# The Magnetic Field Produced by the Heart and Its Influence on MRI

Authors: Dan Xu; Bradley J. Roth
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

Background. Action currents in the heart produce a magnetic field, which could provide a way to detect the propagation of electrical activity through cardiac tissue using magnetic resonance imaging. However, the magnetic field produced by current in the heart is small. The key question addressed in this study is are cardiac biomagnetic fields large enough to be detectable by MRI?Results. A spherical model is used to calculate the magnetic field inside the heart, which has a magnitude of about 14 nT. This field implies a phase shift in the MRI signal of about 0.2°.Conclusion. Phase shifts associated with cardiac action currents will be difficult to detect using current MRI technology but may be possible if motion artifacts and other physiological noise can be suppressed.


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## Body

## 1. Introduction

Action currents in the heart produce a magnetic field, which could create an artifact in a magnetic resonance image (MRI). Viewed from another perspective, in theory MRI could be used to image action currents by detecting their magnetic field. Indeed, if one could image action currents using MRI, it would be a fundamental advance that might make magnetocardiography and even much of electrocardiography obsolete. However, the magnetic field produced by current in the heart is small. The key question addressed in this study is are biomagnetic fields large enough to be detectable by MRI? The main goal of this paper is to compute how large this magnetic field is and explore the possibility of detecting the magnetic field by magnetic resonance imaging.Over the last ten years, much work has been done to detect neuronal activation by measuring the magnetic field produced by biocurrents using MRI [1–8]. In general, these magnetic fields are so tiny that they are difficult to measure [3]. However, the heart is the source of the largest biocurrents in the body, so it may be easier to detect cardiac action currents using MRI compared to neural action currents [9]. In order to interpret and predict the MRI signal produced by cardiac action currents, we must know the magnetic field distribution within the heart.In general, the best starting place for calculating the magnetic field is a simple model that clearly illustrates mechanisms. In this study, we examine a “spherical heart” model [10] with a simple transmembrane potential distribution and calculate the resulting action currents and magnetic field. A simple model allows us to obtain analytical solutions that provide insight into the qualitative behavior and allows us to estimate the quantitative magnitude of the magnetic field.There are two sources that produce a magnetic field in cardiac tissue: one is the intracellular current in the tissue with the “return” loops through an adjacent volume conductor, and the other is the anisotropy of the tissue [11]. In this study we account for the first of these mechanisms: the heart is surrounded by a volume conductor that provides a path for return currents and results in a magnetic field.


## 2. Methods

We consider a spherical shell of cardiac tissue surrounding a blood cavity and surrounded by an unbounded conducting bath [10] (Figure 1). We use spherical coordinates (r, θ, and ϕ) to specify position. The inner and outer radii of the shell are r1 and r2, respectively. The blood (subscript “b”) conductivity is gb and the bath (subscript “o”) conductivity is go.Figure 1
The model of a spherical heart. Part of the spherical shell has been cut out to show the heart wall. The black curves indicate the fiber orientation. The pink tissue has a transmembrane potential of 20 mV, and the blue tissue has a transmembrane potential of −80 mV. The green curve shows the magnetic field. The endocardial (inner) surface has radiusr1, and the epicardial (outer) surface has radius r2.We assume the transmembrane potential has azimuthally symmetry. Therefore, it does not depend on the angleϕ around the heart. With an analogy to earth, the transmembrane potential only depends on latitude but not longitude. The fiber geometry in the heart is complex, and we must simplify it significantly in order to obtain an analytical solution. Here the fiber axis is parallel to the meridians, that is, along lines of constant longitude (lie along θ direction) [10]. With this fiber model, we can calculate analytically the most significant information that we want, that is, the magnitude of the magnetic field B.We use the bidomain model to represent the electrical properties of cardiac tissue, which accounts for anisotropy both inside and outside the myocardial cells [12]. The bidomain model is a continuum model, in the sense that it represents the electrical properties of the tissue averaged over many cells [13]. The electrical conductivities in cardiac tissue (gir, giθ, ger, and geθ) are different in r and θ directions (perpendicular and parallel to the fibers) and in the intracellular (subscript “i”) and extracellular (subscript “e”) spaces.We assume the tissue is quasistatic: the governing equations do not depend on time. The transmembrane potential will depend on time as the action potential propagates through the tissue, but, given the transmembrane potential at any moment, the other potentials, currents, and magnetic field can be calculated at that instant quasistatically.


### 2.1. Calculation of the Electrical Potential

In the cardiac shell,r1<r<r2, there are two potential fields, Vi and Ve, which are functions of r and θ. The transmembrane potential is the difference of the intracellular and extracellular potentials:(1)Vm=Vi-Ve.The fundamental equation governing the current density J is the equation of continuity (2)∇·Jb=0r<r1(3)∇·Ji+Je=0r1<r<r2(4)∇·Jo=0r>r2.Since we only require the sum of Ji and Je be divergenceless, Ji and Je individually need not have vanishing divergence. Thus, current can flow from one domain to another.Using Ohm’s lawJ=g~E and the relationship between the potential and the electrical field E=-∇V, we find that Ji=-g~i∇Vi, Je=-g~e∇Ve, Jb=-gb∇Vb, and Jo=-go∇Vo, with g~i=gir00giθ and g~e=ger00geθ. If we use the above four current density expressions, then (2)–(4) become (5)∇2Vb=0(6)1r2∂∂rr2∂∂rgirVi+gerVe+1r2sin⁡θ∂∂θsin⁡θ∂∂θgiθVi+geθVe=0(7)∇2Vo=0.In order to simplify (6), we do a linear transformation from Vi and Ve to Vm and ψ [10, 14]:(8)Vm=Vi-Ve,ψ=Vi+geθgiθVe,with the inverse transformation given by(9)Vi=giθgiθ+geθψ+geθgiθVm(10)Ve=giθgiθ+geθψ-Vm.If we substitute (9) and (10) into the expressions in (6), we get (11)girVi+gerVe=giθgiθ+geθgir+gerψ-gir1-girgeθgergiθVm(12)giθVi+geθVe=giθψ.If we make the additional assumption that Vm is not a function of r, the second term in the square brackets of (11) does not contribute to (6), and we obtain(13)1λ21r2∂∂rr2∂ψ∂r+1r2sin⁡θ∂∂θsin⁡θ∂ψ∂θ=0,where(14)λ=giθ+geθgir+ger.


### 2.2. Boundary Conditions

The behavior at the interface between cardiac tissue and an adjacent volume conductor is determined by boundary conditions [15, 16]. At the surfaces of the shell, r=r1 and r=r2, we require that the normal component of the current density is continuous:(15)Jir+Jer=Jbrr=r1Jir+Jer=Jorr=r2and the extracellular potential is continuous(16)Ve=Vbr=r1(17)Ve=Vor=r2.The boundary conditions in (15)–(17) give at r=r1(18)∂ψ∂r=λ2gbgiθ∂Vb∂rψ=Vm+giθ+geθgiθVb.For the boundary r=r2, we replace Vb by Vo in (18).


### 2.3. Solution of the Equations for the Potential

If we expandψ in terms of Legendre polynomials [17], Pl(cos⁡θ), where l=0,1,2,3,…, then the general solution to (13) is [10](19)ψ=∑Alrν1+Blrν2Plcos⁡θ,where(20)ν=λ2ll+1+14,ν1=ν-12,ν2=-ν-12.The general solutions to Laplace’s equation in the blood and bath are(21)Vb=∑lClrlPlcos⁡θ,(22)Vo=∑lDlr-l+1Plcos⁡θ.The boundary conditions determine the constants Al, Bl, Cl, and Dl for each l. We expand Vm in terms of Legendre polynomials(23)Vm=∑lFlPlcos⁡θ,where(24)Fl=2l+12∫0πVmθPlcos⁡θsin⁡θdθ.Solving the four linear equations from the boundary conditions, we obtain the constants Al, Bl, Cl, and Dl for each l:(25)Al=-b1+b2b2a1-a2b1Fl,(26)Bl=a1-a2b2a1-a2b1Fl,(27)Cl=Alν1r1ν1+Blν2r1ν2λ2gb/giθlr1l,(28)Dl=-Alν1r2ν1+Blν2r2ν2λ2go/giθl+1r2-l+1,where(29)a1=r1ν11-gir+gergbν1l,b1=r1ν21-gir+gergbν2l,a2=r2ν11+gir+gergoν1l+1,b2=r2ν21+gir+gergoν2l+1.


### 2.4. Calculation of the Magnetic Field

Using Ohm’s law, we obtain the current density distribution(30)Jbθ=-1rgb∂Vb∂θr<r1,Jθ=Jiθ+Jeθ=-1rgiθ∂ψ∂θr1<r<r2,Joθ=-1rgo∂Vo∂θr>r2.Ampere’s law in its integral form is(31)∮B→·ds=μ0∬J→·dS=μ0Ienclosed,where Ienclosed is the current enclosed by the Amperian loop. We choose this loop to be the edge of a right circular cone with the apex at the center of the heart sphere and the integral surface is the surface of the right circular cone: (32)J→=-Jz^=-Jcos⁡θr^-sin⁡θθ^dS=-rsin⁡θdrdϕθ^.The magnetic field is in ϕ direction, tangential to this loop. By varying r and θ of this cone, we can get the magnetic field in all the space(33)B=μoIenclosed2πrsin⁡θ,where (34)Ienclosed=-∫02π∫0rJθrsin⁡θdrdϕ.In order to compute Ienclosed and thereby the magnitude ofB, we need to substitute the potential ((19), (21), and (22)) into the expression of current density (30), and then replace Jθ in (34) by these expressions of current density in each region r<r1, r1<r<r2, and r>r2, respectively. We then sum up the integrals in a piecewise manner for the three spaces in blood, bath, and tissue.In the bloodr<r1, (35)Vb=∑lClrlPlcos⁡θ,Jθ=Jbθ=-1rgb∂Vb∂θ=-gb∑lrl-1ClPl1,Ienclosed=-∫02π∫0rJθrsin⁡θdrdϕ=gb∑lClPl12πsin⁡θ∫0rrldr=gb∑lClPl12πsin⁡θrl+1l+1,Bϕ=μ02πrsin⁡θIenclosed=μ0gb∑lClPl1rll+1,(note: Pl1=dPl0/dθ).In the bathr>r2, (36)Vo=∑lDlr-l+1Plcos⁡θ,Jθ=Joθ=-1rgo∂Vo∂θ=-go∑lr-l+2DlPl1.Since the total current integrated over all the space is zero, we deduce(37)Ienclosed=-Ioutside=∫02π∫r∞Jθrsin⁡θdrdϕ=-go∑lDlPl12πsin⁡θ∫r∞r-l+1dr=-go∑lDlPl12πsin⁡θr-l-lr∞=-go∑lDl2πsin⁡θPl1r-ll,Bϕ=μ02πrsin⁡θIenclosed=-μ0go∑lDlPl1r-l+1l.In the tissue r1<r<r2(38)Vm=∑lFlPlcos⁡θ,ψ=∑Alrν1+Blrν2Plcos⁡θ,Jθ=Jiθ+Jeθ=-1r∂giθVi+geθVe∂θ=-1rgiθ∂Vi+geθ/giθVe∂θ=-1rgiθ∂ψ∂θ.The integral interval of (34) is split into two pieces: 0→r1 is I1 and r1→r is I2. (39)I1=-∫02π∫0r1Jθrsin⁡θdrdϕ=gb∑lClPl12πsin⁡θr1l+1l+1,I2=-∫02π∫r1rJθrsin⁡θdrdφ=giθ2πsin⁡θ∑lPl1Al∫r1rrν1dr+Bl∫r1rrν2dr.The total enclosed current is then (40)Ienclosed=I1+I2=2πsin⁡θPl1∑lgbClr1l+1l+1+giθAl1ν1+1r2ν1+1-r1ν1+1+giθBl1ν2+1r2ν2+1-r1ν2+1+goDlr2-ll.Using (27) and (28), we get(41)Ienclosed=giθ∑lPl12πsin⁡θAlν1+1rν1+1+Blν2+1rν2+1Bϕ=μ02πrsin⁡θIenclosed=μ0giθ∑lPl1Alν1+1rν1+Blν2+1rν2.An advantage of this formulation is that by varying r and θ the magnetic field is determined everywhere including within the heart wall.


## 2.1. Calculation of the Electrical Potential

In the cardiac shell,r1<r<r2, there are two potential fields, Vi and Ve, which are functions of r and θ. The transmembrane potential is the difference of the intracellular and extracellular potentials:(1)Vm=Vi-Ve.The fundamental equation governing the current density J is the equation of continuity (2)∇·Jb=0r<r1(3)∇·Ji+Je=0r1<r<r2(4)∇·Jo=0r>r2.Since we only require the sum of Ji and Je be divergenceless, Ji and Je individually need not have vanishing divergence. Thus, current can flow from one domain to another.Using Ohm’s lawJ=g~E and the relationship between the potential and the electrical field E=-∇V, we find that Ji=-g~i∇Vi, Je=-g~e∇Ve, Jb=-gb∇Vb, and Jo=-go∇Vo, with g~i=gir00giθ and g~e=ger00geθ. If we use the above four current density expressions, then (2)–(4) become (5)∇2Vb=0(6)1r2∂∂rr2∂∂rgirVi+gerVe+1r2sin⁡θ∂∂θsin⁡θ∂∂θgiθVi+geθVe=0(7)∇2Vo=0.In order to simplify (6), we do a linear transformation from Vi and Ve to Vm and ψ [10, 14]:(8)Vm=Vi-Ve,ψ=Vi+geθgiθVe,with the inverse transformation given by(9)Vi=giθgiθ+geθψ+geθgiθVm(10)Ve=giθgiθ+geθψ-Vm.If we substitute (9) and (10) into the expressions in (6), we get (11)girVi+gerVe=giθgiθ+geθgir+gerψ-gir1-girgeθgergiθVm(12)giθVi+geθVe=giθψ.If we make the additional assumption that Vm is not a function of r, the second term in the square brackets of (11) does not contribute to (6), and we obtain(13)1λ21r2∂∂rr2∂ψ∂r+1r2sin⁡θ∂∂θsin⁡θ∂ψ∂θ=0,where(14)λ=giθ+geθgir+ger.


## 2.2. Boundary Conditions

The behavior at the interface between cardiac tissue and an adjacent volume conductor is determined by boundary conditions [15, 16]. At the surfaces of the shell, r=r1 and r=r2, we require that the normal component of the current density is continuous:(15)Jir+Jer=Jbrr=r1Jir+Jer=Jorr=r2and the extracellular potential is continuous(16)Ve=Vbr=r1(17)Ve=Vor=r2.The boundary conditions in (15)–(17) give at r=r1(18)∂ψ∂r=λ2gbgiθ∂Vb∂rψ=Vm+giθ+geθgiθVb.For the boundary r=r2, we replace Vb by Vo in (18).


## 2.3. Solution of the Equations for the Potential

If we expandψ in terms of Legendre polynomials [17], Pl(cos⁡θ), where l=0,1,2,3,…, then the general solution to (13) is [10](19)ψ=∑Alrν1+Blrν2Plcos⁡θ,where(20)ν=λ2ll+1+14,ν1=ν-12,ν2=-ν-12.The general solutions to Laplace’s equation in the blood and bath are(21)Vb=∑lClrlPlcos⁡θ,(22)Vo=∑lDlr-l+1Plcos⁡θ.The boundary conditions determine the constants Al, Bl, Cl, and Dl for each l. We expand Vm in terms of Legendre polynomials(23)Vm=∑lFlPlcos⁡θ,where(24)Fl=2l+12∫0πVmθPlcos⁡θsin⁡θdθ.Solving the four linear equations from the boundary conditions, we obtain the constants Al, Bl, Cl, and Dl for each l:(25)Al=-b1+b2b2a1-a2b1Fl,(26)Bl=a1-a2b2a1-a2b1Fl,(27)Cl=Alν1r1ν1+Blν2r1ν2λ2gb/giθlr1l,(28)Dl=-Alν1r2ν1+Blν2r2ν2λ2go/giθl+1r2-l+1,where(29)a1=r1ν11-gir+gergbν1l,b1=r1ν21-gir+gergbν2l,a2=r2ν11+gir+gergoν1l+1,b2=r2ν21+gir+gergoν2l+1.


## 2.4. Calculation of the Magnetic Field

Using Ohm’s law, we obtain the current density distribution(30)Jbθ=-1rgb∂Vb∂θr<r1,Jθ=Jiθ+Jeθ=-1rgiθ∂ψ∂θr1<r<r2,Joθ=-1rgo∂Vo∂θr>r2.Ampere’s law in its integral form is(31)∮B→·ds=μ0∬J→·dS=μ0Ienclosed,where Ienclosed is the current enclosed by the Amperian loop. We choose this loop to be the edge of a right circular cone with the apex at the center of the heart sphere and the integral surface is the surface of the right circular cone: (32)J→=-Jz^=-Jcos⁡θr^-sin⁡θθ^dS=-rsin⁡θdrdϕθ^.The magnetic field is in ϕ direction, tangential to this loop. By varying r and θ of this cone, we can get the magnetic field in all the space(33)B=μoIenclosed2πrsin⁡θ,where (34)Ienclosed=-∫02π∫0rJθrsin⁡θdrdϕ.In order to compute Ienclosed and thereby the magnitude ofB, we need to substitute the potential ((19), (21), and (22)) into the expression of current density (30), and then replace Jθ in (34) by these expressions of current density in each region r<r1, r1<r<r2, and r>r2, respectively. We then sum up the integrals in a piecewise manner for the three spaces in blood, bath, and tissue.In the bloodr<r1, (35)Vb=∑lClrlPlcos⁡θ,Jθ=Jbθ=-1rgb∂Vb∂θ=-gb∑lrl-1ClPl1,Ienclosed=-∫02π∫0rJθrsin⁡θdrdϕ=gb∑lClPl12πsin⁡θ∫0rrldr=gb∑lClPl12πsin⁡θrl+1l+1,Bϕ=μ02πrsin⁡θIenclosed=μ0gb∑lClPl1rll+1,(note: Pl1=dPl0/dθ).In the bathr>r2, (36)Vo=∑lDlr-l+1Plcos⁡θ,Jθ=Joθ=-1rgo∂Vo∂θ=-go∑lr-l+2DlPl1.Since the total current integrated over all the space is zero, we deduce(37)Ienclosed=-Ioutside=∫02π∫r∞Jθrsin⁡θdrdϕ=-go∑lDlPl12πsin⁡θ∫r∞r-l+1dr=-go∑lDlPl12πsin⁡θr-l-lr∞=-go∑lDl2πsin⁡θPl1r-ll,Bϕ=μ02πrsin⁡θIenclosed=-μ0go∑lDlPl1r-l+1l.In the tissue r1<r<r2(38)Vm=∑lFlPlcos⁡θ,ψ=∑Alrν1+Blrν2Plcos⁡θ,Jθ=Jiθ+Jeθ=-1r∂giθVi+geθVe∂θ=-1rgiθ∂Vi+geθ/giθVe∂θ=-1rgiθ∂ψ∂θ.The integral interval of (34) is split into two pieces: 0→r1 is I1 and r1→r is I2. (39)I1=-∫02π∫0r1Jθrsin⁡θdrdϕ=gb∑lClPl12πsin⁡θr1l+1l+1,I2=-∫02π∫r1rJθrsin⁡θdrdφ=giθ2πsin⁡θ∑lPl1Al∫r1rrν1dr+Bl∫r1rrν2dr.The total enclosed current is then (40)Ienclosed=I1+I2=2πsin⁡θPl1∑lgbClr1l+1l+1+giθAl1ν1+1r2ν1+1-r1ν1+1+giθBl1ν2+1r2ν2+1-r1ν2+1+goDlr2-ll.Using (27) and (28), we get(41)Ienclosed=giθ∑lPl12πsin⁡θAlν1+1rν1+1+Blν2+1rν2+1Bϕ=μ02πrsin⁡θIenclosed=μ0giθ∑lPl1Alν1+1rν1+Blν2+1rν2.An advantage of this formulation is that by varying r and θ the magnetic field is determined everywhere including within the heart wall.


## 3. Results

To perform calculations of the magnetic field, we need to consider a specific transmembrane potential distribution and values for the model parameters. We set the transmembrane potential to beVm = V2 = 20 mV for θ < 89° and Vm = V1 = −80 mV for θ > 91°. For the interval 89° < θ < 91°, Vm decreases linearly from 20 to −80 mV. The heart size is r1 = 30 and r2 = 40 mm [10]. Furthermore, we set the blood and the bath conductivities to gb= go = 1 S/m. The electrical conductivities of cardiac tissue are anisotropic: gir = 0.02, giθ = 0.2, ger = 0.08, and geθ = 0.2 S/m [18].Using the given transmembrane potential, we need to calculate its expansion in terms of Legendre polynomials.


### 3.1. The Calculation of Transmembrane Potential Expansion

To simplify the calculation, we change variables by lettingx=cos⁡θ. We let x1=cos⁡91°, x2=cos⁡89°, cos⁡0°=1, and cos⁡180°=-1, and then (24) becomes (42)Fl=2l+12∫-11VmPlxdx=∫-1x1+∫x1x2+∫x21VmPlxdx.For these integral intervals, ① refers to -1→x1, ② refers to x1→x2, and ③ refers to x2→1. ① is(43)∫-1x1VmPlxdx=V1∫-1x1Plxdx.Using the equation of a line, Vm-V2/x-x2=V1-V2/x1-x2, we get Vm=V2+V1-V2/x1-x2(x-x2). Therefore, ② is(44)∫x1x2VmPlxdx=∫x1x2V2+V1-V2x1-x2x-x2Plxdx=V2∫x1x2Plxdx-x2V1-V2x1-x2∫x1x2Plxdx+V1-V2x1-x2∫x1x2xPlxdx.We need to obtain ∫Pl(x)dx and ∫xPl(x)dx:(45)∫x1x2xPlxdx=12n+1xPn+1x-∫Pn+1xdxx1x2-xPn-1x-∫Pn-1xdxx1x2.③ is(46)∫x21V2Plxdx=V2∫x21Plxdx.The final result is the sum of integrals of the three segments. Figure 2 is a comparison of the calculated approximate transmembrane potential with the original transmembrane potential, showing excellent agreement.Figure 2
The transmembrane potential as a function of position over a cross section of the heart, (a) calculated forl = 200, and (b) the specified transmembrane potential. The color bars are in mV.
(a)
(b)Figure3 contains plots of the extracellular potential, calculated using (10), (16), and (17) with (19), (21), and (22). The potential falls off rapidly with distance into the bath or blood but is relatively large inside the cardiac tissue.Figure 3
The extracellular potential over a cross section of the heart. The dashed curves indicate the heart inner and outer surfaces. An area 40 mm by 40 mm is shown.


### 3.2. The Upper Bound of l of the Approximated Sums

The magnetic fieldB is calculated in terms of Legendre polynomials. The infinite sums given above (l from zero to infinity) must be approximated as finite sums. Figure 4 shows the change of the calculated peak value of the magnetic field (Bmax) as the total number of terms, l, in the sum increases. Bpeak changes by less than 3% for l > 200, and therefore we use an upper bound l = 200, which is enough to determine the magnitude of B accurately and is still easy to compute. When we choosel = 200, the point with the maximum magnitude, Bmax, is about 14 nT.Figure 4
The peak magnetic field versus the maximum number of terms in the Legendre polynomial expansion,l.Figure5 displays the ϕ-component of the magnetic field over a cross section of the heart. The field is largest near the inner and outer surfaces of the tissue and is relatively small throughout much of the heart wall. It has a peak magnitude, Bmax, of about 14 nT. Figure 6 shows the total current density (intracellular plus extracellular).Figure 5
The magnetic field over a cross section of the heart. The dashed curves indicate the heart inner and outer surfaces. An area 40 mm by 40 mm is shown.Figure 6
The current density (intracellular plus extracellular) over a cross section of the heart. The dashed curves indicate the heart inner and outer surfaces. An area 40 mm by 40 mm is shown.


## 3.1. The Calculation of Transmembrane Potential Expansion

To simplify the calculation, we change variables by lettingx=cos⁡θ. We let x1=cos⁡91°, x2=cos⁡89°, cos⁡0°=1, and cos⁡180°=-1, and then (24) becomes (42)Fl=2l+12∫-11VmPlxdx=∫-1x1+∫x1x2+∫x21VmPlxdx.For these integral intervals, ① refers to -1→x1, ② refers to x1→x2, and ③ refers to x2→1. ① is(43)∫-1x1VmPlxdx=V1∫-1x1Plxdx.Using the equation of a line, Vm-V2/x-x2=V1-V2/x1-x2, we get Vm=V2+V1-V2/x1-x2(x-x2). Therefore, ② is(44)∫x1x2VmPlxdx=∫x1x2V2+V1-V2x1-x2x-x2Plxdx=V2∫x1x2Plxdx-x2V1-V2x1-x2∫x1x2Plxdx+V1-V2x1-x2∫x1x2xPlxdx.We need to obtain ∫Pl(x)dx and ∫xPl(x)dx:(45)∫x1x2xPlxdx=12n+1xPn+1x-∫Pn+1xdxx1x2-xPn-1x-∫Pn-1xdxx1x2.③ is(46)∫x21V2Plxdx=V2∫x21Plxdx.The final result is the sum of integrals of the three segments. Figure 2 is a comparison of the calculated approximate transmembrane potential with the original transmembrane potential, showing excellent agreement.Figure 2
The transmembrane potential as a function of position over a cross section of the heart, (a) calculated forl = 200, and (b) the specified transmembrane potential. The color bars are in mV.
(a)
(b)Figure3 contains plots of the extracellular potential, calculated using (10), (16), and (17) with (19), (21), and (22). The potential falls off rapidly with distance into the bath or blood but is relatively large inside the cardiac tissue.Figure 3
The extracellular potential over a cross section of the heart. The dashed curves indicate the heart inner and outer surfaces. An area 40 mm by 40 mm is shown.


## 3.2. The Upper Bound of l of the Approximated Sums

The magnetic fieldB is calculated in terms of Legendre polynomials. The infinite sums given above (l from zero to infinity) must be approximated as finite sums. Figure 4 shows the change of the calculated peak value of the magnetic field (Bmax) as the total number of terms, l, in the sum increases. Bpeak changes by less than 3% for l > 200, and therefore we use an upper bound l = 200, which is enough to determine the magnitude of B accurately and is still easy to compute. When we choosel = 200, the point with the maximum magnitude, Bmax, is about 14 nT.Figure 4
The peak magnetic field versus the maximum number of terms in the Legendre polynomial expansion,l.Figure5 displays the ϕ-component of the magnetic field over a cross section of the heart. The field is largest near the inner and outer surfaces of the tissue and is relatively small throughout much of the heart wall. It has a peak magnitude, Bmax, of about 14 nT. Figure 6 shows the total current density (intracellular plus extracellular).Figure 5
The magnetic field over a cross section of the heart. The dashed curves indicate the heart inner and outer surfaces. An area 40 mm by 40 mm is shown.Figure 6
The current density (intracellular plus extracellular) over a cross section of the heart. The dashed curves indicate the heart inner and outer surfaces. An area 40 mm by 40 mm is shown.


## 4. Discussion

The calculation of the magnetic field throughout the heart indicates that the peak magnetic field has a magnitude of about 14 nT. This is a similar order of magnitude as calculated by Roth [19] for a cylindrical strand of cardiac tissue, such as a papillary muscle, and is several times larger than the magnetic field at the apex of the heart (2 nT) [20, 21], in planar tissue samples (1 nT) [22], or outside the body during measurements of the magnetocardiogram (0.05 nT) [23]. The magnetic field is much larger at the tissue surface than just a few millimeters away from the surface, either in the surrounding bath or deeper in the tissue, which is again similar to the results found in a tissue strand [19]. This behavior arises because deep in the tissue the intracellular and extracellular currents are in the opposite directions with almost the same magnitude, and therefore they tend to cancel each other.The calculation is based on a piecewise linear model for the transmembrane potential. The upstroke is distributed over 2° inθ, which is 0.035 radians. In the middle of the heart wall, the radius is 35 mm, implying that the upstroke occurs over a distance of 1.22 mm. This is a slightly slower upstroke than is typically observed in cardiac tissue. The rapid rise of the action potential occurs in about 1 ms, and the conduction velocity is about 0.5 m/s parallel to the fibers, implying an upstroke distributed over about 0.5 mm. We performed a calculation with the upstroke over 1° and found the peak magnitude of the magnetic field is about 16 nT, a 13% increase compared to 2° case.The cardiac tissue is assumed to have unequal anisotropy ratios [18, 24]. However, this condition is not essential for producing the magnetic field. We repeated the calculation for equal anisotropy ratios, gir = 0.032, giθ = 0.2, ger = 0.032, and geθ = 0.2 S/m [18, 24], and found that Bmax changed only slightly, to 12 nT.Our calculation is based on a simplified model. The heart is spherical, the fiber geometry and wave front are idealized, and the transmembrane potential is uniform across the heart wall. These assumptions are not physiologically accurate but are necessary in order to calculate the biomagnetic field analytically. More realistic calculations would require numerical methods.


### 4.1. How Action Currents Affect MRI

The groundbreaking work of Joy and his colleagues demonstrated that MRI can be used to measure current by detecting its resulting magnetic field [25–27]. However, much of this work has examined applied current, such as during defibrillation [28]. Detecting biocurrents is a greater challenge, because they are so small.In living neural or cardiac tissues there exist various physiological processes such as blood flow, diffusion, and chemical exchange. All these physiological processes alter the magnetic field surrounding the atomic nuclei in ways that affect the magnetic resonance behavior of the nuclei [29]. Therefore, special MRI acquisition methods were designed or need to be designed accordingly to observe different physiological functions.Functional magnetic resonance imaging (fMRI) makes use of the Blood Oxygenation Level Dependent (BOLD) signal changes [30]. Briefly, deoxyhemoglobin in blood is paramagnetic which will distort the external magnetic field of nearby hydrogen nuclei spins and further affect the MR image. When part of brain is active, the vascular system provides more blood and thereby reduces the fraction of blood that is converted from oxyhemoglobin into deoxyhemoglobin, and the magnetic distortion decreases. BOLD fMRI provides a method to measure oxygenation perfusion and thereby to indirectly estimate brain activity. However the spatial and temporal distribution of perfusion may differ from the distribution of electrical activity. Therefore, researchers would like to measure electrical activity directly using MRI.Can the magnetic field produced by bioelectric currents be used as the gradient field for MRI? Theoretically, it is possible. The question is if the resulting phase shift is large enough to detect by current MRI systems. The phase shiftΦ of the MRI signal is given approximately by [5](47)Φ=γBΔt,where γ = 267 × 106 radians/(s T) is the gyromagnetic ratio for protons and Δt is the time over which the phase accumulates (the duration of the magnetic field for brief biomagnetic signals or a duration related to the MRI pulse sequence such as the echo time for long duration biomagnetic signals).The phase shift determines if the electrical activity can be detected by the MRI system. In the brain, although the phase shift produced by a single neuron firing is not detectable, neural firings are not isolated events; in fact they are correlated with each other. There are two ways to increase the MR signal so that it is detectable. One is doing spatial and temporal integrals of neuronal firings over an MRI voxel according to the morphologies and the physiology of the neuronal events; the other one is altering the acquisition method such as using special pulse sequences to increase phase shift. Many researchers have been doing this both experimentally and theoretically [1–8, 31–34]. The MRI signals they measure are very small, and it is still unclear if they are detectable, whether the results are “fantasy, possibility, or reality” [3]. Recently, Sundaram et al. [5] performed neural current imaging in an intact turtle cerebellum using MRI. They were able to detect biomagnetic fields of strength of up to one nanotesla and phase shifts of about a third of a degree. These measurements required signal averaging with thousands of repetitions, imaged long duration metabotropic-induced neural signals lasting nearly a second, and had no interference caused by the vascular response (BOLD signal), tissue motion related to pulsatile blood flow, or other sources of physiological noise. Under these idealized conditions, phase shifts on the order of a third of a degree are detectable.


### 4.2. Phase Shifts by Action Currents in the Heart

The brain is not the source of the largest biocurrents in the body; the heart is, so it may be easier to detect cardiac action currents using MRI compared to neural action currents. Deep in the heart, the magnetic field is small since the intracellular and the extracellular currents have almost the same value but are in opposite directions. However near the surface of the heart the extracellular current leaves the heart with the “return” loops through an adjacent volume conductor which produces a significant magnetic field [11].From this analysis, we have achieved our original goal of computing the maximum value of the magnetic field produced by the action currents near the surface of the heart, 14 nT, which is significantly larger than any magnetic field produced by neuronal activation. For a firing time of 1 ms, this magnetic field can produce a maximum phase shift of 0.21° by (47). This phase shift is similar to that measured by Sundaram et al. [5], implying that the cardiac signal may be just measurable using MRI. However, these signals may be smaller than artifacts caused by heart motion and blood flow effects in a beating heart. Therefore, in vivo measurements of cardiac biocurrents in humans remain a challenge. Perhaps more promising would be to search for such signals in a Langendorff-perfused heart with an electromechanical uncoupler to eliminate contractions. In such an in vitro system—somewhat analogous to the in vitro turtle cerebellum model used by Sundaram et al. [5] to study signals from the brain—biocurrents in the heart may be measureable. Our conclusion is that the magnetic field in the heart may be detectable under idealized conditions but that using MRI to measure action currents in the humans will be difficult.


### 4.3. The Potential of Detecting Cardiac Arrhythmias Using MRI

“Arrhythmia” is a term describing conditions in which the heart is beating too fast (tachycardia), too slow (bradycardia), or irregularly. Abnormal electrical activity exists in the heart in each of these conditions.The usual way to detect a cardiac arrhythmia is using an electrocardiogram (ECG). Magnetic fields are also sometimes measured (the magnetocardiogram [23], or MCG). In MCG measurements, the biomagnetic field is measured outside the body and a difficult inverse problem must be solved to obtain the current sources that produce the magnetic field [35]. Simplifying assumptions are required to make the inverse problem solvable and stable, but these assumptions make the resulting current source distribution less precise and detailed. MRI would provide a way to measure the magnetic field inside the heart directly, avoiding the inverse problem of deducing the field inside the heart from measurements far outside the heart. Much research has been done that attempts to measure the magnetic field of a nerve using MRI. However, the heart, which produces a much larger magnetic field than nerves, may be a better place to start when attempting to use MRI for measurement of biocurrents. Of course, many challenges exist, not the least of which is the motion of the beating heart and the differences between the paramagnetic susceptibility of the blood within the heart, the cardiac tissue, and the tissues outside the heart [36, 37]. However, if successful, obtaining biocurrents from an MRI of the heart would provide a closer look at the function of the heart’s electrical conduction system and should help to diagnose a large number of cardiac arrhythmias.


## 4.1. How Action Currents Affect MRI

The groundbreaking work of Joy and his colleagues demonstrated that MRI can be used to measure current by detecting its resulting magnetic field [25–27]. However, much of this work has examined applied current, such as during defibrillation [28]. Detecting biocurrents is a greater challenge, because they are so small.In living neural or cardiac tissues there exist various physiological processes such as blood flow, diffusion, and chemical exchange. All these physiological processes alter the magnetic field surrounding the atomic nuclei in ways that affect the magnetic resonance behavior of the nuclei [29]. Therefore, special MRI acquisition methods were designed or need to be designed accordingly to observe different physiological functions.Functional magnetic resonance imaging (fMRI) makes use of the Blood Oxygenation Level Dependent (BOLD) signal changes [30]. Briefly, deoxyhemoglobin in blood is paramagnetic which will distort the external magnetic field of nearby hydrogen nuclei spins and further affect the MR image. When part of brain is active, the vascular system provides more blood and thereby reduces the fraction of blood that is converted from oxyhemoglobin into deoxyhemoglobin, and the magnetic distortion decreases. BOLD fMRI provides a method to measure oxygenation perfusion and thereby to indirectly estimate brain activity. However the spatial and temporal distribution of perfusion may differ from the distribution of electrical activity. Therefore, researchers would like to measure electrical activity directly using MRI.Can the magnetic field produced by bioelectric currents be used as the gradient field for MRI? Theoretically, it is possible. The question is if the resulting phase shift is large enough to detect by current MRI systems. The phase shiftΦ of the MRI signal is given approximately by [5](47)Φ=γBΔt,where γ = 267 × 106 radians/(s T) is the gyromagnetic ratio for protons and Δt is the time over which the phase accumulates (the duration of the magnetic field for brief biomagnetic signals or a duration related to the MRI pulse sequence such as the echo time for long duration biomagnetic signals).The phase shift determines if the electrical activity can be detected by the MRI system. In the brain, although the phase shift produced by a single neuron firing is not detectable, neural firings are not isolated events; in fact they are correlated with each other. There are two ways to increase the MR signal so that it is detectable. One is doing spatial and temporal integrals of neuronal firings over an MRI voxel according to the morphologies and the physiology of the neuronal events; the other one is altering the acquisition method such as using special pulse sequences to increase phase shift. Many researchers have been doing this both experimentally and theoretically [1–8, 31–34]. The MRI signals they measure are very small, and it is still unclear if they are detectable, whether the results are “fantasy, possibility, or reality” [3]. Recently, Sundaram et al. [5] performed neural current imaging in an intact turtle cerebellum using MRI. They were able to detect biomagnetic fields of strength of up to one nanotesla and phase shifts of about a third of a degree. These measurements required signal averaging with thousands of repetitions, imaged long duration metabotropic-induced neural signals lasting nearly a second, and had no interference caused by the vascular response (BOLD signal), tissue motion related to pulsatile blood flow, or other sources of physiological noise. Under these idealized conditions, phase shifts on the order of a third of a degree are detectable.


## 4.2. Phase Shifts by Action Currents in the Heart

The brain is not the source of the largest biocurrents in the body; the heart is, so it may be easier to detect cardiac action currents using MRI compared to neural action currents. Deep in the heart, the magnetic field is small since the intracellular and the extracellular currents have almost the same value but are in opposite directions. However near the surface of the heart the extracellular current leaves the heart with the “return” loops through an adjacent volume conductor which produces a significant magnetic field [11].From this analysis, we have achieved our original goal of computing the maximum value of the magnetic field produced by the action currents near the surface of the heart, 14 nT, which is significantly larger than any magnetic field produced by neuronal activation. For a firing time of 1 ms, this magnetic field can produce a maximum phase shift of 0.21° by (47). This phase shift is similar to that measured by Sundaram et al. [5], implying that the cardiac signal may be just measurable using MRI. However, these signals may be smaller than artifacts caused by heart motion and blood flow effects in a beating heart. Therefore, in vivo measurements of cardiac biocurrents in humans remain a challenge. Perhaps more promising would be to search for such signals in a Langendorff-perfused heart with an electromechanical uncoupler to eliminate contractions. In such an in vitro system—somewhat analogous to the in vitro turtle cerebellum model used by Sundaram et al. [5] to study signals from the brain—biocurrents in the heart may be measureable. Our conclusion is that the magnetic field in the heart may be detectable under idealized conditions but that using MRI to measure action currents in the humans will be difficult.


## 4.3. The Potential of Detecting Cardiac Arrhythmias Using MRI

“Arrhythmia” is a term describing conditions in which the heart is beating too fast (tachycardia), too slow (bradycardia), or irregularly. Abnormal electrical activity exists in the heart in each of these conditions.The usual way to detect a cardiac arrhythmia is using an electrocardiogram (ECG). Magnetic fields are also sometimes measured (the magnetocardiogram [23], or MCG). In MCG measurements, the biomagnetic field is measured outside the body and a difficult inverse problem must be solved to obtain the current sources that produce the magnetic field [35]. Simplifying assumptions are required to make the inverse problem solvable and stable, but these assumptions make the resulting current source distribution less precise and detailed. MRI would provide a way to measure the magnetic field inside the heart directly, avoiding the inverse problem of deducing the field inside the heart from measurements far outside the heart. Much research has been done that attempts to measure the magnetic field of a nerve using MRI. However, the heart, which produces a much larger magnetic field than nerves, may be a better place to start when attempting to use MRI for measurement of biocurrents. Of course, many challenges exist, not the least of which is the motion of the beating heart and the differences between the paramagnetic susceptibility of the blood within the heart, the cardiac tissue, and the tissues outside the heart [36, 37]. However, if successful, obtaining biocurrents from an MRI of the heart would provide a closer look at the function of the heart’s electrical conduction system and should help to diagnose a large number of cardiac arrhythmias.



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*Source: 3035479-2017-05-10.xml*