Three-dimensional finite element modeling of guided ultrasound wave propagation in intact and healing long bones Vasilios C. Protopappas Department of Medical Physics, Medical School, University of Ioannina, GR 451 10 Ioannina, Greece, and Unit of Medical Technology and Intelligent Information Systems, Department of Computer Science, University of Ioannina, GR 451 10 Ioannina, Greece
Iraklis C. Kourtis Unit of Medical Technology and Intelligent Information Systems, Department of Computer Science, University of Ioannina, GR 451 10 Ioannina, Greece
Lampros C. Kourtis Biomechanical Engineering Division, Mechanical Engineering, Stanford University, Stanford, California 94305
Konstantinos N. Malizos Department of Orthopaedics, Medical School, University of Thessaly, GR 412 22 Larissa, Greece
Christos V. Massalas Department of Material Science and Engineering, University of Ioannina, GR 451 10 Ioannina, Greece
Dimitrios I. Fotiadisa兲 Unit of Medical Technology and Intelligent Information Systems, Department of Computer Science, University of Ioannina, GR 451 10 Ioannina, Greece
共Received 19 May 2006; revised 19 August 2006; accepted 21 August 2006兲 The use of guided waves has recently drawn significant interest in the ultrasonic characterization of bone aiming at supplementing the information provided by traditional velocity measurements. This work presents a three-dimensional finite element study of guided wave propagation in intact and healing bones. A model of the fracture callus was constructed and the healing course was simulated as a three-stage process. The dispersion of guided modes generated by a broadband 1-MHz excitation was represented in the time-frequency domain. Wave propagation in the intact bone model was first investigated and comparisons were then made with a simplified geometry using analytical dispersion curves of the tube modes. Then, the effect of callus consolidation on the propagation characteristics was examined. It was shown that the dispersion of guided waves was significantly influenced by the irregularity and anisotropy of the bone. Also, guided waves were sensitive to material and geometrical changes that take place during healing. Conversely, when the first-arriving signal at the receiver corresponded to a nondispersive lateral wave, its propagation velocity was almost unaffected by the elastic symmetry and geometry of the bone and also could not characterize the callus tissue throughout its thickness. In conclusion, guided waves can enhance the capabilities of ultrasonic evaluation. © 2007 Acoustical Society of America. 关DOI: 10.1121/1.2354067兴 PACS number共s兲: 43.80.Ev, 43.80.Qf, 43.80.Jz 关CCC兴
I. INTRODUCTION
Quantitative ultrasound has gained significant interest in the assessment of osteoporosis and the evaluation of fracture healing. The so-called axial transmission technique has been used to examine the properties of long bones, such as the tibia and radius. Typically, a transmitter and a receiver are placed in contact with the skin 共percutaneous application兲 along the long axis of the bone. The emitted ultrasonic pulse 共typically in the 0.2– 2-MHz frequency range兲 propagates along the bone and the first arriving signal 共FAS兲 at the rea兲
Author to whom correspondence should be addressed. Electronic mail:
[email protected]
J. Acoust. Soc. Am. 121 共6兲, June 2007
Pages: 3907–3921
ceiver is used to determine the ultrasound propagation velocity. To compensate for the effect of the overlying soft tissues, methods using either multiple percutaneous transducers 共Bossy et al., 2004a; Saulgozis et al., 1996兲 or transducers implanted directly into the fracture region 共Protopappas et al. 2005; Malizos et al., 2006兲 have been proposed. In the assessment of osteoporosis, experimental 共Moilanen et al., 2003; Njeh et al., 1999b; Tatarinov et al., 2005; Raum et al., 2005兲 and simulation studies 共Bossy et al., 2002, 2004b; Camus et al., 2000; Nicholson et al., 2002兲 have demonstrated that the propagation velocity of the FAS is related to the bone mineral density, the thickness of the cortex, and the elastic properties of the bone, and is also a significant discriminator of osteoporotic fracture risk. In the
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context of bone healing, the aim of the ultrasonic evaluation is to monitor the course of healing, detect any complications 共e.g., delayed unions or nonunions兲 and early assess functional bony union. Animal 共Abendschein and Hayatt, 1972; Gill et al., 1989; Malizos et al., 2006; Protopappas et al., 2005兲 and clinical studies 共Cunningham et al., 1990; Gerlanc et al., 1975; Maylia and Nokes, 1999兲 have shown that the FAS propagation velocity in fractured bones gradually increases over the healing period as a result of the fracture callus consolidation process. Experiments on acrylic plates immersed in water have also been performed to investigate the dependence of the FAS velocity on the soft tissue thickness, cortical thickness, and fracture gap width and depth 共Lowet and Van der Perre, 1996; Njeh et al., 1999a兲. However, when the wavelength of the transmitted wave is comparable to or smaller than the thickness of the cortex, the type of wave that contributes to the FAS corresponds to a lateral 共also known as P-head兲 wave 共Rose, 1999兲. Lateral waves propagate only along the bone’s subsurface at the bulk longitudinal velocity and therefore FAS velocity measurements reflect mainly the periosteal region 共outer layer兲 of the bone. Conversely, when the wavelength is larger than the cortical thickness, the bone acts as a waveguide and the received signal waveform is a superposition of multiple guided wave modes. The use of guided waves has recently drawn increased attention in the ultrasonic evaluation of bone because guided waves propagate throughout the cortical thickness and are thus sensitive to both mechanical and geometrical properties. Studies focusing on osteoporosis have investigated guided wave propagation by making use of acrylic plates and tubes of varying thickness 共Lee and Yoon, 2004; Nicholson et al., 2002; Tatarinov et al., 2005兲 and of two-dimensional 共2D兲 finite difference simulations on isotropic bonemimicking plates 共Bossy et al., 2002; Nicholson et al., 2002兲. It has been shown that by incorporating the theory that describes guided modes in plates 共Lamb theory兲 generally two dominant modes could be detected in signals; the fastest one corresponds to the lowest-order symmetric Lamb mode 共denoted as S0 mode兲, whereas the slowest one to the lowest-order antisymmetric Lamb mode 共denoted as A0 mode兲. Assuming that guided waves in a plate agree closely to the type of waves propagating in the cortical shell, ex vivo and in vivo studies 共Lee and Yoon, 2004; Lefebvre et al., 2002; Moilanen et al., 2003; Nicholson et al., 2002; Tatarinov et al., 2005兲 demonstrated that the velocity of the S0 and A0 modes was able to reflect structural changes in the cortex and thus provide an enhanced approach for characterizing healthy and osteoporotic bones. In a recent study, we investigated guided ultrasound propagation in a 2D model of a healing bone 共Protopappas et al., 2006兲. Using time-frequency 共t-f兲 signal analysis techniques, it was made possible to represent the dispersion of the velocity of the fundamental as well as of higher-order Lamb modes. We demonstrated that the propagation of Lamb modes was sensitive to both the material and geometrical properties of the fracture callus tissue during a simulated healing process. However, analysis of ex vivo measurements from an intact bone showed that the Lamb wave theory could 3908
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FIG. 1. Reference coordinates and dimensions of the hollow circular cylinder.
not sufficiently describe the dispersion of the propagating guided modes 共Protopappas et al., 2006兲. This was further supported by a recent three-dimensional 共3D兲 finite difference study 共Bossy et al. 2004b兲 reporting on the effect of the tubular geometry and anisotropy of the bone on wave propagation. In this work, we extend our previous computational study by addressing more realistic conditions. The 3D geometry and anisotropy of the bone and the fracture callus tissue are taken into account. The objectives of this work are 共a兲 to elucidate the parameters 共irregular geometry and anisotropy兲 that affect wave propagation in intact bones and 共b兲 to investigate the influence of callus formation and consolidation on the characteristics of propagation. In this respect, we first examine a simplified geometry of the bone 共hollow circular cylinder兲 for two cases of material symmetry: isotropy and transverse isotropy. Next, the curvature of the cortical bone is considered and the callus tissue is modeled as an inhomogeneous material consisting of several ossification regions. The velocity of the FAS wave is determined using traditional time-domain techniques. The propagating guided modes are characterized in the t-f representation of the signal by incorporating theoretical velocity dispersion curves. The present study is the first to address guided wave propagation in 3D healing bones.
II. GUIDED WAVES IN AN ELASTIC HOLLOW CIRCULAR CYLINDER
The theory of elastic wave propagation in a hollow circular cylinder satisfying traction-free boundary conditions on the inner and outer surfaces will be briefly described in this section. Dispersion curves of the velocity of guided modes in a transversely isotropic medium and their degeneration to the case of isotropy will be presented and compared to those derived by the Lamb wave theory. Let there an infinitely be long linear elastic hollow circular cylinder 共Fig. 1兲. The three-dimensional stress equations of motion in cylindrical coordinates are 共Rose, 1999兲
2u r rr 1 r rz 1 + + + 共rr − 兲 = 2 , t r r z r 2u r 1 2 r + = 2 , + r + t z r r r Protopappas et al.: 3D modeling of ultrasound in bones
2u z zz 1 z 1 rz + = 2 , + rz + t z r r r
共1兲
where r, , and z are the cylindrical coordinates; rr, , zz, z, zr, and r are the stress components; ur, u, uz are the particle displacement components; t is the time; and is the density. The strain-displacement relations are given by 共Fotiadis et al., 2006兲
ur , rr = r
1 u ur + , = r r
冉 冉
冊
rz =
1 ur uz + , 2 z r
z =
1 u 1 uz + , 2 z r
r =
uz zz = , z
冋 冉冊
册
1 u 1 ur r + , 2 r r r
冊
共2兲
where rr, , zz, z, zr, and r are the strain components. The stress-strain relations for a transversely isotropic medium, when the symmetry axis is in the z direction, are given by 共Fotiadis et al., 2006兲
rr = C11rr + C12 + C13zz , = C12rr + C11 + C13zz , zz = C13rr + C13 + C33zz , z = 2C44z,
rz = 2C44rz,
r = 2C66r ,
共3兲
where C11, C12, C13, C33, and C44 are the five independent elastic constants required to characterize the transverse isotropy of the material, whereas the constant C66 is given as C66 = 共C11 − C12兲 / 2 共Fotiadis et al., 2006兲. For the propagation of free harmonic waves, substituting Eqs. 共2兲 and 共3兲 into Eq. 共1兲, the assumed displacement components are 共Mirsky, 1965兲 ur = Ur共r兲cos n cos共t + z兲, u = U共r兲sin n cos共t + z兲, uz = Uz共r兲cos n cos共t + z兲,
共4兲
where is the wave number and is the circular frequency. The functions Ur, U, and Uz are the corresponding components of the radial distribution of the displacement amplitudes. These functions are composed of Bessel functions 共or modified Bessel, depending on the arguments兲 and contain six 共unknown兲 amplitude coefficients 共Mirsky, 1965兲. The index n = 0 , 1 , 2 , 3 , . . . is called the circumferential order 共Rose, 1999兲 and specifies the order of symmetry around the axis of the cylinder. For traction-free boundary conditions 共free motion兲, the stresses must vanish on the inner and outer surfaces of the hollow-cylinder, i.e.,
rr = rz = r = 0
at r = a and r = b,
共5兲
where a and b is the inner and outer radii of the hollowcylinder, respectively. The stress components in Eq. 共5兲 can 3909
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be represented in terms of the six amplitude coefficients by applying Eqs. 共2兲 and 共3兲 to the displacements in Eq. 共4兲 共Mirsky, 1965兲. Satisfaction of the boundary conditions results in a system of six linear equations with respect to the amplitude coefficients. For nontrivial solutions, the determinant, D, of the system must vanish 共Mirsky, 1965兲: 兩Dij兩 = 0
共i, j = 1 to 6兲.
共6兲
Equation 共6兲 is called the characteristic frequency equation. The element Dij is analytically expressed in terms of the elastic constants of the material, the dimensions of the hollow cylinder 共i.e., the inner and outer radius兲, the frequency, and the wave number 共Mirsky, 1965兲. For given material and geometry, Eq. 共6兲 is a transcendental equation of the frequency and wave number. The roots of Eq. 共6兲 provide the dispersion curves of the guided modes, i.e., the wave number as a function of frequency. In the context of this work, we are practically interested in the dispersion of the group velocity 共cgr兲 of the guided modes, which is defined in terms of the derivative of the dispersion curves: cgr =
.
共7兲
The group velocity dispersion curves express the velocity at which the energy of a guided mode propagates as a function of frequency. The characteristic frequency equation 关Eq. 共6兲兴 can be degenerated to the case of an isotropic material 共Gazis, 1959兲 by reducing the number of elastic constants from five to two using the following relationships: C33 = C11,
C13 = C12,
C44 = C66 =
共C11 − C12兲 . 2
共8兲
The different guided modes that propagate in the z direction of a hollow circular cylinder are commonly notated by their type, circumferential order, and consecutive order 共Rose, 1999兲. For n = 0 关from Eq. 共4兲兴, the particle displacement of the modes is axisymmetric. In this case, there exist two types of modes: longitudinal modes, denoted as L共0 , m兲, consisting of axial and radial displacement components, and torsional modes, denoted as T共0 , m兲, containing only displacement in the circumferential direction. A third type of mode called flexural modes, F共n , m兲, n = 1 , 2 , 3 , . . ., corresponds to modes with non-axisymmetric displacements and contains all three possible displacements. The index m = 1 , 2 , 3 , . . . denotes the order 共numbering兲 of the mode. Group velocity dispersion curves of the L共0 , m兲 and F共1 , m兲 modes for a bone-mimicking hollow circular cylinder with wall thickness 4.08 mm and outer radius 8.61 mm are shown in Fig. 2. Two cases are presented: the first for isotropic material 共dashed lines兲 and the second for transversely isotropic material 共solid lines兲 whose properties are provided in Table I. It can be seen that the symmetry of the material has a significant effect on the dispersion of the higher-order modes and almost no effect on the fundamental L共0 , 1兲 and F共1 , 1兲 modes. When the wall thickness is thinner than half the outer radius, the dispersion of the F共1 , 1兲 and the higher-order modes is almost identical to the correProtopappas et al.: 3D modeling of ultrasound in bones
FIG. 2. 共Color online兲 Group velocity dispersion curves of the longitudinal and flexural modes for a free hollow circular cylinder 共wall thickness 4.08 mm and outer radius 8.61 mm兲 for the case of material isotropy 共dashed lines兲 and transverse isotropy 共solid lines兲.
sponding Lamb modes in a plate of equal thickness, whereas the L共0 , 1兲 mode is different than the S0 mode only for very low frequencies 共Lefebvre et al., 2002; Protopappas et al., 2006兲. III. MATERIALS AND METHODS A. Finite element model 1. Geometry of cortical bone
The cross section of the cortical bone was determined by Computer Tomography transverse scans 共Phillips Secura, acquisition parameters: 140 kV, 220 mA, slice thickness 2 mm, slice distance 1 mm兲 obtained from the middle diaphysis region of a sheep tibia. The periosteal 共external兲 and endosteal 共internal兲 contours of the cortex were determined using a threshold-based region-growing segmentation technique 共Protopappas et al., 2005兲. The endosteal contour of the cortex, however, was simplified to the circle 共with radius 4.53 mm兲 that fits to the extracted endosteal contour in a least-squares sense 共Fig. 3兲. The average thickness of the
cortex was 4.08 mm 共min: 3.36 mm, max: 4.74 mm兲, which is within the range found in some types of human long bones, such as tibia and radius 共Njeh et al., 1999b兲. The 3D model describes a diaphyseal segment of the bone with length L = 50 mm. As opposed to the convex curvature of the cortex in the transverse plane, the curvature along the bone axis at the level of diaphysis may reasonably be neglected 共Bossy et al., 2004b兲. Therefore, the 3D bone model was considered uniform in the long axis direction with constant cross section.
2. Equivalent hollow circular cylinder
We also modeled a hollow circular cylinder with inner and outer radii 4.53 and 8.61 mm, respectively, and length L = 50 mm. The outer diameter corresponds to the circle that best fits to the periosteal contour of the cortex, in a leastsquares sense. This model corresponds to a simplified geom-
TABLE I. Elastic constants for isotropic and transversely isotropic cortical bone.
Material properties Elastic constants 共GPa兲 C11 = C22 C33 C12 C23 = C31 C44 = C55 C66 Density 共Kg/ m3兲 Bulk longitudinal velocity 共m/s兲
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Isotropy
Transverse isotropy
24.76 24.76 14.54 14.54 5.11 5.11 1500 4063
17.50 24.76 10.15 10.69 5.11 3.67 1500 4063
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FIG. 3. 共Color online兲 Cross section of a sheep tibia obtained from computer tomography scans along with the extracted periosteal and endosteal contours of the cortex. Protopappas et al.: 3D modeling of ultrasound in bones
FIG. 4. 共Color online兲 One quarter of the callus model 共left side corresponds to endosteal regions兲 and the types of tissues involved in 共a兲 Stage1, 共b兲 Stage2, 共c兲 Stage3, and 共d兲 the “hypothetical” Stage0. The geometry of the healing bone in a transverse plane at the middle of the model’s length is shown in 共e兲.
etry of the bone and was used to investigate the effects of the convex curvature of the cortex on the propagation of guided waves. 3. Model of fracture callus
The secondary 共indirect兲 type of fracture healing involves the formation and gradual consolidation of a soft tissue called fracture callus. Secondary healing takes place when small axial motion is allowed between the bone fragments, e.g., in the case of external fixation device treatment. Spatial and temporal sequences of tissue differentiation and ossification are fundamental processes in this type of healing. The geometry of the callus tissue was described by a periosteal region bulging out of the cortex and an endosteal region bulging into the cortex. Following a previous computational study of fracture healing 共Claes and Heigele, 1999兲, we modeled the callus tissue as an inhomogeneous material segmented into six different ossification regions. The healing course was simulated as a three-stage process. At each stage, the properties of the callus regions evolved corresponding to various types of soft tissues involved in the healing process. We assumed five soft tissue types, namely, initial connective tissue 共ICT兲 describing nonmineralized connective tissue, soft callus 共SOC兲, intermediate stiffness callus 共MSC兲, and stiff callus 共SC兲 representing the phases of new bone formation; and finally ossified tissue 共OT兲. More specifically, at the first stage 共Stage1兲, the callus consisted of regions of MSC along the endosteal and periosteal surfaces of the cortex at some distance from the fracture gap, of SOC adjacent to them, while the remainder consisted of ICT 关Fig. 4共a兲兴. At the second stage 共Stage2兲, ossification has progressed in the direction of fracture gap and the callus tissue contained ICT, SOC, MSC, and SC 关Fig. 4共b兲兴. At the third stage 共Stage3兲, bone formation has taken place and only a small region of 3911
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SOC separates the bone margins 关Fig. 4共c兲兴. According to Claes and Heigele 共1999兲, Stage1 and Stage2 correspond to the fourth and eighth week after fracture, respectively, whereas Stage3 reflects the phase before bone remodeling. We also incorporated a “hypothetical” zero stage 共Stage0兲, in which the callus region consisted only of cortical bone 关Fig. 4共d兲兴. Although there is no physical meaning of this stage, it allows for the investigation of the effect of the callus geometry itself on wave propagation. The geometry of the callus remained the same for all the stages.
4. Mesh generation
The geometrical model of the intact bone was constructed using a 3D structured volume meshing routine implemented in MATLAB 共The Mathworks, Inc., MA兲. The nodes of the mesh have been seeded in three dimensions according to analytical expressions. More specifically, the cylindrical coordinates, rijk,ijk,zijk, of the ijk node were defined as rijk = Re共 j兲 + 关R p共 j兲 − Re共 j兲兴ri,
ijk = j,
zijk = zk , 共9兲
with ri 苸 关0,1兴,
j 苸 关0,2兴, zk 苸 关0,L兴,
ri+1 ⬎ ri,
i = 1, . . . ,M ,
j+1 ⬎ j, zk+1 ⬎ zk,
j = 1, . . . ,N, k = 1, . . . ,K,
共10兲
where M, N, and K are the number of nodes in the radial, circumferential, and axial directions, respectively, and Re共 j兲 Protopappas et al.: 3D modeling of ultrasound in bones
TABLE II. Material properties of the types of soft tissues involved in the healing process. Young’s modulus 共MPa兲
Density 共kg/ m3兲
Tissue type Initial connective tissue 共ICT兲 Soft Callus 共SOC兲 Intermediate stiffness callus 共MSC兲 Stiff callus 共SC兲 Ossified tissue 共OT兲
1050 1100 1200 1250 1400
3 1000 3000 6000 10 000
and R p共 j兲 are the endosteal and periosteal radii of the cortex at angle j, respectively. Note that since the endosteal contour was assumed circular, Re共 j兲 = Re. A number of Q points 共usually peaks and values兲 were selected along the periosteal contour extracted from the CT scan, with R pq and pq, where q = 1 , . . . , Q, representing their polar coordinates. The R p共 j兲 was determined from the selected points according to R p共 j兲 = F共 j, pq,R pq兲,
共11兲
Re⬘共 j兲 =
R⬘p共 j兲 =
冦 冦
兩zk − L/2兩 ⬎ Le/2,
冋 冉
Re − 共Re − Rec兲 · 0.5 1 + cos 2 R p共 j兲,
兩zk − L/2兩⬎L p/2,
zk − L/2 Le
冋 冉
R p共 j兲 + 关R pc − R p共 j兲兴 · 0.5 1 + cos 2
冊册
,
兩zk − L/2兩 艋 Le/2,
zk − L/2 Lp
冊册
,
c3 = The cortical bone was modeled as a linear elastic homogeneous material. In a first series of experiments, cortical bone was considered isotropic, whereas in a second series it was considered transversely isotropic with properties shown in Table I. Although the cortical bone is generally anisotropic, transverse isotropy is a realistic approximations observed experimentally 共Reilly and Burstein, 1975; Rho, 1996兲 and used in models 共Bossy et al., 2004b兲. The elastic constants were derived from longitudinal and shear bulk velocity values typically used for bone 共Bossy et al., 2004b; Protopappas et al., 2006; Rho, 1996兲. The cortical density, = 1500 kg/ m3, represents the average value that has been measured in a previous animal study 共Protopappas et al., 2005兲 from quantitative CT-based densitometry performed on the midshaft cortical region of sheep tibiae. Table I also contains the resulting bulk longitudinal 共compressional兲 velocity, c3, for propagation in the z direction, given by 共Fotiadis et al., 2006兲 J. Acoust. Soc. Am., Vol. 121, No. 6, June 2007
冧
兩zk − L/2兩 艋 L p/2.
5. Material properties
3912
0.4998 0.47 0.45 0.43 0.40
Bulk longitudinal velocity 共m/s兲 1543 2337 3079 3697 3912
The mesh of the hollow circular cylinder model was constructed in a similar manner by defining R p共 j兲 = R p. The model of the healing bone incorporated the callus at the middle of the model’s length, i.e., at L / 2. The cross sections of the endosteal and periosteal callus regions were assumed circular with their radii varying along the length of the callus according to a Hanning function. By denoting with Rec and R pc the radii of the endosteal and periosteal callus regions at L / 2, respectively 关Fig. 4共e兲兴, and with Le and L p the lengths of the endosteal and periosteal callus regions, respectively 关Fig. 4共d兲兴, the radial position of the nodes given in Eq. 共9兲 was modified at the callus area as follows:
where F is a piecewise cubic interpolation continuous function.
R e,
Poisson’s ratio
冑
共12兲
冧
C33 .
共13兲
共14兲
All soft tissues types incorporated in the callus model were assumed isotropic. Their elastic properties have been assessed by indentation tests on tissue sections obtained from different callus regions 共Augat et al., 1998; Claes and Heigele, 1999兲. Table II contains their material properties in the form of Young’s modulus, E, and Poisson’s ratio, , and also their resulting bulk longitudinal velocity 共cL兲, given by 共Fotiadis et al., 2006兲 cL =
冑
E共1 − v兲 . 共1 + v兲共1 − 2v兲
共15兲
We have modified the values of the Poisson’s ratio from those given in Claes and Heigele 共1999兲 so the bulk longitudinal velocities will correspond to realistic values. For instance, in Claes and Heigele 共1999兲, the Poisson’s ratio for the ICT was 0.4, which would result in cL = 78 m / s 关Eq. Protopappas et al.: 3D modeling of ultrasound in bones
FIG. 5. 共Color online兲 The model of the diaphyseal segment of cortical bone incorporating the fracture callus 共sagittal section兲. The transmitterreceiver configuration is also illustrated.
共15兲兴, whereas the value commonly used in the literature for soft tissues is close to 1500 m / s 共Bossy et al., 2004b兲. Similarly, by experimenting with the Poisson’s ratio for the SOC, MSC, SC, and OT, we achieved a smooth transition of their bulk velocities to the cortical bone bulk longitudinal velocity.
6. Element properties
We used eight-node linear hexahedral continuum elements 共element type C3D8R in ABAQUS version 6.4兲 with reduced integration and hourglass control 共hourglassing is a numerical phenomenon by which zero-energy modes propagate兲 共ABAQUS, 2003兲. The spatial discretization of the model is a critical issue when simulating wave propagation. The internodal distance in the i direction, delemi, must be smaller than the smallest wavelength in that direction, i, such that the propagating waves are spatially resolved 共Moser et al., 1999; Zerwer et al., 2003兲. This can be expressed as delemi 艋
min共i兲 , g
共16兲
where g is a factor indicating the minimum number of nodes per smallest wavelength and must be larger than 6 共ABAQUS, 2003兲. The average internodal distances in our model were approximately delem1 = 0.10 mm in the radial direction, delem2 = 0.44 mm in the circumferential direction, and delem3 = 0.18 mm in the z direction. The resulting total number of degrees of freedom was 4 022 040. 3913
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7. Simulation of axial transmission
The 3D configuration and location of the transmitter and receiver relative to the bone and callus are shown in Fig. 5. The excitation signal consisted of a transient loading in the y direction applied to the bone surface over a circular area of 5-mm diameter. The amplitude of the excitation was a threecycle Gaussian-modulated 1-MHz sine 共0.55-MHz bandwidth at −6 dB兲. The time histories of the y displacements of the nodes located at the bone surface over a circular area of 5-mm diameter were recorded. The receiver’s signal waveform was the average time history of the nodal y displacements. The transmitter and receiver were equidistant from the fracture gap and their center-to-center distance was 36 mm, which is in the range typically used in ultrasonic studies of bone 共Bossy et al., 2002; Protopappas et al., 2005兲. Two sites of transmitter-receiver positioning were investigated. The first 共Site1兲 corresponded to a region where the cortical shell has average local thickness 3.4 mm and exhibits an irregular curvature 共Fig. 6兲. Conversely, Site2 corresponded to a region where the cortical shell has local average thickness 3.97 mm and its curvature resembles a circular cylindrical shell 共Fig. 6兲. In this respect, the influence of the anatomical site of measurement on guided wave propagation can be evaluated. 8. Boundary conditions
The bone was considered free and thus the influence of the surrounding soft tissues and the bone marrow was not taken into account. Also, infinite elements 共element type CIN3D8 in ABAQUS version 6.4兲 were attached at the ends Protopappas et al.: 3D modeling of ultrasound in bones
C. Ultrasound signal analysis in the time-frequency domain
FIG. 6. 共Color online兲 The anatomical sites of measurement, Site1 and Site2, and the corresponding average local thicknesses of the cortex 共in mm兲. The circle that fits to the external contour of the cortex is also illustrated 共dashed line兲.
of the model to absorb the energy of the incoming waves and thus simulate an infinitely long model. The infinite elements introduce additional normal and shear stress components, called boundary damping stresses, which eliminate all the reflections from the longitudinal and shear waves that impinge normally on the boundary between the finite and infinite elements 共Lysmer and Kuhlemeyer, 1969兲. However, this formulation does not provide perfect transmission of energy out of the ends of the model for waves arriving from other directions 共i.e., other than the z direction兲. 9. Numerical solution
Solution to the elastic wave propagation problem was performed using the explicit elastodynamics finite element analysis 共ABAQUS/Explicit, version 6.4兲. Explicit analysis requires the integration time increment 共⌬t兲 to be smaller than the stability limit 共ABAQUS, 2003; Hughes, 2000兲:
冉冑
⌬t 艋 min
2 共delem 1
+
2 delem 2
ci
+
2 delem 兲/3 3
冊
,
共17兲
where ci denotes the bulk longitudinal velocity of the material in the i direction. The integration time increment was automatically set by ABAQUS. We obtained nodal displacements at 0.05-s time points corresponding to a 20-MHz sampling frequency. The time history of the nodal displacements was recorded for 50 s and thus the length of the signal was 1001 points. The problem was solved using a 64-CPU, Origin 2000 supercomputer 共SGI, Mountain View, CA兲 with 32 Gbytes shared memory. Typical computational time was 40 min.
Time-frequency analysis has previously been employed in studying velocity dispersion of Lamb waves in nondestructive applications of flaw detection and localization in aluminum and composite plates 共Niethammer et al., 2001; Proseer et al., 1999兲. As opposed to traditional time domain 共Lee and Yoon, 2004; Moilanen et al., 2003; Nicholson et al., 2002兲 and 2D fast Fourier transform techniques 共Lefebvre et al., 2002; Moser et al., 1999兲, which require the collection of multiple signals recorded at equally spaced distances, t-f analysis can represent the dispersion of multiple wave modes using only a broadband signal. In our previous 2D study, three different t-f distribution functions were investigated. It was shown that the reassigned smoothed-pseudo Wigner-Ville 共RSPWV兲 energy distribution provided sufficient t-f resolution and signal localization ability 共Protopappas et al., 2006兲, and therefore in this study we used this distribution function. The RSPWV function belongs to the Cohen’s class of energy distributions, in which the distributions are covariant by translation in time and frequency. A function in the Cohen’s class, C共t , f兲, is given as 共Auger et al., 1995兲 C共t, f兲 =
ei2v共u−t兲g共v, 兲x* u −
⫻x u +
−i2 f e dv du d , 2
冉 冊
Detection of the FAS in the signal waveform was performed using a threshold equal to 10% of the amplitude of the first signal extremum. Such a detection criterion minimizes erroneous estimation of the transition time as opposed to other criteria based on constant thresholds, zero-crossings, signal extrema, etc., which are affected by frequencydependent attenuation, mode interference, etc. 共Bossy et al., 2002; Nicholson et al., 2002; Protopappas et al., 2005兲. 3914
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2
共18兲
where x*共t兲 is the complex conjugate of the signal and g共v , 兲 is an arbitrary function called the kernel. The kernel for the smoothed-pseudo Wigner-Ville distribution SPWV共t , f兲 is 共Auger et al., 1995兲 g共v, 兲 = G共v兲h共兲,
共19兲
where G共v兲 is a frequency-smoothing window and h共兲 is a time-smoothing window. In order to improve the timefrequency resolution of the SPWV共t , f兲, the reassignment version SPWV共r兲共t , f兲 can be used 共Auger et al., 1995兲: SPWV共r兲共t⬘, f ⬘兲 =
冕 冕 +⬁
−⬁
+⬁
SPWV共t, f兲␦共t⬘ − ˆt共t, f兲兲
−⬁
⫻␦共f ⬘ − ˆf 共t, f兲兲dt df ,
共20兲
where ␦共t兲 is the Dirac function. The reassigned coordinates 共tˆ , ˆf 兲 for each 共t , f兲 in the original SPWV are ˆt共t, f兲 = t −
B. Ultrasound signal analysis in the time domain
冉 冊
冕冕冕
SPWVTh共t, f兲 2SPWVh共t, f兲
ˆf 共t, f兲 = f + i
,
SPWVDh共t, f兲 2SPWVh共t, f兲
,
共21兲
where SPWVh, SPWVTh, and SPWVDh are the SPWV with window functions h共t兲, th共t兲, and dh共t兲 / dt, respectively. In this study, the frequency- and time-smoothing windows were W / 10 point Hamming windows, where W denotes the number of points of the signal. Protopappas et al.: 3D modeling of ultrasound in bones
FIG. 7. Signal waveforms obtained from the intact isotropic models of 共a兲 hollow circular cylinder, 共b兲 bone at Site1, 共c兲 bone at Site2, and intact anisotropic models of 共d兲 hollow circular cylinder, 共e兲 bone at Site1, and 共f兲 bone at Site2.
IV. RESULTS A. Analysis in the time signal domain
The signal waveforms obtained from the intact hollow cylinder and from the bone at Site1 and Site2 in the case of isotropy and anisotropy are illustrated in Fig. 7. It can be seen that the waveforms were significantly influenced by the model geometry, anatomical site, and material symmetry. On the other hand, the arrival time of the FAS was slightly affected only by the material symmetry. By dividing the inbetween distance of the transducers 共i.e., 31 mm兲 by the FAS time, the calculated propagation velocity of the FAS was 3954 m / s for the isotropic and 4042 m / s for the anisotropic models. Since the FAS wave propagated at a velocity close to the bulk longitudinal velocity of the bone 共4063 m / s兲, it did not correspond to a guided wave but rather to a lateral wave. Typical snapshots of wave propagation in the anisotropic healing bone at Stage2 are presented in Fig. 8. The propagation of the lateral wave along the surface of the cortex just before the end of the excitation is illustrated in Fig. 8共a兲. Wave reflections at the inner and outer boundaries cause the formation of guided modes that propagate in the axial and circumferential directions 关Fig. 8共b兲兴. When the waves propagate in the callus region, the wavelengths are smaller depending on the properties of the soft tissues 关Figs. 8共c兲–8共f兲兴. Besides the axially propagating waves, additional waves arrive at the receiver due to reflections from the callus geometry 关Fig. 8共d兲兴 and propagation along circumferential paths. The variation of the FAS velocity over the simulated 3915
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healing stages of the bone model is presented in Fig. 9. The FAS velocity decreased at Stage1, remained the same up to Stage2, and then increased at Stage3. The velocity values at each stage were again higher for the anisotropic case, but no difference was observed between the measurements from Site1 and Site2. The fact that the velocity did not change from Stage1 to Stage2 possibly indicates that the propagation of the FAS wave was only affected by the ICT material that filled the fracture gap during these stages 关Figs. 4共a兲 and 4共b兲兴. B. Analysis in the time-frequency domain
The t-f representations of the signals obtained from the intact hollow-cylinder, in the case of isotropy and anisotropy, are shown in Figs. 10共b兲 and 10共d兲, respectively. The t-f representations are shown in the form of pseudo-color images, where the color of a point represents the amplitude 共in dB兲 of the energy distribution. In order to identify the propagating guided modes in the t-f representation, we used the theoretical frequency-group velocity 共f , cg兲 dispersion curves of the free hollow circular cylinder. Using the center-to-center distance between the transmitter and receiver, the corresponding 共f , cg兲 dispersion curves were converted to 共t , f兲 curves in which the theoretically anticipated arrival time of each mode is plotted as a function of frequency. The 共t , f兲 curves were then superimposed on the t-f representations. For the isotropic case, it can be seen that the dispersion of the L共0 , 5兲 was clearly characterized in the 0.55– 0.8-MHz range and of the L共0 , 4兲 mode in the 0.7– 0.85-MHz range. Furthermore, the L共0 , 8兲 mode was identified at its cutoff frequency Protopappas et al.: 3D modeling of ultrasound in bones
FIG. 8. Snapshots of wave propagation in the anisotropic healing bone at Stage2 for excitation applied to Site2. Time instances at 共a兲 2.5 s, 共b兲 5.0 s, 共c兲 8.0 s, 共d兲 10 s, 共e兲 14 s, and 共f兲 20 s. We display the amplitude of the y displacements 共amplified by a 104 scale factor兲, in which the black color corresponds to maximum positive value and white to minimum negative value.
共1.05 MHz兲 and the fundamental L共0 , 1兲 and F共1 , 1兲 modes in the 0.05– 0.15-MHz range. When the material was assumed anisotropic, the L共0 , 5兲 mode was again described by the new dispersion curves 共in the 0.5– 0.85-MHz range兲, whereas the L共0 , 4兲 mode did not propagate. The L共0 , 8兲 mode was identified from its cutoff frequency 共0.92 MHz兲 up to 1.05 MHz and appeared to carry most of the signal energy. The fundamental modes remained similar to those in the isotropic hollow cylinder. In both cases, the t-f representations contained additional waves that did not correspond to any theoretical modes and can be possibly attributed to circumferential waves that arrived later at the receiver. The t-f signal representations obtained from the intact anisotropic bone at Site1 and Site2 are shown in Figs. 11共a兲 and 11共b兲, respectively. Poor agreement exists between the higher-order modes propagating in the hollow cylinder and those in the bone. However, the t-f signal representation from
Site2 contained two modes with dispersion close to that of L共0 , 5兲 and L共0 , 8兲. On the other hand, the irregularity of the bone either at Site1 or Site2 had almost no effect on the dispersion of the fundamentals modes. Similar conclusions can be drawn for the isotropic case. Figures 12共a兲–12共d兲 illustrate the t-f signal representations from the simulated healing stages measured at Site2 of the anisotropic healing bone. At the hypothetical Stage0, where only the geometry of callus was taken into account, the L共0 , 5兲 and L共0 , 8兲 modes were different from those in the intact bone. On the other hand, the fundamental modes were less sensitive to this geometrical change. From Stage1 to Stage3, the properties of the callus tissue significantly affected the propagating guided modes. However, at Stage3, the fundamental modes and the L共0 , 5兲 mode started to approach the behavior observed at Stage0. V. DISCUSSION
FIG. 9. Evolution of the FAS propagation velocity over the healing stages for the isotropic 共䉱兲 and anisotropic 共䊏兲 bone models. 3916
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In this work, a finite element study of guided ultrasound wave propagation in intact and healing long bones was presented. We first examined wave propagation in a structure with idealized geometry and made comparisons with analytical solutions and then we proceeded to investigate the parameters that affect the characteristics of propagation under more realistic conditions. The bone was modeled as a tubular solid with constant cross section describing the diaphysis of a sheep tibia. We neglected the small variations in the dimensions and shape of the cortex along the long axis. Our primary interest was to examine whether the irregularity of the cortical shell has an effect on wave propagation rather than provide an accurate model of a long bone. It should be noted that the shape of a human tibial cortex is different from that of the sheep; nevProtopappas et al.: 3D modeling of ultrasound in bones
FIG. 10. 共Color online兲 The t-f signal representations obtained from the hollow circular cylinder with 共b兲 isotropic and 共d兲 anisotropic properties along with the corresponding signal waveforms 共a兲 and 共c兲, respectively. The analytical dispersion curves of the longitudinal and flexural modes are also superimposed.
ertheless, the mesh of the model is parametric and can be easily used to construct any arbitrary tubular geometry. We extended to the context of fracture healing by incorporating a model for callus. The callus tissue consisted of 3917
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several regions, each holding different material properties through time according to Claes and Heigele 共1999兲. The simulated stages represent critical phases of the healing course. However, the early stage of haematoma development Protopappas et al.: 3D modeling of ultrasound in bones
FIG. 11. 共Color online兲 The t-f signal representations obtained from the intact anisotropic bone at 共a兲 Site1 and 共b兲 Site2.
and the gradual reduction in the dimensions of callus as a result of the bone remodeling process were not taken into account. We examined two cases of material symmetry. In both cases, the elastic constant in the axial direction 共C33兲 was identical and thus the bulk longitudinal velocity was the same, allowing for direct comparisons between the FAS velocity measurements. The anisotropic constants represent effective elastic constants at the macroscopic level and they do not only reflect the elastic properties at the material level, but also the cortical microporosity 共Bossy et al., 2004b; Raum et al., 2005兲. However, the porosity network in the cortical bone 共consisting mainly of Haversian longitudinal canals 3918
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with average pores diameter of 50– 100 m兲 as well as the biphasic nature of the callus tissue 共i.e., a solid and a viscous component兲 introduce wave attenuation, which was neglected in our study. Attenuation, may have a significant influence on the characteristics of guided waves and can affect the measurements. Efficient modeling of elastic wave propagation necessitates a detailed discretization of the spatial and temporal domains. For 1-MHz excitation, the smallest wavelengths were 1.54 and 2.34 mm in the ICT and the SOC materials, respectively. Thus, the internodal distance in the circumferential direction 共delem2兲 was not sufficient, according to Eq. 共16兲, Protopappas et al.: 3D modeling of ultrasound in bones
FIG. 12. 共Color online兲 The t-f signal representations obtained from Site2 of the healing bone at 共a兲 Stage0, 共b兲 Stage1, 共c兲 Stage2, and 共d兲 Stage3.
for the stages that involved regions filled with those two soft tissue types. This was a compromise in our study in order to keep low the memory requirements and computational time. However, those materials occupy relatively small regions and mainly are away from the recording region. For 1-MHz excitation, the wavelength in the bone 共approximately 4 mm兲 was comparable to the cortical thickness. Using traditional velocity measurements, we showed that the FAS wave propagated in the intact models as a nondispersive lateral wave, which is in agreement with previous studies 共Bossy et al. 2002, 2004b; Nicholson et al. 2002; Njeh et al., 1999b兲. Therefore, the propagation velocity of the FAS was not affected by the geometry of the model or the site of measurement and changed only by 2.2% between isotropy and anisotropy. When we investigated the variation of the FAS velocity over the simulated healing stages, we found that the propagation of the FAS wave was sensitive only to the tissue that filled the fracture gap and thus its velocity remained constant between Stage1 and Stage2. This is due to the fact that lateral waves propagate only along a 1.4-mm-deep layer at 1 MHz 共Bossy et al. 2004b; Raum et al., 2005兲 and thus cannot characterize the medium throughout its thickness. However, our broadband excitation supports the generation of multiple wave modes that are sensitive to the geometrical properties of the bone. These modes could not be identified in the waveform of the signal due to their temporal superposition and thus we used t-f signal analysis techniques. 3919
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We first examined the hollow circular cylinder for which there exist analytical dispersion curves. The simulated dispersion of the fundamental modes and of some higher-order longitudinal modes was in agreement with the theoretical dispersion, in both the isotropic and anisotropic cases. The simulated dispersion of each mode was identified within specific frequency ranges. In our previous study on an isotropic 2D plate with the same isotropic properties 共Protopappas et al., 2006兲, the dispersion of the S2 and A3 Lamb modes was similar to that of the L共0 , 5兲 and L共0 , 8兲 modes, although the solution methods were different and the excitation applied to the plate was more broadband. This is justified by the theory for the specific wall thickness and the outer radius values 共Lefebvre et al., 2002; Protopappas et al., 2006兲. Direct comparisons with other 2D bone studies cannot be made since they used low-frequency transducers 共250 kHz兲 that excite only the fundamental modes. In the intact bone model, the high-order guided waves were significantly different from the modes described by the simplified geometry. However, when the geometrical characteristics of the site of measurement were similar to those of the circular cylinder, we identified two modes with dispersion close to that of the L共0 , 5兲 and L共0 , 8兲. This geometrydependent behavior of the higher-order modes may also be observed in real bones where the cross section is not uniform in the direction of the long axis. On the other hand, we found that the fundamental tube modes were almost unaffected by Protopappas et al.: 3D modeling of ultrasound in bones
the irregularity of the bone. Similar behavior is expected in real bone geometries. Furthermore, our results demonstrate that anisotropy has an impact on higher-order mode propagation even if, in the case of isotropy, the properties along the propagation axis are kept constant. However, our finding that the dispersion of the fundamental modes remained practically unchanged between the two elastic symmetries possibly explains why previous ex vivo and in vivo studies reported on the detection of the isotropic A0 and S0 Lamb modes. For the model of the healing bone, we found that the propagating guided modes were influenced by both the geometry and the properties of the callus. The effect of the geometrical disturbance induced by the formation of callus on mode propagation was evaluated separately at Stage0. It was also shown that mode dispersion was sensitive to the transition from Stage1 to Stage2 and, as the callus properties increased to the bone values, the dispersion of the modes gradually returned to that observed at Stage0. Nevertheless, we were not able to quantify our observations and select significant ultrasonic features that can provide monitoring capabilities. We made use of the theory that describes elastic wave propagation in circular cylinders since there are no analytical solutions for complex geometries. Recently, FE and boundary element methods have been developed to numerically obtain the dispersion relations for bars of arbitrary cross section 共Gunawan and Hirose, 2005; Hayashi et al., 2003兲. Thus, the computation of the dispersion curves for the bone under examination could make feasible the interpretation of the signals obtained from real experiments. A parameter not provided by the dispersion relations is the amplitude with which a mode is excited at a specific frequency. The excitation of each mode depends on source loading conditions, such as the size of transducers, their pressure distribution, angle of incidence, etc. This justifies why the detected modes exhibited dispersion within narrow frequency ranges, whereas most of the modes were not excited despite that the spectral content of the excitation was sufficiently broad. The amplitudes of each mode in such transient partial loading problems can be determined using integral transform techniques or the Normal Mode Expansion technique 共Ditri and Rose, 1992兲. One major simplification made in our study was the application of unrealistic boundary conditions. The soft tissues provide leakage paths for the ultrasonic energy resulting in the so-called leaky guided waves and in this case the dispersion curves are modified 共Berliner and Solecki, 1996兲. The soft tissues can be considered as a layer on top of the bone and thus a bilayer model 共Rose, 1999兲 can be used for the analysis of the signals measured in practice. The soft tissues also play an important role in coupling to the percutaneous transducers and in the excitation of specific modes. Free boundary conditions are only met in ex vivo experiments on excised bone specimens. A further consideration regarding the boundary conditions is that, despite the attachment of infinite elements at the ends of the model, some energy may not be transmitted out of the model. However, the boundary between the finite and infinite elements is normal to the 3920
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dominant direction of wave propagation, which optimizes the transmission of energy out of the finite element mesh 共ABAQUS, 2003兲. Since the use of infinite elements is the suggested solution by ABAQUS for simulating infinitely long domains, we extended the mesh to some reasonable distance 共7 mm兲 from the recording region. VI. CONCLUSIONS
In this work, we presented a 3D computational study of guided wave propagation in intact and healing long bones. We observed that the velocity of propagation determined by the FAS wave was not affected by the convex curvature of the cortex and remained almost the same between different material symmetry assumptions, provided that the cortical thickness is comparable to the wavelength. On the other hand, the irregular bone geometry, as a whole, as well as the anatomical characteristics of the site of measurement had a significant impact on the propagation of the higher-order modes. The effect of the complex geometry and anisotropy of the bone was less pronounced on the dispersion of the fundamental tube modes. Therefore, 2D and 3D simulations on idealized geometries have limited efficiency in interpreting wave-guidance phenomena in real bones; however, they can be reliably used to assess the dependence of the FAS velocity on the cortical thickness. We also demonstrated that traditional velocity measurements cannot reflect the material and mechanical changes that take place during a simulated healing process. Although guided waves were sensitive to both the geometry and the properties of callus, it was not made feasible to provide quantitative results. This study can be proved useful for the interpretation of clinical measurements. We suggest that monitoring of bone fracture healing can be enhanced by analyzing the characteristics of the higher-order guided modes, such as L共0 , 5兲 and L共0 , 8兲. ABAQUS, version 6.4, 共2003兲. ABAQUS Analysis User’s Manual, and ABAQUS Theory Manual 共Hibbitt, Karlsson and Sorensen, Inc., Pawtucket兲. Abendschein, W., and Hayatt, G. W. 共1972兲. “Ultrasonics and physical properties of healing bone,” J. Trauma 12, 297–301. Augat, P., Margevicius, K., Simon, J., Wolf, S., Suger, G., and Claes, L. 共1998兲. “Local tissue properties in bone healing: influence of size and stability of the osteotomy gap,” J. Orthop. Res. 16共4兲, 475–481. Auger, F., Flandrin, P., Goncalves, P., and Lemoine, O. 共1995–1996兲. TimeFrequency Toolbox Tutorial 共CNRS-RICE University兲. Berliner, M. J., and Solecki, R. 共1996兲. “Wave propagation in fluid-loaded, transversely isotropic cylinders. Part I. Analytical formulation,” J. Acoust. Soc. Am. 99共4兲, 1841–1847. Bossy, E., Talmant, M., and Laugier, P. 共2002兲. “Effect of cortical thickness on velocity measurements using ultrasonic axial transmission: a 2D simulation study,” J. Acoust. Soc. Am. 112共1兲, 297–307. Bossy, E., Talmant, M., Defontaine, M., Patat, F., and Laugier, P. 共2004a兲. “Bidirectional axial transmission can improve accuracy and precision of ultrasonic velocity measurement in cortical bone: A validation on test materials,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 51共1兲, 71–79. Bossy, E., Talmant, M., and Laugier, P. 共2004b兲. “Three-dimensional simulations of ultrasonic axial transmission velocity measurement on cortical bone models,” J. Acoust. Soc. Am. 115共5兲, 2314–2324. Camus, E., Talmant, M., Berger, G., and Laugier, P. 共2000兲. “Analysis of the axial transmission technique for the assessment of skeletal status,” J. Acoust. Soc. Am. 108共6兲, 3058–3065. Claes, L. E., and Heigele, C. A. 共1999兲. “Magnitudes of local stress and strain along bony surfaces predict the course and type of fracture healing,” J. Biomech. 32共3兲, 255–266. Cunningham, J. L., Kenwright, J., and Kershaw, C. J. 共1990兲. “BiomechaniProtopappas et al.: 3D modeling of ultrasound in bones
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