multipole analysis in magnetopneumography-a model study

IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. BME-32, NO. 6, JUNE 1985

Multipole Analysis in Magnetopneumography-A Model Study

FRIEDEMANN BRAUER AND GERHARD STROINK

Abstract-We compare several methods of calculating the magneticdipole moment of the field of a magnetized human lung. This dipolemoment is proportional to the total magnetic dust load in the lung,hence, it is an important parameter in occupational health studies ofminers and welders exposed to magnetic respirable dust. To test theaccuracy of the several methods explored in this paper, we use a uni-formly magnetized ellipsoidal lung model to simulate the magnetic fieldof the lung. Two commonly used magnetic detectors are compared inthis analysis: a single second-order gradiometer above the lung, and asystem of two first-order gradiometers with one above and one belowthe lung. The field measured by the detectors is analyzed using a mul-tipole expansion of the field. We find that an algorithm using dipole andquadrupole coefficients is the most successful at determining the dipolemoment of the lung. This performs best with the two first-order gra-diometers, and is acceptable with the second-order gradiometer 20-40cm away 'from the center of the lung-the distance used in practicalmeasurements.

INTRODUCTIONINCE the introduction of magnetopneumography in1973 [1], several groups have used this technique to

estimate the dust load in lungs of workers exposed to mag-netic dust such as coal, asbestos, or welding fumes [2],[3].The method consists of exposing the subject's chest

briefly to a magnetic field and subsequently measuring theremanent field of the magnetic dust particles in the lungby scanning the chest under a stationary magnetometer.Only the remanent field component in the same directionas the magnetizing field is measured. From the distribu-tion of this field component, the total magnetic moment,which is proportional to the dust load, has to be calcu-lated. Various methods have been tried to solve this prob-lem. In a forward solution the measured field is comparedto that calculated from a cylindrical [4], ellipsoidal [5], ormore elaborate model of the lung [6]. Because one needsto make assumptions about the shape of the lung, the dis-tribution of dust in the lung, and in particular, the distancebetween lung and magnetometer [5], this method is notvery accurate in calculating the total moment.An inverse solution using a multipole analysis of the

measured field was suggested by Aittoniemi et al. [7]. Inprinciple, such a solution yields the dipole moment, andhence the dust load, together with higher order moments

Manuscript received April 24, 1984; revised December 21, 1984. Thiswork was supported in part by DCIEM, Toronto, Ont., Canada, and theDepartment of Health of Nova Scotia, N.S., Canada.

The authors are with the Department of Physics, Dalhousie University,Halifax, N.S., Canada B3H 3J5.

describing the nonuniformity of the dust distribution. Themultipole method does not depend on the choice of a spe-cific lung model and does not require the knowledge ofthe distance between lung and magnetometer since this isapproximated during the calculation. Also, it is not nec-essary, in principle, to assume a uniform distribution ofthe dust in the lungs. For these reasons, the multipoleanalysis of the magnetic field of the lung seems preferableto the model-based methods, however, previous attemptsat this type of analysis have apparently not been very suc-cessful and no results are available [8].

In this study, we assess the value of the multipole anal-ysis for lung measurements by using test field data calcu-lated from the ellipsoid lung model. Various algorithmswith different orders of multipole moments are tested.Also, two different magnetic sensor arrangements arecompared: a second-order gradiometer common inSQUID-based systems, and a system of two first-ordergradiometers, one above and one below the lung. This lat-ter system can be realized with fluxgates [7].

METHODS

TheoryWe first review the essentials of the multipole analysis

of magnetic measurements 19] as it applies to lung mea-surements.

Let H(r) be the static magnetic field of a body with themagnetization M(r). Outside a sphere enclosing the body,H(r) is irrotational and can be written as the gradient ofa potential 4D

H(r) = -V4t (r). (1)Since there are no sources outside the sphere, we haveV H(r) = 0, hence, from (1), V24) (r) = 0. This meansthat 4) (r) is a harmonic function and can be expanded as

1 o n

)(r) =- E E [AnmYmn(,0 )4-r n=O m=O

+ Bn Yomn(0, 4)]Irn1 (2)

with the even and odd spherical harmonics: Yen=cos (mO) Pm (cos 0) and Y° = sin (mO) Pn (cos 0), re-spectively. The P' (cos 0) are the associated Legendrefunctions of the first kind. The Anm and Bnm are the evenand odd multipole coefficients, respectively. In analogy tothe electrostatic case [10], we can write

0018-9294/85/0600-0386$01.00 © 1985 IEEE

386

BRAUER AND STROINK: MULTIPOLE ANALYSIS IN MAGNETOPNEUMOGRAPHY

Anm = (n m)! M(r')n in(n +m)!(3V '(Yn(O', ')r') d3r'

with E0 = 1 and em = 2 for m > 0. The expression forBnm is similar, using the odd spherical harmonics Y,mn.Note that Aoo = 0 and Bno = O for n - 0. The three

lowest nonvanishing coefficients are the Cartesian com-ponents of the dipole moment p:

A11 = M, d r' =p, B I = My d py

Alo = M, d3r' = p.

Inserting (2) into (1) gives the multipole expansion ofthe magnetic field outside the sphere

1 00 n

H(r) = - E [AnmH n(r) + BnmHon(r)I (4)41r n=O m=O

with the even and odd unit fields, He0 and Ho,, respec-tively. The z components of the unit fields are

Hez(r) = r-n-2(n - m + 1) cos (mO) P m+I(cos 0)

and

Hmn,z(r) = r 2(n - m + 1) sin (mO) Pm I(cos 0).

(5)These components form a complete orthogonal set on asphere enclosing all the sources and therefore determinethe coefficients Anm and Bnm. The z components of thedipole unit fields (n = 1) are explicitly H =r53xz, H = r-53yz, and H I = r5(3z2 - r2). Form < n, the five quadrupole and seven octupole. compo-nents can be calculated for n 2 and n = 3, respectively[11].The quantity measured in magnetopneumography is the

z component of the induction by Bz = t0oHz. Using (4),we can write Bz = T * m with the vector of unit fieldsT = (toI47r) (HeI,z(r), HOI,z(r), Hel,z(r), * , and themultipole vector m = (Ajj, Bj1, Ajo* )T. Extendingthis to M field values measured at locations ri, i = 1,M, we get the system of equations

Bz= Tm (6)

where the matrix of unit fields T has M rows Ti = (puJ47r) (HI,, z(ri), - * * ), i = 1, * * * , M, and Bz = (Bz(rl),*** , Bz(rm)) is the vector of field values. If (6) is limitedto dipole or quadrupole order, the dimension N of the mul-tipole vector becomes N = 3 or N = 8, respectively. IfM > N, the system of equations (6) is overdetermined andcan be solved for m by a least-squares method, after cal-culating the matrix T from the M locations ri. Since theunit fields are independent in an x-y plane, it is sufficientto measure B, at locations with constant z, as is done inthe lung scans.To suppress background noise, the field is measured

with a gradiometer. A second-order gradiometer might usethree groups of coils located at ri, ri + b, and ri + 2b,respectively, where b is the baseline vector. The centergroup, at ri + b, has twice the number of windings as theouter ones and is wound in the opposite direction. Themeasured field at location ri can be approximated by thesum of the fields at the center of each coil group

(r-) = k(Bz(ri) - 2Bz(ri + b) + Bz(ri + 2b))

with a constant k. Accordingly, the matrix T of unit fieldshas to be replaced by the sum of three matrices T, whichthe gradiometer response matrix. Its rows are given byTi = Ti(ri) - 2Ti (ri + b) + Ti(ri + 2b). Consequently,(6) is replaced by

measBHz Tm (7)

It has to be noted that, except for the dipole moment,the multipole coefficients in (4) are not uniquely deter-mined by the magnetic field H alone; they depend, throughthe unit fields, on the origin of the coordinate system. Ifthe dipole and quadrupole coefficients are known for a par-ticular origin re, a new origin r,, for which the quadrupolecontribution is minimized, is given by the shift

rI-r = 4= 4(p2I - ppT) Qp (8)

wherep is the dipole momentp = IIP|, I the unit matrix,and Q a symmetric matrix formed from the quadrupolecoefficients [12], [13]. In practice, if th-e multipole coef-ficients are estimated from a set of field values using (7),the dipole coefficients will also depend on the origin be-cause of the limitations in number and position of the mea-surement points.

Computer Program

The multipole coefficients of the field are computed byfirst finding a suitable origin and then calculating the coef-ficients for this origin up to the desired order. The twoproblems are connected. Given an origin ro, the multipolecoefficients m are evaluated so as to minimize the errorbetween the calculated field values T(r0)m and the mea-sured ones Bz eas. This error is further decreased by vary-ing rO.

In more detail, our program is as follows.1) Make an initial estimate of the origin ro to start the

iterations.2) Determine the gradiometer response matrix T at the

measurement positions relative to this origin, ri = (xi, yi,z) - rO-

3) Solve the system of equations (7) for a set of multi-pole coefficients m up to a desired order, using a least-squares error method and calculate the residual Tm -gneas.

4) Search for an origin which minimizes the norm ofthe residual by iterating steps 2 and 3.

5) For this origin, calculate the final multipole coeffi-cients up to the desired order.

387


4z

D

C~~~~~B

-A* -- X A ----|---

b

C

(a) (b)Fig. 1. Geometry of magnetic detectors. (a) Second-order gradiometer of

baseline b. (b) Two-gradiometer system with baseline b and separationd.

Variations of this procedure include using multipolecoefficients up to dipole, quadrupole, or octupole order,respectively, in step 3. In the section of the tests, they arereferred to as dipole, dipole and quadrupole, and dipole,quadrupole, and octupole origin search, respectively.Moreover, the origin calculated by any of these methodscan be modified using the shift in (8). The combinationstested in this study are listed at the end of this section asmethods A through G.The geometry of the gradiometer enters only through

the response matrix T, which is evaluated in a subroutinethat can be changed to suit each gradiometer geometry.Equation (7) in step 3 is solved by means of a singularvalue decomposition of the matrix T [14]. The iterativesearch for the optimal origin in step 4 applies a finite dif-ferences Levenberg-Marquardt algorithm [15] executed bya subroutine from the library MINPACK [16]. Both rou-tines proved to be stable and fast.

GradiometersTwo second-order gradiometers common in lung mea-

surements are shown in Fig. 1. The noise rejection, con-trolled by the baseline b and the distance d between thecenter coils, is better in the standard second-order gradi-ometer [17] shown in Fig. l(a). The "two-gradiometersystem" in Fig. l(b), with the lung in between two first-order gradiometers, has the advantage of being less sen-sitive to the z positioning of the lung when it is close tothe center of the two gradiometers. This can be seen fromFigs. 2 and 3, depicting the magnetic field of a point dipolesensed by the two different systems, as a function ofx andz. Near the center, the field varies slowly with z (Fig. 3),in contrast to the steep slope of the second-order gradi-ometer (Fig. 2). The field change in the x direction is notsignificantly different between the systems, although theslope is somewhat less for the two-gradiometer system.These results are similar to those obtained in the sensitiv-ity analysis of a two-magnetometer system for magneticbrain measurements [18].

icz(cm)

Fig. 2. The normalized response of a second-order gradiometer of baselineb = 10 cm, to the magnetic field of a dipole. The response is shown asa function of x and z, normalized to the value at z = -25 cm, x = 0.

Fig. 3. The normalized response of a two-gradiometer system of baselineb = 10 cm and separation d = 50 cm. The response is shown as a func-tion of x and z, normalized to the value at the center z = -25 cm, x =0.

In the numerical tests, we assumed b = 4 cm, a baselinecommon in the SQUID-based second-order gradiometersused in biomagnetism. For the two-gradiometer system,realized with the less sensitive fluxgates [7], we chose

388


b = 10 cm and d = 50 cm. Any spatial extension of thecoils or fluxgates was neglected.

Methods TestedHaving outlined the computer program and discussed

the gradiometers, we can now list the various combina-tions. The first parameter is the number of multipole mo-ments that are used in step 3 of the program. Second, thegradiometer geometry enters the calculation of the re-sponse matrix in step 2. And last, an origin shift given by(8) can be introduced before computing the final multipolemoments. Therefore, the following methods were tested.Method A: Search for a dipole origin, for a second-order

gradiometer.Method B: Search for a dipole origin, for a two-gradi-

ometer system.Method C: Search for a dipole origin, for a second-order

gradiometer, followed by an origin shift (8).Method D: Search for the origin using dipole and quad-

rupole terms (dipole and quadrupole origin), for a second-order gradiometer.Method E: Search for a dipole and quadrupole origin,

for a two-gradiometer system.Method F: Search for a dipole and quadrupole origin,

for a second-order gradiometer, followed by an origin shiftas in method C.Method G: Search for dipole, quadrupole, and octupole

origin, for a second-order gradiometer.Method A has been used previously in magnetocardiog-

raphy [9], and a method similar to F, although without anorigin search, was employed in a simulation study of theelectric field of the heart [19].The test fields are that of a dipole and an ellipsoidal lung

model magnetized in the z direction. The field values arecalculated on a grid of 11 x 5 points in an x-y plane abovethe object simulating 5 lung scans with 11 measuringpoints each. The spacing of the points on the grid is 4 cmin the x direction and 5 cm in the y direction, independentof the z distance. Random errors in x and y are added tosimulate measurement errors in the position. Field noisewill be discussed in the section on the ellipsoidal lungmodel. The z components p, of the dipole moment calcu-lated by the various methods are compared below in termsof their accuracy relative to the true value. The compo-nents p, and py of the two test objects, the dipole and theellipsoidal lung, are zero.

RESULTSThe Dipole FieldAs an example, we show in Table I some results for the

field 9f a dipole pointing in the z direction, located 20 cmbelow the probe-a distance typical for the center of thechest in lung measurements. Both the two-gradiometersystem (method B) and method D perform well, whilemethod A is sensitive to positioning errors. At its centerz = -25 cm, the two-gradiometer system shows the high-est accuracy, with a relative error of less than 1 percenteven for random position errors up to 0.5 cm.

TABLE IRELATIVE ERROR IN THE CALCULATED DIPOLE

COMPONENT PZ FOR A DIPOLE IN THE z DIRECTIONAT Z = -20 cm ASAFUNCTION

OF THE POSITION ERROR

x-y Error (cm)

Method 0 0.2 0.5

A 10-6 0.1 0.5B 10-7 0.03 0.07D I0-7 0.05 0.06

E

E-.

0.3

00

v

0.6 2,0 20 40 60 80 100

z-dislance (cm)Fig. 4. Calculated magnetic dipole moment p, for the ellipsoid lung model

using methods A-F in the search of the origin of the multipole expansion.The dotted lines are the results for the two-gradiometer system, the solidlines for the second-order gradiometer. The distance z for the two differ-ent systems is defined in Fig. 1.

Because of the recent interest in source localization, wemention here the accuracy of the programs in determiningthe origin of the moment as well. With no position error,methods A, B, and D determine the origin better than 0.01cm. With positioning errors, the origin error is of the or-der of the maximum position error when method A is used.For method B, the two-gradiometer system, the origin er-rors are approximately twice the maximum position error.For method D with position errors of 0.2 cm and 0.5 cm,the origin error is up to 1.8 cm and 3.5 cm, respectively.

The Ellipsoid Lung FieldA more realistic field is that of a lung model consisting

of two ellipsoids with their lower ends cut off [5]. Thedimensions are chosen so as to approximate the shape andsize of the standard lung defined by Cohen [6]. The mag-netization is the result of 4654 dipoles pointing in the zdirection located on a cubic lattice with a spacing of 1 cm.The measurement grid extends from -20 to +20 cm in

the x direction (across the ellipsoids), and from -5 to + 15cm in the y direction (parallel to the longest axis of theellipsoids), with zero at the center of symmetry betweenthe full ellipsoids.The field values were first calculated from this model

for exact probe positions. Fig. 4 shows the wide differ-

389


2.0

a 1.5

a)

0E

la00

0.5

o.o I I I I

0 20 40 60 80 100

z-distance (cm)

Fig. 5. Calculated magnetic dipole moment p, as in Fig. 4 for a set of ran-

dom errors of less than 0.2 cm in the x-y measuring positions.

ences between the moments p, resulting from the variousmethods. A search for a dipole only (curve A) causes thelargest deviation of p, from the true value, if the probe isclose to the lungs. At a probe distance z = 40 cm, thecalculated solution is still 25 percent too high. Far fromthe center of the lungs, this method gives a good approx-

imation, e.g., less than 1 percent error at z 60 cm. Ob-viously at this rather impractical distance the field startsto look more dipolar. The result can be improved some-

what by using a gradiometer of longer baseline. For ex-

ample, using b = 10 cm we get only a 10 percent error ata distance of z = 40 cm. This curve is not shown. A betterchoice is a dipole search with an additional origin shift(curve C). Again, for the second-order gradiometer ofbaseline b = 4 cm, the solution for pz is within 16 percentof the true value at z = 30 cm, and within 4 percent at z

= 40 cm.

The best results are achieved with a dipole and quad-rupole origin search. At z = 25 cm the solution is alreadywithin 3 percent of the true value, and at z = 40 cm it iswithin 1.5 percent (curve D). With an origin shift addedthe results deteriorate (curve F). Including the octupolecoefficients in the origin search (method G, not shown inFig. 4) gives rather poor results. For example, forz = 40 cm the calculated pz is off by 10 percent. Addingan origin shift to this method does not improve it much.The methods A and D are shown to perform much better

with the two-gradiometer system (curves B and E). Bothcurves converge to an error of 0.2 percent for the lung inthe center of symmetry, z = 25 cm. For an off-center dis-placement of the lung, the error increases sharply for thedipole origin search (B), while in the case of the dipoleand quadrupole search (E), the error in pz remains below1 percent.

Positioning errors were simulated by adding random er-

rors to the x-y positions used to calculate the test fielddata. The results for two different sets of random errors

with a maximum of 0.2 cm are shown in Figs. 5 and 6. Itis to be noted that the calculated results for pz for the sec-

ond-order gradiometer (curves A, C, and D) no longershow convergence toward the true value with increasing

&1.5

Eo BE

5 1.0 EB- /

0.5

0.0 10 20 40 60 80 100

z-distance (cm)

Fig. 6. Calculated magnetic dipole momentp, as in Fig. 5 but for a differentset of random errors of less than 0.2 cm in the x-y measuring positions.

z. This is due to the fact that the field values Bz decreaserapidly with z, whereas the position errors remain con-stant. In addition to this, our measuring grid does not in-crease with z so that relatively less spatial information isavailable. The dipole and quadrupole origin search (D) stillshows by far the best results, particularly in the near fieldfrom 25 to 40 cm. For example, at z = 40 cm, p, is correctwithin 6 percent.

In contrast, the two-gradiometer system still displaysconvergence of pz toward the true value as the lung posi-tion approaches the center of the system (curves B and E).For the dipole and quadrupole search, the relative error inPZ is below 1 percent, for both sets of random errors (curveE, Figs. 5 and 6).

Using 55 as the number of measurement points in thegrid seemed to be a reasonable compromise between ac-curacy and computation time. For example, reducing thenumber of grid points to 44 (11 x 4) increased the errorin pz for method B at 18 cm by 50 percent.

In practical measurements, field noise also has to beconsidered aside from position noise. Moreover, a lungwith a uniform magnetization is not very realistic. There-fore, we selected the two most successful methods, D andE, for a series of tests under the following conditions. Thez distance was chosen close to the lung, z = 20 cm. Ran-dom x-y position errors with a maximum of 0.2 cm wereassumed as before. Random magnetic field noise with amaximum of 6 percent was added. Three different typesof magnetization of the ellipsoid lung were considered: 1)uniform, as above; 2) linearly increasing from zero at thelower end of the ellipsoids to a peak value at the top; and3) uniform, with the magnetic moment of one ellipsoidonly 75 percent of that of the other one. For each of thesemodels and for both gradiometer configurations, an en-semble of 40 different field data sets was calculated usingthe same measurement grid as above. The mean valuesand standard deviations of the components pz obtainedfrom these data sets by our computer program are shownin Table II. It can be seen that the dipole and quadrupoleorigin search methods D and E perform well even underthese conditions, with the two-gradiometer system (E)

390


TABLE IIMEAN AND STANDARD DEVIATION a OF THE DIPOLE MOMENTSP,

CALCULATED FROM 40 DIFFERENT FIELD DATA SETS (WITH MAXIMALLY 6PERCENT FIELD NOISE AND 0.2 cm x-y POSITIoN ERRORS), FOR THREE

DIFFERENT MAGNETIZATIONS OF THE ELLIPSOID LUNG

Second-Order Two First-OrderGradiometer Gradiometers

Magnetization pZ a pz e

Uniform 1.00 0.05 0.99 0.03With Linear 1.05 0.04 1.00 0.03

GradientLeft Lung 1.14 0.05 1.09 0.03Reduced to75 Percent

slightly better than the second-order gradiometer (D). Theaccuracy of these methods in actual measurements will bediscussed in a separate paper [20].

DIsCUSSION

We have shown that the dipole and quadrupole originsearch (methods D and E) is the most accurate in deter-mining the dipole moment of the field of the ellipsoidallung model. This search gives the best results with a sen-

sor system consisting of two first-order gradiometers andis slightly less accurate with a single second-order gradi-ometer system in the near region from 20 to 40 cm. It isalso the algorithm which is least sensitive to small randomposition errors as they occur in practical measurements.The results confirm that because of the dimensions of thelung, quadrupole terms have to be considered in the near

field. Only beyond about 60 cm can the field of the lungbe considered more dipolar. This is in contrast to the near

field of the heart, whose dipolar moment can be calculatedreasonably well using a dipole origin search [9].Our results show that even for a nonuniform magneti-

zation, and in the presence of field noise, the dipole mo-

ment can be calculated with good accuracy from the lungmodel data, using a multipole analysis, for both types ofgradiometers considered in this work. This suggests thatwithout using the distance to the lung, the lung size, or

any particular lung model, this method would be efficientin calculating the dipole moment from field measurementsof human lungs.

ACKNOWLEDGMENT

We thank P. Krieger for executing some of the computercalculations, and R. A. Dunlap and C. Purcell for theircritical comments during the preparation of this manu-

script.

REFERENCES

[1] D. Cohen, "Ferromagnetic contamination in the lungs and other or-

gans of the human body," Science, vol. 180, pp. 745-748, 1973.[2] P. L. Kalliomaki, K. Aittoniemi, and K. Kalliomaki, "Occupational

health applications and magnetopneumography," in Biomagnetism, S.J. Williamson, G. L. Romani, L. Kaufman, and 1. Modena,Eds. New York: Plenum, 1983, pp. 559-568.

[3] G. Stroink, D. Dahn, and J. Holland, "Magnetopneumographic esti-mation of lung dust loads and distributions in asbestos miners andmillers," Amer. Rev. Resp. Dis., vol. 123, p. 144, 1981.

[4] K. Kalliomaki, P.-L. Kalliomaki, V. Kelha, and V. Vaaranen, "In-strumentation for measuring the magnetic lung contamination of steelwelders," Ann. Occup. Hyg., vol. 23, pp. 175-184, 1980.

[5] G. Stroink, D. Dahn, and F. Brauer, "A lung model to study themagnetic field of lungs of miners," IEEE Trans. Magn., vol. MAG-18, pp. 1791-1793, 1982.

[6] D. Cohen, "The magnetic field of the lung," Nat. Tech. Inform. Serv.,Springfield, VA, Rep. MIT/FBNML-78/1, 1978.

[7] K. Aittoniemi, K. Kalliomaki, T. Katila, and T. Varpula, "Practicalmagnetopneumography using fluxgate magnetometers," in Bio-magnetism, S. N. Erne, H.-D. Hahlbohm, and H. Lubbig,Eds. Berlin, Germany: DeGruyter, 1981, pp. 475-484.

[8] K. Kalliomaki, P.-L. Kalliomaki, K. Aittoniemi, and M. Moilanen,"Magnetic technique for measuring lung contamination," in Bio-magnetism, S. J. Williamson, G. L. Romani, L. Kaufman, and 1.Modena, Eds. New York: Plenum, 1983, pp. 545-557.

[9] P. J. Karp, T. E. Katila, M. Saarinen, P. Siltanen, and T. V. Varpula,"The normal human magnetocardiogram. I. A multipole analysis,"Circ. Res., vol. 47, pp. 117-130, 1980.

[10] P. M. Morse and H. Feshbach, Methods of Theoretical Phys-ics. New York: McGraw-Hill, 1953, p. 1276.

[11] G. Arfken, Mathematical Methods for Physicists. New York: Aca-demic, 1970.

[12] D. B. Geselowitz, "Two theorems concerning the quadrupole appli-cable to ECG," IEEE Trans. Biomed Eng., vol. BME-12, pp. 164-168, 1965.

[13] R. M. Arthur and D. B. Geselowitz, "Effect of inhomogeneities onthe apparent location and magnitude of a cardiac current dipolesource," IEEE Trans. Biomed. Eng., vol. BME-17, pp. 141-146, 1970.

[14] G. E. Forsythe, M. A. Malcolm, and C. B. Moler, Computer Methodsfor Mathematical Computations. Englewood Cliffs, NJ: Prentice-Hall, 1977.

[15] J. J. More, "The Levenberg-Marquardt algorithm," in NumericalAnalysis (Lect. Notes Math.), vol. 630, G. A. Watson, Ed. Berlin,Germany: Springer-Verlag, 1977, pp. 105-167.

[16] B. S. Garbow, K. E. Hillstrom, and J. J. More, "MINPACK Fortransubroutines," Argonne Nat. Lab., Argonne, IL, Mar. 1980.

[17] G. L. Romani, S. J. Williamson, and L. Kaufman, "Biomagnetic in-strumentation," Rev. Sci. Instrum., vol. 54, pp. 1815-1845, 1982.

[18] J. A. V. Malmivuo, "Distribution of MEG detector sensitivity: Anapplication of reciprocity," Med. Biol. Eng. Comput., vol. 18, pp.365-370, 1980.

[19] P. Savard, G. E. Mailloux, F. A. Roberge, R. M Galrajami, and R.Guards, "A simulation study of the single moving dipole representa-tion of cardiac electrical activity," IEEE Trans. Biomed. Eng., vol.BME-29, pp. 700-707, Oct. 1982.

[20] G. Stroink, F. Brauer, C. Purcell, and P. Krieger, "An inverse solu-tion in magnetopneumography," in- Biomagnetism, Application andTheory, H. Weinberg, G. Stroink, and T. Katila, Eds. Elmsford, NY:Pergamon, to be published.

Friedemann Brauer was born on April 9, 1947.He studied physics and mathematics at the Uni-versity of Gottingen, West Germany, from 1966 to1973, and received the Diploma in physics in 1973.He was a Research Assistant at the Klinikum

Charlottenburg of the Free University, West Ber-lin, from 1973 to 1980, working on the computer

X analysis of the EEG and evoked response audi-ometry. Since 1981 he has been with the Bio-magnetismGroup at Dalhousie University, Halifax,N.S., Canada.

Gerhard Stroink was born on October 6, 1942.He received the engineering degree in physics fromthe Technical University of Delft, The Nether-lands, and the Ph.D. degree in physics fromMcGill University, Montreal, P.Q., Canada.

He joined the Department of Physics, Dalhou-sie University, Halifax, N.S., Canada, in 1973 andis presently an Associate Professor in that De-partment. He is also an adjunct Associate Profes-

ia @ | sor of the Technical University of Nova Scotia. Hispresent research activities are in biomagnetism as

applied to the heart and lung, and in the magnetic properties of amorphousalloys.

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multipole analysis in magnetopneumography-a model study

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