Boundary element model for simulating sound propagation and source localization within the lungs M. B. Ozer Baxter Healthcare Corporation, Deerfield, Illinois 60015
S. Acikgoz and T. J. Roystona兲 University of Illinois at Chicago, Chicago, Illinois 60607
H. A. Mansy and R. H. Sandler Rush Medical University, Chicago, Illinois 60612
共Received 19 April 2006; revised 14 February 2007; accepted 15 February 2007兲 An acoustic boundary element 共BE兲 model is used to simulate sound propagation in the lung parenchyma. It is computationally validated and then compared with experimental studies on lung phantom models. Parametric studies quantify the effect of different model parameters on the resulting acoustic field within the lung phantoms. The BE model is then coupled with a source localization algorithm to predict the position of an acoustic source within the phantom. Experimental studies validate the BE-based source localization algorithm and show that the same algorithm does not perform as well if the BE simulation is replaced with a free field assumption that neglects reflections and standing wave patterns created within the finite-size lung phantom. The BE model and source localization procedure are then applied to actual lung geometry taken from the National Library of Medicine’s Visible Human Project. These numerical studies are in agreement with the studies on simpler geometry in that use of a BE model in place of the free field assumption alters the predicted acoustic field and source localization results. This work is relevant to the development of advanced auscultatory techniques that utilize multiple noninvasive sensors to construct acoustic images of sound generation and transmission to identify pathologies. © 2007 Acoustical Society of America. 关DOI: 10.1121/1.2715453兴 PACS number共s兲: 43.80.Qf, 43.80.Vj, 43.80.Cs 关FD兴
I. INTRODUCTION A. Background
Passive listening 共auscultation兲 has been used qualitatively by physicians for hundreds of years to aid in the monitoring and diagnosis of a wide range of medical conditions, including those involving the pulmonary system 共breath sounds兲, the cardiovascular system 共e.g., heart sounds and bruits caused by partially occluded arteries and arteriovenous grafts兲 and the gastrointestinal system. There may be unique and diagnostically important information in audible frequency sound since characteristic times for many physiological processes and anatomical structural resonances are in that range.1 This approach offers several potential advantages including noninvasiveness, safety, availability, prompt results, and low cost, making it suitable for in-office check-ups, outpatient home monitoring, the emergency room, and field operations following natural or man-made catastrophes. Simple stethoscopic use is skill-dependent, provides qualitative rather than quantitative information at only a single location, and suffers from inherent limitations of human ability to discern certain acoustic differences. In recent years, many researchers have applied more quantitative measurement and analysis techniques to increase the diagnostic utility of this
a兲
Author to whom correspondence should be addressed. Electronic mail:
[email protected]
J. Acoust. Soc. Am. 122 共1兲, July 2007
Pages: 657–671
approach, utilizing electronic sensors and applying computational signal processing and statistical analyses to the measured signals to discern trends or biases correlated with pathologies.2–14 To reap the full potential of the inherently rich source of diagnostic information within the audible frequency regime will require a better fundamental understanding of: 共1兲 the acoustic source and its relation to pathology, 共2兲 the acoustic path from the source to the sensor, which can be far more complex at sonic than ultrasonic frequencies due to the potential for multiple reflections, multiple propagating wave types, and multipath behavior, and 共3兲 the use of more accurate and multiple measurement sensors. 共4兲 It could also require more sophisticated and spatially resolved computational processing of the measured signals that considers multipath propagation of the acoustic event from its source to the sensor location to reconstruct a sonic image ingrained with quantitative information. Alterations in the structure and function of the pulmonary system that occur in disease or injury often give rise to measurable changes in lung sound production and transmission. Lung sounds are known to contain spatial information that can be accessed using simultaneous acoustic measurements at multiple locations. It has been shown that lung consolidation, pneumothorax, and airway obstruction, to name a few conditions, alter the production and/or transmission of sound with spectrally and regionally differing effects that, if properly quantified, might provide additional information
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about the severity and location of the trauma or pathology.7–10 Indeed, simultaneous, multisensor auscultation methods have been developed to “map” sounds on the thoracic surface by several groups.9,11–14 Beyond this mapping process, Kompis et al.10 attempted to form a three-dimensional 共3D兲 acoustic image of the likely sound source共s兲 location共s兲 by using multiple sensors and assuming “ray acoustic,” i.e., “free field,” models for how sound propagated away from these sources. In their study, they noted that a useful imaging system for the human lung should: 共1兲 be robust with respect to acoustic properties, especially speed of sound which varies and is not precisely known; 共2兲 provide 3D data sets and resulting images that are intuitively interpretable; and 共3兲 be robust with respect to missing sensors or noisy data in individual sensors. The algorithm employed assumed spherically symmetric sound radiation away from a source and propagation throughout the thorax without reflections or standing waves, only dependent on sound speed and a per unit length damping factor; it then triangulated with redundancy on the likely sources of the sound. In five human subject studies the algorithm indicated differences in acoustic source location between inspiration and expiration and in one case, the presence of severe consolidation in the lower left lung. Parametric computer simulation studies showed that their imaging algorithm lacked some robustness with respect to acoustic properties, specifically speed of sound. In their idealized mechanical phantom study the algorithm demonstrated a resolution of nominally 2 cm. In modeling the transmission of sound throughout the pulmonary system and chest region, the system may be viewed as having two main components: 共1兲 transmission in air through the tracheobronchial tree and 共2兲 coupling to and transmission through the surrounding biological tissues to reach the chest surface—namely the parenchyma, free air, or water/blood region 共in the case of a pneumothorax or hydro/ hemothorax兲, surrounding muscle and rib cage regions, and outer soft tissues. Many studies have focused on the transmission of sound in the respiratory tract, the tracheobronchial airway tree, with some also considering coupling to modes of wave propagation in the parenchyma. See Royston et al.15 for a review of this literature. The focus of the present study is related to the second component, transmission through the surrounding biological tissues to reach the chest surface. Previous studies of this part of the problem have assumed simplified geometries and homogenized material properties.15–17 Wodicka et al.16 assumed an axisymmetric cylindrical geometry, with the outer tissue regions of the chest treated simply as a mass load on the parenchyma. In Vovk et al.17 an axisymmetric layered model for the torso region is used that includes annular regions for the parenchyma, rib cage region, soft outer tissue and skin. In previous work of the authors15 simplifications of both airway and tissue structures were imposed that resulted in an axisymmetric assumption or two-dimensional planar model assumption that could be easily handled with finite element analysis. For frequencies above 100 Hz and neglecting the larger segments of the bronchial airway tree, it has been proposed that parenchymal tissue can be modeled as a homogenous 658
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TABLE I. Acoustic properties of foam phantom and human lung tissue.
Density 共kg/ m3兲
Phase speed 共m/s兲
Attenuation in 10 cm 共dB兲
160 160 160 160
27.5 27.5 27.5 27.5
0.053 0.56 5.1 18
250 250 250 250
23 23 23 23
0.16 1.7 15 54
300 300 300 300 ¯
32 32 32 32 22–24
4.3 9.4 20 30 ¯
¯
23–33
¯
¯
62–82
¯
Foam at:a 100 Hz 220 Hz 460 Hz 700 Hz Lung theory atb 100 Hz 220 Hz 460 Hz 700 Hz Lung model atc 100 Hz 220 Hz 460 Hz 700 Hz Lung experiment atd 70– 140 Hz Lung experiment ate 125– 500 Hz Lung experiment atf 75– 2000 Hz a
Experimentally determined by authors. Theory of Wodicka et al. 共Ref. 16兲. c Empirical model of Vovk et al. 共Ref. 17兲. d Experiments of Paciej et al. 共Ref. 14兲. 共Sound introduced at supraclavicular space, measured on thoracic surface.兲 e Experiments of Kraman 共Ref. 42兲. 共Sound introduced through mouth, measured on thoracic surface.兲 f Experiments of Mahagnah and Gavriely 共Ref. 43兲. 共Sound introduced through mouth, measured on thoracic surface.兲 b
isotropic material supporting acoustic compression waves only, i.e., a lossy fluid.16,18 The parenchymal region is defined by its density p and a complex wave number k p, whose real part is linked to phase speed and whose imaginary part defines attenuation. Typical property values provided in the literature are given in Table I and used to generate Fig. 1. Theoretical values of Wodicka et al.16 are based
FIG. 1. 共a兲 Real and 共b兲 imaginary parts of complex wave number for: lung parenchyma 共Ref. 16兲 共쎲兲; soft tissue 共Ref. 15兲 共䊊兲; and air 共⫻兲. Ozer et al.: Boundary element model of lungs
on calculating effective elasticity, viscosity and density by assuming the parenchyma can be approximated as a closed cell porous material, and by knowing the individual properties of the solid tissue and air that comprise it. Compression wave numbers for the parenchymal material are quite different from those of the two components of which it is comprised, soft tissue and air; correspondingly, in the lung parenchyma sound travels much slower and attenuates rapidly. To model acoustic compression wave propagation in the surrounding tissues of the torso, the soft tissue regions composed of fat, muscle, and connective/visceral material, can be defined by a density t and a complex wave number kt, with t ⬇ 1 g / cm3, and the real and imaginary parts of the wave number shown in Fig. 1.15,16 In these nonparenchymal soft tissue regions, shear waves may also be present. Their behavior is primarily governed by the density and the shear viscoelastic moduli. Shear wave lengths and propagation speeds at a given frequency are typically 3 orders of magnitude smaller than that of compression waves in the same soft tissue medium.19 Boundary element 共BE兲 or coupled finite element/BE methods have been used by researchers to simulate the inverse problem of magnetocardiography and electrocardiography to aid in diagnosis.20–25 In other words, given an array of noninvasively recorded torso surface potentials or extracorporal magnetic field measurements, reconstruct epicardial potentials or myocardial activation times. In these and other studies, the theory behind the application of a BE method involves first expressing the governing field equations including source terms, here having to do with electromagnetics, i.e., Maxwell’s equations, in terms of boundary integral equations by use of Green’s theorems. Then, the boundary integral equations are approximated as a summation of coupled linear algebraic equations by replacing the integral over the boundary surfaces with a finite summation of simplified expressions over a discretized version of the surface 共boundary elements兲. By beginning with the governing acoustic field equations, Helmholtz equations, including source terms, the BE method can be applied to the solution of comparable acoustic problems. A BE model for lung acoustics is preferred over a finite element model due to the computational burden of meshing the entire parenchyma region. The speed of sound in parenchymal tissue is estimated to be ⬃23 m / s in the frequency range of interest. This translates into wavelengths that are roughly ten times shorter than the acoustic waves in air for the same frequency. If the nominal element dimension needs to be ⬃1 / 6 of the wavelength for accuracy, the number of nodes and elements in a 3D finite element solution would be enormous. For example, at 220 Hz, one would need ⬃120 000 degrees of freedom 共DOF兲 in a finite element model for just the left lung of the visible human male 共VHM兲.26 Unlike the finite element method, a BE analysis requires only the surface of the parenchymal region to be meshed 共with comparable demands on nominal element dimension兲, which drastically reduces the number of elements and degrees of freedom in the analysis. 共Only ⬃4000 DOF are needed for 220 Hz for the left lung of the VHM.兲 J. Acoust. Soc. Am., Vol. 122, No. 1, July 2007
B. Objectives
The first objective of this study is to adapt an acoustic BE model to simulate sound propagation in the lung parenchyma. This BE model is first computationally validated and then compared with experimental studies on lung phantom models with simple geometries. The type of BE formulation that is used is based on the Burton–Miller integral equations.27 The discretization and the solution of the integral equations are performed as presented in the literature on direct acoustic BE modeling.28–30 Another objective of this study is to be able to locate a localized acoustic source within the lung. Acoustic source localization is common to a number of engineering applications, including room acoustics, earthquake epicenter localization, and underwater sonar. Advanced sonar techniques use an analytic acoustic wave propagation model that represents the propagation and reflection of the acoustic waves from the ocean floor and the sea–air interface. The acoustic responses at sensors are predicted using this model and acoustic responses are calculated due to the acoustic source whose position is varied in a grid-like fashion. Sensor predictions for each hypothesized source location are compared with the actual sensor measurements in order to assign a probability value to a particular source location. This method is called matched field processing.31–35 There are similarities between the lung acoustics problem posed in the present study and the source localization problem in underwater sonar research. There are studies in underwater sonar research which try to quantify and minimize the errors due to incorrect wave number assumptions 共due to variations in water temperature and salinity兲 and imprecise knowledge of the acoustic and mechanical properties of the ocean floor.33–35 Analogous error sources in the present application are imprecise wave number estimations of the parenchyma tissue and imprecise knowledge of the thickness and content of the biological tissue between the parenchyma and the skin surface. In this study, source localization will be performed using matched field processing principles; but, differing from underwater sonar studies, the field prediction will be calculated using an acoustic BE model.
II. BOUNDARY ELEMENT MODELING A. Basic theory
The boundary element formulation that was developed and used to generate the numerical results presented in this article is primarily based on the work of Kirkup.28,29 Using Green’s first and second theorems, the Helmholtz equation, which governs compression wave propagation throughout the volume of a finite acoustic domain, can be expressed as a boundary integral equation: 兵H其S关p兴 + 21 关p兴 = 兵Lv其S关p兴 + 关p兴inc ,
共1兲
where 兵H其S关p兴 ⬅
冕
G关p,q兴 关q兴dS, nq S
Ozer et al.: Boundary element model of lungs
共2兲 659
兵Lv其S关p兴 ⬅
冕
G关p,q兴v关q兴dS,
共3兲
S
v关q兴 =
关q兴 . nq
共4兲
Here, S denotes the surface or boundary of the domain and the integrals are confined to this surface, p and q are points on the boundary, G关p , q兴 is the Green’s function that relates how the acoustic response at point p is affected by an acoustic point source 共monopole兲 located at q, and nq is a unit vector outward normal to the surface at point q. The velocity potential field at a point p is denoted by 关p兴. The term 关p兴inc denotes the incident field that would exist at point p due to a distribution of sources within the domain if no boundary were present; i.e., it is the free field solution at point p. And, v关q兴 is the normal velocity component on the surface at point q directed outwards. The Green’s function for the acoustic problem is G关p,q兴 =
1 ikR e , 4R
共5兲
R = 兩p − q兩,
共6兲
where R is the distance between the point of observation and the source location. Boundary conditions on the domain surface S may be specified as either 关p兴 or / n p关p兴 or a linear combination of these. The surface S is approximated with a finite set of planar triangles 共boundary elements兲 ⌬S j, j = 1 , . . . , N, and it is assumed that and v do not change within each triangle; this is referred to as C0 collocation. Errors due to this approximation of the boundary and property values on the boundary become smaller as the number of elements used is increased. Thus: N
S ⬇ 兺 ⌬S j ,
共7兲
关p兴 ⬇ j ,
共8兲
j=1
v关p兴 ⬇ v j
if p 苸 ⌬S j .
共9兲
Then, Eq. 共1兲 can be approximated as follows: N
兺 j=1
再 冎 1 H+ I 2
N
⌬S j
关p兴 j ⬇ 兺 兵L其⌬S j关p兴v j + 关p兴inc j=1
共p 苸 S兲.
共10兲
The previous expression is for a single location point p. For the complete solution of the problem one should calculate this for a point on each of the boundary elements, i = 1 , 2 , . . . , N. This leads to
共关H兴 + 21 关I兴兲 ⬇ 关L兴v + inc , 关L兴ij ⬅ 兵L其⌬S j关pi兴 = 660
冕
⌬S j
Gk关pi,q j兴dS,
J. Acoust. Soc. Am., Vol. 122, No. 1, July 2007
共11兲 共12兲
关H兴ij ⬅ 兵H其⌬S j关pi兴 =
冕
⌬S j
Gkpi,q j dS. n j
共13兲
The subscript k is added to emphasize that the k value in the Green’s function is nonzero. It will be shown that the case where k is zero will be used to evaluate certain singular conditions. The previous integrals have Green’s function or its derivative in the integrand. From Eq. 共5兲 one can see that the R value in the Green’s function will tend to zero as the observation element and source get close to each other. The integrand value will increase and the variations of the integrant within integration limits will be considerable. Therefore, high order quadrature rules should be used for integration on these elements. As the integrand values are big these elements would affect the result much more than the elements where the source and the observation elements are far away from each other. Considerable attention should be paid in using sufficient numbers of quadrature points in boundary element codes. Further, the R value is exactly zero when evaluating the diagonal elements 共source and the observation point are on the same element兲. The integrals of the diagonal elements become singular and cannot be evaluated using regular quadrature methods. A common approach to this problem is to subtract out the singularity 共reducing the strength of the singularity兲 by subtracting out and adding another integrant. The below equation shows how one can evaluate the diagonal elements in the 关L兴 matrix: 关L兴ii =
冕
⌬Si
共Gk关pi,qi兴 − G0关pi,qi兴兲dS +
冕
⌬Si
G0关pi,qi兴dS. 共14兲
Here, G0 denotes that the k value in the Green’s function is zero. Therefore, G0 can be expressed as G0关pi,qi兴 =
1 . 4R
共15兲
This expression will tend to infinity as R gets smaller. But, when two large numbers are subtracted from each other, a smaller value will be obtained and the first integral in Eq. 共14兲 becomes regularly integrable. The second integral though is still singular and is hard to evaluate numerically; but, it has an analytical solution.28 Therefore, it is possible to evaluate the integrals corresponding to diagonal elements through a combination of numerical and analytical methods. In the present application a sound generated within the parenchyma region, such as due to breathing, is approximated via a finite number of fundamental acoustic sources. The incident field that would be created by these sources at the locations pi of the i = 1 , . . . , N centroids of the boundary elements that approximate the actual boundary is denoted by inc, a vector of length N. This vector contains the analytically predicted values of the incident field caused by these sources assuming they radiated into an infinite medium. That is, the ith element of this vector represents the incident field on the ith element due to sources in the medium. In Eq. 共11兲 and v are vectors of length N. For the problem to be solvable one needs N boundary conditions Ozer et al.: Boundary element model of lungs
relating and v. This can be mathematically stated as follows, where ␣i, i, and f i are constants that define the boundary conditions:
␣ i i +  iv i = f i,
i = 1,2, . . . ,N.
共16兲
Once the boundary condition assignments and the calculations of 关H兴 and 关L兴 matrices are completed the problem can be solved using Gauss elimination or any other linear system solving method. Once the velocity potential and normal velocity components are calculated one can also find the acoustic pressure at the surface elements using the relationship: p = i. Although there is tremendous advantage in not needing the mesh inside of the domain, there is still a computational burden, the majority of it coming from the need to use higher order quadrature methods and fully populated system matrices. For elements that are far from each other single point Gauss quadrature is sufficient; but, for elements that are close to each other as much as 171 Gauss quadrature points have been used for accurate evaluation 共asymptotic convergence兲 of surface integrals in some of the examples cases described below. B. Coupled boundary conditions for surrounding shell-like structure
A viscoelastic shell-like structure surrounding the BE domain can be approximated in a discretized form using, e.g., a finite element 共FE兲 approach. If the FE degrees of freedom are coincident with the BE degrees of freedom 共motion normal to each boundary element at its centroid兲 then we can write in place of Eq. 共16兲 that:
冉
− iA + i
冊
K + C − iM v = F.
共17兲
Here, K, M, and C are structural stiffness, mass, and damping matrices, respectively, for the surrounding shell. The N ⫻ N matrix A is diagonal, with 关A兴ii being the area of the ith boundary element. The length N vector F represents forces applied externally to the shell, such as percussive excitation at the skin surface. Generally, K, M, and C can be nondiagonal matrices that couple the N degrees of freedom. Simplifying assumptions though can lead to decoupled boundary conditions. For example, consider the case of an inertia load only on the elements. Suppose a layer of material of density s and thickness hi resided on the ith element. Then, M is diagonal with 关M兴ii = 关A兴iihis and K and C are neglected. If F = 0, Eq. 共17兲 reduces to: − i − hisvi = 0,
i = 1,2, . . . ,N.
共18兲
III. COMPUTATIONAL VALIDATION STUDY
Comparison of the BE solution to a FE solution was performed for a cylindrical parenchymal region with different boundary conditions. Parenchymal material property values are given in Fig. 1 and Table I. The radius of the cylinder is 41.7 mm and its axial length is 83.3 mm, with 2352 elements used in the BE simulation. An axisymmetric FE model J. Acoust. Soc. Am., Vol. 122, No. 1, July 2007
FIG. 2. Comparison of FE and BE results for cylindrical volume with a finite monopole located along the central axis at 2 / 3 of the height of the cylinder. Vertical velocity 共m/s兲 on surface at bottom center of cylinder is shown for different uniform boundary conditions. Key: BE 共– – –兲 and FE 共䊊兲 results for free boundary; BE 共---兲 and FE 共⫻兲 results for 4 mm “soft” tissue layer covering the entire parenchyma surface with s = 1 g / cm3 and Young’s modulus E = 7.5 kPa; BE 共– - –兲 and FE 共*兲 results for 4 mm “hard” tissue layer covering the entire parenchyma surface with s = 1 g / cm3 and Young’s modulus E = 750 kPa.
was constructed in ANSYS® 9.0 with 5000 Fluid79 elements. A monopole source of radius 2.4 mm was located on the central axis of the cylinder at a height of 55.5 mm 共at 2 / 3 the length of the cylinder兲. Figure 2 shows the vertical velocity as a function of frequency for a point located at the bottom center point of the cylinder using the BE and FE methods for the following boundary conditions over the entire cylinder: a free boundary condition; and a 4 mm thick uniform “tissue” layer covering the parenchyma with density of 1 g / cm3 and Young’s modulus of 7.5 or 750 kPa. This density and the smaller modulus approximate biological soft tissue, whereas the larger modulus may approximate a composite of soft and hard 共bone兲 tissue. For the cases of the tissue layer covering the parenchyma, mass 共M兲, stiffness 共K兲, and damping 共C兲 matrices were developed for Eq. 共17兲 as follows. The mass matrix, M, was diagonal with 关M兴ii = 关A兴iihis, as described in the text before Eq. 共18兲. Linear viscous proportional damping was used such that C = 0.001ⴱ K. The stiffness matrix K was developed as follows. The BE node locations and meshing information were imported into ANSYS 9.0 FE software. A FE mesh was generated using three-node Shell63 shell elements with nodes coincident with the BE nodes. The stiffness matrix for this mesh KA, which includes three translational DOFs per node, was exported from ANSYS as a text file, which was then read into MATLAB® after some minor format editing. Motion normal to a boundary element at its centroid can be approximated using a linear interpolation of the motion of each of the three nodes that form the element. Motion of each of these nodes in the direction of the element normal can be expressed as a weighted sum of the global Cartesian coordinates with which the Ansys FE model is constructed via direction cosines. Consequently, a transformation matrix Ozer et al.: Boundary element model of lungs
661
T can be constructed that approximates motion of the N degrees of freedom of the BE model, in terms of the motion of the NA degrees of freedom of the ANSYS FE model with NA ⬎ N. 共In this case study N = 2352 and NA = 3534.兲 The T matrix is N ⫻ NA in size, with potentially nine nonzero entries per row 共three for each DOF of each of the three nodes that form an element兲. Via singular value decomposition, using the “pinv” command in MATLAB a pseudoinverse of T is calculated and referred to as 关T兴−1. It has dimensions NA ⫻ N. The stiffness matrix K of dimension N ⫻ N is then approximated as follows: K = TKA关T兴−1 .
共19兲
Note that this same transformation from KA to K could be applied to approximate the mass matrix M given MA obtained from ANSYS in the same way as the stiffness matrix KA was obtained. In the present case study the diagonal M is easily determined directly. Minor differences between the developed BE code predictions and those of the FE simulation for all cases considered are likely due in part to the fact that the BE method treats the monopole as a point source whereas it has a finite radius of 2.4 mm and represents an additional boundary in the FE case. Discrepancies may also be due to an insufficient number of elements used in the FE and/or BE simulation, though comparable sized elements enabled the BE model to exactly match theory for a case study with a spherically symmetric parenchymal geometry 共not shown兲. FE and BE predictions disagree similarly for the free and covered surface case with 4 mm of “soft tissue” with density 1 g / cm3 and Young’s modulus E = 7.5 kPa. Indeed, simulations with E = 0 共mass loading only兲 agreed with this case indicating that the inertia effect of the 4 mm of soft tissue was much more dominant than the stiffness effect. This was also nearly true for E = 75 kPa 共not shown兲. For the case where E = 750 kPa stiffness did have a significant effect and agreement between the BE and FE simulations is not as good. Clearly, the approximation involved in replacing KA with K per Eq. 共19兲 needs improvement or an alternative approach may be needed to account for the stiffness of the ribcage in the eventual application. Next, a more complex boundary condition was considered. The surface of the cylindrical parenchymal region was covered with 4 mm of the soft tissue with density 1 g / cm3 and Young’s modulus E = 7.5 kPa everywhere except along three circumferential bands, each of thickness 4 mm and width ⬃4 mm at heights on the side of the cylinder of ⬃10, ⬃32, and ⬃52 mm. In these regions bone was simulated with a density of 1.9 g / cm3 and Young’s modulus E = 10 GPa. An axisymmetric ANSYS FE model was constructed as described earlier with material properties altered for the Shell63 elements within these three bands. For the BE model boundary conditions, the diagonal mass matrix M was again easily determined directly based on the density and thickness of the material overlying each boundary element of known area. A very sparse and diagonal stiffness matrix K was directly constructed by recognizing axisymmetry, neglecting the stiffness of the soft tissue per the previous dis662
J. Acoust. Soc. Am., Vol. 122, No. 1, July 2007
FIG. 3. 共Color online兲 Comparison of 共a兲 FE and 共b兲 BE results for cylindrical volume with a finite monopole located along the central axis at 2 / 3 of the height of the cylinder and with a composite soft and hard tissue layer. Axisymmetric radial velocity 共m/s兲 on the side of the cylinder as a function of excitation frequency is shown. In both plots the color bar unit is decibels reference 1 mm/ s.
cussion and by assuming that the thickness of the boundary layer is small relative to the cylinder radius. Under these conditions the stiffness value associated with a boundary element within one of the rib bands is given by Young’s modulus of the bone multiplied by its thickness and area, and divided by the square of the radius of the cylinder. In Fig. 3, the axisymmetric response on the side of the cylinder from 100 to 700 Hz is shown, based on the FE and BE models. Agreement is good though not exact. The three dark bands of neglible motion are where the ribs were located. In the FE model, in addition to the rib regions themselves, there is a discernable attenuating effect near to these regions. In the BE model, the strong attenuating effect of the ribs seems to be confined to the rib region only. Though in both the FE and BE models, the overall acoustic pattern on the surface of the parenchyma is altered by the presence of the ribs. Ozer et al.: Boundary element model of lungs
FIG. 5. 共Color online兲 Lung phantom. 共Left兲 Photograph showing LDV measurement points and 共right兲 close-up of a piezoelectric disk transducer.
FIG. 4. 共Color online兲 Top and side view diagrams showing the locations of the piezoelectric disk transducers in the lung phantom.
IV. EXPERIMENTAL EVALUATION USING A MECHANICAL PHANTOM MODEL A. Setup
The acoustic response to an internal source in a mechanical model with acoustic properties similar to the lung parenchyma was investigated in order to: 共1兲 determine if the acoustic field that is generated depends significantly on reflections from the finite boundaries and standing wave patterns or, due to the high rate of attenuation, is this field reasonably approximated by only accounting for outgoing waves from the internal source; 共2兲 determine if the modeling assumptions that were made in formulating the BE modeling approach are reasonable for a compliant porous material like lung parenchyma; and 共3兲 validate the accuracy of the developed BE model experimentally, which is related to and dependent on the second reason. The experimental phantom setup is shown in Figs. 4 and 5. The foam material is Flex Foam-IT X 共Smooth-On Inc., Easton, PA兲. In terms of phase speed and density, the foam material is comparable to lung tissue.15,16 A theoretical estimate of the phase speed cph, which is nondispersive over the frequency range of interest, is based on cph = 冑1 / K where is the effective density of the porous foam and K is the volumetric compliance given by K = 共1 − f v兲Kg + f vKs, where J. Acoust. Soc. Am., Vol. 122, No. 1, July 2007
Kg is the entrapped gas compliance, Ks is the solid material compliance, and f v is the volume fraction of solid material. The compliance Ks is unknown, but it is reasonably assumed that, like in the lungs, this term can be neglected relative to the gas compliance, Kg ⬇ 1 / P, where P is the atmospheric pressure and is the polytropic constant 共a value of 1.0 is assumed within the frequency range of interest16兲. The content of the entrapped gas is unknown; it is a by-product of the exothermic process of forming the foam by mixing two components. But, given atmospheric pressure and based on a measured foam density of 160 k / m3 and volume fraction of solid material of f v = 1 / 6, we have cph = 27.5 m / s. This value was consistent with experimental measurements of phase speed conducted using a 1.5 m length, 10 cm inner diameter rigid tube filled with the foam material with embedded piezoelectric disk sensors along its length at 15 cm intervals and driven by a piston actuator at one end. The resulting real part of the wave number kR for this material is given by kR = 2 f / cph = 0.2285⫻ f m−1, where f is the frequency in hertz. Recall, a typical value used for lung parenchyma is cph = 23 m / s. Regarding attenuation, below the resonant frequencies of the entrapped gas bubbles dissipative losses are expected to roughly be a function of the frequency f cubed and to be dependent on the foam material properties as well as the dimensions and spacing of the entrapped gas bubbles; this is difficult to measure. Consequently, experimental measurements of attenuation as a function of frequency within the same foam-filled tube were used to approximately fit an attenuation curve, yielding an imaginary component to the wave number of ki = 6 ⫻ 10−8 ⫻ f 3. For the lung parenchyma, Wodicka et al.16 and Royston et al.15 have used ki ⬇ 18.1 ⫻ 10−6 ⫻ f 3. Thus, the foam damping is about one third that of parenchymal tissue. In the experimental phantom setup depicted in Figs. 4 and 5, there are two buried piezoelectric disk transducer pairs put in place prior to curing the foam 共locations shown in Fig. 4兲. A single “piezodisk” transducer is shown in Fig. 5. Two of these are glued back-to-back to form each internal acoustic source. During the experiment a band-limited random noise is supplied to the transducers through a high voltage amplifier 共P0623A, Trek, Medina, NY兲. In the BE simulations, it was assumed that these sources radiate with spherical symmetry, like a monopole, into the foam phantom. Normal velocity at the outer surface of the foam was measured at discrete points using a laser doppler vibrometer 共LDV兲 共CLV 800/ 1000, Polytec, Tustin, CA兲 with data acquisition and spectral analysis performed using a two-channel spectrum Ozer et al.: Boundary element model of lungs
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analyzer 共35670A, Agilent, Palo Alto, CA兲. The geometry of the foam material is a tapered cylinder. The bottom of the cylinder has a diameter of 12 cm and diameter linearly increases to 15 cm over a cylinder height of 11 cm. Note, although there are two acoustic sources buried at two different locations in order to test different methods for source localization of the multiple acoustic sources, the work reported here is confined to localization of and response to a single source. Also, a 2 mm thick layer of CF-11 silicone 共Nusil Technologies, Carpinteria, CA兲 was adhered to the conical portion of the foam phantom and experimental measurements were repeated. Due to the translucence of the silicone, vibrometry measurement points were taken at the same location depicted in Fig. 5共a兲 on the now-covered foam. B. Results and discussion
Normal velocity measurements on the radial surface of the tapered foam cylinder were taken at 216 locations. Acoustic source 1 was used 共Fig. 4兲. The BE model predicted the acoustic response due to a sinusoidal acoustic input. The BE model has 2352 elements. A contour plot of the experimental velocity measurements is shown in Figs. 6共a兲, 7共a兲, and 8共a兲 for frequencies of 300, 500, and 700 Hz, respectively. The horizontal axis denotes angular location, as established in Fig. 4. The vertical axis is the height of the measurement point. Corresponding BE predictions at the experimental locations are plotted in contour form in Figs. 6共b兲, 7共b兲, and 8共b兲. Also, predictions based on assuming free field propagation, neglecting boundary reflections, are shown in Figs. 6共c兲, 7共c兲, and 8共c兲. 共Note, for this experimental phantom it is not possible to generate a plot of response at a point as a function of frequency that can be directly compared to the BE simulation, as the strength of the piezodisk source is unknown. It will vary with frequency and will be dependent on the coupling of the source to the surrounding foam material. Hence, the piezodisk source cannot be calibrated separately and then implemented in the phantom.兲 It is clear from comparing these plots that the assumption of free field propagation is not justified. The frequencydependent wave number k based on theory and experiments described in Sec. IV A is used for BE and free-field simulations reported here. It is expected that as the excitation frequency increases damping will increase and the velocity distribution will look more similar to the free field propagation predictions. However, lung sounds and other sounds of interest in diagnosis will have a dominant component at the lower end of the audible frequency range that will diminish as the frequency increases beyond 1000 Hz or so. Although neither the free field assumption nor the BE simulation precisely match experiment for any of the frequencies, in all cases 共those shown here and at other frequencies兲 the BE simulation is more accurate, capturing standing wave patterns that tend to increase the spatial variation of the amplitude at the surface as compared to the free field assumption. The wider range of amplitude values predicted by the BE simulation is closer to that of the experiment, as compared to the free field assumption. 共Note, the simulated amplitude level depends on an assumed piezoelectric disk source strength; this value is 664
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FIG. 6. 共Color online兲 Velocity field amplitude on lateral 共curved兲 surface of lung phantom at 300 Hz. 共a兲 Experiment, 共b兲 simulated using BE model, and 共c兲 simulated using free field assumption. In all plots color bar unit is millimeter per second.
only an approximation with frequency-dependent accuracy. However, the range of amplitude values, maximum minus minimum, is independent of the source strength, as this is a linear system.兲 Ozer et al.: Boundary element model of lungs
FIG. 7. 共Color online兲 Velocity field amplitude on lateral 共curved兲 surface of lung phantom at 500 Hz. 共a兲 Experiment, 共b兲 simulated using BE model, and 共c兲 simulated using free field assumption. In all plots color bar unit is millimeter per second.
There are significant differences between the assumptions in the simulations and the experimental system that would account for discrepancies in numerical predictions and J. Acoust. Soc. Am., Vol. 122, No. 1, July 2007
FIG. 8. 共Color online兲 Velocity field amplitude on lateral 共curved兲 surface of lung phantom at 700 Hz. 共a兲 Experiment, 共b兲 simulated using BE model, and 共c兲 simulated using free field assumption. In all plots color bar unit is mm/s.
measurements. First, the simulations assume a homogeneous medium. In the process of curing the foam, which expands to ⬃6 times its original volume, it was found that air bubbles generated near the bottom of the mold were smaller in size Ozer et al.: Boundary element model of lungs
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FIG. 10. Hypothetical point source locations using source localization algorithm. 共a兲 Top view and 共b兲 three-dimensional view. FIG. 9. 共Color online兲 Velocity field amplitude on lateral 共curved兲 surface of lung phantom at 500 Hz with 2 mm silicone coating. 共a兲 Experiment and 共b兲 simulated using BE model. In all plots color bar unit is millimeter per second.
than those generated at the top. Second, the embedded acoustic source is not a point monopole. The piezodisk transducer has a diameter of 24 mm, more than 1 / 5 of the phantom model diameter. The transducers directivity pattern is not spherically symmetric. Next, a 2 mm thick layer of CF-11 silicone 共Nusil Technologies, Carpinteria, CA兲 was adhered to the conical portion of the phantom and experimental measurements were repeated as described earlier. A contour image of the experimental measurements is depicted in Fig. 9共a兲. Results of a BE simulation treating the silicone layer as a thin shell per Sec. II B are shown in Fig. 9共b兲. Material properties for the silicone material are: density= 1049 kg/ m3, Young’s modulus= 72 kPa, Poisson’s ratio= 0.495 and linear viscous damping proportional to the stiffness matrix was assumed via a proportionality constant  = 0.001. For this 2 mm thick layer, the silicone is expected to primarily behave as a mass load. The BE simulation results match experimental mea666
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surements as closely as simulation matched experiment in the previous uncoated cases. Although specific regions of high and low response amplitude are not precisely matched, the range of amplitudes and general distribution of amplitudes are comparable. V. SOURCE LOCALIZATION A. Matched field processing
The simple beamformer equation for the Bartlett processor is36,37 B关兴 =
兩aH关, 兴y关兴兩2 . aH关, 兴a关, 兴
共20兲
Here, a is a vector of length N of the predicted responses at N sensor locations due to a monopole source located at at frequency ; y is a vector of the measured responses at those N sensor locations. Here, superscript H denotes the Hermitian of the vector. First, a grid is created within the volume where the source will be searched. A hypothetical monopole source is placed at Ns grid locations and the Bartlett processor is evaluated. The grid locations are shown in Fig. 10. The BE model is computationally efficient for this type of grid Ozer et al.: Boundary element model of lungs
TABLE II. Distance between the actual source location and estimated source locations at 300 Hz. Distance 共mm兲
Baseline case with 216 sensors Baseline case with 9 sensors With −10% error in k and 216 sensors With −10% error in k and 9 sensors
BE code
Free-field
18.7 18.7 21 21
33.8 33.8 33.8 33.8
search algorithm, as it is easy to calculate the incident field created by a source at each grid location, which then only alters inc in Eq. 共11兲. The location that results in the maximum value of B关兴 indicates the most likely location for the source. The case considered here was that of free boundary conditions on the small experimental phantom model of Sec. IV. For this case, Eq. 共11兲 reduces to 0 = 关L兴v + inc .
共21兲
For a hypothetical source located at we then have a关, 兴 = 兩 − 关L兴−1兩subinc ,
共22兲
where 兩关L兴 兩sub refers to a subset of N rows of 关L兴 that correspond to N boundary elements that are coincident with the N sensor locations. If a free field assumption is used, then in place of Eq. 共22兲 we simply have −1
a关, 兴 = vinc ,
−1
共23兲
inc
where v is the calculated particle velocity amplitude normal to the boundary surface due to a source located within the boundary at , assuming no reflections at the boundary. B. Experimental results
Likelihood estimates B关兴 are calculated at each node of the grid geometry based on the BE model and based on a free-field assumption using Eqs. 共20兲, 共22兲, and 共23兲. The vector y of the measured responses is taken from experimental measurements summarized in Sec. IV 关Fig. 6共a兲兴. The actual source location is at x = 10 mm, y = −15 mm, and z = 50 mm. Grid spacing is such that the nearest grid point is 4.7 mm from this location; thus, this is the closest possible source location prediction. Table II summarizes case studies done at 300 Hz, in terms of what the distance was from the predicted source location 共grid point resulting in maximum value for B关兴兲 to the actual source location. Although predicted and actual locations do differ in all cases, the prediction using the BE model consistently outperforms the prediction based on the free field assumption. As the frequency was increased to 700 Hz, differences between predictions using the BE model and free field assumption did become less but were still present. This is due to the fact that the free field assumption becomes more realistic as frequency increases. Although Table II only indicates the difference between the location of maximum value of B关兴 and the actual source location, 3D graphical depictions in Fig. 11 provide more detail for two cases. Here, the boundary element mesh is shown, the actual source location is at the intersection of the J. Acoust. Soc. Am., Vol. 122, No. 1, July 2007
FIG. 11. 共Color online兲 BE surface mesh is shown; actual source location is at the intersection of the three thick 共black兲 lines; iso-surfaces are shown 共in red and green兲 for B关兴 calculated using the BE model and free field assumption 共respectively兲. The calculation using the BE model results in the smaller 共and darker in grayscale兲 single iso-surface closer to the source in both cases. Each iso-surface encompasses the interpolated 3-dimensional region of theta within which B关兴 exceeds 90% of its maximum value.
three thick lines, and isosurfaces are shown for B关兴 calculated using the BE model and free field assumption. The calculation using the BE model results in the smaller 共and darker in grayscale兲 single isosurface closer to the source. Specifically, the isosurface encompasses the interpolated 3D region of within which B关兴 exceeds 90% of its maximum value. These isosurfaces illustrate the relative levels of precision, as well as accuracy, of the Bartlett beamformer based Ozer et al.: Boundary element model of lungs
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FIG. 12. 共Color online兲 Visible human male. 共a兲 Both lungs meshed with high resolution. Each lung is composed of ⬃10 000 triangular elements. 共b兲 Left lung meshed with 4052 elements. The location of the 247 sensing locations on the lung parenchyma used in the source localization study are indicated by the darker grayscale marks. Positive y denotes anterior direction.
on the BE or free-field approaches. In general, use of the BE model results in a more precise, or focused, prediction for the source location given the same number of surface sensors. Generally, increasing the number of surface sensors better focuses the predicted source location. The material properties of the lung parenchyma are frequency dependent, are not known precisely, and will be subject dependent to some degree. Consequently, source localization sensitivity to inaccurate values for the complex wave number was investigated. Real and imaginary parts of the wave number calculated at 300 Hz per Sec. IV A were reduced by 10% and then used to determine a关 , 兴 in Eqs. 共22兲 and 共23兲. Experimental measurements y at 300 Hz were used with these calculations of a关 , 兴 to determine B关兴 using Eq. 共20兲. Table II shows the resulting difference in predicted and actual source location values under these conditions for different numbers of sensors. In this case, both BE and free-field approaches seem robust to wave number errors, and the BE approach still provides a better estimate of the source location. VI. NUMERICAL STUDY USING ACTUAL LUNG GEOMETRY FROM VISIBLE HUMAN PROJECT A. Meshing and acoustic simulations
Simulation results using the boundary element model are presented for actual lung geometry derived from the visible human male26 共Visible Human Project, National Library of Medicine兲. Lung geometry was extracted based on x-ray CT images available online. Image processing was performed using MATLAB. External nodes on the lung surface were selected using a threshold technique and meshed within the MATLAB environment; Fig. 12 shows sample meshes. Acoustic simulations were performed on the left lung only with an internal acoustic source. After the mesh was obtained node and connectivity information were input into the BE code. 668
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FIG. 13. 共Color online兲 Velocity field due to point acoustic source at 220 Hz calculated using 共a兲 BE model 共two different views兲 and 共b兲 free field model 共two different views兲. Positive y denotes anterior direction.
The wave number value used was k = 60+ 2i, which corresponds to 220 Hz per Fig. 1 and Table I. The main reason for not considering higher frequencies was that a finer mesh would be needed resulting in an increased number of elements. All the simulations were performed in MATLAB 7.0 in Windows XP. The usable memory limit of the 32 bit processor Pentium IV system was 2 GB; however, it was observed that the maximum memory limit was reached when MATLAB variables occupied 800 MB. Therefore, going over 6000 elements was not possible due to computational limitations. However, these memory limitations would not apply for a system with a 64 bit processor. Although 6000 elements is not very many for finite element simulations, the matrices in finite elements are sparse and real. The system matrices in boundary element analysis are fully populated and complex. The number of finite elements that would be needed to obtain the same resolution as ⬃4000 boundary elements with ⬃4000 DOF is on the order of 20 000 finite elements with ⬃120 000 DOF. BE simulation results in Fig. 13共a兲 show the velocity field due to a monopole acoustic source located at the centroid of the lung. Figure 13共b兲 shows the predicted velocity field for the same case but using a free-field assumption; results differ substantially from the BE prediction. The acoustic wavelength at 220 Hz is more than 10 cm, compaOzer et al.: Boundary element model of lungs
FIG. 14. 3436 hypothetical point source locations using source localization algorithm. 共a兲 3D view and 共b兲 side view. Positive y denotes anterior direction.
rable to the size of the geometry. As frequency increases, one would first expect the field to become more complex as standing wave shapes become more complicated, further reducing the accuracy of the free field assumption. But, as frequency increases even further, ultimately damping will diminish standing wave patterns and the free field assumption will become more reasonable.
FIG. 15. 共Color online兲 Source localization results at 220 Hz using Bartlett beamformer coupled with 共a兲 BE model and 共b兲 free-field propagation assumption. Positive y denotes anterior direction. Each isosurface encompasses the interpolated three-dimensional region of within which B关兴 exceeds 87% of its maximum value.
exact as the boundary element simulation results were also used to generate the “actual” acoustic response. Figure 15共b兲 shows the results of the source localization process when free-field propagation model was used to simulate the velocity field. The estimate is 68 mm away from the actual source location where the minimum possible error due to discretization is 3.8 mm. VII. DISCUSSION AND CONCLUSION
B. Source localization studies
Unlike the previous source localization study on the mechanical phantom model, experimental results are not available here. Consequently, the BE simulation was used to calculate the “actual” acoustic response. In other words y = a关 , 兴, calculated based on Eq. 共22兲. Figure 14 shows the internal points where hypothetical sources were placed from two different views for the source localization algorithm. A total of 3436 internal points were selected at 16 height levels in the source localization algorithm. Each point is spaced 7.5 mm apart from its nearest node. A total of 247 sensing points are used and their distribution on the lung mesh is shown in Fig. 12共b兲. The procedure for source localization is the same as in Sec. V. Figure 15共a兲 shows the result of the source localization process when the BE model is used for predicting the velocity field. Of course, the source location is J. Acoust. Soc. Am., Vol. 122, No. 1, July 2007
An acoustic BE model was used to simulate the propagation of sound waves in compliant, porous material like the lung parenchyma. The BE model compared well with numerical 共finite element兲 simulations for fixed, free and massloaded 共soft tissue with minimal stiffness兲 boundary conditions. Initial approaches and case studies for stiffer shell structures at the surface of the BE domain implemented as complex coupled boundary conditions derived from a reduced-order finite element model of the shell compared less favorably with results from a finite element analysis of the entire coupled system. Some improvement is needed for accurate application to human anatomy in order to appropriately account for the rib cage structure. Experimental studies were performed on a phantom composed of a foam material that has similar acoustic properties to healthy lung parenchyma. Numerical predictions usOzer et al.: Boundary element model of lungs
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ing the developed BE model had better agreement with experimental measurements as compared to predictions based on a free field assumption. The experimental studies and numerical simulations indicate that the acoustic response field caused by an internal acoustic source is different than that of free field acoustic propagation away from the source, at least in the low audible frequency range, which is the range within which most lung sounds occur. This is relevant to the development of new diagnostic techniques and is an advancement over prior lung acoustic imaging studies that used multiple sensor locations either to map out the sound field that is observed on the torso surface or, by assuming a free field propagation model, to estimate the location of a sound source. Results reported here do support the widely used assumption that acoustic compression waves dominate the response and the shear and surface waves can be neglected for frequencies in the hundreds of hertz. 共Note the good comparison between the compression wave only BE model and the FE model simulations, which account for all wave types.兲 The source localization studies suggest that the developed BE model coupled with a Bartlett beamformer 共matched field processing兲 yields more accurate results than when the free field assumption is coupled with the same matched field processing algorithm. Source localization results are accurate even when significant errors in the phase speed and attenuation rate are used in the BE-based localization algorithm. Accuracy is dependent on the number of sensors used and becomes more robust to errors in phase speed, attenuation and size when the number of sensors is increased. The case studies performed with an actual lung geometry that is extracted from the Visible Human Project showed that the velocity field of the boundary element model is different than the velocity field predicted by free-field propagation model. This difference had a significant effect on the source localization problem. In Sec. IV it was noted that the phase speed and density of the foam material used in the phantom study were similar to those of human parenchymal tissue when compared to what has been reported in the literature based on experiments and theory. In terms of attenuation the foam was about 1 / 3 as damped as parenchymal tissue based on studies available in the literature. Few researchers have attempted to derive or measure a value for the attenuation rate. In vivo measurement of this value, as well as phase speed, is difficult given that it is impossible to know exactly what path the sound may have taken from a source to a receiver. This path may have involved airborne propagation within the bronchial tree at much faster speeds and will be influenced by the soft and hard tissues of the thorax surrounding the lungs and the impedance mismatches at tissue interfaces. Additionally, in many attempted measurements stethoscopes are used, which can be highly nonlinear in their amplitude response and spectral weighting.3 The model of Wodicka et al. derived from first principles16 and the empirical model of Vovk et al.17 yield nondispersive phase speed calculations for the frequency range of interest. The attenuation rate derived by Wodicka et al. based on a bubble swarm analogy increases more rapidly with frequency than the “loss factor” model of Vovk et al., which, according to that article, was based on 670
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analysis of experimental data given in another reference 共in Russian兲. Careful measurements of phase speed by Paciej et al.14 that avoid the trachea and bronchial airways by inputting sound at the supraclavicular space with multiple measurement points on the thoracic surface seem to more closely support the bubble swarm model. Experimental work of Wodicka and associates38–40 do not conflict with their theoretical model’s predictions of phase speed and attenuation for the frequency range of 100– 600 Hz, though the ability to experimentally determine the attenuation of the parenchymal material alone is difficult, as noted earlier. Extrapolating the theory of Wodicka et al. to say 1000 or 2000 Hz results in attenuation rates of 8 and 60 orders of magnitude, respectively, which may seem unlikely given that measurements of lung sounds have been recorded up to at least 2000 Hz.3,41 In summary, it seems that there is some range of uncertainty as to what the actual damping level within the parenchyma is as a function of frequency. Regardless, in this study it was found that both in terms of predicting the acoustic field and in terms of acoustic source localization, the BE method yielded accurate results even when the assumed attenuation rate was in error. Additionally, simulation studies 共not shown兲 with the geometry of the mechanical phantom but the higher attenuation rate of Wodicka et al. for lung tissue indicated that at least up to 700 Hz there was a significant difference between the free field model and the BE model. Extending the presented technique to the envisioned application, lung diagnosis in vivo, will require accounting for the presence of multiple, coherent and distributed sound sources as a part of the source localization problem. It may be necessary to use more sophisticated source localization algorithms. Parametric methods, such as maximum likelihood methods, may be required to deal with this complex source localization problem since simple methods, like the Bartlett beamformer used in this study, are not as effective when dealing with multiple coherent sources.37 ACKNOWLEDGMENT
Support for this research from the National Institutes of Health is acknowledged 共Grant Nos. HL061108 and EB003286兲. 1
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