Forward and Inverse Potential Field Solutions for ... - IEEE Xplore

IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. BME-32, NO. 8, AUGUST 1985

566

Forward and Inverse Potential Field Solutions for Cardiac Strands of Cylindrical Geometry NIRMALA GANAPATHY, JOHN W. CLARK, JR., OWEN B. WILSON,

Abstract-This paper deals with the classical forward and inverse volume conductor field problems associated with the isolated active cardiac muscle preparations of cylindrical geometry. Specifically, these are the Purkinje fiber, the atrial trabeculum, and the (idealized) single atrial cell. The electrical behavior of the multicellular preparations (Purkinje strand and atrial trabeculum) is modeled in terms of the electrical activity of an equivalent single cell, with a representative membrane that separates an anisotropic intracellular medium from an isotropic extracellular medium. The isolated single atrial fiber is considered an interesting special case anid is modeled in an idealized sense as a long cylindrical cell with an isotropic internal medium. A model based on potential theory is developed for the equivalent cardiac cell; it is based on a solution of Laplace's equation in the media of interest, subject to appropriate boundary conditions. The solution for potential at an arbitrary point in the extracellular medium is in the form of a Fourier integral; the equation is subsequently reformulated into a more convenient computational form using a discrete Fourier transform (DFT) method. Implementation of this method, using a fast Fourier transform (FFT) technique, results in a fast and efficient numerical algorithm for the calculation of volume conductor potentials. A benefit of this approach is that the classical forward and inverse problems in electrophysiology may be viewed as equivalent filtering problems. Thus, not only can volume conductor field potentials at various distances from the strand be easily and rapidly computed, but given field potential data, good estimates of the action potential waveform can also be obtained provided the signal-to-noise ratio is adequate.

INTRODUCTION

THE relationship between the cellular transmembrane 1 potential and the potential at any point in surrounding extracellular space is a fundamental problem in cardiac electrophysiology. This relationship is very difficult to study experimentally in the single active cardiac cell since it is so small. Many investigators, however, have anatomically and morphologically studied multicellular structures that form into relatively long strands such as Purkinje fibers of the specialized conduction system l1], [2] and atrial trabeculae [3]. The geometry of each of these strands may be approximated by a cylinder with a radius a that is much, much smaller than its length. Both strands are clearly multicellular in nature, but they function electrically as a syncytium. To a significant degree, an action potential recorded from one component cell of the strand is very similar to that recorded from any other cell in the strand. Therefore, to a first approximation, a given strand Manuscript received May 11, 1984; revised January 29, 1985. This work was supported in part by the National Science Foundation under Grant BNS 8014052. N. Ganapathy, J. W. Clark, Jr., and 0. B. Wilson are with the Department of Electrical Engineering, Rice University, Houston, TX 77001. W. Giles is with the Department of Medical Physiology, University of Calgary, Alta., Canada.

AND

WAYNE GILES

may be considered to be electrically equivalent to a single cylindrical cell with a membrane of radius a and essentially infinite length (Fig. 1). Also, to a first approximation, the bulk intracellular medium of the equivalent cell may be considered to be uniform, homogeneous and anisotropic with a preferential conduction in the z direction (Fig. 1), and smaller, equal specific conductivities in the radial and angular (0) directions. The specific conductivity in the z direction is denoted as oiz (S/cm),while that in the transverse direction is labeled a,, (S/cm). The extracellular bathing medium is considered uniform, homogeneous, and isotropic, and is characterized by the specific conductivity a,, (S/cm). This single "equivalent cell" approach to the multicellular preparation has been taken previously by Spach et al. [4] and Harman et al. [5] for the case of an isolated, active Purkinje strand with an isotropic internal medium. The action potential waveform assumed for the equivalent cell is considered to propagate at constant velocity along the length of the cardiac strand. This type of model is used to predict extracellular volume conductor field potentials due to the passage of the action potential, and little attention is paid to the collective, idealized intracellular medium, although the potential distribution in this medium could be computed as well. Spach et al. [4] and Harman et al. (5] have shown that the model-generated extracellular potential waveforms, produced by using potential theory models of this type, yield good fits to experimentally recorded extracellular potential data, and are useful in that sense. If structural complexities of the Purkinje strand are introduced, such as regions of nonuniformity and inhomogeneity in the cellular composition of the strand, local changes in the axial and transverse resistance will occur which will have an effect on current flow and propagation velocity [6]. Although these effects may be important, model complexity would also have to increase significantly in order to study them. As a compromise, we will consider the bulk properties of the internal medium of the equivalent Purkinje cell to be anisotropic,' and characterized by a transverse specific conductivity ui, and a longitudinal specific conductivity vi. The objective of the present model, however, is to focus on rapid forward-inverse solutions for the potential distribution in the extracellular volume conductor for the lumped equivalent cardiac cell. These methods will hopefully prove to be useful in the future in obtaining volume conductor field potentials as-

0018-9294/85/0800-0566$01.00 © 1985 IEEE

567

GANAPATHY et al.: SOLUTIONS FOR CARDIAC STRANDS OF CYLINDRICAL GEOMETRY

I

verse specific conductivities ai,, and ai,,, respectively; 4) the only source for the potential field resides in the membrane of the "equivalent cardiac cell;" 5) field propagation effects are neglected (quasi-stationr arity is assumed) and; 6) angular symmetry (a/la = 0) prevails. One may obtain scalar potential in the extracellular mez,oIt dium cV(r, z) by solving the Laplace equation subject z to appropriate boundary conditions. A solution for the potential everywhere in the intracellular medium of the Fig. 1. Fiber geometry. The external medium is considered uniform, ho- equivalent cardiac cell could likewise be obtained by solvmogeneous, and isotropic with a specific conductivity a, (S/cm); the in- ing Laplace's equation in that medium. However, the reternal medium is assumed to be uniform, homogeneous, and anisotropic, and it is characterized by a longitudinal specific conductivity u and a sult would have dubious significance due to the gross modtransverse specific conductivity ai,, (S/cm). eling assumptions made regarding the composition of the Purkinje strand or the atrial trabeculum. Rather, the focus sociated with more complex equivalent cellular models, of this study is the extracellular potential field arising from such as those based on Hodgkin-Huxley-type models of the cardiac membrane source, which may be affected by the bulk resistive anisotropy of the strand itself. cardiac strands (e.g., [7]). It is also possible to isolate individual cardiac cells of different types (e.g., ventricular [8]-[10], atrial [11], [12], THE FORWARD PROBLEM SA and AV node (13], and frog sinus venosus cells [14]) The expression for extracellular potential Vb at an arusing enzymatic dispersion techniques. Recent studies bitrary field point P(r, z) from an arbitrary isolated cylinhave successfully applied special voltage clamp tech- drical cell of radius a and essentially infinite length is niques to the electrophysiological study of the membrane given as properties of these isolated cells, yielding quantitative data I that describe the voltage and time dependence of the com4°(r, z) = 2w (1) ponent membrane currents. A relatively simple procedure J-O0 Ko(IkIr)Fm(k)e7-kzdk a(XlkIaIKo(IkIa) in which fragments of the right atrial wall of the frog heart are enzymatically digested, yielding single atrial cells, has where been developed in the laboratory of one of us (W. G.). The + a (2) x(Xlkla)-= typical isolated cell has a diameter of approximately 6 gm Ko(lkIa)Io(XlkIa) and a length of 200 1tm (see [11, Fig. 1]). Action poten11/2 tials recorded from this type of cell were of normal wav(3) eshape and resting potential (- -90 mV). As a crude apUi,t proximation, one could neglect tapering and end effects and consider the single atrial cell to be of essentially in- and finite length-in essence, a smaller version of the Purkinje U0 (T _ (TO0 a iand atrial trabecular strands mentioned previously. In this (4) i,t case, the intracellular medium would be uniform, homogeneous, and isotropic. We will include the single, iso- Details of the derivations of (1) are given in the Appendix. lated atrial cell as a third type of cardiac fiber in our study. Here, Fm(k) is the Fourier transform of the transmembrane potential c'-m(Z), and the I, and Kn functions are THE MODEL modified Bessel functions of the first and second kind, orThe approach taken in studying the forward and inverse der n (see [15], [16]). One will note that the general form aspects of this problem is based on previous "equivalent of the expression for extracellular potential (1) is of the source" modeling studies of Clark and Plonsey [15], [16] same form as that described previously by Clark and Plonthat develop and extend a field-theoretic model of the sey [16] for the case of the single unmyelinated nerve fiber intra- and extracellular potential distributions about an with an isotropic internal medium. The internal anisoisolated active nerve fiber. The present cardiac model as- tropy ratio X given by (3) appears only in the function sumes that: ci(Xlkla) defined by (2), where, for a value of X = 1 (iso1) the action potential is conducted uniformly along the tropic case), t(Xlkla) is identical to the x function destrand with a conduction velocity v; Therefore, an arbi- fined in [16]. Noting that (1) has the form of a Fourier trary field quantity I varies as I(z - vt); integral for some constant value of radial distance (r), let 2) the extracellular medium is purely passive and char- us define acterized by a specific conductivity parameter ao; F°(kr) H(Iklr)Fm(k) (5) 3) the intracellular medium is purely passive, is anisotropic, and characterized by the longitudinal and trans- where

-LI

58IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. BME-32, NO. 8, AUGUST 1985

568

H(IkIr) H(Iklr)

~Ko(IkIr)(Ikla) _W(IkIr) -a(XIkIa)KO c(XIk1a)

Therefore, with regard to the forward problem, the poten-

(6) tial distribution in z at an arbitrary radial distance r from

and

W(IkIr)

7Ko(IkI)

Here, H(Iklr) is referred to as the overall or "combined" filter function, ca -(XI kIa) is the membrane filter function, and W(Iklr) is the "bathing medium" filter function according to the terminology employed earlier in [17]. Substituting Fm(k) as determined by (5) into (1), one obtains the following Fourier transform pair at radius r in the external medium:

V'(r, z)

F° (kr)e-ikzdk

=

=2i 9' H(Iklr)Fm(k)e-jkzdk F°(kr) = =

(8)

0(r, z)eikzdz

-00o

H(Iklr)Fm(k).

(9)

At the outer membrane surface of the fiber, the potential 4'o(z) is given by

4bs(z)

40(a,z)

=-

27

- c.

FO(ka)e-jkzdk.

(10)

Defining the Fourier transform of the outer membrane surface potential as

Fso(k)= F0(ka)

H(Ikla)Fm(k)

=

=

( ) (11) ox(XIkIa)

we establish,via (10), the following Fourier transform pair at the membrane surface of the equivalent cell: 00

bso(z)

Fso(k)eJkzdk

=

aQm(ki)

I

e

-jkzdk

(12)

00

Fso(k) =

-

00

4so(z)eJkzdz.

(13)

Note here that at r = a, the medium filter function W( Ikia) is equal to one. Since via (11), Fm(k) = ac(XIkja)FsO(k) and H(Iklr) = a-c(XIkIa)W(tkIr) in (6), the potential at an arbitrary field point in the extracellular medium (8), may also be expressed in terms of the Fourier transform pair

4'(r, z) = F°(kr)

3 Fso(k)W(ikIr)e-jkzdk 21r - oo ck

= -

(r,z)eJ zdz

=

(14)

F50(k) W(Ikl r). (15)

the cylindrical cell (or fiber) may be computed by specification of the Fourier transforms of either the transmembrane or outer membrane surface potential distributions, according to (8) and (14), respectively. Thus, the Fourier transform of 4°(r, z) may be regarded as the result of a two-stage filtering process applied to the transmembrane potential 4m(Z) [17]. This is shown diagrammatically in Fig. 2 where the first stage is associated with the membrane [i.e., (11)], where the filter function is [oz-1(XIkja)], and the second stage is characteristic of the external medium [i.e., (14), where the filter characteristic is W(Iklr)]. The characteristics of the individual filters are shown in Figs. 3 and 4. Fig. 3 is a plot of the membrane filter characteristics c-C'(XIkIa) for a 36 ,tm diam atrial trabeculum as a function of spatial frequency k for various different conductivity ratios 6, where 6 is defined by (4). The individual filter characteristics have the shape of a family of parabolas, and a-'(XIkIa) goes to zero at k = 0. As is well known from cable theory [15], the transmembrane current per unit length (im) entering the extracellular medium is related to the second spatial derivative of the transmembrane potential distribution d24 m(Z)IdZ2. Since in the passive extracellular medium, current is directly related to potential, the extracellular surface potential 4',,(z) will also be directly related to the second derivative of the transmembrane potential distribution. The membrane, therefore, acts as a second derivative filter that has a parabolic characteristic in the k domain. Since a-'(XIkIa) goes to zero at k = 0, the dc component of the input transmembrane potential 4m(Z) is completely attenuated by the membrane filter. Consequently, any dc offset potential associated with the 4.m(Z) waveform (e.g., the resting potential) cannot contribute to either the current traveling through the membrane (im) or the outer membrane potential distribution 4),(z). It should also be noted that the dc resting potential associated with 'm(Z) will not be able to be recovered in the inverse problem. One will also note from Fig. 3 that the overall gain of the filter varies with the conductivity ratio 6 which is a function of X [see (4)]. Specifically, as 6 decreases, the gain of the filter ax- (XIkIa) increases. The modified Bessel functions Io and I1 in the expression for a(Xlkla) (2) have arguments that are functions of X, k, and a. Table I shows the range on Xlkla (for a specific value of X = 4) for all the three cases studied here. The maximum value of XI kIa is well below 2 for both types of fiber studied and, therefore, the modified I and K Bessel functions may be approximated by polynomial expressions of their arguments XIkIa and Ikla, respectively [18], over the range 0 < Xjkla < 2. Subsequently, the ratio I0(XIkIa)K1(IkIa)/ Id(XIkla)Ko (Ikla) in (2) can be simplified into a ratio (R) of second-order polynomials given as

Io(Xlkla)Kl(lkla) IR(Xlkla)Ko(lkla)

R _

(16)

569


F [ki

80k

1/anlkia]

F0[k,r]

W[Iklr] Medium Filter

Membrane Filter

.8

Fig. 2. Representation of the forward problem as an equivalent two-stage filtering problem. Fm,,, F'o, and F° are the Fourier transforms of the transmembrane, outer membrane surface, and field potential distributions, respectively. The membrane filter is ax-(Xjkja), and W(Iklr) is the medium filter-see text.

r=3a

-.8

8 =.71

CO

O x

-

-V

-.4F

0

11

5)

Fig. 4.

8=5

O0_ 0

2

1

I

I

3

4

,

I

5

klrad/cmi Fig. 3. The membrane filter function et (XIkIa) versus k for various specific conductivity ratios (5 =00;/,). The case chosen is for a cylindrical atrial trabeculum having a radius of 18 ,um.

1+

o1X

[375 2

--

2

+

61

+4

+

ln(IkIa/2)1 ik12a2 2

Ik12a2 [ln(IkIa/2)

+

'ye]

where (xl, 'yo, and 61 are constants associated with the polynomial expansions of Io, Ko and Kl, respectively. From [18], the constants (xl, 'yo, and 61 are 3.5156229, 0.57721566, and 0.15443144, respectively. The approximation (16) agrees very well with table values [18] over the range of argument values used (Table I). A sensitivity analysis of the ratio (16) was performed to determine its sensitivity to variations in X. The relative sensitivity function (XO3RIRoAX) was found to vary over a range less than 0.002 with variations in X (for the worst case where XIkla = -0.6 (see Table I). Here, Xo represents a nominal value of the parameter X, and Ro is the corresponding ratio function R (XoIk1a). Clearly, the ratio is fairly insensitive to changes in X. One can therefore conclude that changes in anisotropy ratio X2 are linked mainly

1

2

3

4

5

k(rad/cml The medium filter function W(Iklr) versus k for four specific field point radii, atrial trabeculum, a = 18 ,lm.

to changes in 6 given by (4), and thereby to changes in the gain of the membrane filter function a1-'(XIkIa). This being the case, it is interesting to note from (4) that, for a constant extracellular specific conductivity (a0), 6 varies inversely with the average specific conductivity 4,;;;. In the case of the filter W(lkjr), associated with the external medium of the trabeculum (Fig. 4), one will note that as r increases, the degree of attenuation of the filter W(Iklr) increases with increasing values of k. The transfer function W(Iklr), therefore, behaves as a low-pass filter; i.e., the further from the fiber the potential distribution is recorded, the more its energy will be concentrated at lower spatial frequencies. The overall combined filter function H(Iklr) is shown in Fig. 5 for the atrial trabecular case (a = 18 ,m). Clearly, the gain of this filter decreases as the field observation point moves further from the surface of the trabeculum. The plot of the membrane filter aC(X IkIa) for various fiber radii is shown in Fig. 6; for sake of comparison, the conductivity ratio 6 is held at a value of 2 in each case. The uppermost curve (a = 100 Mm) represents the membrane filter for the Purkinje fiber, while the lower (a = 3 Mm) represents the membrane filter characteristic of the single atrial fiber. One will note that there is a very dramatic increase in filter gain with increasing fiber radius. Thus, in the forward problem, one is given the filter characteristics W(Iklr) and. a-1(XIkIa) and the transmembrane potential distribution cIm(z); the Fourier transform of 4m(Z) is Fm(k), and (5) specifies F"(kr). The inverse transform of F°(kr) yields the desired result-the potential distribution h10(r,z) at an arbitrary radial distance from the fiber.


570

TABLE I RANGE ON THE ARGUMENT OF THE MODIFIED I BESSEL FUNCTIONS FOR THE THREE CASES CONSIDERED IN THIS STUDY USING A WORST-CASE ANISOTROPY RATIO (X2) OF 16

Tissue Type

Atrial single cell Atrial trabeculum Purkinje fiber

Range on X 1k I-a 0-0.018 0-0.108 0-0.6

CO

Y-

J9

-1

Fig. 6. The membrane filter function a 1(Xlkta) as a function of radius a. The conductivity ratio 6 is again chosen to be equal to 2.

klrad/cml Fig. 5. The combined filter function H(IkIr) for the atrial trabeculum as a function of k for three specific field-point radii; a = 18 ym and 6 is chosen to be equal to 2.

THE IDEALIZED INVERSE PROBLEM

The procedure for solving the inverse problem in the k domain is analogous to the procedure outlined in the previous section for the forward problem. One is given the same filter characteristics and the potential distribution 4(r, z). The objective is to solve for Fm(k) via (5), i.e., Fm(k) = a(Xlkla) W-'(IkIr)FO(kr). (17) Similarly, to obtain Fs,(k), one employs (15) so that (18) Fso(k) = W1(Iklr)FO(kr). The validity of these equations depends on the existence of the filter functions a(Xlkla) and W-1(Iklr). From the definitions given in (2) and (7) and Figs. 3 and 4, it follows that (19) a(Xlkla) oo as k 0 as k -X oo. (20) Fig. 7 is a plot of the ideal overall inverse filter function H 1(Ik r) computed at three specific field radii r = 3a,

W-1(Iklr)

-0

1

k(rad/cml

2

Fig. 7. Inverse filter function for the 18 ym atrial trabeculum at three specific field radii.

7a, 15a, for the 18 ,u.m radius trabeculum. Since, as previously discussed, the dc component of the transmembrane potential distribution cannot be recovered, the dc component of the inverse membrane filter c4(Xlkla) is set equal to zero. Consequently, H-1(Iklr) is zero at k = 0. The nature of the ideal inverse filter H- (Ik r) is generally

571


TABLE II the same for all three types of fibers (Purkinje, atrial traOF THE INVERSE FILTER FOR THE BANDWIDTH AND GAIN MAXIMUM beculum, and single atrial fiber). All peak in the very lowTHREE CASES frequency range, but have different gains. The gain of the Bandwidth inverse filter function for the single atrial fiber is much Magnitude (rad/cm) of Gain Fiber greater than that for the Purkinje fiber as can be seen from Table IL which is constructed using a field radius r = 9a; Atrial Cell 1.17 1.85 * 10'0 the bandwidth, however, is essentially constant for all three Atrial Trabeculum 1.27 6.23 * 108 0.137 2.54 * 107 cases. The importance of the particular form of these ideal Purkinje fiber inverse filter functions to the overall recovery process, especially when the input field potential cb(9a, z) to the inverse filter is corrupted with measurement noise, will be discussed in greater detail in the Results section.

DISCRETE FOURIER METHODS OF SOLUTION OF THE FORWARD AND INVERSE PROBLEMS The continuous z-domain expressions for the various potentials [4?m(Z), I(r, z), and 4,0(z)I and their Fourier transforms [Fm(k), F0(kr), and FSo(k)] can be reformulated in the discrete k domain for implementation on a digital computer by introducing the'following DFT pair (transmembrane potential is used as an example): N-1

Fm(Pq)

=

bm(Zn)

=

E

n =O,

where

-m(Zn)e 2tnqlN

DFT[4tm(Zn)]

1N-1

E F (Pq)e J2TnqlN = N q==om -

(21)

IDFT[Fm(Pq)] (22)

z(cm] (a)

P-2irNZ.

(23) Here, Z and P are the sampling intervals in the z and k domains, respectively, and n and q are integers. The function 1%m(Z) is normally limited in both the z and k domains, meaning that 4m(Z) (see Fig. 8) is nonzero for a small finite range of z values (-Z1 < z < Z2) and essentially zero outside this range. Similarly, Az(z) is limited with respect to frequency content; therefore, Fm(k) is nonzero only within a small range Iki < M (a constant) and zero elsewhere. Thus, the discrete functions 4.m(Zn) and Fm(Pq) approach zero as Zn and Pq, respectively, become large. The sampling interval Z is chosen to be small enough so that no aliasing occurs in the k domain, and the number of samples or sampling duration NZ is chosen to include the entire signal. Relationships similar in form to the DFT pair of equations [i.e., (21) and (22)] exist for 4bS(z) and F,,(k) as well as cV°(r,z) and F°(kr). The forward solution, as defined previously, can now be written in terms of the products of the DFT's (equivalent to- linear convolution). With the transmembrane potential given [4?m(Z)] the DFT of the field potential at a given radius r is [analogous to (5)]:

F°(Pq,r) where q given by

=

0,

l,-*

,

4f(Zn,r)

=

H(Pq,r)Fm(Pq)

N and the field = IDFT

(24)

potential itself is

[F°(Pq,

r).

(25)

z[cmj (b)

Fig. 8. Typical action potential distribution 4.m(Z) from (a) a single bullfrog atrial cell and (b) a dog Purkinje fiber.

In the inverse problem in discrete space, the field potential cV°(r, z) is given and the source potential 4Pm(Z) is calculated. Thus, given F°(kr) = DFT [V'(r, z)], one may compute its transform and form the following function analogous to (5):

(26) Fm(k) = F0(kr)/H(Iklr) where the hat notation on Fm(k) refers to the fact that the dc value of Frn(k) is not recoverable. Taking the inverse transform of Fm(k) yields

)m(Z) = IDFT [Fm(k)].

(27)

572


The impulse response for the forward filters H(Iklr), cx_1(XIkja) and W(Iklr) is not finite, however, several techniques exist for obtaining a finite impulse response (FIR) filter, given a frequency response [19], [20]. We have employed the well-known Hanning window [19], [20] in this study. The resulting FIR filter approximates the actual impulse response with minimum mean-square error.

V%V

A .

r=7aa

COMPUTATIONAL ASPECTS The Purkinje fiber and atrial transmembrane potential waveforms utilized in this study are shown in Fig. 8. The typical atrial action potential used in this study [Fig. 8(a)] has a resting potential of -90 mV, an upstroke velocity of 40 V/s, and a conduction velocity that is assumed to be 0.1 m/s, a value commonly reported for atrial tissue. The (c) (b) (a) data were obtained in the laboratory of one of us (W.G.) Fig. 9. Computed extracellular potential waveforms at several radial disunder experimental conditions similar to those described tances from the fiber (r = a, 3a, 7a, Ila, and 15a) for the (a) single atrial fiber, (b) atrial trabeculum, and (c) Purkinje fiber. For compariin references [111, [12]. Fig. 8(b) shows a typical action son's sake, the conductivity ratio a for all three cases is chosen to be 2. potential distribution 4m(Z) recorded from an isolated dog Purkinje fiber positioned in a large Ringer-Tyrode solution volume conductor that was maintained at a temperaTABLE III ture of 37°C. These data have been reported previously MODEL PARAMETER VALUES [5, Fig. 9]. It has a resting potential of -88 mV, an Purkinje Atrial Atrial upstroke velocity of 128 V/s, and a conduction velocity of Fiber Cell Trabeculum 2 m/s. The frequency spectra Fm(k) of these input data waveforms 4'm(z) were obtained using a fast Fourier trans100 a 3 18 6 2.0 2.0 2.0 form (FFT) algorithm and were found to be bandlimited 2 0.1 0.1 v to spatial frequencies below k = 1.92 rad/cm for both types Z 0.4 0.04 0.04 of fiber. The signals were sampled at rates. that ensured 256 L 128 128 512 N 256 256 that all of the significant frequency components in the sigbe would that no and aliasing would be nal represented, introduced by the sampling process. The atrial waveform a = fiber radius (nm) transient in had a capacitive shown Fig. 8(a) originally = specific conductivity ratio _ xUi,t associated with it due to stimulus artifact; this has been v = conduction velocity (m/s) removed to yield a typical conducted action potential waveZ = sampling interval in z domain (cm) L = signal duration (number of samples) form. Table III provides a summary of the' model paramN = signal periodicity eter values utilized in this study for each of the three cases of interest: 1) the single atrial fiber, 2) the atrial trabeculum, and 3) the Purkinje strand. The atrial action potential waveform shown in Fig. 8(a) is used as the trans- potential of the diphasic extracellular waveform (normalmembrane potential waveform in cases 1) and 2) ized scale), with increasing radial distance from the fiber surface, as a function of the size of the source fiber (i.e., mentioned above. with radius a). The uppermost curve (a = 3 ,um) is repRESULTS resentative of the single atrial fiber, while the lowermost With the data waveforms of Fig. 8 as input to the model, curve (a = 100 ,um) is representative of the Purkinje fiber. extracellular field potentials 4(r, z) are calculated for sevThus, in a relative sense, the falloff in potential with eral values of radial distance according to (8). The results distance normal to the fiber surface is much more rapid are shown in Fig. 9(a), (b), and (c) (at the relative radial for larger rather than -smaller fibers. Furthermore, the distances r = a, 3a, 7a, lla, and 1Sa) for the single atrial falloff in frequency content of the field potential waveform fiber, atrial trabeculum, and Purkinje fiber, respectively. with increasing radial distance is much more pronounced The internal media for the Purkinje fiber and atrial tra- in the case of larger fibers, due to the nature of medium beculum are considered'to be anisotrqpic, while that for filter W(Iklr) as a function of source fiber radius (a); i.e., the single fiber is considered isotropic. It can be seen that as the radius increases, W(tklr) becomes progressively the field potential is approximately diphasic in nature and more low pass in'nature as can be seen from Fig. 11. These falls off in amplitude and frequency content, with increas- effects may also be observed by comparing the plots in ing radial distance from the surface of the cell. Fig. 10 is Fig. 9(a), (b), and (c), particularly at larger radial disthe plot of the falloff in the magnitude of the negative peak tances.from the fiber surface. 0

r=15a

.2cm

CD

2cm


573 TABLE IV PEAK-TO-PEAK OUTER SURFACE POTENTIAL 4?s, AS A FUNCTION OF INTERNAL ANISOTROPY RATIO X2

.2ai,

Purkinje Fiber

Atrial Trabeculum

LV

AV

i

1.0 3.0 5.0 7.0 9.0

N

57.12 170.74 283.56 395.59 506.85

1.85 5.56 9.27 12.97 16.67

E.4

z

.2-

0= a 5a la Field radius

15a

Fig. 10. Normalized magnitude of the negative peak potential of the field potential waveform as a function of increasing radial distance for fibers of different radii.

1.,

atrial trabeculum as the anisotropy ratio 2 is varied, while the specific conductivity a0 is held constant. The peak-topeak value of 4'S(z), in general, varies linearly with anisotropy ratio X2. In order to study the inverse problem, a model solution of the forward problem (i.e., a solution for the extracellular potential distribution in z at a particular value of r) is first corrupted by various amounts of additive random measurement noise, Fourier transformed, and then fed as input to the inverse reconstruction filter G(Ikjr). This concept is expressed in Fig. 12, where it is implied that the inverse filter function H-l( klr) is modified in some manner in order to facilitate proper recovery. The modified filter is called the reconstruction filter (kI r). Assuming that the signal 4m(Z) and the noise n(z) are statistically independent, one may employ a Wiener filter of the form

Smm(k)

1

1

GkI)-SMM(k) + Snn(k) Sav H(IkIr)

.8

where Smm(k) is the spectral density funciton of the extracellular field potential, Snn(k) is the spectral density function of the noise, and Sav is the average weighting factor over the entire frequency range [21], [22]. The filter G(Iklr) is essentially the same as a Wiener filter [21] with the exception of the additional term 1/Sav. Say is an average weighting factor that specifies the extent to which the ideal filter function H -(kI r) is scaled down over the entire frequency range. It is obtained by averaging Smm(k)/[Smm(k) + Snn(k)] over the entire range of spatial frequency k. Since the noise corrupting the signal is broad-band white noise Snn(k), the spectral density function of the noise is nowhere zero in the frequency range of interest. Hence,

.6!

.2 .4~ ~~~~ -

0

1

2

3

k[rad/cm]

a=I00A

4

(28)

5

Fig. 11. A plot of the medium filter W(Iklr*) evaluated at r* - 15a as a function of k for fibers of different radii a.

Snm (k) +

Snm(k)

+

Snn(k)l