ACOUSTIC PROPERTIES OF CANCELLOUS BONE AND EVALUATION OF BONE DENSITY

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1 ACOUSTIC PROPERTIES OF CANCELLOUS BONE AND EVALUATION OF BONE DENSITY PACS REFERENCE: 43.8.Q OTANI Takahiko; SHIMOI Yuichi Department o Electrical Engineering, Doshisha University 1-3 Miyakodani, Tatara Kyotanabe-shi Japan Tel.: (81) Fax: (81) totani@mail.doshisha.ac.jp ABSTRACT The propagation o both ast and slow longitudinal waves in bovine cancellous bone in vitro was experimentally examined by the pulse transmission technique in relation to the bone density. Experimental results show that the propagation speed and the amplitude o the ast wave increase greatly with the bone density and those o the slow wave decrease reversely with the bone density. I. INTRODUCTION Osteoporosis is a disease characterized by decrease in bone density and deteriorations in bone microstructure, which cause bone ragility and increase in racture risk. Bone tissues are elastic and can be classiied into two types depending on macroscopic and microscopic eatures. Bone with a low volume raction o solid (less than 7%) is called cancellous (spongy) bone and above 7% cortical (compact) bone. Whereas cortical bone is a dense hard solid, cancellous bone is comprised o a porous network o numerous rod-like and plate-like solid trabecular elements with sot tissue (bone marrow) in the pore space. The eect o decreasing bone density (or a symptom o osteoporosis) is stronger or cancellous bone than or dense cortical bone, because cancellous bone is metabolically more active. Accordingly, it is adequate to measure bone density at the position with a high volume raction o cancellous bone to estimate the onset o osteoporosis. X-ray method or X-ray absorptiometry is now widely used or non-invasive measurement o bone mineral density or the assessment o osteoporosis. Ultrasonic method or Ultrasonic Osteodensitometer has been proposed as an alternative in the diagnosis o osteoporosis. Ultrasonic method takes several advantages over X-ray bone densitometry as it is radiation-ree and inexpensive technique which has the potential to evaluate the elastic or mechanical properties o bone. However the reproducibility and accuracy o ultrasonic measurements are a problem peculiar to the ultrasonic wave propagation phenomenon due to the complexity o bone structure. Ultrasonic evaluation o bone density is based on the correlation between the bone mass density measured by X-ray densitometry and the slope o requency dependent attenuation (or broadband ultrasonic attenuation, BUA) and/or the ultrasonic wave propagation speed (or speed o sound, SOS). In our previous study on the acoustic properties o cancellous bone [1-3], it is shown that both ast and slow longitudinal waves propagate through cancellous bone along the trabecular orientation, which correspond to waves o the irst and second kind as predicted by Biot s theory. Thereore, the evaluation o bone density based on the acoustic properties o both ast and slow

2 waves is more reasonable than the conventional method by BUA and SOS. In the present study, the eects o the bone density (bone porosity) on the amplitude and the propagation speed o transmitted both ast and slow waves through bovine cancellous bone are experimentally examined. II. ULTRASONIC WAVE PROPAGATION IN CANCELLOUS BONE II.1. Biot s Equations Maurice A. Biot initially proposed a general theory o elastic wave propagation in a system composed o a porous elastic solid saturated with a viscous luid. Biot s theory is basically developed by considering a dynamic coupling between the solid and luid, which is the relative motion o the luid in the pore spaces to the solid rame. The average motion o the solid and luid are separately described. The wave equations or compressional (longitudinal) waves are given as [4-6] ( He - Cz) = ( re - r z), (1) t F( k) h z ( Ce - Mz ) = ( r e m ). - z - () t k t Here, e is the dilatation o a volume element attached to the solid. z is the luid volume that lows into or out o that element called increment o luid content. The harmonic solutions o eq.(1) and () yield the dispersion relation or the longitudinal waves at high requencies [4-7] H / z - r C / z - r C / z - r M / z -ar = / b where z=v. Equation (3) has two roots corresponding to two speeds o propagation, v. Two longitudinal waves have the orm as slow v ast ( Mr + Hm - Cr ) 1 = ( HM - C ) Ø ( Mr + Hm - Cr ) ø v m slow Œº - 4( HM - C )( mr - r. ) œß (3) v ast and The terms H, C and M are generalized elastic coeicients deduced by Biot and expressed in terms o the bulk moduli o the solid K s, the pore luid K, the bulk modulus K b and the shear modulus m o the skeltal rame and the porosity b [4-7]. According to Gibson, the bulk modulus K b and shear modulus m o the skeltal rame o cancellous bone are expressed as a unction o porosity b [8]. K b Es(1 - b) =, (5) 3(1 - n ) b n n E s (1 - b) m =, (6) (1 + n ) where E s is the Young s modulus o the solid bone, n b is the Poisson s ratio o the skeltal rame and n is a variable depending on the geometrical structure o the cancellous bone. In terms o the porosity b and the density o the solid r s and luid r, the total density r o the luid-saturated medium is given by b r = 1 - b) r + br. (7) ( s The term a is the structure actor, and is obtained rom the relation determined by Berryman [9,1] : a = 1- r (1-1 b) (8) where r being a variable determined rom a microscopic model o the solid rame moving in the luid. (4)

3 II.. Experimental Results [1] The experimental arrangement or transmission o ultrasonic pulse is shown in Fig.1. Cancellous bone specimens, -3 mm in size and 9 or 7 mm thickness, were cut rom the distal epiphysis o bovine emora, with sot tissue in situ. A single sinusoidal pulse wave o 1 MHz, transmitted by a wide band PVDF transmitter, was used in order to observe the ast and slow waves separately. Figure shows the pulsed waveorm travelling in water, which is applied to the specimens. Figure 3 shows typical waveorms travelling through the cancellous bones in the direction o trabecular alignment. Figure 3(a) is a waveorm or the high density (r=1 kg/m 3 ) specimen and (b) is a low density (r=11 kg/m 3 ). In both Fig.3(a) and (b), the ast and slow longitudinal waves can be clearly observed in the time domain. As the density increases (i.e., as the volume raction o the solid bone increase), the amplitude o the ast wave becomes greater. At the same time, the amplitude o the slow wave decreases. Accordingly, it can be deduced that the ast wave is associated with solid core in cancellous bone, and the slow wave with the propagation in sot tissue. Power Amp. Transmitter Function generator Specimen Preamp. Hydrophone Trigger Oscilloscope Voltage [mv] Time [µs] 8 1 Figure 1. Experimental arrangement or transmission o ultrasonic pulse. Figure. Pulsed waveorm traveling in water. Voltage [mv] 1-1 Fast wave Slow wave Voltage [mv] Time [µs] Time [µs] (a) (b) Figure 3. Pulsed waveorm traveling through cancellous bone: (a) high density; (b) low density The propagation speeds and attenuations o the ast and slow waves were measured using pulse spectrum analysis. Figure 4 shows the propagation speeds o the ast and slow waves in cancellous bone at 1 MHz as a unction o porosity b. The speed o the ast wave varies rom 7 to m/s as the porosity increases. The slow wave remains constant at about 14 m/s, which is close to the propagation speed o 145 m/s in bone marrow. Thus the ultrasonic properties o the sot tissue in cancellous bone can be expected to be similar to the bone marrow, and it can be thereore assumed that the slow wave is mainly dominated by sot tissue. The propagation speed -7 m/s o the ast wave is much slower than the propagation speed 34-4 m/s o cortical bone (solid bone). This can be explained by the act that the cancellous bone is not solid but has a spongy or porous structure.

4 Propagation speed [m/s] 4 3 Fast wave Slow wave Measured Calculated Propagation speed [m/s] 4 3 Fast wave Slow wave Measured Calculated Porosity β Frequency [MHz] Figure 4. Propagation speeds o ast and slow waves in cancellous bone at 1 MHz as a unction o porosity b. Figure 5. Propagation speeds o ast and slow waves in cancellous bone at a porosity o b =.81 as a unction o requency. Attenuation [neper/mm] β=.75 β=.81 β=.83 Attenuation [neper/mm] Frequency [MHz] (a) Frequency [MHz] (b) Figure 6. Attenuation o ast and slow waves in cancellous bone at three porosity o b =.75,.81 and.83 as a unction o requency. Figure 5 shows the requency dependence o the propagation speeds or a specimen o porosity b=.81 (density r=114 kg/m 3 ). No data or the ast wave at requencies over 1.5 MHz was obtained because the weak amplitude o the signal made urther measurement diicult. In Fig.5, the propagation speeds o both the ast and slow waves are considered nondispersive in the range.5-5 MHz. Figure 6 shows the attenuation or the three specimens o b=.75,,81 and.83 (density r=1, 114 and 11 kg/m 3 ). From Fig.6(a) it can be seen that the attenuation o the ast wave at porosity b=.75,.81 and.83 are about the same. The attenuation o the slow wave (Fig.6(b)) increases with requency and the variation o the slow wave with the porosity b is large. The attenuation in cortical bone and bone marrow were also obtained as about [neper/mmmhz] and [neper/mmmhz]. Both the ast and slow waves in cancellous bone show much higher attenuation than the bulk wave in cortical bone (r=196 kg/m 3 ) or bone marrow (r=93 kg/m 3 ). Accordingly, both the ast and slow waves are attenuated not only by the solid core or sot tissue component in the cancellous bone but also by the porous structure o the cancellous bone. III. BONE DENSITY AND ULTRASONIC WAVE TRANSMISSION THROUGH CENCELLOUS BONE In Fig.6 o the ultrasonic wave attenuation in cancellous bone, the attenuation o the ast wave is almost independent o the bone density. However, observed pulse waveorms through

5 cancellous bone in Fig.3 show that the amplitude o the ast wave becomes greater as the bone density increases. The act that the ast wave is associated with porous solid core in cancellous bone and the slow wave with sot tissue in the pore space, means the cancellous bone as an acoustic medium has two dierent characteristic impedances or the ast wave and or the slow wave. The characteristic impedance or the ast wave is essentially dominated by the mass o solid core (bone density) and the characteristic impedance or the slow wave is dominated by the sot tissue. Consequently, the transmission coeicient o the ast wave is strongly dependent on the bone density o the cancellous bone. Thus, it would be deduced that the amplitude o transmitted ast wave depends greatly on the transmission coeicient at the boundary o cancellous bone or on the bone density, and that the amplitude o the slow wave depends mainly on the attenuation coeicient o cancellous bone. As or the propagation speed through cancellous bone, the dependence on bone density is high or the ast wave and very small or the slow wave as shown in Fig.3. The amplitudes and propagation speeds o the ast and slow waves through cancellous bone in vitro were measured in a water tank. A ocused PVDF transmitter was driven by a single sinusoidal impulse voltage o 5 V peak to peak with a requency o 1 MHz. Pulse waves propagating through the water/specimen/water system was detected by a wide band non ocused PVDF hydrophone and the output was ampliied by a 4 db preampliier. A cancellous bone specimen o about 1 mm thickness was cut rom the distal epiphysis o bovine emur with sot tissue in situ as shown in Fig.7. The specimen was mounted between the transmitter and the hydrophone at normal incidence. Ultrasonic wave path was scanned in an area o mm and transmitted waves were measured at 1 mm intervals or 5 points. The scanned area was taken in the nearly middle region o the specimen, where the trabecular alignment was approximately parallel to the thickness direction or the direction o ultrasonic wave propagation. The local bone density corresponding to the measured points was obtained by use o a micro ocus X-ray CT system. The dependence o measured peak to Figure 7. Sectional view o a distal epiphysis o bovine emur. 4 Amplitude [mv] Amplitude [mv] Bone volume raction [%] Bone volume raction [%] Propagation speed [m/s] Propagation speed [m/s] Bone volume raction [%] Figure 8. Amplitude and propagation speed o the ast wave through bovine cancellous bone as a unction o bone volume raction. Bone volume raction [%] Figure 9. Amplitude and propagation speed o the slow wave through bovine cancellous bone as a unction o bone volume raction.

6 peak amplitudes and propagation speeds is shown or the ast wave in Fig.8 and or the slow wave in Fig.9. The bone density has a strong positive correlation with both amplitude and propagation speed or the ast wave and a clear negative correlation or the slow wave. IV. CONCLUSION Ultrasonic wave propagation through cancellous bone has been experimentally examined. It was shown that both ast and slow longitudinal waves propagate through cancellous bone in the direction o the trabecular alignment, and that the ast wave is associated with solid core o cancellous bone. The slow wave is associated with sot tissue in the pore spaces. The propagation speed o the ast wave depends strongly on the bone density and that o the slow wave depends slightly and negatively on the bone density. The amplitude o the ast wave is mainly determined by the transmission coeicient and consequently depends strongly on the bone density. The amplitude o the slow wave is mainly given by the attenuation coeicient o the sot tissue and depends on the visco-elastic characteristics o the sot tissue and pore shape. In order to reveal the above acts, the wave propagation phenomenon at the boundary o cancellous bone and the characteristic impedance or both the ast and slow waves should be examined. REFERENCES 1. A. Hosokawa and T. Otani,: J. Acoust. Soc. Am. 11, (1997).. A. Hosokawa, T. Otani, T. Suzaki, Y. Kubo and S. Takai: Jpn. J. Appl. Phys. 36, Pt.1, No.5B, (1997). 3. A. Hosokawa and T. Otani: J. Acoust. Soc. Am. 13 (5), Pt.1, (1998). 4. M. A. Biot: J. Acoust. Soc. Am. 8, (1956). 5. M. A. Biot: J. Acoust. Soc. Am. 8, (1956). 6. M. A. Biot: J. Acoust. Soc. Am. 34, (196). 7. R. D. Stoll: Physics o Sound in Marine Sediments, ed. L. Hampton, Plenum, New York, (1974). 8. L. J. Gibson: J. Biomech. 18, (1985). 9. J. G. Berryman: Appl. Phys. Lett. 37, (198). 1. J. G. Berryman: J. Acoust. Soc. Am. 69, (1981). ACKNOWLEDGEMENT Part o the study was supported by a Grant -in-aid or University and Society Collaboration rom the Ministry o Education, Sports and Culture.