivative are normalized thereare only small changes in the AF function due to MC design [51, 52, 70]. Roth is the only investigatorwho has modelled nerve activation by combining the Hodgkin-Huxley equations with MC model results(i.e. the AF function) [49].Although the magnetic field is greatest along the axis of a round MC [48], the induced E-field is nullalong the same axis. For a horizontal MC the greatest induced E-fields are located directly below thewindings of the MC, flowing in concentric rings about the MC axis. Clinical studies find that magneticstimulation is difficult if a long straight target nerve is aligned symmetrically below a horizontal MC. In thiscase the EF and AF functions for the axon are zero along its entire length. Conversely clinical studiessuggest that stimulation is optimal if the horizontal MC is somewhat laterally offset, this corresponds to16conditions which maximize the AF (Fig. 1.4). A figure-8 MC design is different in that both the EF and AFfunction peaks are located directly below the center bar of the MC (Fig. 7.6). This MC design producesa more focused E-field stimulus than a round MC [48, 36, 76, 50].2.2.2 Tissue EffectsOne goal of magnetic stimulation modelling studies is to obtain realistic results. Conversely, themodelled material characteristics of the "tissue volume", have been simplified to the extreme by definingtissue as a homogeneous, isotropic conductor with a planar surface and infinite volume. Although thesesimplifications have revealed several fundamental EM field characteristics, further understanding willrequire more realistic modelling methods.The results of Tofts are among the first to clearly indicate a surprising characteristic of the E-fieldinduced by MCs [72]. The boundary condition 04)/an = n • EA implies that, at the tissue surface, thenormal component of the electrostatic field cancels the normal component of the induced E-field EA. Lessobvious is the fact that this cancelation is present at all points within certain tissue volumes regardlessof the MC orientation or MC design. For example, the E-field within a planar tissue with homogeneousconductivity is everywhere parallel to the tissue surface, and no normal component exists. Tofts alsoclearly illustrates the closed-path nature of eddy currents which lie in planes parallel to the tissue surface.Analysis by Branston similarly indicates that a MC placed over a spherical tissue volume is incapable ofgenerating currents in the radial direction regardless of MC shape or orientation [55, 77]. This characteristicwas confirmed by other investigators [50, 58] and was analytically demonstrated [77, 51].Regardless of MC orientation, E-field levels within a planar volume are greater than in finite conductingvolumes where more surface charges [53, 54, 50] are present. Durand demonstrated that such E-fieldreductions may be more pronounced for vertical rather than horizontal MCs [53]. Finally, several papersstate correct boundary conditions at the air/tissue interface, yet their plotted E-field results contain variousdiscrepancies [80, 36, 50, 72].The E-field at a given point within a planar tissue volume is not a function of the distance betweenthe MC and the planar surface but is simply a function of the distance from the measurement point to theMC [72, 36, 51]. This characteristic cannot be generalized to other conductor shapes. For example, theinduced E-field at the center of a spherical volume is always zero [48]. Finally a theoretical study by Hellerindicates that the peak E-field must be located at the surface of a homogeneous conducting volume [82].Studies modelling induced E-fields from MCs have been limited in their scope of complexity relativeto the actual biological systems they attempt to simulate. Little attention has been payed to how theE-field may be affected by the shapes and sizes of human limbs or by the anisotropic and inhomogeneousconductivities found within these tissues. Nevertheless, a few studies have given these factors someconsideration:CI Roth modelled the human arm as a cylinder of infinite length with a homogeneous, isotropic conductivity[54].170 Roth also modelled the human head as three concentric layers with different isotropic conductivities [50].0 Guidi modelled the pulsed E-fields in an inhomogeneous cortex but only presented the E-field at onepoint as a function of time (not space) [78].0 Tay measured E-fields in cat brain tissue generated by MCs, but these results were not modelled [56].Further detailed discussion of these studies is presented in later chapters.18Chapter 3Theory and Methods in FEM ModellingPrevious modelling studies have explored the fundamental characteristics to be expected from mag-netically induced E-fields, as listed in Section 2.2. Assuming that the modelled results are correct thepractical relevance of the models can only be measured by the degree to which they represent the actualconditions under which clinical magnetic stimulation takes place. So far, magnetic stimulation modellingstudies have only considered isotropic conducting volumes. Thus the anisotropic conductivity, known toexist in muscle and brain tissues, are not taken into consideration and little data is available to estimatethe potential importance of this tissue characteristic. Also, in previous studies inhomogeneous materialcharacteristics have been modelled only to the extent that the air/tissue conductivity interface can be con-sidered as such. The problems posed were so simplified that, within a large range, the conductivity valuechosen for the modelled tissue doesn't affect the E-field distribution or its magnitude. Realistic modellingof magnetic stimulation must account for: tissue anisotropy, inhomogeneity and the geometric complexityof the conductivity interfaces.A critical inspection of the previous modelling studies (Sections 2.1 and 2.2) suggests that there hasbeen more interest in developing new algorithms which produce similar results to a limited range of modeldiversity rather than trying to improve the realism or relevancy of the modelled results. If one is to abandonmost of the assumptions used in the previous modelling algorithms then more versatile and generalizedsolution methods must be employed.Finally, it appears that the software used in previous studies was, in every case, developed in-houseand consequently little time and effort was allocated to the visual presentation of the model results. Theuse of line plots or surface plots is usually adequate for simple symmetric tissue volumes but is notadequate for more spatially-complex problems. Thus, to model realistic material properties, more realistictissue geometries, and to present the results in a useful fashion, it was decided to use, in our work, ageneralized electromagnetic finite element analysis package supported by advanced visualization software.This chapter presents the theoretical basis for the FEM analysis software called MSC/EMAS®, [83]. Thevisualization software package we used is called MSC/XL®.19A5 gB (A) The domain and range of aoo^functional F.b (x)^(B) The extremal functiong(x), the varied paths -4(x)g(x)^and incremental changebetween them 6g(x).3.1 Variational SolutionProblems in applied physics can be specified in one of two ways. In the first, differential equationsgoverning the behavior of a typical, infinitesimal region are given, e.g. Maxwell's equations. In the second,a variational extremum principle valid over the whole region is postulated and the correct solution is theone minimizing some quantity F which is defined by suitable integration of the unknown quantities overthe whole problem domain D. Such an integral quantity as F, which is a function of unknown functions, isknown as a 'functional'. The two approaches are equivalent in that a purely mathematical and manipulativetransition from one to the other form is possible [84].Whether the problem is one of mechanical elastics or electromagnetic fields, the integral quantity Fis often defined as a quantity which is proportional to the total energy of the system, and this is thenminimized [85]. The minimum of the functional is found by setting the derivative of the functional, withrespect to the variational parameters, to zero. Thus the basic equation for finite element analysis isag^0 (9)where F is a functional defined on the class of functions g(x), for example:g(x) = x2^.7" (g) = f g2 dx = f x4dx .^ (10)a^aThe domain of F is g(x) and its range is a set of numbers (Fig. 3.1A).A small, arbitrary change in the dependent variable g(x) for a fixed value of the independent variablex is given by:61g(x)] = i(x) — g(x) ,^ (11)where g(x) is the extremal function and 4(x) is any one of its varied paths. The symbol (6) is called thedelta operator. At the indicated position x, in Fig 3.1B, each of the three varied paths g (x) is associatedwith a particular variation in g (4); we show only one example of bg. Note that bg is not associated withdx (unlike dg). Thus Og is the vertical distance between points on different curves at the same value x.We define a functional using the varied path -O(x) (Equ. 12). The variation of the functional is thengiven by equation 13. If the variation of g(x) approaches zero then the second integral term in equationFigure 3.1 Functionals and Path Variations20(B)(A) Two dimensional electrostaticproblem with Dirichlet andNeumann BCS.Surface plot of variationalsolution using second orderpolynomial trial function.v 2 d0 dn0z= 3 xy40408.7.8.05,04.09.02.01,04^ xA^d (1)1 ^dn 13 approaches zero and functional variation is then given by equation 14. The last term in equation 14gives a general expression for the variation of a functional in terms of partial derivatives with respect tothe extremal function g(x).(12)=^bg)— „FM =Sg dx^(Sg)2dxa^adY(X Lintog,,0^= f Sg dx —^dg' Sg dxa^aIf a function -4(x) gives a minimum or maximum to a functional, then a necessary condition forextremum is:d.TSY(1) = 0 =^6g dx .dg(15)aSince 8g is an arbitrary quantity the solution is given by Euler's equation:^ — 0 .^ (16)OgThus if we want to solve Euler's equation, and we know the functional, the function which minimizes itwill be the solution of equation 16.3.1.1 Variational Solution for ElectrostaticsWe present a variational solution for the two dimensional electrostatic problem depicted in Fig. 3.2.The Euler Equation is solved using the Rayleigh-Ritz Procedure where the functional is minimized in adirect way by using the coefficients which define the function g(x).Figure 3.2 Electrostatic Example(13)(14)21Electrostatic energy is proportional to the square of the potential gradient; Energy = f €E2dxdythus a commonly used two dimensional electrostatic functional is given by an integral expression overthe domain D:..r(0)= (v0)2 dxdy —2 I kb ds ,^ (17)where the second term on the right accounts for the Neumann boundary conditions (BCS): h= 0010n.The electrostatic potential o is defined as a superposition of linearly independent polynomial functions.This selected function is called a trial function:cb(x,y)= + o2s + coy + c4x2 csy2 c6xy .^ (18)The Dirichlet boundary conditions 0 = 0 along the x and y axes are imposed on the trial function giving:+ o2x + o4s2 = 0 for 0 <x <1,andc3y c5y2 = 0 for 0 < y <1 .^ (19)Thus 4)(x, y) = csxy is the trial function which satisfies the BCS where c6 is a variational parameter.Note that O(x,y) does not need to satisfy the Neumann BCS. Substituting the trial function into equation17 gives:h= a(blan =1vcb= (iy+ jx)c6(v)2 _ (x2 + y2)(c6)2^ (20)11^ 1T= (c6)2 I I (x2 + y2) dxdy — 211 • coxYdY lx=1 •00The functional is solved and minimized with respect to the variational parameter (c6) to give the finalapproximate solution 0(x,y) for the electrostatic potential (Fig. 3.2B):..rq(c6)2 — 2 ((o6))O.T 4= (c6) — 1 = 0cs = 3/40(x, y) = xy4(21)223.2 Finite Element MethodFor very simple problems the variational solution can be equivalent to the analytic solution; however,this is generally not the case. Note how, in the example presented in Section 3.1.1, the polynomial equationwas assumed to be valid throughout the entire domain. For more complex problems, it is not possible toapproximate electrostatic or electromagnetic field distributions using a single polynomial equation.The FEM method of analysis consists of representing (subdividing) the device or problem domain D asan assemblage of many smaller finite elements defined by nodes and straight line boundaries, (Fig. 3.3).Each finite element (FE) is associated with its own polynomial expression and thus model complexity is nolonger a limiting factor. A general, two-dimentional, homogeneous, electrostatic example is presented hereas an example. This example is relatively simple but it is similar to the full electromagnetic FEM analysis.The functional is once again defined as in equation 17. Each FE is triangular and defined by threevertices located at i(x,y), j(x,y) and m(x,y) with potentials Oi,^and Om respectively, (Fig. 3.3).Within each FE a linear equation is assumed to describe the potential as a function of location:0(x, = co + cix + c2Y •^ (22)Rather than solving for these three coefficients c0,12, it is convenient to use the three node potentials0i,j,k as variational parameters such that the continuity between adjacent triangles is automatically satisfied(Rayleigh-Ritz Procedure). The potential in triangle C2t in terms of its node potentials is given by:Cb(x^Ni(x,Y)Oi + Ni(x, y)cbi Nm(x, y)cbm ,^ (23)where the first order interpolatory or shape functions Ni j,m(x,y) are strictly in terms of node locations. Forexample:Ni(x , = {(xjyrn — xmki) + (Yj — Ym)x + (x m — x i)y} /27?. ,^ (24)where for instance node m is located at coordinates (x,„ ym) and R. is the surface area of the triangle.Note that each shape function has value of unity at its associated node and zero at the two other nodes,e.g.Ni(xi, yi) = 1 and Ni(xm, ym) = 0 , Ni(xj, yi) = 0 .^ (25)A FEM model consists of a problem domain D,subdivided into finite elements a Each element isdefined by nodes (i,j,m) where node potentials cbare defined. Dirichlet boundary conditions (V.) aremarked along a section of the domain boundary.Figure 3.3 Basic FEM Model Components23If there are L nodes and Z Dirichlet nodes in the problem space D the total number of unknownsis (L-Z)= w,^02, ...^. The functional to be minimized is F = F(01, 02,^Ow) andthe minimizing condition is: ay-lacbi = 0 , for (1=1, 2, ... w). Thus, each triangle contributesterms derived from functional minimization at its three vertices, and each of these three partial differentialequations generates three S terms, which are components of the final FEM matrix. For exampleaFrdooi =^+ sijoi +simcbm] ,^ (26)where the S terms are once again simply functions of node coordinates; for instanceSii = 1Z [(yi — ym)2 + (xi — xm)21 .^ (27)Thus each FE triangle contributes a 3x3 matrix to the generally much larger wxw model matrix describingthe entire problem space D. These nine contributions from triangle Ot are shown in the matrix below:-SSSSS SS SS S_ SSSSSiiSjiSmi SSSSijSjjSmjS^S^S-0-S^SSim^SSjm^SSmm SSi0jm'bbibjbm(28)All electrostatic problems have BCS where the node potential is forced (e.g. 0=2 Volts), called a Dirichletcondition, thus the square matrix [S] is of dimension wxw. Triangles with such Dirichlet vertices, on themodel boundary, will generate non-zero components for the second term on right side of the functionalequation 17 and contribute to the b terms in the FEM matrix above. These boundary triangles will contributefewer S terms to the FEM matrix, for example a triangle with one Dirichlet node will only contribute a 2x2rather than a 3 x 3 matrix. Natural BCS (e.g. &plan = 2 Vim , where n is a vector normalto boundary), are called Neumann BCS. Nodes with such BCS will contribute S terms and not b termsto the FEM matrix.243.3 FEM Method for ElectromagneticsWe now present the basic theory for the electromagnetic analysis software called EMAS.3.3.1 Converting Maxwell's EquationsMaxwell's equations describe the four basic laws of classical electromagnetics, namely that the sourceof electric flux density D is free charge e , that magnetic field density B has no sources, that E-fieldsE are induced by time-varying magnetic fields and that the sources of magnetic field strength H areconduction current J and displacement current b . These four relations are respectively given byV-D=ev • B = 0vxE=--B^ (29)vxH=13+JThe response of materials to electromagnetic (EM) fields are specified by three macroscopic char-acteristics: permittivity, permeability and conductivity (e, ji, o-). Conduction current density is proportionalto the E-field and describes the average drift velocity of free electrons as they collide with obstructionswithin a lossy material: J = a-E . The volume charge density at some time T is the value eo at someinitial time To plus the total amount which has flowed into the volume since the initial time. This can bewritten in a time integral form as:^= eo — f V • Jdt .^ (30)ToThe response of fixed (orbital) electrons to E-fields causes a dielectric response (microscopic atomicdipole moment) which is detected as an equivalent macroscopic charge separation (polarization) and canbe described in terms of permittivity e (F/m):D = foE P = focrE = EE . (31)The response of bound microscopic currents to magnetic fields is significantly more complex and diversein nature but can be similarly reduced to a net magnetization M (A/m) and can be described in terms ofmagnetic reluctivity v (m/H)H = voB — M = vB . (32)These equations are extremely difficult to solve analytically except for very simple situations. EMASuses a differential equation method which generates sparse, banded and symmetric matrices whichincrease computation efficiency. EMAS also uses the finite element method. The finite elements canhave standard shapes which allows efficient and accurate solutions algorithms. All physical behavior25described by Maxwell's equations can be modeled using EMAS. Rather than solving for the E and B fieldsdirectly EMAS transforms Maxwell's equations using the following relations. Since the divergence of B iszero, B must be the curl of some vector called the vector magnetic potential A^B=VxA .^ (33)The E-field can also be expressed in terms of the vector magnetic potential A and the gradient of thescalar potential 0 or its time integral 0:E =^— vcd =^—^.aA^aAat (34)Thus the conventional scalar potential 0 is, to within a constant C, the time derivative of the newscalar potential 0.^f 0dt=0-1-C .^ (35)The four basic unknowns, i.e. degrees of freedom (D0Fs) which are solved at every grid point by EMASare the three spatial components of the vector magnetic potential A x, y, z and the unconventional scalarpotential 0. Combining all the above equations and using the potential substitutions 33 and 34 gives twoequations which are equivalent to all four of Maxwell's equations:V x [i.](v x A) =^— [ir] (v't ±Ä)v • e(vi.k +A)^— (v • cf(vi4 + A.))dr .^ (36)toEMAS separates the work associated with a problem volume into several components:Electric Field^= D SE = (SA b (70) e + v10Magnetic Field^814 = H • 6B =^x A) v(v x A)Charges^bOe =^ (37)Dissipation^S4,„ = —(b(V0)+ SA) • J = —(8(vik)d- SA) • o-(—À— v4)Penalty Term^bilop = —pv S(V • A) • (v • A) .The penalty term is used to maintain the Coulomb gauge condition (v - A = 0) , where p has the sameunits as magnetic reluctivity v. Work is also associated with the boundary conditions of the FEM model:^initial electrostatic potential, Solisi = —^D)Toterm associated with A, 64).32 =^To dt bA (H xii)^(38)^term associated with V), 6.1,53= —^dt 6.0 •^• (1. + to)) ,26where i is the outward directed unit normal to the surface S. The energy functional consists of summingthe first five work related terms, integrating them over the problem volume V, and over time. The threesurface terms are added and integrated over the problem boundary surface S:= (joy at (oe+ Ob + OR + + Op)) (f dS (Os]. + 4).92 ± s3))V^To^ (39)If there are w nodes in the problem space D the total number of unknowns is 4xw, the functional to beminimized is F = Y(Azi,^= 1...w and the minimizing condition is: ayjaci= 0,for^Ci = (Azi, Ayi, AzioGi)^and i =1, 2, ... w .3.3.2 Finite Element ApproximationThe finite element method obtains approximations of EM field solutions by minimizing an energyfunctional F [86-88, 84, 83]. The functional is much more complex then that for the electrostatic caseand involves the integration of both space and time, (Equ. 39). Finite elements can be one, two or threedimensional. The number of nodes n associated with a FE is 2, 3, 4, 6 or 8 depending on the shape anddimension of the FE. As with the electrostatic case the EM fields within a FE (with n nodes) are descibedby a set of known shape functions 1\11 and the unknown node potentials Azi,A1,Azi and thi^Ax(r, =^(r)Azi (t) N2(t)A,2(t)^Nn (t)A„ (t)^Ay (r, t) =^(r)Ayi (t) + N2 (t)Ay 2 (i)^Nn (t)Ayn (t)^(40)^Az (r, t) =^(r)Azi (t) + N2 (t)Az 2 (t) •^N(t)A(t)^0(r, t) =^(r)1•14 (t) + N2 (t)02 (t)^Nn (t)ikn (t) .Note that the shape functions depend only on position and the node potentials depend only on time. Thefour equations above, which define the potential fields within one finite element can be written in matrixform1/1(r,t)1^rN(r)][Aiki((tt))1^[N][u]^ (41)kb(r, jwhere the solution matrix [ A(r,t) l,b(r,t) ]T is a 4x1 column vector, the shape function matrix [ N(r) ] is a4x4n matrix which contains element geometry as well as material properties. [ N (r) ] has the form:[N]^9[0^I^N1 0^0^0 I N2 0^0^0 IU^0 Ni 0 1 0 0 N20^0^0 N1 I 0^0^0 N2 I, N,N1 0^0 I 0 N2 0^0 I (42)where the number of partitions is equal to the number of nodes in the FE. The DOF matrix [u] is a 4n x1column vector[u] = [A1, Ay 1 Az1 011 Ar2 A,2, Az 2 021 • • • I Axn, Ayn Azn^iT •^(43)The matrices [N] and [u] are independent of time and position respectively. Thus when solving equations36 the curl, divergence and gradient operations only concern the [N] matrix while time differentiation and27variational analysis only concern the [u] matrix. The shape function matrix [N] is considered constantwith respect to variations. This separation of time and position combined with the fact that FE volumeshave nearly ideal geometries allows the use of very efficient algorithms to evaluate the volume integralsassociated with [N], generating reduced matrices called the dielectric [M], conduction [B] and reluctivity [K]matrices. Once the virtual work expressions have been developed for each FE, a single matrix equation isassembled containing work contributions from all FEs. This expression has the same form as for individualelements but now represents the entire problem volume. This expression can be written as follows:[lu] [iil+ [N[iL] + [nu] = [P] (44)where [P] is the assembled load vector which contains all volume and surface loads. Note that equation44 is of the same form as the equation describing a resonant circuit with a current source, where [M] iscapacitance, [B] is resistance, [K] is inductance, [it] is node potential and [P] is sourced current. Oncethese DOFs are solved at all grid points all other field quantities (E, H, B, D, J, etc.) are recovered byinterpolation and differentiation.3.4 Generating a FEM ModelFinite element analysis with the user interface and visualization software MSC/XLe and the analysissoftware MSC/EMAS© consists of six main stages: model geometry, finite elements, BCS, excitations(loads), matrix resolution, and results verification. XL is the general user interface software which generatesthe FEM model and displays the final EM results. EMAS is the analysis software which takes the FEMmodel and then generates and solves an equivalent matrix representation of the problem.3.4.1 Model GeometryModel geometry determines the location, size, and dimension of objects within the FEM model. It iswithin these objects that FEs will be defined in the next modelling step (see Fig. 3.4). Objects are furtherdivided into smaller geometric regions for various reasons such as different material properties, differentnode meshing requirements, and odd shapes. All spaces between objects (air) must be modeled as well.The FEM geometry can consist of any combination of one, two and three dimensional geometric entities(lines, surfaces, volumes) and these will accordingly determine the dimensions of the finite elementsgenerated in the next step.283.4.2 Finite ElementsRegardless of the dimension(s) used in generating model geometry, the basic unit of finite elementanalysis is the grid point or node. The four basic EM variables resolved by EMAS analysis are calleddegrees of freedom (D0Fs). Every node has four associated DOFs, consisting of the three spatialcomponents of the vector magnetic potential A(x,y, 4 and the scalar potential (0). From these fourDOFs all other EM fields and forces are calculated. Geometric lines, surfaces, and volumes are usedas skeletal structures onto which and into which a much finer FE mesh is defined, (Fig. 3.4). For examplea one—dimensional (line) finite element is defined by two nodes, two-dimensional finite elements (trianglesand rectangles) are defined by 3 and 4 nodes respectively, and various three-dimensional finite elementvolumes (tetra, penta, and hexa) are defined by 4, 6, and 8 nodes respectively.Finite elements are not just frames onto which nodes can be placed at vertices. They also defineregions of homogeneous material properties such as conductivity, permittivity, permeability, anisotropy,and non-linearity. Common EM field parameters such as E-fields and current density fields (J-fields) arenot defined at nodes but rather are defined at the center (centroid) of each finite element. Each node canhave up to three different coordinate systems associated with it, one to define its location, one to defineits anisotropic material properties, and one to define field constraints at the boundaries of the FEM model.Node (mesh) density determines solution accuracy. Generally, higher mesh densities are requiredwhere EM field values change rapidly with position. Examples are sources, abrupt changes in materialproperties, and singular geometries such as sharp corners. Other reasons for higher mesh densities arefine geometric details and regions of interest to the user. Usually uniform mesh density cannot be usedsince solution costs increase roughly as the sixth power of the linear mesh density. Thus doubling thenode density in all three directions would require about 64 times more time and/or memory!The work horses of FEM are the three-dimensional hexa (cube) and the two-dimensional quad (square)finite elements. These finite elements offer the most accurate and cost-efficient results. Other finiteelement types (shapes) are required when reducing the mesh density; however, these mesh transitionsare avoided in areas of interest since they tend to generate lower magnetic fields within this transitionzone. Mesh density, element type and model geometry have to be compatible such that severe finiteelement distortions do not occur causing poor accuracy. For example an ideal hexa element is a cube(ie: 900 corners and equal length sides), an ideal penta element is an equilateral right prism and an idealtetra element is an equilateral tetrahedron.Figure 3.5A shows the finite elements used to model a vertically oriented MC over a tissue volumeas depicted in Figure 4.1. Figure 3.5B shows the remaining "air' finite elements required for the completemodel. Note that the small space devoid of finite elements in Figure 3.5B corresponds to the tissuevolume finite elements in Figure 3.5A.29Relationship between model geometry and finiteelement and node generation.Figure 3.4 Generation of a Finite Element Mesh3.4.3 Boundary ConditionsMaxwell's equations only involve the vector magnetic potential A through space and time derivatives.Since A is specified only through specification of its curl, it is possible to add the gradient of any scalarfield without changing the physical fields (E, D, B, H) (The curl of a gradient is zero). Thus Maxwell'sequations do not uniquely specify the A vector. In fact only the curl of A is defined by Maxwell's equations.Analytic solutions can be structured to avoid this problem, but numerical methods, such as EMAS, producesingular matrices when confronted with non-unique solutions (singularities). To remove these singularitiesthe divergence of A must be defined (The translation, and shear of A are also defined [83]). If A istransformed to At:At = A ± (45)there will be no change in the resulting magnetic fields, even for an arbitrary choice of the function e.EMAS selects the (gauge) function e such that v • A = 0. This transformation causes the electrostaticpotential to no longer obey the wave equation, and dynamic E-field effects are now contained solely in A.The translation, and shear of the A vector [83] are defined by constraining (setting to a constant) certainDOFs at the boundary nodes. These constraints are called Dirichlet boundary conditions. Translationsingularities are removed by constraining all three components of A to some particular value at one node.Shear singularities are removed by constraining DOFs in such a way that the five independent shearmotions are set to zero without affecting the resulting field. One example of the five shear motions is:aAz aAytiyz -- a ±y ^az •In more common engineering terms, "setting constraints" at selected boundary nodes involves definingthe orientation of the A-fields and electrostatic E-fields at the model boundaries. In particular, boundaryconstraints are utilized to define far-field and symmetric-field behaviors. The most common type ofconstraint is the single point constraint (SPC) where a DOF is set to an independent predetermined(46)30Top Side(B)Lateral SideB) Boundary elements of FEMmodel. All these finite elementsconsist of "air. Note that panels Aand B are not scaled equally.constant value (usually zero). The result of setting constraints along entire boundary surfaces can bepredicted in a general and gross fashion as follows: 1) setting Atangent=0 gives B„rmal =0 and inducedEtangent=0, 2) setting Anormal=0 gives Btangent=0 and induced Enermal=0, 3) setting 0=0 gives electrostaticEtangent=0, 4) if no A SPCs are constrained along a boundary surface then H will be normal to the surface,and 5) if no I SPCs are constrained along a boundary surface then D will be tangent to the surface andno current will cross the boundary.A) FEM mesh for a verticallyoriented MC over a finite planartissue volume. The MC is modelledas a quarter-circle of line elementsand the tissue is modelled as threedimensional slab quadrant below it.No finite elements modelling air areshown here.Medial SideFigure 3.5 FEM Model Symmetry31Only at large distances from a source is it possible to simplify the EM fields in such a crude manner.This is why the EMAS method requires that the finite element model extend out far beyond the area ofinterest, as exemplified by the back side of the FEM model in Fig. 3.5B. The second condition whichallows these simplified EM fields is when symmetry is present. This is seen along the top, left and rightsides of the same figure.In particular, in Figure 3.5B, the tangential component of A is constrained to zero along the left,top and back sides. This is equivalent to the setting the magnetic flux density to be tangential at thesesurfaces. Similarly the normal component of A is constrained to zero along the right side. From previousstudies it is known that the total E-field is normal to the surface of the left side. It is also known that themagnetically induced E-field has no tangential component there thus the electrostatic field cannot have atangential component long that surface. Consequently, the scalar potential is set to zero along the leftside such that the electrostatic E-field will be normal to that surface.3.4.4 ExcitationsSome type of energy source must be supplied to excite non-zero field conditions. This energy sourceis referred to as a load. These loads are independent of the fields they produce, (ie: the outside world willsupply whatever energy is required to maintain the excitation). Loads are generally produced by imposinginhomogeneous Neumann boundary conditions specified by setting non-zero values for the H, D or Jfields along boundary surfaces. These and all other types of loads are internally reduced to equivalentcurrents applied at specific nodes. The time profile of excitations can be static, sinusoidal or arbitrarilydefined (e.g. pulse).The load type most used in our study is a surface current density (JSURF) applied to the ends of theMC model. This load defines the constraints set on the ik degree of freedom during dynamic excitationsand represents the sum of the conduction and displacement currents. Due to the high conductivity of theMC and the low frequencies involved, the contribution of the displacement current to the load is negligible.3.4.5 FEM Analysis and ValidationThere are two basic types of FEM errors: 1) those that create physically implausible models and2) those that create plausible models but which do not model (or do not accurately model) the desiredphysical process. Type one errors are often automatically pointed out by the EMAS software or theyare evident by gross EM field errors. Type two errors are not nearly as simple to detect and require amulti-pronged approach, verifying the solution's validity and accuracy using several independent methods,if possible. Creation and validation of FEM model is an iterative process during all stages of development.The following are important aspects of this process:I. The critical physical characteristics must be identified (i.e. geometry and material properties). The relativesize, shape and orientation of the MC and the tissue volume are critical. The E-field within the tissueis determined by the MC shape and by surface charges along the air/tissue interface. A magnetic coil32consists of many turns of copper wire. It is not necessary to model each turn explicitly, only the overallMC dimensions are modelled. The conductivities of the MC and the (homogeneous) tissue volume neednot be accurately modelled since the induced E-field is insensitive to conductivity as long as the skineffect and other dynamic phenomena do not become significant.II.The simplest, but realistic, mode of analysis should be chosen (i.e. electrostatic, sinusoidal steady-state,magnetostatic, or transient, etc.). Magnetic stimulation is a pulsed, single-shot event. Thus a full analysiswould require a very memory intensive transient analysis. This option was neither necessary or possibledue to a lack of hard disk space. Several quasi-static approximations can be made: 1) the displacementcurrent in tissue is negligibly small compared to the conduction current, 2) the penetration (skin) depth ismuch greater than the distances of interest, and 3) time-retardation of electromagnetic fields is negligible.Thus a sinusoidal analysis offers all the pertinent data required. The spatial distribution of the E-field isunaffected and its magnitude is proportional to the instantaneous time derivative of the MC current dVdt.III.The required degree of spatial precision determines the mesh density and memory requirements. Aforeknowledge of the general electromagnetic field behavior is critical for allocating a proper nodedensity in those areas where a high field gradient is expected. The general E-field gradient can bepredicted by referring to previously published results. The assignment of constraints at the modelboundary surfaces is guided by basic far-field electromagnetic principles.Once the first draft of the model is complete EMAS will run a basic diagnostic which identifies basicerrors such as incoherent executive/analysis commands, faulty mesh generation (distorted elements,dimension tolerances, missing elements etc.). When the program runs to completion and a solutionfile is generated the results are checked to verify that the desired field constrains and that other featuressuch as the loads were satisfied. At this stage it is often possible to determine if the model results arerealistic by using basic EM knowledge but the solution accuracy can only be determined by comparing theresults to those obtained by experimental measurement and/or by other theoretical methods which dealwith similar situations. The general validity of the EMAS modelling technique was tested by comparingresults with previously published results, basic EM theory, and use of analytic solutions (see Chapter 4).33(A) Horizontal MCTissue VolumeChapter 4Validation of FEM Results4.1 IntroductionThe value of a modelling procedure is dependent on the realism and accuracy of the model results.Before applying the EMAS approach to unknown magnetic stimulation problems, we first investigatesituations which have been already described using other methods. This section presents results whichdemonstrate that EMAS modelling results are in good agreement with: basic EM theory, consisting ofboundary conditions, phase shifts, general field distribution etc; analytic solutions, namely the magneticfield along the MC axis; and published experimental and theoretical results, dealing with induced E-fielddistributions.4.1.1 Standard MC Orientations and Coordinate SystemHorizontal and vertical MC orientations (Fig. 4.1) were investigated using the EMAS finite elementmethod. The induced E-fields results presented in this chapter are plotted along standard paths whichfollow the tissue volume surface (axes X and Y) and paths normal to this surface (axes Z and Z').The inner and outer MC radii are marked as Ri and Ro respectively. The MC height is markedas H, the minimum MC/tissue separation is marked as "s". The path denoted with Z' is defined asx = LaTL), Y = 0, and Z > 0, i.e. directly below the MC windings.Three well documented MC arrangements are solved using the EMAS method; these three MCcases are called N1H, N1V and N250H (Table 4.1). In each of the three models the MC is over a(A) Axes [X, Y, Z, and Z ] for the horizontal MC orientation. The MC plane is parallel tothe tissue surface.(B) Axes [X, Y, and Z] for the vertical MC orientation. The MC plane is normal to thetissue surface.See text for details.Figure 4.1 MC Orientations and Coordinate System34planar tissue and the MC current derivative is 108 Ns (steady state solution). The N1V and N1H MCshave identical designs (0D=ID=10cm, H=0) and both are 1cm above the tissue surface (s=1cm) but theN1H is in the horizontal orientation and N1V is in the vertical orientation. Field distributions and FEMboundary constraint techniques differ significantly between horizontal and vertical MC orientations, thusvalidation is presented for both models. Most clinical MCs are bulky and cannot be accurately modelledusing one—dimensional line elements, thus a three—dimensional MC model (N250H) is also validated.The N250H MC is horizontally oriented 1cm above the tissue surface and has dimensions: OD=71mm,ID=57mm and H=17mm.42 Initial ValidationAll FEM results were initially validated by noting that the modelled EM fields agreed with some basicrules:0 The current density in the MC was uniform as specified in the EMAS input data file.0 The induced sinusoidal E-field had a 900 phase lag with respect to the MC current and its magneticfield (Equation 5).0 The magnitude of the induced E-field is proportional to the frequency of the MC current (Equation 5).0 Within the tissue, the induced E-field is parallel to the air/conductor interface [72].0 The induced E-fields form closed circular paths associated with eddy currents [72].Although EM fields around MCs cannot be described using closed-form analytic solutions there issuch a solution for the magnetic flux density along the axis of an ideal current-loop given by,po/r213, _2(r2 ± z2)31'2(47)where, I= total MC current (A), r = MC radius (m), z=measurement point along axis of MC (m) and Bz=axial magnetic flux density (T) [89]. Figure 4.2 plots the axial (along z-axis) magnetic flux density for theN1H and N1V MC models and compares them with the analytic solution. There is a good agreementbetween the EMAS and the analytic results with an error of less than 6% at the MC's center.35oo 2^4^6^8^10Distance Along MC Axis (cm)The magnetic coil axis is definedby the z-axis. The total MC currentis 105 A1.4[....] N1H max. = 1.26 Tesla[ o N1V max. = 1.33 TeslaTheory max. = 1.26 Tesla1.2a)1a)0.83,0.62"5a 0.4-0al 0.2Figure 4.2 Axial Magnetic Flux Density for the N1H and N1V MCs4.3 Validation Using Published ResultsTwo studies provide E-field data for the horizontal and vertical MC orientations [77, 72]. Thesedata are both reliable and plotted in a useful manner. The authors compared their modelled results withexperimental data. This data is used here as a standard to determine the accuracy of our EMAS modellingtechnique. In all cases the time derivative of the total MC current is 108 Ns.4.3.1 Validation of Horizontal MC ModelFigure 4.3A plots the E-field magnitude along the X-axis (1cm below and parallel to the MC plane,along the tissue surface). The E-field is parallel to the MC plane and is normal to the figure page. Thepeak E-field value (directly below the MC winding) is only 2.3% below the published results and generallythere is good agreement along the entire X-axis length. Figure 4.3B plots the E-field magnitude along theZ'-axis, i.e. as a function of tissue depth directly below the MC winding. Again there is good agreementwith the measured results of Tofts [72].Since the MC plane is parallel to the tissue plane, no surface charges are present. The direction andmagnitude of the induced E-field mirrors that of the vector magnetic potential field A. In turn, the A-fieldhas been described as "like the MC-current distribution but fuzzy around the edges, or like a picture ofthe MC-current out of focus" [90].3620154030[---] EMAS max. 33.2 V/m[o] Tofts max. = 34.0 V/m-30-4Q-20^-15 -10^-5^0^5^10Distance Along X Axis (cm)[---] EMAS max. = 33.2 V/m[ 0] Tofts max. = 34.0 V/m104^6Distance Along E Axis (cm)35300 Upper panel, E-Field along x-axisO Lower panel, E-Field along z'-axisReference data taken from Tofts,1990[72]. See Table 4.1 for test conditions.Figure 4.3 Horizontal N1H MC Results4.3.2 Validation of Vertical MC ModelFigure 4.4A plots the E-field magnitude along the X-axis. The peak E-field (at X=0) is 3.9% belowthe published results of Tofts [72]. At points further away from the origin (lxj>0) the error increasesconsiderably. Figure 4.4B plots the E-field magnitude along the Z-axis, directly below the MC winding;again the error increases considerably at deeper levels within the tissue volume. These discrepanciesare examined in Section 4.4. As with the horizontal MC results, vector plots for the vertical MC modelshowed that no significant component of the E-field within the tissue was oriented normal to the air/tissueinterface (results not shown).3782^4^6Distance Along Z Axis (cm)1000 Upper panel, E-Field along x-axis0 Lower panel, E-Field along z-axisReference data taken from Tofts,1990[72]. See Table 4.1 for test conditions.15 20-20^-15^-10^-5^0^5^10Distance Along X Axis (cm)20[---] N1V max..19.7 V/m[ 0 ] Tofts max. = 20.5 V/mFigure 4.4 Vertical N1V MC Results4.3.3 Validation of Three-Dimensional-MC ModelThe N1H and N1V models and results discussed in Sections 4.3.1 and 4.3.2 are typical of publishedstudies in that they reduce all MC designs to an ideal current loop, i.e. a single-turn of thin wire. Sucha simplification is unrealistic for any clinical MCs. Only two published studies account for all three MCdimensions: OD, ID and H [76, 47]. To validate our three-dimensional MC modelling we refer to theexperimental results of Tay [56]. Figure 4.5 plots the E-field magnitude of the N250H MC, calculated usingthe EMAS FEM method, along the Z'-axis (directly below the windings). The peak E-field, at the tissuesurface, is 5.4% below the published experimental results and the error remains low at deeper tissue levels.384.4 DiscussionWe found that our modelled MC field results are in good agreement with analytic expressions. Themodelled electric and magnetic fields are out of phase, the E-field magnitude increases with the frequencyof the MC current, and the modelled axial magnetic flux density is described by equation 47. In addition,our modelled results agree with previously published experimental and modelled results. The E-field isparallel to the air/tissue boundary throughout the planar tissue volume for both the vertical and horizontalMC orientations, no significant vertical E-field component exists [72]. Other investigators showed thata MC placed over a spherical tissue volume is incapable of generating currents in the radial directionregardless of MC shape or orientation [55, 77]. This basic E-field characteristic casts doubt on someclinical mechanisms presented in clinical magnetic stimulation studies [78, 8, 5] which postulate thatradially oriented E-fields are induced within the head.Excellent agreement was obtained between our EMAS results for the horizontal MC and previouslypublished experimental data. In contrast, our EMAS results for a vertical MC are less conclusive (Fig.4.4). Tofts presents results for a vertical MC where the isopotential contours are incorrect since the peakE-field is not at the tissue surface [72]. However, their modelled peak E-field value is in good agreementwith our results and have been confirmed experimentally by Branston [55]. Unfortunately, lofts publishedthe only study which provides well defined and thus useful plots of the three-dimensional characteristic ofthe E-field distribution for a vertical MC.The presentation of MC field results in the literature is worthy of comment. Many theoretical andexperimental studies have been published concerning E-fields induced by various MC designs andorientations. However, not one is totally satisfactory as a standard by which to judge our EMAS results.Many reports use surface plots which only describe the E-field along a single plane, and more importantly,Figure 4.5 Validation of N250H MC Results39these surface plots do not offer a precise data base [81, 53, 48, 52, 51, 57, 49, 76]; some papers presentincongruent data/plots [72, 56]; while others do not state critical MC specifications [57, 48].To achieve accurate E-field results in all three of our EMAS models a very fine node mesh must bepresent between the MC and the location of the E-field peak. In this zone the FE thickness is less thanlmm. Although the axial magnetic flux density may be accurate (Figure 4.2) experience indicates that thisis not necessarily an indication of the accuracy of the E-field results.Table 4.1 lists our modelled results together with results from various other studies, where the E-fieldsare normalized for a MC current derivative of dl/dt = 108 Ns. Our EMAS results for the E-field peak areconsistently lower but in good agreement with theoretical, as well as experimental results (compare cases{1 and 2], {4 and 51 and {8 and 91 ). In contrast to a loop-MC, a true MC with an identical MC currentderivative dl/dt induces a significantly lower peak E-field and a much less focused E-field distribution.For example, by comparing our three-dimensional MC model (N250H) to a similarly sized loop type MCdesign it is evident that by dispersing the MC current the induced E-field is also dispersed and that thepeak E-field is reduced (compare cases 4, 5 and 6). Since there are no published theoretical results forthis three-dimensional MC design it is unclear whether the EMAS or the experimental results are moreaccurate for the N250H MC.Table 4.1 also indicates:0 a smaller MC induces a weaker E-field at a given distance below the MC plane (cases 1 and 7).0 that a horizontal MC generates a greater E-field (cases 1 and 8).0 that there is good agreement among the results of investigators (cases 10-13)0 that the E-field results are independent of the isotropic tissue conductivity; this was confirmed with ourmodels (results not shown).40Table 4.1 Published MC Modelling Results(A) Results For Horizontal MC Orientation. "Emax" is measured 1cm below the MC plane(B) Results For Vertical MC Orientation. This Table indicates the minimal vertical distance fromthe MC to the measurement point (Dist.) and the maximal E-field measured at that point.All results are for loop or spiral MC designs over a planar tissue volume. These Tablescompare the maximal E-field value (experimental and/or theoretical) given in various referencesto that calculated using the EMAS FEM method. The Table indicates the reference, theoretical(T) or experimental (E) results, the MCs' dimensions [OD, ID, Hj and the maximal inducedE-field ( Emax). A dash ( —) signifies that the information is not available. All results havebeen normalized such that the total contribution from all MC windings generate dl/dt = 1E8 A/s.Note that these E-field results are independent of the distance between the MC and theair/tissue interface.(A) Horizontal MC Model ResultsCase Reference T I E MC Dimensions (mm) Emax(Vim)OD ID H1 [72, 55] T&E 100 100 0 34.02 EMAS [N1H MC] T 100 100 0 33.2, 3 [48] E 90 — — 26.24 [56] E 71 57 17 16.75 EMAS [N250H MC] T 71 57 17 15.86 [36] T 70 70 0 277-[54] E 40 40 0 16(B) Vertical MC Model ResultsCase Ref. T I E MC Dimension ( mm ) Dist.( mm )Emax( Vim )OD ID H8 [72, 55] T&E 100 100 0 10 20.59 EMAS [N1V MC] T 100 100 0 10 19.710 [72] T 50 50 0 15 8311 [54] T 50 50 0 15 8.812 [55] T 47 — — 15 8.5713 [55] E 47 — — 15 8.2541Chapter 5Modelling Commercial MC Designs5.1 IntroductionThe two basic goals of MC design are to maximize the AF function magnitude and to minimize the AFfunction's zone of influence. These two characteristics allow supramaximal stimulation of the target nervewithout co-stimulation of other nearby nerves. We present in this chapter an investigation of the importantrole played by the inner and outer diameters of the MC and their effect on the EF and AF functions. Wealso investigate how these functions may be affected by surface charges generated at the boundaries ofa finite tissue volume [91].Basic EM principles suggest that if an MC could be reduced to an ideal current-loop, then the EFand AF functions would increase for a given MC current derivative. Thus, an idealized current loop wouldinduce greater fields than a similar sized spiral MC design (Fig. 2.2). No published study has specificallyinvestigated this aspect of the construction of MCs. Most studies of magnetic coil designs have modelledthe MC as a single turn of one-dimensional line-elements, (Table 4.1), where OD=ID, yet no practicalMC can achieve such a concentrated coil-current path. A larger outer diameter of an MC allows for adeeper stimulus penetration, but nowhere is it discussed why all commercial manufacturers produce their"standard" MCs with a relatively small inner diameter (Table 5.2) thereby diffusing the vector magneticpotential field and reducing the maximum induced E-field. Many clinical studies are performed using thesecommercial MCs. This preference for an MC diameter ratio of OD//D ,=::: 2 is not as prevalent inrecent prototype MC designs which are often characterized by a lower OD/ID ratio [5, 8].Only a few studies have modelled the inner and outer diameters of MCs and these were not concernedwith the same issues described above. For example, Mouchawar [76] and Durand [47] did not investigatedifferent OD/ID ratios, and Cohen [48] did not normalize or state the MC current derivative (dVdt) for anyof his results. Roth presented results for MCs with different OD/ID ratios but did not normalize the results,did not investigate the AF function and complicated the issue of MC design by modelling E-fields withinthe head where extraneous factors such as surface charges will affect the results [50].42Reference OD[nun]ID[min]BMr[us][5, 57] t 93 50 2.0 50[11] $ 116 54 2.0 100[9211' 140 40 — 150Three commercially available MC designs are iden-tified by:f = Novametrix Magstim 200t.-- Cadwell MES-10 stimulator1(= DantecThe table headings are as follows: OD and ID = theinner and outer MC diameters, B = peak magneticfield density at MC center, and r = pulsed MC-current rise time. A dash (—) indicates that noinformation was given in reference.Table 5.2 Commercial Magnetic Coil Designs5.2 FEM ModellingFour related FEM models were analyzed. Each model consists of about 2200 nodes and requireson average 25 minutes of CPU time on a SUN SPARC station 2. Four configurations were investigatedusing one spiral (MC1) and one round (MC2) MC design and two tissue volumes (Table 5.3). MC1 wasmodeled using N=6 turns of line-elements evenly spaced between the inner and the outer MC diameters.MC2 was modelled with a single turn of line-elements. Both MCs are vertically oriented over two types oftissue volumes: a planar tissue volume defined by z 0, and a finite cylindrical tissue volume with a 10cm radius and a 5 cm depth, Fig. 5.1. The plane of the MC is coplanar with the X-Z plane. The minimumMC/tissue separation is 10 mm and the total MC current is increasing at a rate of 108 A/s. A typical clinicalsituation might have a nerve axon along the X-axis.Figure 5.1 Vertical MC Over Finite Tissue43From top to bottom are panels A and B.oo^spiral coil (MC1) over planar tissuespiral coil (MC1) over finite tissue++^loop coil (MC2) over planar tissueloop coil (MC2) over finite tissueSee Table 5.3 for details30—E 2520-01510^-15^-10^-5^0^5^10^15Distance Along Nerve Length [ X Axis ] ( cm )20,.-Cr0 A 0, al-5^0^,5 • so tt * 10Distance Lateral to MC Plane [Y-axis] ( cm )30—E 255.3 FEM Model ResultsThe induced E-field is parallel to the X-Y plane at all points within the tissue volume. The onlyexception is for the finite tissue volume where the E-field deviates from this plane at the vertical air/tissueboundaries. In particular, the E-field along the YZ and XZ planes always points in the X-direction, exceptnear jxj =10 cm for the finite tissue volume. The E-field magnitudes along the X and Y axes are plottedin Fig. 5.2. The finite tissue volume distorts the E-field distribution in three ways: a) the E-field peak isreduced, b) the E-field approaches zero at fx1=10 cm (Fig. 5.2A ), and c) large return currents flow lateralto the MC at 4 .10 cm (Fig. 5.2B).Figure 5.3 shows the AF function for a nerve path defined by the X-axis. The position and distancebetween the virtual poles (anode/cathode) remain constant regardless of MC design and tissue volume.Figure 5.2 Induced E-fields Along the X and Y Axes44-5^0^5Distance Along Nerve Length [X-axis] (cm)100.080.06Y.,0.04'46– 0.020 00000Lf -0.02Coil-0^oo --0^spiral coil (MC1) over planar tissue^-0.08^— —0^spiral coil (MC1) over finite tissue -10++ --+^loop coil (MC2) over planar tissue--,^loop coil (MC2) over finite tissueSee Table 5.3 for detailsFigure 53 Induced Activation FunctionsThe AF function profiles seem to change only in magnitude for each of the MC designs. The tissue volumehas no apparent effect on the AF function profile.The model results are summarized in Table 5.3. The loop-MC compared to the spiral MC, inducedpeak E-fields that were 134% greater in a planar tissue and 149% greater in the finite volume tissue.Similarly the loop MC induced an AF function 200% greater than the spiral MC, regardless of the tissuevolume considered. The effect of the tissue volume on the E-field and AF function is much less pronounced.The E-fields in the planar tissue (compared to the finite tissue volume) are 17% and 25% greater for theloop and spiral MCs respectively. In addition, the AF function remained essentially constant for any givenMC design regardless of the tissue volume.45Table 5.3 MC Dimensions and Model ResultsThis table defines the symbols used in Figures 5.2 and 5.3. The peak E-field and peak AF functioncorrespond to values at the tissue surface.Case Symbol MC DesignOD/ID (mm)Tissue Volume Peak E-Field(V/m)Peak AFFunction(V/cm/cm)1 o o()spiral MC1, OD=120, ID=50planar volume 12.7 0.022 ___spiral  MC1, OD=120, ID=50finite volume 10.2 0.023 + +Oloop MC2, OD=120, ID=120planar volume 29.8 0.064 _ _Oloop MC2, OD=120, ID=120finite volume 25.4 0.0654 DiscussionEsseIle compared the AF functions generated by loop, square, quadruple and figure-8 MC designswhere each MC was modelled using idealized one-dimensional line-elements of current [51]. The mostsignificant difference between all these MC designs was that the figure-8 MC induced a more focused AFfunction. This focusing ability is supported by clinical studies [5]. When their results are normalized forA) the number of coil turns, B) the total MC current derivative, C) spatial scaling and D) MC orientationthere is no major difference in the peak AF function values associated with any of these MC designs,(maximum difference was 12%). In contrast, our results indicate that a 200% gain in the AF function ispossible by reducing the OD/ID ratio of a MC towards unity. Thus, regardless of the MC shape, to increaseor maintain a large AF function (for normalized conditions) one effective design option is to minimize thecross sectional area of the MC windings, thereby increasing the coil-current density and peak E-fields.It is expected that the larger AF function and/or E-field associated with loop-MC designs will resultin more effective nerve stimulation; however the clinical data at present is unclear. Maccabee stimulatedhuman peripheral (median) nerves in the arm [8]. Two MC designs were used: a spiral MC with OD=92mmand ID=50mm and a very thin tear-drop MC with OD 90mm. Their data suggests that the total MCcurrent rise time (dl/dt) was similar for both MC designs. They found no marked differences in the clinicalproperties of either MC, and went so far as to not even mention from which MC design their publishedclinical data was derived.No investigators have addressed the design process which led to the commercially available spiral MCdesign which is commonly used in clinical studies. There is little well documented clinical data to supportor reject the value of thinner MC designs. Amassian stimulated the human cortex with three MC designs:spiral, tear-drop and figure-8 [5]. The tear drop design was relatively thin with OD/ID 1. The46tear-drop and figure-8 MC designs were found to offer a more "focused" stimulus but many test parameterscomplicated any further analysis: a) the MC current was not defined, b) the MC sizes varied considerably,c) the MC orientations were not consistent, and d) only focality was mentioned rather than the thresholdstimulus levels. Note that this type of disregard in defining and designing the test conditions is foundthroughout the clinical literature of magnetic stimulation.Two obvious and practical advantages of the spiral MC design are increased heat dissipation andphysical ruggedness. Thicker copper wires are less lossy thus the heat generated during the capacitordischarge is lower. Secondly the high magnetic field intensity between the MC windings generates strongrepulsive forces which tend to stress and distort the MC shape.Clinical magnetic stimulation is often used at the wrist or the cortex. Both these situations involvefinite tissue volumes where surface charging will reduce the magnitude and change the direction of theE-fields from those that would be found in a planar tissue [53, 50]. Our results suggest that although afinite tissue volume may significantly distort the E-field distribution and magnitude, it does not necessarilyfollow that the associated AF functions will be equally distorted. The next two chapters investigate morerealistic tissue volumes (shapes) and tissue conductivity characteristics.47(A) tendon,(B) perimysium,(C) epimysium,(D) intercellular space,(E) fascicle,(F) muscle cellCChapter 6Modelling Anisotropic Tissue in the Arm6.1 IntroductionBody tissues often have some degree of directional organization which can result in marked anisotropiccharacteristics. One common and well established example are the cell fibers which are oriented alongthe axis of skeletal muscle. Muscle tissue consists of bundles (fascicles) of muscle cells (fibers) wrappedin connective tissue sheaths (epimysium, perimysium) (Fig. 6.1). The resistance to current flowinglongitudinally through the cytoplasm of a muscle fiber is always assumed to be much less than theresistance to current flow transversally across the fibers or through the cell membrane.Burger measured the specific conductance of whole-limb segments (arm, leg) and isolated musclefibers [93]. In both cases the conductance was relatively constant form 20Hz to 5kHz. For human forearmthe conductivity parallel to the arm axis was up = 0.417 S/m, and in the human upper leg the conductivitytransverse to the leg axis was o-r=0.148 S/m. These results give a parallel/transverse conductivity ratioof 2.8. An earlier study by Burger using DC current on whole-arm body segments reported a lowerconductivity ratio of 2.0 with up = 0.43 S/m and c--r=0.21 S/m [94].These whole-limb specific-conductivity measurements include the effects of non-muscle tissues suchas blood, connective tissues etc. In addition, and possibly more important, whole-limb samples containmany muscles which are not oriented in a parallel fashion thus reducing the anisotropic conductivityratio as compared to measurements made on a single muscle sample. For example, Burger measuredsingle, extirpated rabbit skeletal muscle and obtained a conductivity ratio of 14.4 with up = 0.80 S/m andu-r=0.056 S/m [93].A more recent study by Zheng measured the impedance rather than resistance of single-muscletissues and obtained a parallel/transverse admittance ratio of 12.6 in monkey tissue [95]. He reported aFigure 6.1 Structured Membrane Geometry in Muscle Tissue48Table 6.4 Admittance Ratio for Monkey Muscle TissueRatio of parallel/transverse admittance ratio,from Zheng, 1984 [95].Frequency(Hz)1,10 100 1k.10k 100k IMAdmittanceRatio12.6 13.0 12.5 11.2 7.0 2.1 1.2frequency independent parallel conductance from 1Hz to 1MHz. The transversal admittance was nearlyconstant up to 1 kHz and then gradually increased at higher frequencies. The frequency dependenceof the admittance ratio (parallel/transverse) for monkey muscle tissue is shown in Table 6.4. At lowfrequencies, the transverse admittance is much lower than the parallel admittance. Currents flowingparallel to the long muscle cells encounter few membranes. In contrast, transverse currents encountermany membranes which constitute distributed capacitors. At high frequencies this membrane associatedcapacitance becomes a high admittance pathway and the anisotropic characteristics disappear.When modelling currents within body segments containing many muscles, the general orientation ofthese muscles must be considered. Rush modelled electrocardiographic E-fields within the human torsoand accounted for muscle anisotropy [96]. In this case it was assumed that the chest and back muscleswere essentially directed parallel to the body surface but otherwise almost uniformly distributed over allangles. Thus the muscle layer conductivity, as a whole, was modelled as essentially isotropic for directionstangent to the body surface but a lower conductivity was defined for currents flowing normal to the torsosurface.Clinical stimulation of the medial and radial nerves at the wrist is a common diagnostic technique.Only one published study has investigated magnetically induced E-fields in the arm [54]. No magneticstimulation study has modelled the effect of anisotropic conductivity on the EF and AF functions or onthe spatial distribution of induced E-fields. Many of the usual relationships do not hold for anisotropicmedia. For example, lines of current are not parallel to the E-fields, and Laplace's equation is no longervalid even though the media may be free of charges. In this chapter we present FEM model results fortwo MC designs: square and figure-8. Both models simulate conditions associated with peripheral nervestimulation at the arm (wrist) and account for the anisotropic conductivity properties of whole-limb. Thegeometry of our square MC model closely resembles the conditions modelled earlier by Roth and is usedhere as a confidence test [54]. Our second FEM model simulates conditions for a figure-8 MC. Bothmodels include an analysis of the EF and AF functions along an axial nerve path.49(A) top view of square MC over arm,(B) end view of square MC over arm, thecylindrical coordinate system (r, 0, z)and the rectangular coordinate system(x,y,z).Only one quarter of this model is explicitlymeshed, see Fig. 6.3Figure 6.2 Model Geometry6.2 FEM Model of Square MC Over the ArmRoth modelled magnetic stimulation of the arm [54]. The arm was modelled as a 5 cm diametercylinder of infinite length containing tissue having a homogeneous, isotropic conductivity. Induced E-fieldswere solved for a 5cm diameter loop MC positioned symmetrically one cm above the arm in a tangentialorientation. Fig. 6.2 shows our model configuration which closely resembles that of Roth except thata 4.8cm square MC design was chosen and the cylinder length was finite (35 cm) [54]. The squareMC option allowed a simpler FEM mesh and a corresponding reduction in the disk space necessary foranalysis. Note that only one quadrant (1/4) of the arm and MC is explicitly modelled and that our fieldresults are presented using four views (angled, end, medial, lateral) and that the lateral view has a 200offset (Fig. 6.3).The arm was modelled as a homogeneous conductor of finite length. Human anisotropic conductivityparameters were taken from previous human whole-limb experimental results [93]. The conductivity [0-]matrix was used to define anisotropic tissue properties using a cylindrical material coordinate system. Theconductivity matrix is given by:a --=[alla21a31al2022032013a23033(48)Angled view of modelled (meshed) portion of arm and MC,and the rectangular coordinate system (x, y, z). Three otherviewing planes (end, medial, lateral) are also indicated.Figure 63 Four Views for Square-MC Results Presentation50where for example the current density along the x-axis Jx is defined as a function of all three vectorcomponents of the E-field:J„ ,--[av—an—axav^avi. (49)— 012 — —Cr13— 1ay^aZMany anisotropic modelling situations, including ours, only require the diagonal elements of the conductivitymatrix to be non zero. These three terms represent conductivity values along three mutually orthogonalaxes giving the more simple expression:av .^av .^av= —a,--1— ae—J— az—Icax^ay^azwhere i, j and k are unit vectors along the three cylindrical axes and where ffr = •11,(50)0-9 =0.22, and Clz = 733. Anisotropy within the whole-arm was defined by az = 0.417 S/m, cr0=0.148S/m and ar=0.148 S/m, (Fig. 6.3). The relative permeability was set at 1.0 and the relative permittivitywas set at 80.0. As a control the arm was also modelled as an isotropic conductor with a = 0.417 S/m. Themodel boundary conditions were defined such that the induced E-field is normal to the xy- and yz-planesof the arm.6.2.1 FEM Model ResultsFigure 6.6, an angled view, shows our results for the overall E-field and J-field distribution throughoutthe arm. Figure 6.7 an end view, Fig. 6.8 a lateral view and Fig. 6.9 a medial view provide supplementaldepictions along the model's symmetry planes and its external surface. Each of these four figures showthree electromagnetic field results: panel (A) the E-field (V/m) for anisotropic tissue, panel (B) the E-fieldand J-field (A/m2) for isotropic tissue, and panel (C) the J-field for anisotropic tissue. Panel (D) shows thecorresponding FEM mesh and the MC position.The figure legends correspond to a peak MC current derivative of 108A/s. The isocontour plots consistof 10% increments in the E-field and J-field magnitudes denoted by the gray scale shown in Fig. 6.4. Notethat in all isocontour figures the large white region represents the 0% to 10% isocontour while the smallblack region represents the 90% to 100% isocontour. For simplicity the isocontour regions are referred toby their upper magnitude limit, e.g. the 0% to 10% isocontour is referred to as the 10% isocontour.The E-field within the tissue resembles, although distorted, a mirror image of the MC shape. Figures6.6 and 6.8 show how the isocontours follow the circular (square) outline of the MC. Since there is nocurrent sink or source, eddy currents must form closed loops, i.e. like the MC shape. Recall that in aavailan   imamIn increasing order, from left to right, are the 10% to 100% isocontour markers for E-field and J-field plots. Thislegend is valid for all isocontour plots in Chapters 5,6 and 7.Figure 6.4 Gray Scale for Isocontour Plots51planar tissue the E-fields are, at all tissue points, parallel to the air/tissue interface. Thus the eddy currentsflow in loops along parallel planes. This is also generally true within the modelled arm and agrees withthe FEM model constraints which define that the E-field be normal to the xy- and yz-planes (Fig. 6.7 and6.9) and that the E-field is always tangent at the air/tissue interface (Fig. 6.8). In the modelled arm thecurrent loops flow along surfaces which follow the curved contour of the arm while also loosely mirroringthe shape of the MC.The E-field in the left half of the arm along the xy-plane depicted in Fig. 6.7 is normal to the figureplane, i.e. pointing into the page and then one half period later pointing out of the page. At any givenmoment the total current passing through the cross-sectional surface of the entire arm must be zero. Thisequilibrium is satisfied by symmetry conditions present in the right half of the arm which was not explicitlymodelled here.The isotropic model shows a peak E-field of 16.1 V/m (see panel B of Figures 6.6-6.9). This valueis in good agreement with the peak value (somewhere between 15.0 V/m and 17.5 V/m) modelled byRoth for a similarly sized loop MC design [54]. In addition our E-field isocontour plot corresponds wellwith that presented by Roth, where the peak E-field is located laterally, just below the MC, along thexy-plane (arrow-1 in Fig. 6.7B). A secondary maxima (12.8 V/m) is located along the yz-plane directlybelow the MC, see arrow-1 in Fig. 6.98. As expected the highest E-field intensities in the arm mirror thecontour of the MC above. As in the two previous chapters, the E-field along the axis of the MC (y-axis)is zero. Although our modelled arm has a finite length (35cm) the MC is small enough that the arm maybe considered effectively infinite in length. One indication of this is that the 10% E-field isocontour is only7 cm from the model's center measured along the z-axis.The locations of the two E-field maxima ( 16.1 and 12.8 V/m) previously mentioned are of interest.In a planar isotropic tissue volume the E-field strength drops off with distance from the MC. In our armmodel the minimum tissue/MC separation is 1cm (Fig. 6.28). Yet the maximal E-field is located where theMC is 1.77cm above the arm (Fig. 6.7B) rather than where the MC is 1cm above the arm (Fig. 6.9B).Surface charges are the likely cause of this anomaly. Without surface charges the E-fields would tendto follow the circular path of the MC on planes parallel to the xz plane; however, currents within the armare restricted in both the x and z directions but more so in the x direction since the arm's diameter isshorter than its length. Thus the E-field directed along x-axis (Fig. 6.98) is significantly more attenuatedby surface charging than are the E-fields directed along the z-axis (Fig. 6.7B).Generally the appearance of the E-field isocontours are similar for both the isotropic and anisotropicmodels: compare panels A and B in Fig. 6.6. However, the locations of the peak E-field and secondarymaxima in the anisotropic model are reversed compared to those in the isotropic model. In particular:1.the anisotropic model has a peak E-field (152 V/m) located along the yz-plane (arrow-2 in Fig. 6.9A).This is a 19% increase relative to the field at the same position in the isotropic model (12.8 V/m).2. and a smaller secondary maxima (13.8 V/m) at along the xy-plane (arrow-2 in Fig. 6.7A). This is a 14%decrease relative to the field at the same position in the isotropic model (16.1 V/m).52This reversal of the locations of the peak E-fields may be explained by anisotropic conductivity. Forthe isotropic model the J-field is directly proportional to the E-field both in magnitude and direction. Thus16.1 V/m corresponds to a current density of 16.1 x 0.417 = 6.7 A/m2 and the appearance of the J-fieldisocontour plot is identical to the E-field one, i.e. panel B of Figures 6.7 to 6.9. This correspondence isnot present in the anisotropic model where conductivity is a function of direction (compare panels A andC in Figures 6.7 to 6.9):I. f The J-field peak (5.9 A/m2) (arrow-1 in Fig. 6.80) is not associated with the E-field peak but ratherwith the smaller E-field secondary maxima (arrow-2 in Fig. 6.8A) .II.The current flow is skewed along the z-axis (Fig. 6.80) rather than mirroring the curvature of the MC(Fig. 6.8A).These observations are consistent with the fact that the parallel/transverse conductivity ratio is 2.8. Thus,for our anisotropic results, the effective resistance associated with the end-view is Cfp = 0.417 S/m (Fig.6.7A and C), while that associated with the medial-view is o-T=0.148 S/m (Fig. 6.9A and C).The EF and AF functions were calculated along the path of an imaginary axon. This facilitatesquantification of the stimulus differences relevant to nerve stimulation which are attributable to anisotropicconductivity. The straight nerve (axon) was positioned along the surface of the arm, parallel to the axis ofthe arm, and the axon center was located where the isotropic E-field is maximum. The corresponding EFand AF functions are shown in Fig. 6.5. The general profiles of the EF and AF functions remain relativelyconstant within both the isotropic and anisotropic tissues; however, note that the EF and AF functionsin anisotropic tissue are slightly more dispersed over a longer length of the axon. Within the anisotropictissue decreases of 14% and 25% are seen in the EF and AF functions, respectively, relative to thoseseen in the isotropic tissue. In the next section, we present results for the figure-8 MC model.5320LTJ5Electric Function-8^-4^0^4^8Distance Along Axon Length (cm)Activation Function0.060.03..g.C)Et.,-a1-0-0.03-0.06-8^-4^0^4^8Distance Along Axon Length (cm)The EF and AF functions for an axon parallel to z-axis, along surface of arm, and centered at the locationof the peak isotropic E-field.(....) results for isotropic media –4. EF max. = 16.0 V/m, AF m ax. = 0.056 V/cm2(—) results for anisotropic media , EF max. = 13.8 V/m, AF max. = 0.042 V/cm2Figure 6.5 Fields Along Axon54FAL E(A) anisotropic E-field (max. = 15.2 V/m),(B) isotropic E-field (max. = 16.1 V/m) and J-field (max. = 6.7 A/m2),(C) anisotropic J-field (max = 5.8 A/m2),(D) finite element mesh of quadrant showing arm and MC (Fig. 6.3).Figure 6.6 Results for Square-MC Over Arm [Angled View]55a—_L1 C^---, -------::::1(A) anisotropic E-field (max. = 15.2 V/m),(B) isotropic E-field (max. = 16.1 V/m) and J-field (max. = 6.7 A/m2(C) anisotropic J-field (max = 5.8 A/m2) and(D) finite element mesh of arm and MC.View along xy plane (Fig. 6.3), fields are normal to page.Figure 6.7 Results for Square-MC Over Arm [End View]561 < (A) anisotropic E-field (max. = 15.2 V/m),^()(B) isotropic E-field (max. = 16.1 V/m) and J-field (max. = 6.7 A/m2),^.IL__(C) anisotropic J-field (max = 5.8 A/m2) and(D) finite element mesh of arm and MC, lateral view (Fig. 6.3).All E-fields and J-fields are tangent to this curved surface. Center of arm is at right sideof each panel (i.e. z=0 in Fig. 6.3).Figure 6.8 Results for Square-MC Over Arm [Lateral View]57/- I =(A) anisotropic E-field (max. = 15.2 V/m).4^(B) isotropic E-field (max. = 16.1 V/m) and J-field (max. = 6.7 A/m2), the 90% and100% isocontours are not present in this view.(C) anisotropic J-field (max = 5.8 A/m2). Only the 10% to 40% isocontours arepresent in this view.(D) finite element mesh of arm and MC.View along yz plane in Fig. 6.3. All E-fields and J-fields are normal to page. Center ofarm is at left side of each panel (i.e. z=0 in Fig. 6.3).Figure 6.9 Results for Square-MC Over Arm [Medial View]58AI^•7.5cm••^4(A) Top view of figure-8 MC over arm(B) end view of figure-8 MC over armThe material coordinate system is (r, 0, z)and the geometric coordinate system is (x,y, z).•1rIv• •5cm• Coil..4.....m4......1...■411....—..... —t- 5mm5cm6.3 FEM Model of Figure-8 MC Over the ArmOur previous model results are in good agreement with published results and suggest that our FEMmodelling correctly accounts for the effect of surface charges at the arm's air/tissue interface; however,such a symmetric MC orientation is not generally used in regular clinical practice. Our following FEMmodel investigates a more standard clinical MC orientation using a novel MC design intensely investigatedsince 1990, [36, 48, 50, 51, 58, 76].The MC consists of two 5cm square MCs with a common side called the center bar (Fig. 6.10 and6.11). The MC current derivative along the center bar is 108 Ns. The arm was modelled as a 5 cmdiameter cylinder with a length of 20 cm containing tissue having a homogeneous conductivity. The MCplane is tangent to the arm surface, with a minimum MC/tissue separation of 5mm, and symmetricallypositioned with the center bar parallel to the arm axis. The material characteristics of the tissue weredefined as in the previous model: az = 0.417 S/m, 0-0=0.148 S/M and cr=0.148 S/m and the relativepermeability and permittivity were set to 1.0 and 80.0 respectively. The model boundary conditions weredefined such that the E-field is normal along the xy-plane and tangential along the yz-plane. Note thatonly one quadrant (1/4) of the arm and MC is explicitly modelled, (Fig. 6.11).Figure 6.10 Model Geometry596.3.1 FEM ResultsOur model results are presented in a fashion similar to the previous section (Fig.6.11). Figures 6.13to 6.16 show four views of the FEM model results: Fig. 6.13 an angled view, Fig. 6.14 an end view, Fig.6.15 a lateral view and Fig. 6.16 a medial view. Each of these four figures show three EM field results:panel (A) the E-field (V/m) for anisotropic tissue, panel (B) the E-field and J-field (A/m2) for isotropic tissue,and panel (C) the J-field for anisotropic tissue. Panel (D) shows the corresponding FEM mesh and theMC position. The figure legends correspond to a peak MC current derivative of 108A/s. The isocontourplots consist of 10% increments in the E-field and J-field denoted by the gray scale described previouslyin Fig. 6.4.The factors affecting the field distributions in our previous square MC model are also valid for ourfigure-8 MC model but are expressed in different fashions and degrees. Consequently our figure-8 MCmodel results are presented by comparing them to our square MC model results.The E-field within the tissue still mirrors the outline of the figure-8 MC; however, the MC's center bar isstrongly represented with high and focalized E-field and J-field levels, (Fig. 6.13). The induced fields formtwo closed loops below each of the adjacent square MC loops and flow along planes which are parallelto the air/tissue interface. The E-field is tangent to the yz-plane, (Fig. 6.16). This is in contrast to thesquare-MC model, where the E-field is normal to the medial-plane.In a spherical tissue volume the E-field has no component normal to the surface and that its magnitudeis always zero at its center, regardless of MC shape and orientation. This behavior is strongly expressedwithin the arm, where only small field magnitudes are present along the z-axis of the arm (arrow-1 inFig. 6.16).In a planar tissue the field strength drops off monotonically with distance from the MC plane. Thisgeneral result is not valid in this model as suggested by the 20% and 30% isocontours at the bottom lefthand corner of Fig. 6.14 (arrow-1). Since the isocontours only indicate field magnitude it follows that the0% to 10% isocontour (arrow-2 in Fig. 6.14) represents the transition zone where currents can be flowingAngled view of modelled (meshed) portion of thearm and the figure-8 MC. Three other viewingplanes (end, medial, lateral) are also indicated.4 Figure 6.11 Four Views for Figure-8 MC Results Presentation60in opposite directions. For example, in Fig. 6.14 the isocontours above the large white 10% isocontourhave E-fields pointed into the page while the isocontours below the 10% isocontour have E-fields pointedout of the page. Although the area above the 10% isocontour is smaller than the area below it, the totalcurrent passing through this cross-sectional surface of the arm at any given moment is zero. These fielddirections are reversed one half period later. Note that this is not entirely the same case as with theprevious square-MC model where all return currents are located in the symmetric half of the arm whichwas not explicitly modelled.The isotropic model shows a peak E-field (16.2 V/m) just below the center bar of the MC (panel B ofFigures 6.13 to 6.16). The anisotropic model shows a peak E-field of 14.2 V/m (panel A of Figures 6.13to 6.16) a 12% drop from the isotropic model results. Similarly, the peak anisotropic J-field is 5.8 A/m2,a 13 % drop from the isotropic model results. Thus, for currents flowing along the z-axis, both modelshave an effective resistance of about 0.417 S/m, but the anisotropic tissue is defined by a greater over-allresistance and has lower field magnitudes, as in the square-MC model results. Unlike the square-MCmodel, their is no secondary E-field or J-field maxima and the E-field intensity quickly disperses at furtherdistances from the MC's center bar.For the isotropic model the J-field is directly proportional to the E-field. Thus the appearance of theJ-field isocontour plot is identical to the E-field one (panel B of Figures 6.13 to 6.16). This correspondenceis not present in the anisotropic model results (Fig. 6.15A and C): the 20% J-field isocontour is larger(arrow-1) and the E-field distribution shows a large lateral bulge in the 40% and 50% isocontours (arrow-2). This isocontour bulge is due to a related transition in the tissue conductivity, where induced currentsflowing parallel to z-axis are defined by the parallel resistance op = 0.417 S/m, while currents flowingcircumferentially are defined by the transverse resistance 0-T=0.148 S/m.The EF and AF functions were once again calculated along an imaginary path. This straight path waspositioned along the surface of the arm, parallel to the axis of the arm, and where the E-field is maximum.The corresponding EF and AF functions are shown in Fig. 6.12. As with the previous model, the generalprofiles of the EF and AF function remain relatively constant within both the isotropic and anisotropictissues; however, again note that the EF and AF functions in anisotropic tissue are slightly more dispersedover a longer length of the axon. Within the anisotropic tissue decreases of 12% and 23% are seen in theEF and AF functions respectively, relative to those seen in the isotropic tissue.6120.0LL00o-5-8^-4^0^4^8Distance Along Axon Length (cm)Electric Function Activation Function0.10.05f-6---^0N1:31-0-0.05-0.1-8^-4^0^4^8Distance Along Axon Length (cm)D The EF and AF functions for an axon parallel to z-axis, along surface of arm, and centeredat the location of the peak E-fieldO (....) results for isotropic media –+ EF max. = 16.2 V/m, AF max. = 0.082 V/cm2O (—) results for anisotropic media --, EF max. = 14.2 V/m, AF max. = 0.063 V/cm2Figure 6.12 Fields Along Axon62L E D_J(A) anisotropic E-field (max= 14.2 V/m),(B) isotropic E-field (max.= 16.2 V/m) and J-field (max=6.7 A/m2),(C) anisotropic J-field (max=5.9 A/m2),(D) finite element mesh of arm and MC.Figure 6.13 Results for Figure-8 MC Over Arm [Angled View]63(A) anisotropic E-field (max. = 14.2 V/m),(B) isotropic E-field (max. = 16.2 V/m) and J-field (max=6.7 A/m2),(C) anisotropic J-field (max. =5.9 A/m2),(D) finite element mesh of arm and MC. View along xl,f plane, all E-fields and J-fields are normal to page.Figure 6.14 Results for Figure-8 MC Over Arm [End View]64AdlFprz(A) anisotropic E-field (max. = 14.2 V/m),(B) isotropic E-field (max. = 16.2 V/m) and J-field (max=6.7 A/m2),(C) anisotropic J-field (max. =5.9 A/m2),(D) finite element mesh of arm and MC. All E-fields are tangent to this curved surface.Center of arm is at right side of each panel (i.e. z=0 in Fig. 6.10).Figure 6.15 Results for Figure-8 MC Over Arm [Lateral View]65•°(Nr -Y.," / /u_ ialatatt •^Isompa.#41.• Tow, Atiiiiii1111111111111111111111E11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111011111111111111111111111111111111111111.„......*^r-V.-1-/-..,^--f--.L.^......:.":-...,--....-__'-------- \ ...'^1 \ \ \\\,,,,„„. NIMinMIMIillEllIII.....MilMIPP.,^ii:::.1,•.it 1, ilir(A) anisotropic E-field (max. = 14.2 V/m),(B) isotropic E-field (max. = 16.2 V/m) and J-field (max=6.7 A/m2),(C) anisotropic J-field (max. =5.9 A/m2),(D) finite element mesh of arm and MC. View along yz plane. All E-fields and J-fields are parallel to page.Center of arm is at left side of each panel (i.e. z=0 in Fig. 6.10).Figure 6.16 Results for Figure-8 MC Over Arm [Medial View]666.4 DiscussionAs electromagnetic models become more complex the interpretation of data becomes equally com-plex. This situation requires more advanced presentation formats for the results, without which a fullcomprehension of the system characteristics may be missed. For example much more information is pro-vided by panel B in our four isocontour Figures (6.6 to 6.9) than was provided by Roth's presentation whichconsisted of the equivalent our end view, i.e. Fig. 6.7B [54]. Roth fails to stress (for a vertically orientedMC) that the great majority of the eddy currents flow along planes parallel to the outer surface of the arm.In addition, Roth stresses the presence of the current loops tangent to the medial yz-plane (Fig. 6.16)which we found to be negligible in magnitude in our model results. It is evident that the isocontour plotalong one symmetrical side of a model is not sufficient to convey a complete sense of the electromagneticphenomena involved. As a general rule, the visual presentation of results may be one of the weakestpoints throughout the entire literature concerning modelled E-fields for magnetic stimulation.Our FEM models are the first to describe how anisotropic tissue characteristics may influence E-fieldand J-field distributions from MCs as well as their EF and AF functions. A general effect seen in our modelsis that anisotropy reduces the global E-field, J-field and EF function magnitudes by 12% to 14% and the AFfunction by 24% relative to the values modelled in isotropic tissue. In contrast, localized regions within theanisotropic tissue may have higher E-fields (see "f" on page 53). These observations are strictly limitedto our model characteristics and the way we defined the tissue conductivity in the isotropic and anisotropicmodels. Basically we defined the anisotropic tissue as having a greater overall resistance relative to theisotropic tissue. For example it is likely that our observations would alter significantly if the values cr, and0-0 were exchanged, or if the isotropic conductivity was defined as 0.148 S/m rather than 0.417 S/m.Although there is a certain amount of arbitrariness in the tissue characteristics chosen for the isotropicmodel, these results do provide useful information. When a homogeneous tissue is modelled with isotropicconductivity the magnitude and distribution of the magnetically induced E-field is independent of the tissueconductivity. Thus, the tissue conductivity used in our particular isotropic model is inconsequential to ourfinal E-field results. In contrast, our anisotropic tissue results demonstrate that both the E-field and theJ-field vary with anisotropic conductivity. In addition, our comparison of anisotropic and isotropic modelresults provides a rough standard by which we may estimate how realistic previous isotropic model resultsmay be. More importantly the proper modelling of tissue conductivities allows the evaluation of absoluteE-field and J-field values and how they may be affected by various other MC designs and orientations.The isotropic E-field results from Chapter 4 are similar to published reports in that the E-field levelsand distributions did not alter as a function of the modelled tissue conductivity. The only necessaryassumption was that the tissue conductivity was isotropic and several orders of magnitude greater thanthe surrounding air. In our isotropic model results the E-fields and J-fields have isocontours which areidentical in appearance. The E-fields and J-fields in anisotropic tissue differ from each other and fromthe isotropic results and are thus dependent on the tissue conductivity. A few related comments are nowmade concerning anisotropic model results.67The J-field will be directly related to the defined anisotropic conductivities. However the E-field mayhave some level of independence from anisotropic conductivities, as seen in isotropic models. Surfacecharges (on a planar tissue or finite tissue volumes) affect the magnitude and direction of E-fields inisotropic tissue. Similarly, anisotropic conductivity further modifies the distribution of surface charges andmay be the cause of those E-field changes seen in anisotropic tissue. Thus, for example, it may bethat our E-field results are only a function of the ratio of the parallel to transverse tissue conductivities,i.e. E-field may be independent of the absolute tissue conductivities. One possible test, to isolate theeffects of anisotropy from those of surface charges, would be to model a horizontal MC over a planartissue. Regardless of tissue conductivity (anisotropic or isotropic) no surface charges are expected, thuswe might expect the J-fields to differ but the E-field to be identical in both models.Our results suggest that, in general, that the spatial distribution of the E-field is more affected byanisotropic conductivity than is the J-field. For example consider the square MC model. The peak E-and J-field in isotropic tissue is found along the end plane Fig. 6.7B, similarly the anisotropic J-field isfound at the same place. In contrast, the peak anisotropic E-field is found along the medial plane (Fig.6.9A). A second example is the figure-8 MC model. The anisotropic E-field along the lateral exteriorsurface of the arm shows distortions (bulges) in the isocontours (Fig. 6.15A) which are not present inthe anisotropic J-field or in the isotropic tissue results. This relative constancy in the J-field relative tothe E-field distribution will be seen in a more extreme context when inhomogeneous tissue conductivityis investigated in the next chapter.EsseIle and Stuchly have investigated MC designs identical to ours [51]. Their MCs were orientedhorizontally above a planar isotropic tissue volume. The AF function peaks were plotted as a function ofthe vertical distance between the axon and the MC plane. No E-field results were presented. Table 6.5compares the results obtained by EsseIle and Stuchly to those we obtained in the modelled arm.Table 6.5 Review of Modelled AF FunctionsMCDesignTissueVolumeTissueResistancePeak AFFunction(VIcm2)% LI4.8cmSquareMCplanarisotropic0.049 0%arm0.056 + 14%anisoliopic 0.042 - 14%5cmFigure-8MCplanarisotropic0.100 0%UM0.082 - 18%anisotropic 0.063 - 37%0 All results correspond to a peak MC cur-rent derivative of 108 A/s, where thesquare MCs are 1.57cm above the axonand the figure-8 MCs are 0.5cm abovethe axon.El All results for planar tissue volume arefrom EsseIle, 1992 [51], using a horizon-tal MC orientation, and are used as thestandard (0% change).1:1 All arm tissue volume results are fromour model results. The plane of squareMC is tangent to arm, and symmetricallypositioned 1cm above it. The plane offigure-8 MC is tangent to arm, symmet-rically positioned with center bar parallelto axis of arm and 5mm above surfaceof arm. For AF function details see ap-propriate sections in this chapter.68If only the isotropic tissue results are considered in Table 6.5 one can isolate the effect of surfacecharges on the AF function. When a MC is horizontal over a planar tissue no surface charging occurs andthus maximal E-fields will be induced relative to any other type of homogeneous tissue volume, e.g. thearm. However, the AF function is the spatial derivative of the E-field component parallel to the arm axis andthis derivative may be increased by sudden drops in the Ez component. One example of such a situationis the square MC over the arm which induces an AF function 14% greater then in a planar tissue. TheAF function peak in both tissue volumes occurs below the 900 corner of the square MC. This AF functionincrease may be associated to the asymmetry of the E-field isocontours along the end and medial sides,see Fig. 6.7B and 6.9B. No such asymmetry exists along the equivalent planes in a planar tissue volume.In contrast to the AF function increase seen with the square MC over an arm, the figure-8 MC overan arm induces an AF function which is 18% smaller than the that seen in a planar tissue (Table 6.5 ).Although surface charges in the arm are the cause for the increased AF function seen in our square MCmodel, they are also the cause of reduced E-field magnitudes in the same model. The effect of surfacecharging increases as the MC dimensions increase relative to the tissue dimensions. An extreme exampleof this is the fact that MCs do not induce significant E-fields within the cytoplasm of an axon (axoplasm)because of the axon's small diameter (2-20 pm). A related example seen in transformer designs are thethin laminar slabs of magnetic cores used to reduce eddy current losses. Our figure-8 MC is significantlylarger than the square MC; it exceeds the diameter of the arm by a factor of two (compare Fig. 6.10 and6.2). We suggest that the decrease of the AF function (for our figure-8 MC), relative to that found in aplanar tissue, is caused by a significant decrease in the E-field level within the arm. Finally, when the armtissue is modelled with anisotropic conductivity the AF function undergoes a further decrease relative tothe values modelled in isotropic tissue.69Chapter 7Modelling Inhomogeneous Tissue in the Brain7.1 IntroductionStimulation of the cerebral cortex and/or brain is an established diagnostic procedure for nervousdysfunctions. Magnetic stimulation has contributed to the clinical ease by which this procedure can beadministered. Unlike electrical stimulation which forces large radial currents through the sensitive scalp(skin), magnetic stimulation induces E-fields more evenly throughout the scalp, skull and brain (cortex)thus minimizing patient discomfort.Only one magnetic stimulation study has been published investigating the effect of conductivityinterfaces within modelled tissue. Roth modelled the head as an inhomogeneous spherical conductor[50]. His study is also one of few which have modeled the induced E-fields associated with magneticcortical stimulation. Roth modelled the scalp and skull as two concentric shells (layers) and the brain as aconcentric sphere at their center. The three tissues were modelled with appropriate isotropic conductivities.Roth states "Charge accumulation not only occurs on the head surface but also at the interface betweenany two tissues with different conductivities". Although this statement is generally valid for variously shapedconductor volumes, the truth of this statement is not evident when dealing with a case involving such ahigh level of spherical symmetry.It is an intrinsic property for magnetically induced E-fields to be everywhere parallel to the surfaceof a homogeneous spherical tissue volume, and to be zero at its center, regardless of MC design or MCorientation [55, 77]. This property forced their modelled induced E-fields to be parallel (at all points) to allthree of the concentric conductivity interfaces: air/scalp, scalp/skull, skull/brain. Simply put, no inducedE-fields or J-fields had a component normal to any of the modelled interfaces. Thus, it is possible that thecharges present at the air/scalp interface account for the parallel orientation of the E-field throughout thespherical conductor and that no surface charges are present along the two internal conductor interfaces.Regardless of the actual surface charge distribution, their E-field results are equivalent to the more trivialcase of the E-field distribution within a homogeneous spherical tissue.Unlike stimulation of peripheral nerves, cortical stimulation (both magnetic and electrical) involvesmany distinct and independent factors which make the interpretation of clinical data particularly difficultand speculative. The following is a partial list of these unique cortical factors and their differences fromperipheral stimulation, e.g. median nerve at the wrist:1. Pyramidal (nerve cell) fibers evenly cover the entire upper surface of the brain rather than being restrictedto a well defined and isolated bundle of fibers. Thus it is not clear which pyramidal cells are stimulated.2. Stimulation of pyramidal cells (the direct pathway between brain and muscle) can be either direct ormediated by any number of secondary cortical and subcortical fibers by way of synapses which add asignificant propagation delay.703. Pyramidal fibers terminate within the cortex where the cell soma is located. This consideration com-plicates the analysis of AP initiation due to end-effects and active influences of discrete functional cellcomponents located near the cell soma.4. Pyramidal fibers follow sharp bends in the cortical and subcortical regions. This bending can greatlyaffect the effectiveness of a stimulus E-field.5. The geometric structure of brain tissue is not similar to concentric tissue layers but rather form aninhomogeneous conducting volume containing a high degree of geometric orderliness combined witha high level of complexity. These tissue types form oddly shaped volumes which are characterized byvarious conductivities. These factors will generate many charged surfaces which may greatly affect thedistribution of E-fields throughout the brain.6. On a microscopic level the high degree of geometric orderliness found within any individual brain tissueaccounts for its anisotropic conductivity. This factor will uncouple the simple vector relationship betweenthe J-field and the E-field and thus add a further level of complexity and may influence the clinicaleffectiveness of any MC design or MC orientation.Our FEM model and its analysis addresses several of the latter concerns for the first time: anisotropicconductivity, inhomogeneous conductivity, realistic tissue shapes and sharp bends in pyramidal fibers. Wepresent a FEM model of the precentral cortical gyrus which accounts for many tissue properties which havebeen experimentally measured over the last few decades. In the next Sections we will discuss, in somedetail, our assumptions and simplifications which allowed the practical use of the FEM method of analysis.7.2 The Central Nervous SystemThe nervous system consists of neurons arranged into a highly integrated central nervous system(CNS) consisting of the brain and spinal cord and a peripheral nervous system (PNS) consisting of bundlesof neurons in the peripheral body parts. Cross-sectional views of the CNS show that it is composed ofwell defined gray and white regions. Gray matter owes its color to a high concentration of cell bodies,dendrites and unmyelinated axons. White matter owes its color to the fatty structures associated withbundles or tracts of myelinated axons. Many of the cell bodies in the gray matter and axons in the whitematter are parts of the same nerve cells as they traverse through one to the other.The CNS is enveloped by fibrous meninges (membranes). The outermost and thickest of thesecoverings is the dura matter. Further below are the arachnoid matter and the very thin pia matter whichlies directly on the CNS. About 100 ml of cerebral spinal fluid (CSF) flows through the subarachnoid spacenext to the pia matter and within the ventricles of the brain (Fig. 7.1). This fluid is extracellular and thusits effective conductivity is relatively high compared to tissues with high cell densities.The brain can be divided into the cerebral hemispheres, diencephalon, medulla, midbrain andhindbrain. The latter four areas are associated with lower body functions: integration of emotionsand sensory experience, reflexes, alertness, wakefulness, equilibrium, muscle tone etc. The cerebralhemispheres (cerebrum) is the focus of this discussion and our FEM modelling. The cerebrum is71CENTRAL SULCUSPOSTCENTRAL GYRUS^ PRECENTRAL GYRUS„e 111111\Pe"fiCentral Sulcus /•••••••'MEW'1••••/••••MEOWEEL•••EMI•••/a.•Cortex —■SubcortexGREY MATTER (CORTEX)GREY MATTER (BASAL  NUCI RI)WHITE MATTERPremotor Areafill^Motor Area(Precentral Gyrus)I • III^IISomesthetic Area(Postcentral gyrus)fr./.4,114 do •4.41.4Er doe VENTRICTcharacterized by an outer cortex of gray matter and subcortical areas containing masses of underlyingtracts (white matter) and nuclei (gray matter) (Fig. 7.1A). The cortex is characterized by fissures (deepgroves), gyri (hills), and sulci (furrows) and is functionally divided into lobes. The cortex is the most highlyevolved area of the brain, and varies from 2mm to 6mm in thickness. A large percentage of it is tuckedaway within the depths of the sulci. Just below the cortex is a large mass of white matter with numeroustracts. Further below are discrete masses of gray matter (basal nuclei) that subserve motor areas of thecortex. Ventricles containing CSF are generally located close to these basal nuclei.(A) View of brain along medial plane.(B) Lateral view of external brain surface.Figure 7.1 The Brain72The movements produced by cortical stimulation are neverskilled (complex) acquired movements but rather consist ofeither flexion or extension of one or, more commonly, severaljoints. Cortical mapping based on electrical stimulation andclinical/pathological data have been the major methods ofdetermining and localizing the functions of the cortex. Therelative size of the motor cortex associated with the face, arm,and leg varies considerably. The muscles which control thedigits are largely represented beneath the surface, within theconvolution of the central sulcus [97].7.2.1 The Motor CortexThe cerebrum can be divided into four lobes: frontal, parietal, temporal and occipital (Fig. 7.1B). Allparts of the cortex are concerned with memory and information exchange with other cortical areas andsubcortical areas. The frontal lobe is the likely area of stimulation during magnetic cortical stimulationstudies. This lobe is concerned with intellectual functions, aggressive behavior, speech, language andmore pertinently the initiation of skilled (voluntary) movement.Electric stimulation of almost any point on the entire surface of the cerebral hemispheres can provokesome type of movement. Low threshold, reproducible, and focused stimulation of skeletal muscles canonly be elicited from the posterior half of the frontal lobe and is most marked in the primary motor cortex(PMC). This narrow cortical region, just anterior to the central sulcus, is mostly comprised of the precentralgyrus from which impulses are conducted down to the corticospinal tract directly to motor neurons of thecranial/spinal nerves.Since the 1870s it has been repeatedly shown that electrical stimulation of the cortex producesmovements of skeletal muscles. In man, the primary representation of movement is located upon theprecentral gyrus with subordinate motor representation in the post central gyrus (Fig. 7.1B). Considerablevariation exists in the fissural pattern within a species, and in the two hemispheres of the same individual.The location of the point from which any given movement can be elicited varies between individuals andbetween hemispheres. The response of a brain segment, within an individual, can remain constant forlong periods of time but can be altered by stimulation of adjacent cortex [67].Figure 7.2 Coronal Section of Cerebral Hemisphere and Motor Cortex Sequence Map73The cerebral cortex consists of six layers:1. Molecular layer: consists of delicate nerve fibers andis an important synaptic field.2. Outer granular layer: contains small pyramidal andstellate cells with many dendrites extending intolayer 1.3. Pyramidal cell layer: contains medium and largepyramidal cells. Apical dendrites extend into layer1 and their axons enter the white matter becomingprojection, association or commissural fibers.4. Inner granular layer: contains closely arranged stel-late cells with complex connections.5. Ganglionic layer: contains large and giant (Betz)pyramidal cells with axons extending into the whitematter as in layer 3. This layer is only found withinthe precentral gyrus of the motor cortex.6. Fusiform cell layer: contains fusiform and other celltypes.Penfield was among the first to clearly map out in detail the motor responses elicited by direct electrodestimulation on the surface of the human brain, (Fig. 7.2) [67]. Motor responses are elicited primarily fromthe cortex adjacent to the central sulcus (fissure of Rolando), 80% from the precentral gyrus and 20%from the postcentral gyrus. On occasion movement was elicited in the anterior portion of the precentralgyrus or the posterior portion of the postcentral gyrus. Most direct stimulation of cortex is done on theouter superficial surface of the gyri. Stimulation deep within the cortical sulcus (along the banks of thecentral sulcus) elicit movements similar to surface stimulation and were located at the proper site in themotor cortex sequence map. Cortical stimulation suggests that the hp of the precentral gyrus adjacent tothe central sulcus constitutes the most important cortical representation of movement of an extremity.Recordings from microelectrodes have shown that the cortex is functionally organized into minutevertical units (traversing the cortex) which include nerve cells from all its six layers (Fig. 7.3) [98]. Eachcortical layer can be distinguished on the basis of their constituent neuronal cell bodies and nerve fibers.All neurons in the unit are activated by the same stimulus. Electrical stimulation of the motor cortex elicitscontraction of muscles predominantly on the opposite (contralateral) side of the body.Early studies of specific tissue resistivities treated cortex as a homogeneous structure. For exampleFreygang used a pulsed current and measured the conductivity of cat cortex as 0.45 S/m [99]. Otherstudies [100] measured cortical conductivity of the superficial layers but treated them as a single isotropicFigure 7.3 Cytoarchitecture of Cerebral Cortex74Table 7.6 Cortical ConductivitiesCorticalLayeroxSlmcf. YSlmClzSlm1 — — 1.72 0.29 0.49 0353 0.29 0.49 0354 0.41 0.29 0.505 — — —6 036 035 035The anisotropic, specific conductivity of catsomatosensory cortex was measured at 10 Hz,[101]. The six cortical layers are illustrated inFig.7.3.The physiologic and anatomic distinction betweenthe primary motor cortex and the primarysensory cortex of the cat is not as clear as inthe primate and is thus commonly referred toas "sensorimotor" cortex. Cat and primate PMCare cytoarchitectonically very similar [97] andanisotropy is directly relate to this cytoarchitecture.homogeneous medium and obtained similar results, i.e. 0.43 S/m at 5kHz. In 1979 Hoeltzell measured theanisotropic conductivity of cat somatosensory cortex in most of the six cortical layers and their results aregiven in Table 7.6 [101]. Their coordinate system is given in Fig. 7.4. In summary, Table 7.6 indicates: thatlayers one and five were difficult to measure, that the cortex is generally most conductive in the directionnormal to its surface (o-z), and that conductivities along the two remaining axes is more variable.The orientation of the gyrus is idealized such thatits central axis (x) is mediolateral, the anteroposteriordirection is along the y axis, and the vertical directionis along the z axis. It is clear that this rectangularcoordinate system is not fully adequate to describe theconvoluted gyrus region where the surface normal canpoint in any direction. Thus for the cortex, az is definedas the conductivity in the direction normal to the gyrussurface, ax is the conductivity parallel to the corticalsulcus, and ay is the conductivity orthogonal to thesetwo the xz-plane. Note that the x-axis corresponds tothe various locations along the cortical map (Fig. 7.2).Figure 7.4 Coordinate System for Cortical Conductivity75Association TractsCorona Radiata^■View along medial plane showing associationand projection fibers in white matter.7.2.2 The SubcortexThe output tracts (efferent fibers) of the motor cortex are the axons of the pyramidal and fusiformcells, (from cortical layers 3, 5 and 6) which descend through the cortical layers and enter the underlyingwhite matter. These axons have various destinations: as projection fibers (e.g. corona radiata) theygo to the basal nuclei, cerebellum or spinal cord; as association fibers they connect the anterior andposterior cortical regions within the same hemisphere; and as commissural fibers (e.g. corpus callosum)they connect the left and right cerebral hemispheres (Fig. 7.5).The pyramidal tract (corticospinal tract) begins at cell bodies within the cortex of the precentral gyrus.Their long axons form part of the corona radiata and pass through the entire length of the spinal cordwithout interruption, i.e. no synapses. This pathway is responsible for generating voluntary, skilled anddiscrete skeletal movements. The basal nuclei largely introduce facilitating and inhibitory influences onother descending motor tracts from the frontal cortex except those of the precentral gyrus (pyramidal tract).As with the cortex, initial studies of white matter did not account for anisotropy and average conductivityvalues were measured, for example Freygang measured a conductance of 0.29 S/m [99]. The subcorticalarea contains many long fiber tracts in various directions throughout the white matter and make itstructurally less homogeneous than the cortex. In contrast, within a given tract, the fiber orientationcan be very homogeneous resulting in marked anisotropic characteristics. Nicholson studied cat internalcapsule (white matter) at 2kHz and measured' the conductance transverse to the fibers as ol-= 0.129 S/mand the conductance parallel to the fibers as ap=1.27 S/M [102].1 It is of interest to note that Nicholson also found a similar admittance ratio (10:1) for the anisotropic reactance(capacitive) of white matter.Figure 7.5 Subcortical Fiber Tracts767.3 FEM ModelOur FEM models rely on three basic simplifications (among many others). These were accepted inorder to minimize the number of nodes and allow FEM modelling without excessive disk space and CPUtime requirements:1.The head was modelled as a planar semi-infinite tissue rather than a sphere. This model simplificationis not expected to greatly affect the characteristics of the E-field and J-field distributions which are ofparticular interest to this study. Our figure-8 MC design is relatively small compared to a human head,consequently if a spherical model were to be used the E-field reductions due to related surface chargeswould be minimal [50]. Thus, it is expected that our E-field magnitudes will be only slightly greater thanfor an equivalent spherical model. Regardless of which brain shape is modelled (planar or spherical)the E-fields in the brain are expected to be everywhere parallel to the air/tissue interface, assuming thebrain was modelled as a homogeneous tissue. In addition, the anteroposterior width of the precentralgyrus is relatively small compared to the curvature of the brain. Thus, relative to a sulcus the entirebrain may be likened to a planar volume.2. The second simplification is that only one gyrus is modelled. Consequently the narrow gap of CSFsurrounding a real gyrus is modelled by a large expanse of CSF. The cortical gyrus has never beenmodelled in previous magnetic stimulation studies and it is expected that our results will be a usefulfirst approximation of the actual field conditions present in the gyrus. It is commonly assumed thatnerve excitation is initiated within the gray and/or white matter of the precentral gyrus during clinicalbrain stimulation.3. The third simplification is that the six layered cortex, each with their unique anisotropic properties, aremodelled as a single 4mm homogeneous but anisotropic layer.These three simplifications were highlighted in a rather arbitrary fashion solely because of their relativeimportance in the context of previous studies and the particular novelty of the modelling geometryintroduced in our investigation. The brain is highly complex and many additional types of simplificationsare required for modelling purposes. Our other simplifications by and large are not unique when comparedto previous magnetic brain stimulation studies and are stated by implication in our FEM model descriptionbelow.The modelled MC is a figure-8 design using line elements as depicted in Fig. 7.6. The MC ishorizontally oriented above the cortex such that the center bar of the MC is positioned in an anteroposteriororientation similar to that used in clinical practice (Fig. 7.8). Our FEM model uses symmetry conditionsalong three orthogonal planes thus only a quadrant section of the MC and gyrus are modelled.Fig. 7.7 depicts views along orthogonal planes of our FEM model using the coordinate system shownin Fig. 7.4. The plane of the MC is 12 cm above the highest point of the gyrus (gray matter). The CSFcovers the cortex and is 4 mm deep at its shallowest point directly above the crest of the gyrus. Thecranium and scalp are approximately 3.5 and 2.5 mm thick respectively [50] and are lumped together77center bar of MC10cm•111Figure 7.6 Figure-8 MC for Cortical Stimulationwith 2mm of air to give a total of 8 mm of nonconducting (air) media between the MC and the planarsurface of the CSF.The cortex varies from 1.5 to 6 mm in thickness [103, 104] and is thicker over the crest of a convolutionthan in the depths of a sulcus. We modelled the cortex as having a uniform, intermediate thickness of 4mm and only extended its coverage to 4 cm from the edge of the gyrus. The gyrus dimensions are thoseof the human precentral gyrus and were measured from scaled, sectional photos [105], with a gyrus depthof 2.4 cm and an anteroposterior width of 2.0 cm. A mediolateral gyrus length of 7 cm was arbitrarilychosen. White matter underlies the cortex and the surrounding CSF. Structural elements such as thelateral ventricles and the basal bodies are not included since they are relatively distant from both the MCand the precentral sulcus.785.0 2.5BCoil04■40-110-10041■10AIR.4^CEREBRO SPINAL FLUID 0.4^CEREBRO SPINAL FLUIDCORTEX °A01.5(A) corona! view,2.1 (B) medial view, analogous viewgiven in Fig. 7.1. A pyramidalfiber path is shown.(C) top view, seen from vertex ofhead.Lengths in cm. Also see Fig. 7.8.2.5^1.5^0.4 0.6.4^COILFigure 7.7 Views of Modelled Precentral Gyrus and MC797.3.1 Tissue ConductivitiesAlthough the scalp is a relatively good conductor (0.45 S/m) [106, 94] and the cranium is nonconducting(0.0056 S/m) [106] we have modelled both as a planar layer of air above the CSF. As discussed previouslywe do not expect this simplification to significantly affect our results relative to a concentric shell model.The conductivity of human CSF is constant between lkHz and 30 kHz and measurements vary from 1.56S/m [106] to 1.96 S/m [102]. In order to investigate the effects of inhomogeneity and anisotropy our FEMbrain model was analyzed using three sets of conductivity parameters:1. Homogeneous model:O CSF ---). cr =1.96 S/mO cortex --+ c =1.96 S/mO white matter —* cr =1.96 S/mAll tissues in the homogeneous model are defined with the conductivity of CSF. This reducesthe model to that of a horizontal MC over a planar tissue volume where no interface charging isexpected and is thus similar to previously published studies.2. Isotropic model:O CSF --+ c =1.96 S/mO cortex —*. c =0.50 S/mO white matter -- a =1.27 S/mThe conductivities assigned to white and gray matter in the isotropic model are somewhat arbitrarysince these tissues are in fact anisotropic. It was decided that the isotropic conductivity shouldbe determined by that component of the anisotropic conductivity which is normal to the tissueinterfaces (cz). This choice may minimize the differences in tissue interface charging and allowsome comparative analysis between our isotropic and anisotropic model results.3. Anisotropic model:O CSF --- c =1.96 S/mO cortex --* ox =0.36 S/m, cy =0.35 S/m, az =0.50 S/mO white matter —> cx =0.129 S/m, 0-y =1.27 S/m, az =127 S/mA study by Nicholson provided the conductivity parameters (Table 7.6) used in this anisotropicmodel [102]. Our anisotropic model lumps all six cortical layers into a single 4mm homogeneousbut anisotropic layer. The cortical conductivity along each axis is the median (intermediate) valuelisted in Table 7.6.Anisotropy within white matter is defined as being either parallel or transverse to the dominantaxon fibers. Within a gyrus the dominant orientation of the fibers is assumed to be along thevertical z-axis (Fig. 7.7) as the fibers make their way down the gyrus towards the main bulk of thewhite matter. A second dominant fiber orientation is normal to the cortical surface as the fiberspass through the cortex/white-matter interface. Thus both ay and az are modelled with equal80parallel conductivities (op). It is assumed that much fewer fibers follow a mediolateral orientationand thus erx is modelled with a transversal conductivity (o-p).The intermingling and ubiquitous extent of the projection and association fiber tracts (Fig. 7.5)make it difficult to model the effective anisotropic conditions present at any point within the whitematter. As a first approximation the modelling of the white matter within the gyrus is givenprecedence over the lower portions of the white matter (the main bulk). Thus in order to avoidconductivity discontinuities the same anisotropic parameters were used throughout the modelledsubcortical white matter.7.3.2 Isocontour PlotsAlthough E-fields are the direct cause of nerve stimulation we also present J-field results to aid abetter understanding of the field characteristics within inhomogeneously conducting media (head). Notethat previous studies had no cause for presenting both these field distributions since they were identicalin appearance. Our E-field results for magnetic brain stimulation are the first to seriously considercomplex tissue shapes and the associated inhomogeneous conductivity. The resulting E-field and J-field magnitudes have similarly complex distributions and are presented along several viewing planes (Fig.7.8). The peak E-field and J-field results correspond to a peak MC current derivative of 108A/s throughthe center bar of the figure-8 MC. The isocontour plots consist of 10% increments in the E-field and J-fieldmagnitudes denoted by the gray scale previously described in Fig. 6.4. Recall that in any isocontourfigure the large white region represents the 0% to 10% isocontour while the small black region representsthe 90% to 100% isocontour and that for simplicity the isocontour regions are referred to by their uppermagnitude limit, e.g. the 0% to 10% isocontour is referred to as the 10% isocontour. The FEM boundaryconstraints were defined such that the E-field is normal along the coronal surface (xz-plane) and tangentialalong the medial surface (yz-plane). Fig. 7.9 shows the FEM mesh of all the FEM model components:the cortex (in black), the underlying white matter, the cerebral spinal fluid and the overlying layer of air(shaded in grey). Our FEM results are presented in three Sections: homogeneous (7.4), isotropic (7.5)and anisotropic (7.6) model results. The E-fields and J-fields are presented within the entire brain usingFig. 7.9 and more detailed results are given within the precentral gyrus using the views in Fig. 7.8. Afourth section presents the EF and AF functions along the nerve path described in Fig. 7.7B for each ofthe three modelled media. A discussion section then concludes this chapter.811,11p071001119 ::11;- simmlimmagA) an angled view, B) a anterior view, C) a posterior view and D) a medial view.The first four layers of finite elements at the crest of the gyms mark the cortical layer which covers theunderlying white-matter.Figure 7.8 Four Views of the FEM Mesh for Precentral Gyrus and MC82zY ■ tif x Angled view of meshed quadrant of modelled brain. Cortexis shaded in black, white matter is below cortex, cerebralspinal fluid is above cortex and planar layer of air is shadedin gray. The quadrant model of the figure-8 MC is shownas a thick black line. The FE mesh of the cortical layer isshown in Fig. 7.8.Figure 7.9 FEM Mesh for Head837.4 Homogeneous Model ResultsWhen brain tissues (CSF, cortex, white matter) are modelled as one homogeneous isotropic media theE-field and J-field isocontours are identical in appearance, (Figure 7.10). Since the head is modelled asa planar conducting volume all E-fields and J-fields are parallel to the air/CSF interface. The E-field peak(23 V/m) is located at the CSF surface directly below the center bar of the figure-8 MC. The correspondingJ-field peak is 46 A/m2.Figures 7.12 and 7.11 show various other views of the cortical gyrus and their corresponding E-fieldand J-field isocontours. The CSF is not shown. The peak fields (17.7 V/m, 35 A/m2) are located in thecortex, again directly below the center bar of the figure-8 MC and no discontinuities in the isocontoursare present. By model symmetry, the fields are tangent to the figure plane in Fig. 7.12B and normal tothe figure plane in Fig. 7.11A.Head modelled as homogeneous isotropic tissue volume. The E-field and J-field isocontours arein 10% increments with E-field max. = 23 V/m and J-field max. = 46 A/m2.Figure 7.10 Angled View of Field Magnitudes in Head840 (A) anterior view and (B) posterior view of gyrus.0 Isocontours are in 10% increments where E-field max. = 17.7 V/m and J-field max. = 35 A/m2. Head ismodelled as homogeneous isotropic tissue volume. Fields are normal to figure plane in panel A.Figure 7.11 Anterior and Posterior Views of Field Magnitudes in Precentral Gyrus850(A) angled view and (B) medial view of gyrus.0 lsocontours are in 10% increments where E-field max. = 17.7 V/m and J-field max. = 35 A/m2. Head is modelled as homogeneous isotropictissue volume. Fields are tangential to figure plane in panel B.Figure 7.12 Angled and Medial Views of Field Magnitudes in Precentral Gyrus7.5 Isotropic Model ResultsThe three brain tissues (CSF, cortex and white matter) are each modelled with their characteristicisotropic conductivities. The resulting inhomogeneous planar tissue (brain) will consequently have chargedsurfaces at the conductivity interfaces and it can no longer be assumed that all induced E- and J-fields willbe parallel to the air/CSF interface. In addition the E-field and J-field isocontours are no longer identicalin appearance and we present each of them side by side.The peak E-field in the CSF increased from 23 V/m in the homogeneous model to 28 V/m, a 22%increase, (Fig. 7.13A). The J-field peak also increased by 22% to 56.9 A/m2, (Fig. 7.13B). Theseincreases may be associated with a bottleneck effect where induced currents are channeled into thenarrow conducting (CSF) region above the less conductive cortical gyrus. As a relative insulator it isexpected that the surfaces of the cortex become charged and thus nearby E-fields in the CSF woulddeviate from their usual planar (x-y) orientation. Our model confirms this, for example in the CSF near thecurvature (lip) of the gyrus (arrow-1 in Fig. 7.15B), J-fields have a significant component which is verticallyoriented (z-axis) reaching a peak level of 8.9 A/m2 (results not shown). This vertical current componentis inferred by the absence of large J-fields within the gyrus and a large J-field gradient in the CSF abovethe gyrus in Fig. 7.13B. Note also how the cortex and white matter within the gyrus are insulated fromthe higher J-field levels, as compared to the homogeneous model.E-field isocontours are greatly affected by the lower conductivity of the cortex. Large E-field gradientsare present across the cortical layer along the medial plane (Figures 7.13A and 7.15A). This is in contrastto the relatively continuous J-field isocontours which cross unimpeded through the lower portions of thecortex, (Figures 7.13B and 7.15B). This difference between the E-field and J-field isocontours is a directreflection of the current/voltage relationship (IR=V) combined with the different conductivities in each ofthe three tissues and the surface charges generated at their interfaces.Figures 7.14 to 7.17 show angled, medial, anterior and posterior views respectively of the corticalgyrus and their corresponding E-field and J-field isocontours. The peak E-field in the gyrus is located inthe upper cortex directly below the center bar of the MC and increased from 17.7 V/m in the homogeneousmodel to 25.6 V/m, a 45% increase. The peak J-field is not located in the cortex but rather in the betterconducting white-matter closest to the MO's center bar (arrow-1 in Fig. 7.14B). The relative discontinuity ofthe E-field compared to the J-field is once again apparent at the cortex/white-matter interface, for examplecompare Figures 7.15 A and B. The E-field is attenuated by up to 50% as it passes from the cortex tothe white-matter. As in the homogeneous model, the fields are tangential to the page in Fig. 7.15 andnormal to the page in Fig. 7.16.87A0(A) E-field (max= 28 V/m) and (B) J-field (max. = 56.9 A/m2).0 Isocontours are in 10% increments. Gyrus modelled as inhomogeneous isotropic tissue volume.Figure 7.13 Angled View of Field Magnitudes in Head0 (A) E-field (max. = 25.6 V/m) and (B) J-field (max. = 18.1 A/m2) in gyrus.0 Isocontours are in 10% increments. Gyrus modelled as inhomogeneous isotropic tissue volume.Figure 7.14 Angled View of Field Magnitudes in Precentral Gyrus0 Panel (A) E-field (max. = 25.6 V/m) and panel (B) J-field (max. = 18.1 A/m2).o Isocontours are in 10% increments. Gyms modelled as inhomogeneous isotropic tissue volume. Fields are tangential to figure plane.Figure 7.15 Medial View of Field Magnitudes in Precentral GyrusA8011030(A) E-field (max. = 25.6 V/m) and (B) J-field (max. = 18.1 A/m2) in gyrus.0 lsocontours are in 10% increments. Gyrus modelled as inhomogeneous isotropic tissue volume.Figure 7.17 Posterior View of Field Magnitudes in Precentral Gyrus910(A) E-field (max. 25.6 V/m) and (B) J-field (max. 18.1 A/m2) in gyrus.0 lsocontours are in 10% increments. Gyrus modelled as inhomogeneous isotropic tissue volume.Fields are normal to figure plane.Figure 7.16 Anterior View of Field Magnitudes in Precentral Gyrus927.6 Anisotropic Model ResultsThe three brain tissues are now modelled with their characteristic anisotropic conductivities. Theresulting inhomogeneous anisotropic planar tissue will thus contain charged surfaces and the orientationof the E-field and J-field are no longer identical within the cortex and white matter. Despite the anisotropicconductivity characteristics, there is little change in the E-field and J-fields within the gyrus as comparedto our isotropic model results.The peak E-field in the CSF increased from 28 V/m in the isotropic model to 28.7 V/m, a 3% increase,(Fig. 7.18A). The J-field peak correspondingly also increased by 3% to 57.3 A/m2, (Fig. 7.18B). The samebottleneck effect seems to be present in both inhomogeneous models where current flow is redirectedtowards the narrow CSF gap just above the gyrus. And as with the isotropic model, the E-field isocontoursare discontinuous (high field gradient) at the CSF/cortex/white-matter interfaces, (Fig. 7.18A).Figures 7.19, 7.20, 7.21 and 7.22 show angled, medial, anterior and posterior views respectivelyof the anisotropic gyrus and their corresponding E-field and J-field isocontours. The peak E-field in thegyrus is located in the upper cortex and increased from 25.6 V/m in the isotropic model to 26.1 V/m, a2% increase. Like the isotropic model, the peak J-field is located in the white-matter closest to the MC,(arrow-1 in Fig. 7.20B).The relative discontinuity of the E-field compared to the J-field is once again apparent at thecortex/white-matter interface, for example compare Figures 720A and B. The E-field is attenuated byup to 50% as it passes from the cortex to the white matter. As in the homogeneous model, the fields aretangential to the page in Fig. 7.20 and normal to the page in Fig. 7.21.One of the few clear differences between the isotropic and anisotropic model results is shown inpanel B of Figures 7.14 and 7.19. In the isotropic model the J-field at the top of the gyrus, within thecortex is 70% of the maximum seen in the CSF while in the anisotropic model the J-field at the samelocation is only 50% of the maximum J-field. This reduction in the J-field for the anisotropic model isexplained by a localized effective decrease in the conductivity at the lip of the cortex. In the isotropicmodel cortex has an conductance of 0.50 S/m; however, in the anisotropic model the effective corticalconductivity at the lip of the gyrus is somewhere between the parallel conductivity ap=.50 S/m and thetransverse conductivity 01—.35 S/m.930 (A) E-field (max. = 28.7 V/m) and (B) J-field (max. = 57.3 A/m2) in gyrus.0 lsocontours are in 10% increments. Gyrus modelled as inhomogeneous anisotropic tissue volume.Figure 7.18 Angled View of Field Magnitudes in Head0(A) E-field (max. = 26.1 V/m) and (B) J-field (max. = 18.6 A/m2) in gyrus.0 Isocontours are in 10% increments. Gyrus modelled as inhomogeneous anisotropic tissue volume.Figure 7.19 Angled View of Field Magnitudes in Precentral GyrusA0(A) E-field (max. = 26.1 V/m) and (B) J-field (max. = 18.6 A/m2) in gyrus.0 lsocontours are in 10% increments. Gyrus modelled as inhomogeneous anisotropic tissue volume. Fields are tangential to figure plane.Figure 7.20 Medial View of Field Magnitudes in Precentral GyrusBA..^.. .......0 (A) E-field (max= 26.1 V/m) and (B) J-field (max. = 18.6 A/m2) in gyrus.0 lsocontours are in 10% increments. Cyrus modelled as inhomogeneous anisotropic tissue volume.Fields are normal to figure plane.Figure 7.21 Anterior View of Field Magnitudes in Precentral Gyrus970 (A) E-field (max= 26.1 V/m) and (B) J-field (max. = 18.6 A/m2) in gyrus.0 lsocontours are in 10% increments. Note panel B only shows 0% to 60% isocontours. Gyrus modelledas inhomogeneous anisotropic tissue volume.Figure 7.22 Posterior View of Field Magnitudes in Precentral Gyrus987.7 Stimulus Fields for a Pyramidal FiberWe investigated the stimulus fields along a typical pyramidal motor neuron path traversing a corticalgyrus. The particular case of the axonal path is described in Fig. 7.7B. This pyramidal cell lies on themedial yz-plane, the proximal portion of the axon (terminated with the cell soma) goes through the cortexand white matter of the gyrus in a radial direction (45° to horizontal) till it reaches the center of the gyrus;at this point the axon bends such that the remaining distal portion of the axon continues vertically alongthe z-axis, exiting the gyrus and continuing further downward into the main bulk of the white matter belowthe gyrus.Figure 7.23 shows the E-field and stimulus function results for all three FEM models (homogeneous,isotropic and anisotropic):El panel (Al) the E-field magnitude along the axon,CI panel (B1) the component of the E-field magnitude which is tangent to the axon, i.e. the EF function,0 panel (A2) the spatial derivative of the E-field magnitude,Opanel (B2) the spatial derivative of the component of the E-field magnitude which is tangent to the axon,i.e. the AF function.The (cortical) AF functions in panel B2 exhibit several differences from those associated with peripheralnerve stimulation (Fig. 6.5 and 6.12)1.The magnitude, span2 and location of the EF and AF functions are determined more by tissue conduc-tivity interfaces and the nerve path than by the MC design.2. The AF function is essentially monopolar and non-symmetric with only a minor negative componentnear the cortical surface3. Two positive peaks occur. The first associated with the cortex/white matter interface (1.5 V/cm2), thesecond associated with the fiber curvature (0.2 V/cm2).4. The larger AF function is more than an order of magnitude greater, i.e. 1.5 V/cm2 versus 0.082V/cm2.5. The AF function has a spatial distribution one order of magnitude smaller. The major AF function peakonly spans an axon length of 2.5 mm, whereas for peripheral stimulation the AF function spans nearly50mm.By comparing panels Al and B1 it becomes apparent that the E-field is not parallel to the axon. Ingeneral, the induced E-fields are oriented along planes parallel to the MC plane, in contrast no portion ofthe pyramidal fibre is oriented parallel to this plane. The effect of this divergence in orientation betweenthe E-field and the axon is most apparent at distances greater than 1cm from the cell soma where theaxon is normal to the MC plane. In particular, the EF function drops from about 7.8 V/m to 0 V/m at adistance of about 1.1 cm from the cell soma, (Fig. 7.23B1). Furthermore, this effect of the axon bendwithin the white matter of the gyrus is apparent in all three models. This rapid drop in the EF function2^We define the "span" of the AF function as the axon length which is exposed to stimulus fields from 50% to 100%of the peak AF function value. See Fig. 8.1A for more details.991Distance from Cell Soma (cm)1Distance from Cell Soma (cm)(++) homogeneous model, (—) isotropic model and (oo) anisotropic model results(Al)^(++) max. = 15.4 V/m, (—) max. = 25.0 V/m, (oo) max. = 24.4 V/m(B1) (++) max. = 9.6 V/m, (—) max. = 20.0 V/m, (oo) max. = 20.9 V/m(A2)^(++) max. = 0.1 V/cm2, (—) max. - 1.2 V/cm2, (oo) max. = 1.3 V/cm2(B2) (++) max. = 0.2 V/cm2, (—) max. = 1.4 V/cm2, (oo) max. = 1.5 V/cm2See text for details.Figure 7.23 E -fields Along a Typical Pyramidal Fiberis mirrored in panel B2 where the AF function shows a secondary peak of about 0.2 V/cm2 in the samelocation. These effects are present regardless of how the tissue is modelled and thus they seem to belargely a function of the axon bend.Smaller but significant differences between results in panels Al and B1 are also seen within thecortex and white matter at distances less than 1cm from the cell soma. Along this proximal portion of thepyramidal cell, the EF function is 20% to 35% lower than the E-field magnitude (panel Al). This differenceis largely due to the angle between the pyramidal axon and the induced E-fields.The FEM results for the isotropic and anisotropic models are nearly identical in all four panels of Fig.7.23. This suggests that the anisotropic characteristics are not critical in determining the field distributionalong the axon. Conversely, the inhomogeneous tissue characteristic is directly associated with some of100the stimulus functions. For example, panels Al and B1 show a discontinuity at the cortex/white matterinterface (0.4 cm from the cell soma) where the E-field drops by 40% to 50%. This large drop in the E-field,at the tissue interface, is mirrored in panels A2 and B2 by a large spike in the E-field spatial derivative. Itis not expected that significant AF functions are present at any of the five cortical tissue interfaces sincethere are only minor variations in cortical conductivity (Table 7.6).Large differences are seen in all four panels when comparing the fields within the homogeneous modelto those in either of the inhomogeneous (isotropic and anisotropic) models. In the homogeneous modelthe E-field magnitude and the EF function increases smoothly across the cortex/white-matter interface,(see panels Al and B1). In contrast the inhomogeneous models show a sharp discontinuity 4mm fromthe cell soma. The inhomogeneous model results show large spikes in panels A2 and B2 associated withthis discontinuity, while these are absent in the homogeneous model results.7.8 DiscussionThis is the first investigation of magnetic stimulation which provides modelled results for the E-fielddistribution where inhomogeneous conductivity is treated in a non-trivial fashion and where anisotropicconductivity is taken into account. Roth previously modelled the head as a set of concentric shells andobtained results equivalent to the case of a homogeneous spherical conductor, i.e. all induced fields wereparallel to the air/tissue interface [50]. Guidi presented a short preliminary report of E-fields modelledusing an FEM method, but only plotted the time-domain profile of a pulsed E-field at a single point withinthe brain; the spatial field distribution was not considered [78].The general characteristics of our homogeneous model results agree with previous related investiga-tions [81, 57, 48, 50]: all tissue E-fields are parallel to air/tissue interface and the peak E-field is locatedbelow the figure-8 MO's center bar. Roth modelled a similar 5cm figure-8 MC tangentially positioned over a92mm diameter conducting sphere [50]. The peak E-field, 22 mm below the MC plane, was 8.9 V/m. Thiscompares favorably with our result of 10.0 V/m at the same MC distance. Our slightly higher result maybe due to the surface charge at the air/tissue interface in Roth's model. Grandori modelled a horizontalfigure-8 MC 1cm above a planar tissue and reported a peak E-field of 21.6 V/m [36]. Our modelled peakE-field, 1cm below the MC plane, was 20.7 V/m, i.e. 4% less.Significant changes are seen in the model results when the head is modelled as an inhomogeneousmedium. A fundamental difference from previously reported results is the presence of EM fields withcomponents normal to the air/CSF interface. Past studies have assumed homogeneously conductingtissue [72, 55, 77, 51, 20] and they report the total absence of a field component normal to thetissue surface. In contrast, we found significant normally oriented fields in the SCF near the lip of thegyrus. Conductivity interfaces are generally associated with surface charging and consequently with theredirection of E- and J-fields. It is stressed that even at the lip of the gyrus the dominant E-field andJ-field component is parallel to the air/CSF interface and that the E-field at the CSF surface is tangent101to this interface. These are not the only changes in the field distributions, relative to those seen in thehomogeneous model, that are related to inhomogeneous tissue conductivity:0 The greatest differences in the appearance of the field-isocontours occur where induced currents havea large component normal to a conductivity interface. In our model this is most evident along the lowerwall of the gyrus (Fig. 7.13A), where large abrupt gradients in the E-field are seen across the highresistance cortex. Similarly, high J-fields are excluded throughout the cortex, particularly within the lipof the cortex (Fig. 7.13B). Roth accounted for conductivity inhomogeneity but not for the complex shapeof the gyrus [50]. The clinical importance of the gyrus shape was speculated on by Ueno and is dealtwith in Section 7.8.1 of this thesis [20].0 Significant (22%) increases in the magnitude of the peak E-field and J-field are seen in the CSF abovethe lip of the gyrus. The previously mentioned vertically oriented fields suggest that this increase is dueto a channeling of induced currents between the overlying layer of air and the underlying cortical layerof the gyrus crest. Possibly, this channelling effect is a model artifact and would not be as pronouncedif other adjacent gyri were modelled.0 The most significant change in E-field magnitude within the cortex (45% increase) occurs at the crestof the gyrus. This E-field increase is associated with a 59% drop in the J-field and is caused by thecombined effect of the channeling of induced currents and the high resistivity of the cortex.There are only small (2-3%) differences in our FEM results between the inhomogenous-isotropic andinhomogenous-anisotropic models. This indicates that the great majority of the distortions in the E- andJ-fields, with respect to the homogeneous tissue model, are due to an inhomogeneous conductivity ratherthan anisotropic characteristics.As mentioned in the discussion of the previous chapter (Section 6.4) our comparison of anisotropicmodel results to isotropic (homogeneous and inhomogeneous) models is only valid for the somewhatarbitrary conductivity values we chose for our isotropic models. However, the comparison of such modelresults does highlight the importance of inhomogeneity relative to anisotropy. The possible importance ofanisotropic tissue conductivity has been mentioned by several magnetic stimulation investigators [36, 50].Our results suggest that such anisotropic characteristics may not be a critical factor for field distributions.1027.8.1 Stimulus FunctionsOur ultimate purpose for MC modelling is to understand the process of magnetically induced nervestimulation and to improve its associated techniques. To this end, it is not sufficient to model the EM fieldswithin tissue, and to account for inhomogeneity and anisotropy. Our modelled E-field results must befurther analyzed in terms of the actual stimulus (AF function) applied to the target nerves. In our analysis,the peripheral nerve is assumed to be straight and long, such as the medial and radial nerves in the arm,and the AF function is solved for various test conditions (Chapter 6). A similar type of stimulus analysishas not been reported in the literature for brain stimulation.The novel feature of the stimulus functions found within the modelled brain is that they are an order ofmagnitude smaller in spatial distribution (span) and an order of magnitude larger in peak magnitude relativeto similar cases for peripheral nerve stimulation, (cases 5 and 6 in Table 8.7). For peripheral (straight)nerve stimulation, the magnitude, location and span of the AF function is largely determined by the locationand diameter of the MC (for a given current derivative dl/dt). In contrast, the magnitude, location and spanof the AF function(s) for our modelled pyramidal fiber are principally determined by the characteristics ofthe conductivity interface and the path taken by the nerve fiber. In fact, similar results for the modelledstimulus functions would be obtained if the J-field had been homogeneous throughout the gyrus!Cortex/White Matter Conductivity Interface The large (1.5 V/cm2) AF function spike seen inFig. 7.23B2 is associated with the conductivity transition at the cortex/white matter interface. How such aconductivity interface may affect the validity or form of the AF function has never been explicitly consideredin previous studies. The AF function AF = dE/dx is a simple expression which was extracted from amuch more complex set of cable model equations. Its significance is not obvious other than to say thatthe larger is its magnitude, span and duration the more likely a nerve will be stimulated. The AF functionsuggests that the spatial derivative of the extracellular current density may be an equally valid expressionAF** = dJ I dx to gauge the effectiveness of a stimulus. The isocontour plots for our inhomogeneousbrain model show no discontinuity in the J-field and thus the J-field would generate AF** 0 at thecortex/white-matter interface. In our introduction (Chapter 1) we developed the AF function using Fig. 1.6and equation 2 which is repeated below:— 21/,. +^Ve,n—i —^ Ve,n+i^ic,n2 dVn^VnAx2 Ri Ax2Ri dt Rm(51)This equation is independent of the tissue conductivity and is only affected by the cytoplasmic (R1) andmembrane resistances (Am). Figure 1.6 in Chapter 1 indicates that the critical role of the extracellular(tissue) potential (Ve) is to pass current across the membrane capacitance Cm [45]. Contrary to the AFfunction's form, there is no significance to the extracellular currents which flow tangent to the nerve path.In conclusion, the AF function is valid for both homogeneous and inhomogeneous tissues.The modelled results presented in Fig. 7.23 are based on E-field values calculated at the FEMelement centroids. In panel B1 of this figure the E-field abruptly drops from about 20 to 8V/m over adistance of about 1.9 mm. This modelled distance is entirely a function of the FEM mesh density used103to model the gray/white tissue interface. The finer the FEM mesh the more abrupt the modelled E-fielddrop since the J-field passes through this interface in a continuous fashion. Similarly, the differential dE/dxwill increase indefinitely as the distance (dx) between the finite element centroids on either side of thegray/white matter interface are modelled closer together. Thus the modelled magnitude of the AF functionat the cortex/white matter conductivity interface is somewhat arbitrary.It is difficult to determine how abruptly the tissue conductivity actually changes at the gray/whiteinterface. Since there is no membrane separating the gray and white matter it is possible that these tissueconductivity changes occur over several 10's or 100's of microns. For example, Hoeltzell measured abruptchanges in cortical conductivity using a microelectrode manipulator capable of 43 urn steps in cortical depth[101]. Thus, it may be more appropriate to refer to it as the gray/white "transition" rather than an interface.The magnitude of the modelled E-field drop across the gray/white transition is relatively independent ofthe thickness of the transition zone from gray to white matter. Hence if the gray/white transition was tentimes narrower than modelled (i.e. 190pm rather than 1900pm) the AF function peak would be increasedby ten, from 1.5 to 15 V/cm2 in panel B2 of Fig. 7.23. The clinical implications of this aspect of FEMmodelling are further investigated in Table 8.7.Since no membrane separates the cortex from white matter it is clear that the actual conductivityinterface is not truly discontinuous. To put the situation into perspective, for a given E-field change acrossan interface the area under the rectangular-shaped AF function (Fig. 7.23) will remain a constant. Onecan interpret this as a relatively large AF function extending over a short fiber length or as a smaller AFfunction over a longer fiber length. It is not clear from present studies which of these two alternatives ismore effective. Finally note that this AF function is dependent on the location of the fiber and its orientationto the E-field. For example, fibers at the convexity of the precentral gyrus are likely to be perpendicular tothe skull and normal to the E-field, thus a null AF function would be present at the gray/white transition.Fiber Curvature The smaller (0.2 V/cm2) AF function spike seen in Fig. 7.2382 is associated with thepyramidal fiber curvature described in Fig. 7.7B. Prior to this investigation, FEM models were not realisticenough to seriously apply the concept of the AF function. A complicating factor concerns the complexityof the cortical folds and the commensurate folding of the longer neurons within it. The orientation of theseneurons relative to the skull surface changes continually down the wall and base of the central sulcus.Unlike the AF function associated with the gray/white transition this AF function is not dependent onthe FEM model meshing. Generally it may be assumed that at some point within the white matter ofthe gyrus the pyramidal fiber will be vertically oriented and that the EF function there will be nearly zero.Thus the AF function magnitude is largely determined by the magnitude of the EF function where the fiberexits the cortex. This EF magnitude is dependent both on the location and orientation of the fiber. At thecrest of the gyrus the orientation of the pyramidal tract neurons are likely to be perpendicular to the skull.There the E-field magnitude is largest but only a small E-field component will be tangent to the nerve thusthe EF function there will be small. Fibers which are horizontally oriented within the gray matter (deeperin the sulcus) will have a large component of the E-field tangent to the fiber, but the E-field magnitude104Several hypothetical mechanisms and locations of brainstimulation are depicted. Direct stimulation of a corticospinalfiber may occur at:(A) its dendrites or cell body [115, 116],(B) at or near the axon hillock [117, 118],(C) at the gray/white matter interface,(D) several centimeters lower, in the white matter [107, 119],white matterLegendaxon termination —IP'cell body and dendrites CDIndirect stimulation of the corticospinal fiber may occur at:cpPp (E) cortical interneurons (one or more neurons) [120],(F) more complex interneuron associations [120],crg),^(G) near the termination of an afferent axon [21],(H) at the gray/white matter interface,(I) a deeper location within the white matter [115],Figure 7.24 Modes of Brain Stimulationrapidly decreases with tissue depth. Pyramidal fibers located within the lower portion of convexity of theprecentral gyrus (i.e. with an angle 00 < 9 < 450 in panel B of Figure 7.7) will have the greatest EF andAF functions. The stimulus functions presented in Fig. 7.23 are for an angle of 45°.We have chosen one specific path for the modelled pyramidal fiber. In particular, the fiber bends0.6 mm from the gray matter. If in fact the fibers bend at lesser distances from the gray matter than it ispossible that the two AF function spikes shown in Fig. 7.23B2 would overlap creating an AF function withtwice the span and, presumably, significantly increasing its effectiveness.Clinical Results for Brain Stimulation We have investigated the modelled E-field within the brainand discussed possible stimulus functions generated by the E-fields within the gyrus. We now introducesome published clinical results to determine if there is any supporting experimental evidence for thepresence and effectiveness of such stimulus functions.Clinical investigations have found that a single pulsed stimulus (electric or magnetic) applied to thebrain can generate several action potentials along a nerve fiber. This series of descending waves (usuallybetween one and four APs) can be measured in the efferent fibers of the corticospinal tract and in theirtarget skeletal muscles. Several tentative explanations have been proposed to account for the presenceand characteristics of this series of waves.When more than one wave is generated the one with the shortest latency is called a direct wave (0-wave) [5]. Any responses which have a delay greater than the 0-wave are called indirect waves (I-waves).It is generally assumed that the 0-wave is caused by direct stimulation of the efferent corticospinal fiberand that the later l-wave(s) are due to its indirect (trans-synaptic) stimulation. In this case smaller corticalinterneurons or large afferent fibers are stimulated and subsequently re-excite the corticospinal tract viaexcitatory post-synaptic potentials EPSPs [107-114]. Several possibilities are depicted in Fig. 7.24.105Many of the possible stimulation sites within the cortex (A, B, E, F and G in Fig. 7.24) cannot bemodelled using the concept of the AF function. When stimulation occurs near the termination of a fiber orwhere the effective axoplasmic conductivity varies considerably then electrochemical diffusion is affectedand the assumption that the axon behaves as an infinitely long cable is no longer valid. We refer to thissituation as an "end effect". Extreme examples of stimulation near fiber terminals are dendritic stimulationand interneuron stimulation where the scale of the critical cell component lengths is measured in microns.Little is known either at the experimental or theoretical level concerning stimulation near fiber terminations.Investigators suspect that end stimulation is common during clinical brain stimulation [21], and that if afiber is exposed to a uniform E-field the nerve response will be initiated at one of the two fiber terminationsdepending on the E-field polarity [51]. In a uniform E-field the AF function is zero along the entire length ofthe fiber and it is thus evident that this stimulus function is not valid under these circumstances. For lackof a better alternative, we suggest that the EF function may be a good indicator of E-field effectivenesswhen end-effects are suspected.Our FEM model results show that the fibers exposed to the greatest E-fields would be located at thetop of the sulcus. In the convexity of the precentral gyrus the orientation of the pyramidal tract neurons islikely to be perpendicular to the skull but the E-field is parallel to the skull [20, 58] and will consequentlystimulate cells which lie in the horizontal plane. It is believed that this situation favors indirect stimulation(I-waves) of the pyramidal fibers by interneurons. In contrast, electrode stimulation is expected to generatecurrents both parallel to and transverse to the long axis of the pyramidal cells in the wall of the centralsulcus. The parallel component will favor direct stimulation of pyramidal cells (D-wave).The deeper stimulation sites depicted in Fig. 7.24, i.e. C, D, H and I, are associated with the AFfunctions we described in Fig. 7.23. These four stimulation sites can account for both direct and indirectstimulation seen in clinical investigations. Voluntary contraction of target muscles during electrode andcortical stimulation reduces the threshold stimulus intensity, increases the CMAP response and shortensthe CMAP latency. This enhanced stimulus response is called "facilitation" and suggests that the initiationsite is within the cortex; however, one cannot further discern direct from indirect activation since the levelof excitability of both the corticospinal neurons as well as the cortical interneurons will be affected by thisconditioning [121, 107].106Chapter 8Discussion and ConclusionIn this chapter we present an analysis to quantitatively determine how meaningful our modelled MCdesigns and their AF functions are in terms of clinical data. We conclude with a summary of the resultspresented in this thesis.8.1 Clinical DataThe focus of previous chapters has been to demonstrate that our modelled E-fields are realistic andaccurate, and to present an analysis of how certain factors (MC design, tissue geometry, anisotropy andinhomogeneity) may affect the stimulus functions. Up to this point, our study has not quantitatively relatedthe absolute values of our MC model results to published clinical data. Failure to associate modelled MCfield results to clinical data is seen throughout the modelling literature. Such an analysis is critical: 1)to rate the strength and effectiveness of the modelled AF functions, 2) to determine if the modelled AFfunction characteristics are compatible with experimental data and 3) to quantize the magnetic stimulator'scomponent parameters necessary for clinical usage.The purpose of the following discussion is to estimate the threshold stimulus requirements for eachof the MC examples presented in the previous chapters. The critical parameters which define thresholdstimulation (magnitude, duration and spatial profile of the AF function) have been experimentally measuredin previous studies. Our problem consists of determining the MC current derivative (dl/dt) which generatesa threshold AF function.Although there are many published clinical studies using MCs, very few fully describe the conditionsassociated with threshold stimulation. In addition it is never clear which nerve fibers are being stimulated orwhere along the fiber length stimulation takes place. Finally, lengthy model analysis would be required todetermine the actual E-fields within the tissues. These questions have been resolved with some success forelectrode stimulation and consequently we use electrode data to determine threshold stimulus conditions.As previously mentioned, nerve stimulation is initiated by E-fields and it is presumed that the basisfor electrode and MC stimulation are identical in nature, i.e. the AF function determines if a nervewill be stimulated. The electric potential generated by a monopolar spherical electrode imbedded in ahomogenous conducting medium is given byp •Iii=47r(52)where V is potential at a distance r from an electrode with current I and p is the medium's resistivity (C2 •m). The equation is modified in terms of rectangular coordinates where z is the smallest distance betweenthe electrode and a fixed straight axon and x is any point along the axon's length,V = p • I (x2 + x2)-1/2 .(53)4ir107The AF function [45] is then derived by solving for the second derivative of the extracellular potential alonga long straight axon:AF =a2v , _ aE , _p •I . (3x2 (x2 + x2)-5/2 _ (x2 ± x2)-3/2) .8X2^OX^47rThis derivation assumes that the nerve is straight and infinite in length. If these conditions are met, Rattay[45] proposes that where the AF function is positive the nerve membrane is depolarized and that where itis negative the membrane is hyperpolarized. The associated EF function is given by:_av _ p •IEF =^x . (Z2 + x2)-31'2 .OX^47r(55)Normalized plots of equations 54 and 55 are given in Fig. 8.1A. The AF function peak corresponds withthe location of the electrode along the length of the axon. Panel A also compares the relative position andspatial profiles of the EF and AF functions. The "span" of the AF function is defined as the axon lengthwhich is exposed to stimulus fields from 50% to 100% of the peak value (Fig. 8.1).Figure 8.1B presents experimental [21] results of the threshold cathodal current magnitude for variouselectrode distances from the axon. Using equation 54, the AF function span is plotted as a function ofelectrode/axon separation, (i.e. Figure 8.1C). Using the boundary values of the threshold cathodal currentin panel B and solving equation 54 provides a range for the peak magnitude of the threshold AF function(Figure 8.1C).Figure 8.1C shows both the span and magnitude of the threshold AF function, for a wide range ofelectrode/axon separations. From our FEM results we can obtain the peak magnitude and the span of theAF function for each of our modelled MCs. Table 8.7 (columns 1-6) summarizes our modelled stimulusfunction results from Chapters 5, 6 and 7. By combining these results with the experimental data inFig. 8.1C we estimate the threshold magnitude for our modelled stimulus functions (column 7). Finally thethreshold MC current derivative (column8) is calculated by taking the ratio of the results in columns 6 and 7.Before discussing the findings in Table 8.7 we consider several assumptions required to directlycompare clinical electrode data and our modelled MC results:1.We assume that all our modelled MCs have a current rise time (i.e. a stimulus duration) of 100us. Thisparameter is a function of the inductance, capacitance and resistance of the MC stimulator circuit. A100 us risetime is a reasonable median value for commercial MC designs. For example the CadwellMES-10 stimulator [5, 122], the Novametrix Magstim 200 [10, 11] and the Dantec stimulator [92] haverisetimes of 50, 100 and 150 us respectively.2. The monopolar electrode generates a tripolar AF function and a bipolar EF function (Fig. 8.1 A). Incontrast, for peripheral nerve stimulation using a MC the EF function is monopolar and the AF functionis bipolar (cases 1 - 4 in Table 8.7). Thus, their are inherent qualitative differences in the spatial profilesof stimulus functions from electrodes and MCs. Both stimulators generate AF functions which may beaffected by anodal block but the MC's AF function generate a stronger hyperpolarization relative to thedepolarization. We assume that since we are concerned with threshold response levels anodal blockshould not be a major distinguishing factor between these two types of stimulus functions.(54)1081 05101109—101^102^103^104Electrode/Cell Distance ( urn )1S10s_1o4g 10.1 02103101^102^103^104Electrode/Cell Distance ( urn )10A:Spatial Profiles of Normalized StimulusFunctions: Relative spatial profiles ofcathodal EF (light line) and AF (dark line)functions. The "span" of the AF functionis defined as the axon length which isexposed to stimulus fields from 50% to100% of the peak value and is markedwith arrows. The scale of the x-axisvaries linearly with the electrode/celldistance, whereas the scale of the y-axisvaries linearly with the magnitude ofelectrode current.B:Summary of results from review ofpublished clinical data by Ranck, 1975[21]. All data corresponds to thresholdcathodal electrode stimulation using100ps current pulse. Results forstimulation along the myelinated axon aremarked with crosses, ( i.e. electrode tipwithin gray and/or white brain tissue). Athreshold stimulus zone is extrapolatedfrom set of data and is marked withshaded region.106104C:3 Magnitude range and span of threshold10 z^AF function for various electrode/cellu_^distances. Span of AF function was2 obtained by solving equation 54.io =Threshold AF function was extrapolatedfrom shaded region in panel B above and10'^by solving equation 54.10°Figure 8.1 Threshold Stimuli and Electrode Distance109This Table combines our modelled MC results with published experimental data. The firstfour columns (1-4) describe the modelled MC design and the modelled tissue which is beingconsidered. Columns 5, and 6 show our modelled results. Cases 5 and 6 deal with stimulusfunctions generated by curved axons $ and by conductivity interfaces t respectively. The AFfunction spans correspond to the conditions described in each of the referenced Tables andFigures. The AF function magnitudes correspond to a MC current derivative of 100MA/s.By corresponding our modelled AF span results to the data presented in panels C in Fig.8.1 the range for the estimated threshold AF functions was defined and are given in column 7.The calculated threshold MC current derivative is shown in column 8. The results in column8 are derived by taking the ratio of the results in columns 7 and 6 and multiplying by 100.Reference1CaseNo.2MCDesign3TissueMedia4AFSpancm5AFMagnitudeV/cm26AF ThresholdV/cm27dl/dtThresholdMA/s8Table 5.3 1 loop planar isotropic 2.66 0.06 0.19-0.65 316 - 1100Table 5.3 2 spiral planar isotropic 2.91 0.02 0.18-0.55 900 - 2750Fig. 6.5 3 square arm anisotropic 2.8 0.042 0.19-0.6 450 - 1430Fig. 6.12 4 figure-8 arm anisotropic 2.3 0.063 0.2-0.7 317 - 1111Fig. 7.23 5 figure-8 headinhomogeneousanisotropic $0.24 0.2 0.9 -7 450 - 3500Fig. 7.23 6 figure-8 headinhomogeneousanisotropic t0.190.0191.5151-105-12067 - 66733 - 800Table 8.7 Threshold Levels for Modelled AF Functions3. Our modelled stimulus fields have spans of up to 7.75 cm. This is an order of magnitude greater thanthe spans associated with the experimental data in Fig. 8.1. We assume that the extrapolated zonesare valid in Fig. 8.1 B.4. We assume that our modelled AF functions for brain stimulation are valid in the sense that end effectsare not significant. This is a reasonable assumption since the AF functions only span short distances(less than 3mm).The peak MC current derivative for two commercially available magnetic stimulators (Cadwell andNovametrix ) is about 3200 MA's [50]. Several brain stimulation studies [9, 121, 123] applied thesestimulators in a standard fashion and measured the stimulus threshold level to be 30% to 40% of itsmaximum output. This corresponds to a threshold MC current derivative of about 960 to 1280 MA's.There is no published threshold data for peripheral nerve stimulation; however studies suggest peripheralstimulation (along axon) may have a lower threshold [4, 8] than cortical stimulation.110We first consider cases 1 to 4 in Table 8.7 since they involve AF functions which are largely determinedby MC design rather than axon curvature or conductivity interfaces. The AF span is relatively constantin all 4 cases (2.3 — 2.9 cm). Case 2 has a small AF function (0.02 V/cm2) which is attributed to thedispersed MC current associated with a spiral MC design. Case 3 also has a small AF function (0.042V/cm2) which is attributed to a larger MC/axon separation. The lower dl/dt threshold for all four cases isbetween 10% and 28% of the commercial stimulator maximum output (3200 MA's). For cases 1, 3 and4 the upper dl/dt threshold is between 34% and 44% of the commercial stimulator maximum output. Theupper dl/dt threshold for the spiral MC (case 2) is 86% of the commercial stimulator maximum output.Case 5 deals with cortical stimulation using a figure-8 MC design. The AF span (0.238 cm) is anorder of magnitude smaller than in cases 1 to 4. This span is attributed to axon curvature within thesulcus and corresponds to a large electrode threshold AF function (0.9 - 7 V/cm2). The AF functionmagnitude generated by the MC with dIdt=108A/s is significantly greater than in cases 1 to 4 (i.e. 0.2V/cm2). Thus the estimated threshold MC current derivative ranges from 14% to 110% of commercialstimulator maximum output.Case 6 deals with the AF function generated at the conductivity transition between gray and whitematter. Two sets of results are given in columns 5 to 8 corresponding to two possible AF function spans(0.19mm to 1.9mm). The estimated threshold MC current derivative ranges from 2% to 21% of commercialstimulator maximum output, when the transition zone is assumed to be 1.9mm; and from 1% to 25%of commercial stimulator maximum output, when the transition zone is assumed to be 0.19mm. Thesethreshold values are significantly lower than those estimated in case 5 (at the nerve curvature) and suggeststhat the gray/white transitions may be a preferred site for direct stimulation of afferent and efferent fibers.In summary, there is a wide range for the experimental threshold data and this in turn results in a widerange (e.g. cases 5 and 6) for our threshold MC current derivative (dl/dt). The intent of this quantitativecomparison between modelled results and clinical data is to estimate the minimum MC performancerequired to achieve nerve stimulation. There are several factors which prohibit this comparison from beingaccurate:most of our MC models used ideal thin-wire type designs rather than more realistic spiral type designs,0 most modelled MC/axon distances were slightly greater than seen in clinical practice,El the modelled MC position relative to the tissue and target nerve was not standard for clinical practice(case 3 in Table 8.7)0 there is a wide variation in the experimental data for threshold stimulus levels.Despite these shortcomings a few useful observations and comments can be made on the resultsof this comparison:o All but one of the six modelled cases have dl/dt threshold ranges which partially overlap the experimentaldl/dt threshold range of 960 to 1280 MA/s.El For these five cases (1 - 5 ) our comparison adds credibility to the notion that electric and magneticstimulation are based on the same fundamental mechanisms as described by the AF function.1110 This comparison also suggests that modelled threshold levels for direct pyramidal fiber stimulation at anerve curvature is in partial agreement with experimental dl/dt levels, notwithstanding the wide rangein the experimental data.0 The modelled AF function at the gray/white matter interface (case 6) has a very small lower limit forits dl/dt threshold (1% to 2% of the commercial stimulator maximum output). This suggests that, forsome reason, this type of AF function may not be effective clinically, for instance, the actual span ofthe AF function may be much smaller than 0.19mm. Other possibilities are that end effects may be acritical factor. The importance of this type of AF function cannot be ruled out since the upper limit of themodelled dl/dt threshold is 800 MA/s which is not far from the experimental dl/dt threshold of 960 MA/s.8.2 SummaryMagnetic stimulation, as an accepted clinical procedure, has existed for almost a decade [2-4] andinvestigators have been modelling MC E-fields for less than half that time [47, 52]. The purpose of thesemodelling studies has been to improve MC designs and to further the understanding of the mechanismsinvolved with this method of nerve stimulation.Previous investigations have modelled MC E-field distributions by expressing the total E-field as thesum of two components: a magnetically induced and an electrostatic component. Variations on this basicsolution strategy have been explored in many studies and their results have elucidated several fundamentalcharacteristics associated with magnetic stimulation. This modelling method offers several advantages:1) each parameter may be varied continuously 2) general conclusions may be drawn from the form of thefinal equations and 3) and results can be used to verify numerical methods.The relevance of modelled results increases as the model characteristics mimic the true clinicalconditions. Common shortcomings seen in many previous studies are: 1) the MC designs are usuallyrestricted to simple line-element representations, 2) geometry of tissue volumes are unrealistic 3) onlytrivial cases of tissue inhomogeneity have been investigated and 4) the anisotropic conductivity is notaccounted for.We have decided to use a commercially available finite element package. The reason for this choicewas to increase the diversity and complexity of the models which could be solved and thus improve therealism and relevancy of our modelled results to actual clinical conditions. The validity of our basic FEMmodelling strategy was verified and we investigated factors which affect the magnetically induced E-fieldfrom a MC.1128.2.1 MC DiameterMagnetic coil design was recognized early by researchers as an important design parameter whichcould be readily modified. The spiral MC design is used as a standard by commercial magnetic stimulatormanufacturers. Investigators have not explored why this MC design is more popular than loop-type MCswhere the MC-current is more concentrated and peak induced E-fields are inherently greater. We modelledtwo MC designs in a vertical orientation 1cm above a planar tissue: a spiral MC with OD=120mm andID=50mm and a loop-type MC with OD=ID=120mm. Our results showed a 200% increase in the AFfunction when the OD/ID ratio is reduced to unity, from that typically seen in commercial MC designs.Previous models with exotic MC shapes indicate that when the test conditions are normalized similarE-field levels are induced [76, 36, 51] and that there is no significant difference (12%) in their peak AFfunction values [51]. Thus we propose that, regardless of the MC shape, to increase or maintain a largeAF function (for normalized conditions) one effective design option is to minimize the cross sectional areaof the MC windings.We find that if two magnetic coils have the same outer diameter than the one with the largest innerdiameter will induce the largest E-field and AF function. This result has direct clinical implications forcommercial magnetic coils, presently available, which are characterized by small inner diameters. Avertical magnetic coil (outer diameter = 12cm) was modelled over a planar tissue and a finite cylindrical(20cm diameter) tissue volume. The electric field induced in the finite volume was 25% smaller than inthe planar volume; however, the AF function magnitude remained essentially constant.8.2.2 Anisotropic Conductivity in the ArmAnisotropic conductivity within tissues has been recognized for more than half a century [93]. OurMC models are the first to account for anisotropic conductivity. Only one published study has investigatedmagnetically induced E-fields in the arm [54]. We presented models for two MC designs tangentiallyoriented over the arm. Results were compared when the arm was modelled with isotropic and anisotropicconductivity.Unlike isotropic tissue the magnitude and distribution of the E-field within anisotropic tissue are bothdependent on the parameter values defining the conductivity. In our anisotropic model the E-field, J-fieldand EF function magnitudes were reduced by 12% to 14% and the AF function by 24% relative to thevalues seen in the isotropic model. These field reductions may be entirely attributable to changes in thedistribution of surface charges. The relationship between the E-field and conductivity is not evident since,in isotropic tissue, they are independent parameters.A second general characteristic seen in anisotropic tissue is that the spatial distribution of the E-fieldis more affected by anisotropic conductivity than is the J-field. Again, this is in contrast to isotropic tissuewhere the J-field is subordinate to the magnetically induced E-field and the tissue conductivity. Examplesof these E-field distribution changes are illustrated by the relocation of the E-field peak (Fig. 6.6) and alocalized isocontour distortion (Fig. 6.9A) when compared to the isotropic model results.113Our AF function results in modelled arm-tissue were compared to those previously published for aplanar tissue [51]. A horizontal MC over a planar tissue generates no surface charges and thus maximal E-fields are expected relative to any other tissue shape. In spite of this fact, our modelled AF function for thesquare MC over the isotropic arm was 14% greater than in the planar tissue. We attribute this AF functionincrease to rapid spatial changes in the E-field magnitude rather than a greater peak E-field magnitude.Our anisotropic model had lower E-field magnitudes and this resulted in an AF function 14% lower thanin the planar tissue. The AF functions modelled for the figure-8 MC were 18% and 37% (isotropic andanisotropic tissue) lower than for a planar tissue. We attributed this decrease to a large amount of surfacecharging associated with the large MC dimension relative to the arm, and the consequent reduction ofthe peak E-fields.8.2.3 Inhomogeneous Conductivity in the BrainThe effect of tissue inhomogeneity on the distribution of induced E-fields has been overlooked byprevious investigators [50]. We modelled the gyrus of the primary motor cortex within a planar volume ofcerebral spinal fluid and a horizontal figure-8 MC positioned above the gyrus. The gyrus (gray and whitematter) conductivity was modelled in three ways: 1) homogeneous, 2) inhomogeneous-isotropic and 3)inhomogeneous-anisotropic.Only minor differences were seen between the two inhomogeneous model results (<3% variation inthe E- and J-field magnitudes and <8% variation in the EF and AF functions). This indicates that theanisotropic conductivity parameters, which have been measured in brain tissue, do not account for mostof the E-field and J-field characteristics seen in our inhomogeneous-anisotropic model. This is in spite ofa 10:1 anisotropic conductivity ratio modelled within the white matter.When the homogeneous and inhomogeneous model results are compared, significant changes in thefield distributions are seen:1.The most significant qualitative change is the presence of E- and J-fields with a component orientednormal to the air/CSF interface. These fields were most prominent, within the SCF, near the lip ofthe gyrus. This field characteristic is attributed to surface charges at the CSF/cortex interface. Thesenormally oriented CSF currents are channelled up above the gyrus thus causing a 22% increase inthe E- and J-field magnitudes, relative to the homogeneous model results. If other adjacent gyri weremodelled this channelling effect might not be as important.2. Large E-field gradients are found at conductivity interfaces. In our model this is most evident at thegray/white interface along the lower wall of the gyrus (Fig. 7.13A). Similarly, high J-fields are excludedthroughout the cortex, particularly within the lip of the gyrus (Fig. 7.13B).3. The most significant change in E-field magnitude (45%) occurs at the crest of the gyrus. This E-fieldincrease is associated with a 59% drop in the J-field and is caused by the combined effect of thechanneling of induced currents and the high resistivity of the cortex.114Stimulus Functions in the Brain Previous investigators have modelled AF functions induced byMCs but only for straight (peripheral) nerve fibers within homogeneous, isotropic tissue. We modelledthe AF function of a curved pyramidal fiber within the gyrus. This AF function had several distinguishingfeatures relative to those typically associated with MCs: 1) its span is an order of magnitude smaller, 2)its strength is more than an order of magnitude larger, 3) it is monopolar, 4) it is non-symmetric and 5)it has two positive peaks.The larger AF function peak is associated with the cortex/white matter interface (1.5 V/cm2), thesmaller peak is associated with the fiber's curvature (0.2 V/cm2). This is in contrast with peripheralnerve stimulation where the magnitude, location and span of the AF function is largely determined by thelocation and diameter of the MC.The presence of an AF function at a conductivity interface is a novel finding and it is still not clearhow wide a span is realistic for this type of AF function. The span is determined by the tissue lengthover which the conductivity makes the transition from gray to white matter and this will directly affect themagnitude of the AF function peak.The possibility of preferential magnetic stimulation at fiber bends has been speculated by investigatorsbut has not been previously modelled. The magnitude of this AF function is largely determined by themagnitude of the EF function where the fiber exits the cortex. For a maximum we suggest that the fibershould be located within the lower portion of convexity of the precentral gyrus.Either of these two types of AF functions (conductivity interface and fiber curvature) can account forboth the D-waves and I-waves seen in clinical investigations. These AF functions could act directly onthe axon of pyramidal fibers or stimulate axons of afferent fibers within the sulcus. D-waves and I-wavesmight also be generated by stimulation at the terminals of fibers (afferent, efferent and interneurons) whereend-effects may play a critical role and the EF function might be a good indicator of E-field effectiveness.8.2.4 Future InvestigationsMuch is known of the structural organization of the brain and its constituent substructures (nervefibers). Clinical investigations have provided data to support a diverse assortment of possible mechanismsfor the initiation of brain stimulation. Previous electromagnetic models have not been able to investigatethe details of these theories, nor have they been able to demonstrate the full implications of how varioustissue properties may affect the electric field distribution.As electromagnetic models become more complex, the interpretation of data becomes equally com-plex. This situation requires more advanced presentation formats for the results, without which a fullcomprehension of the system characteristics may be missed. The powerful graphics display offered bythe EMAS/XL software will be a useful tool for the analysis and presentation of modelled results.Several point of interest remain to be explored: 1) do anisotropy and inhomogeneity affect electrodeand magnetic stimulation in different ways? 2) will our results change significantly if the head is modelled115as a sphere or if several adjacent sulci are modelled? 3) what level of E-field reduction is to be expectedfrom MC's with a large height (H)?Regardless of the model's complexity it must never be forgotten that there is a difference between 1)getting the correct and accurate results for what was modelled and 2) getting results which are realisticwith respect to the true biological conditions.116Bibliography[1] J.B. Reswick F.T. Hambrecht. Functionalelectrical stimulation. Marcel Dekker Inc.,NY., NY., U.S.A., 1976.[2] A.T. Barker et al. The design,constructionand performance of a magnetic nerve stim-ulator. In IEEE Intern. 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The influence ofstimulus type on the magnetic excitation ofnerve structures. E.E.G. Clin. Neurophy.,75:342-349, 1990.121Appendix AcronymsAcronyms for Physiology:AF — activation function (dE/dx)EF — electric function (dV/dx)AP — action potentialCMAP — compound motor-unit action potential (muscle activity recording)CNS — central nervous systemPNS — peripheral nervous systemAcronyms for Electromagnetics:• — permittivity (F/m)• — permeability (H/m)g — conductivity (S/m)co — permittivity in free space 8.854 x10-12(F/m)• — permeability in free space 47 x10-7(H/m)g — conductivity (S/m)vo — magnetic reluctivity (m/H)A — vector magnetic potential field (Wb/m)M — magnetization (A/m)P — polarization or average dipole moment per unit volume (C/m2)H — magnetic field strength (Aim)B — magnetic flux density (W/m2 = T)E — E-field (V/m)D — electric flux density (Q/m2)J — surface current density (A/m2)— scalar potential (V)E-field — electric field (Vim)J-field — current density field (A/m2)A-field — vector magnetic potential field (Wb/m)Acronyms for VectorsVA — gradient of vector A —^Peviv• A — divergence of vector A — ta-• x A — curl of vector A —_ L.4.2]^2-11.1 • + { OA,^14.21kay^az Oz^Oz 13^Oz^OyA = -Qat — time derivative of vector AA =^— second time derivative of vector A[ B ] — matrix B[B] T — transpose of matrix BOther Acronyms:MC — magnetic coil0 FEM — finite element model0 FE — finite element

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