Numerical and experimental simulation of the effect of long bone fracture healing stages on ultrasound transmission across an idealized fracture

Transcription

1 Numerical and experimental simulation of the effect of long bone fracture healing stages on ultrasound transmission across an idealized fracture S. Gheduzzi, S. P. Dodd, and A. W. Miles Department of Mechanical Engineering, Centre for Orthopaedic Biomechanics, University of Bath, Bath, Avon BA2 7AY, United Kingdom V. F. Humphrey Institute of Sound and Vibration Research (ISVR), University of Southampton, Southampton, Hampshire SO17 1BJ, United Kingdom J. L. Cunningham a Department of Mechanical Engineering, Centre for Orthopaedic Biomechanics, University of Bath, Bath, Avon BA2 7AY, United Kigndom Received 3 March 2008; revised 3 June 2009; accepted 3 June 2009 The effect of various stages of fracture healing on the amplitude of 200 khz ultrasonic waves propagating along cortical bone plates and across an idealized fracture has been modeled numerically and experimentally. A simple, water-filled, transverse fracture was used to simulate the inflammatory stage. Next, a symmetric external callus was added to represent the repair stage, while a callus of reducing size was used to simulate the remodeling stage. The variation in the first arrival signal amplitude across the fracture site was calculated and compared with data for an intact plate in order to calculate the fracture transmission loss FTL in decibels. The inclusion of the callus reduced the fracture loss. The most significant changes were calculated to occur from the initial inflammatory phase to the formation of a callus with the FTL reducing from 6.3 to between 5.5 and 3.5 db, depending on the properties of the callus and in the remodeling phase where, after a 50% reduction in the size of the callus, the FTL reduced to between 2.0 and 1.3 db. Qualitatively, the experimental results follow the model predictions. The change in signal amplitude with callus geometry and elastic properties could potentially be used to monitor the healing process Acoustical Society of America. DOI: / PACS number s : Ev, Jz, Qf, Vj DLM Pages: I. INTRODUCTION This paper describes a numerical study of the transmission of ultrasonic waves across a simulated transverse bone fracture as might be observed when using the axial transmission technique and, in particular, how the progressive healing of the fracture affects the transmission. It indicates that the formation of the callus has a significant effect on the transmitted wave and that the transmission changes in a complex manner through the healing process. Experimental measurements using a simulated fracture in Sawbones plates are used to confirm the qualitative features of the simulations. The use of ultrasound for the assessment of musculoskeletal conditions is appealing due to its quantitative potential. Through in-vitro experimentation and numerical modeling, advances have been made in the understanding of ultrasound wave propagation along cortical bone. These studies are important for devising techniques for monitoring fracture healing 1 5 and for detecting the onset of osteoporosis In the case of fracture healing monitoring attempts have been a Author to whom correspondence should be addressed. Electronic mail: j.l.cunningham@bath.ac.uk made to utilize the change in velocity, measured for a specified transducer separation, to distinguish between ultrasound waves traveling through bone with and without a fracture present. As the velocity is expected to gradually return to the value of the solid bone as the fracture heals this technique could potentially be used to monitor intermediate stages in the repair process 1,11 and, ultimately, predict the end-point of healing. In the axial transmission technique, the waves propagating along a plate/bone reduce in amplitude as a result of both the absorption in the material and wave energy leaking out of the plate/bone into the surrounding fluid/tissue, a process known as re-radiation. As a consequence of both these effects the re-radiated waves, detected by a receiving ultrasound transducer, reduce in amplitude with distance. Measurements of the amplitude of the re-radiated wave with propagation distance can be used to evaluate the effective attenuation of the ultrasound along the plate. In addition, if there is a discontinuity in the plate/bone, such as a fracture, then the amplitude of the wave transmitted across the discontinuity is reduced. By measuring the re-radiated wave, the fracture transmission loss FTL resulting from the discontinuity i.e., the fracture can be measured. J. Acoust. Soc. Am , August /2009/126 2 /887/8/$ Acoustical Society of America 887

2 Numerical modeling has proved a useful tool for understanding how the velocity and attenuation of ultrasound vary with bone geometry and acoustic frequency, 4,12 14 when measured using an axial transmission technique. In recent twodimensional 2D and three-dimensional studies by Protopappas et al., 2,3 the different stages of fracture healing inflammatory to hard callus and their effect on ultrasound velocity were simulated, thus providing predictions for the behavior of this particular acoustic parameter during the healing process. It has recently been demonstrated that ultrasound signal loss has the potential to be a good quantitative indicator of the presence of idealized fractures of varying geometries in bone A number of fracture geometries have been modeled for a particular plate thickness, starting with simple transverse fractures surrounded and filled by water. The water simulates the hematoma present at the inflammatory stage of the fracture healing and the soft tissues surrounding the bone. Numerical models, approximating cortical bone as an isotropic flat plate, compared reasonably well with experimental results 5,12 and simulations provided a good understanding of the relationship between first arrival signal FAS velocity and attenuation, plate thickness, and the acoustic frequency. These show that for a small ideal gap the transmitted signal may be reduced by 6 db or more as a result of the fracture presence. This work is designed to complement these studies, with experimental and numerical modeling being employed to simulate the axial transmission technique and the amplitude of the ultrasonic waves transmitted across fractures at different stages of the healing sequence. As the fracture heals, the mechanical and geometric properties of the fracture site change; 18 the formation of an external callus and the increasing stiffness of the material in the fracture gap will markedly influence the propagating wave. Hence, the more advanced stages of healing have also been studied using models aimed at replicating a bridging symmetric callus. To simulate the progress of healing, the mechanical and acoustical properties of the callus bridging the fracture have been varied based on reference data gathered from the literature. The changes in ultrasound propagation resulting from the remodeling stage of healing have also been investigated in this study. In order to verify the performance of the numerical model, experimental results were also obtained using Sawbones plates to simulate the cortical bone and bone cement to simulate the callus. II. METHODS AND MATERIALS The modeling studies described here used an axial geometry and parameters based on an experimental facility that has been used previously to study the axial transmission technique. 5,15 This system was also used to obtain the experimental results described in this work. A. Axial transmission model The axial transmission technique was modeled using a 2D finite difference code WAVE2000 PRO, Cyberlogic, the simulation details adopted are tailored to replicate as closely FIG. 1. A geometric representation of the axial transmission apparatus showing the water and test sample, where T and R represent the transmitting and receiving transducers, respectively. x 0, x, d, h, and w are the initial transducer separation, transducer separation, test plate thickness, height of transducer face above plate, and fracture gap width, respectively. The dotted line above point A represents the starting position of the measurements. as possible the experimental geometry. In the finite difference numerical models, a rectangular region with dimensions mm 2 was used to simulate the water surrounding the cortical bone plate; water was used as an ultrasound mimic for soft tissue and blood in a similar manner as the conditions in the experimental set-up. One ultrasound source T and an array of receivers R with widths of 20 mm were positioned within the water region at a constant height, h =5 mm, above the plate see Fig. 1, an initial transducer separation, denoted by x 0, was chosen to avoid signal interference from direct water borne waves. The time and space resolution of these simulations were mm and s, respectively. The source T used in the models was configured to output a 4-cycle Gaussian-modulated pulse with 200 khz central frequency, which simulated the characteristics of the output produced by the experimental transducers. Since the simulated receiving transducers R gave an output proportional to the average pressure over the face of the receiver at a particular position and did not have any effect on the wavefield, it was possible to create an overlapping transducer array spaced 1 mm apart to provide data for the FAS amplitude V x over a specified propagation distance x. The FAS amplitude was determined from the peak of the first half cycle of the signal. B. Fracture geometries for healing study For the healing process, seven stages were simulated using a 4 mm thick, isotropic, flat bone plate with a 1 mm wide transverse fracture. Figure 2 a shows the intact bone plate and Figs. 2 b 2 g show the geometries used for each of the main stages of the healing processes. 18 Stage 1 Fig. 2 b represents the initial fracture of the plate, which provided information on the inflammatory stage of fracture healing. In this case the gap is filled with water to simulate blood. Stage 2 Fig. 2 c simulates the initiation of the bridging and ossification process with a symmetric external callus forming to bridge the gap and stabilize the fracture, while cartilage forms at the fracture site. A Gaussian profile with a peak height of 3 mm and a width of 20 mm was used to represent the callus. In this study, height and width refer to the dimensions of the callus and not the medical terms relating to measurement of the whole callus. Subsequently, stages 3 and 4 Figs. 2 d and 2 e show the continued ossification 888 J. Acoust. Soc. Am., Vol. 126, No. 2, August 2009 Gheduzzi et al.: Ultrasound transmission across a healing fracture

3 FIG. 3. Modeling pictures representing the three fracture geometries used in the in-vitro experiments and 2D simulations. A 6 mm, thick plate, surrounded by water and a a 4 mm transverse fracture, b bone cement filling/bridging the fracture gap, and c a callus is added to the top surface. FIG. 2. Modeling pictures of the six stages of secondary healing: a an intact bone light gray. b Stage 1: The initial transverse fracture. Stages 2 4: A bridging callus with degrees of ossification, where c shows the hard callus material dark gray forming the bridge with cartilage filling the gap black, d shows the hard callus material connecting to bridge the gap and cartilage remaining within the fracture gap, and e shows the hard callus material throughout the fracture site. Stages 5 and 6: Remodeling, where f represents the hard callus material replaced by cortical bone within the fracture gap and g the reduction in callus size as the bone returns to original state. Water is used to represent the blood in these simulations. of the callus and fracture site materials, resulting in a hard callus. In stage 3 the gap is filled with cartilage while in stage 4 it is replaced with stiffer callus material. Finally, stages 5 and 6 Figs. 2 f and 2 g simulate the remodeling stages of healing where the cortical bone replaces the hard callus material and the external callus reduces in size to return the bone to its original state. C. Experimental configuration The in-vitro experiments to simulate the healing stages used 6 mm thick Sawbones plates immersed in a water tank. A baseline measurement was performed over a specified distance on an intact plate; then a 4 mm transverse gap was introduced and the measurement was retaken. While the fracture gap width was maintained, the plate discontinuity was filled using Palacos R bone cement and the FAS attenuation, over the same distance, was determined. Finally, bone cement was added to the top surface of the plate to simulate a callus. The callus was approximately dome shaped with a height of 4 mm at the center and extended 7.5 mm either side of the center of the fracture site width of 15 mm. This geometry was maintained across the width of the plate. Figures 3 a 3 c provide a representation of the geometrical and mechanical changes. In the in-vitro experiments, the receiving transducer R was moved over the distance x at 1 mm intervals using a linear stepper motor rig with the separation being measured by an optical encoder accuracy of 1%. The received signals were filtered, amplified, and captured using a digital oscilloscope and averaged over 128 waveforms. As in the simulations, the amplitude of the first peak at each separation interval was used to calculate the FAS attenuation, with the experimental data recorded for three separate runs and then averaged. These experimental measurements were also simulated numerically using an appropriate callus geometry. D. Simulated material, elastic, and acoustic properties Material and acoustic properties used in the simulations were estimated from data in the literature and are shown in Table I. In the healing stage models, human cortical bone was represented by the properties, =1850 kg/m 3, V L =4000 m/s, and V S =1800 m/s E=16.5 GPa, which were used in previous studies. 5,13 Cartilage properties =1050 kg/m 3, V L =1775 m/s, and V S =946 m/s, giving E =2.45 GPa were estimated using a velocity range for hu- TABLE I. Material, elastic, and acoustic properties used in numerical simulations. Data taken from the literature are referenced. The remaining values were estimated or calculated from referenced data using standard equations. Material kg m 3 E GPa V L m s 1 V S m s 1 Cortical bone 1850 a a 1800 a Water/blood 1000 a 1500 a Cartilage 1050 b c 1775 b 946 b Callus material c Callus material c Callus material c Sawbones 1700 d d 1600 d Palacos R bone cement 1281 e a Reference 5. b Reference 19. c Reference 20. d Reference 15. e Reference 23. J. Acoust. Soc. Am., Vol. 126, No. 2, August 2009 Gheduzzi et al.: Ultrasound transmission across a healing fracture 889

4 man specimens, 19 and Poisson s ratio given by Claes and Heigele. 20 A bovine cartilage density of 1050 kg/m 3 reported by Joseph et al. 21 was assumed to be a good estimate for the human equivalent, since data on human cartilage are scarce. Human and bovine cartilages have similar constituents 60% 80% water by weight 22, therefore resulting in similar densities. Material damping values for these materials have not been included due to the lack of reliable data at 200 khz. The density and acoustic properties of the callus were difficult to predict due to the ossification process. 20 Therefore, a range of Young s modulus values 5 15 GPa were modeled for this callus material Table I, with Poisson s ratio based on data by Claes and Heigele. 20 These values lie in between the simulated mechanical properties of cartilage and cortical bone. In a similar manner, the density of the hard callus material will lie between cartilage and cortical bone; however, for this study it was estimated to be closer to cortical bone 1600 kg/m 3. For the experimental section of the study, Sawbones plates and Palacos R bone cement were used. The modeling of these experimental results used appropriate data based on values from the literature. Sawbones properties were taken from a similar study, 5 while the density of the bone cement has been measured previously by Armstrong et al. 23 Due to the amount of acrylic present in this particular bone cement 24 and the measured density, the acoustic velocities were estimated to be similar to Perspex. Therefore, these values were based on the material database in the WAVE2000 PRO software package. E. Fracture loss calculation The amplitude data for the FAS was plotted as a sound pressure level SPL, in decibels, versus transducer separation m using SPL x = 20 log V x /V x 0, 1 where V x 0 is the signal amplitude at x=x 0 and V x is the signal amplitude at each measurement position x. The difference in SPLs at a given measurement position x was used previously 15,16 to represent the change in FAS amplitude caused by a discontinuity and was expressed as a FTL, calculated using FTL x = SPL F x SPL I x, 2 where the subscript I denotes an intact specimen and F denotes a fractured specimen. These parameters are used throughout the paper. With the notation of Eq. 2, a reduction in signal as a result of the fracture will give rise to a positive FTL. III. RESULTS A. Effect of different healing stages on the FAS amplitude Typical experimentally measured and numerically simulated signals are presented in Figs. 4 a and 4 b, respectively. Figure 4 b shows the complex nature of the received signals with more than one wave contributions resulting in FIG. 4. Typical experimental a and simulated b signals. interference effects between the different contributions. The nature of the waves received when using the axial transmission technique has been discussed previously. 13,5,10 This has shown that the first arrival travels at the velocity of a compressional wave or S 0 mode depending on the range and plate thickness. In this work the amplitude of the first peak of the transmitted waveform Fig. 4 a is used for simplicity. Figure 5 a shows the effect of the simulated healing stages 1 6 on the FAS amplitude of an ultrasound wave propagating along a4mmbone plate with a1mmtransverse fracture gap. With reference to Figs. 5 a 5 c, the first fracture interface is positioned at a transducer separation of 0.07 m. In this particular study, the net signal loss, or FTL, was considered. This parameter gave the difference in SPL between the intact bone plate and the subsequent healing stages of the fractured bone plate at a particular transducer separation. Figure 5 a shows the results obtained using callus material 1 E=5 GPa ; the geometry and mechanical properties of each healing stage resulted in a different shape to the corresponding SPL curves. At the maximum transducer separation 120 mm, the difference in signal amplitude between the subsequent healing stages and the baseline value varied significantly. These differences expressed as both a percentage change from the baseline data and as a FTL are presented in Table II. The simulation of stage 1, which represented the inflammatory stage of healing, produced a significant drop in the signal amplitude compared to the intact plate Fig. 5 a, resulting in a large FTL. When the bridging callus was introduced in stage 2, the FTL value was reduced relative to stage 1, i.e., the signal amplitude increased. Stages 2 and 3 represented a bridging callus and little change in FTL was seen at stage 3 as the external callus formed to bridge the gap. A subsequent change to a hard callus stage 4, i.e., when the ossification had occurred throughout the fracture site, resulted in a small decrease in the loss of signal amplitude, and hence FTL, relative to stage 3. However, the beginning of the remodeling stage stage 5 caused no significant change in the FTL value compared to stage 4. Only when the callus reduced in size in stage 6 did the FTL change significantly. 890 J. Acoust. Soc. Am., Vol. 126, No. 2, August 2009 Gheduzzi et al.: Ultrasound transmission across a healing fracture

5 TABLE II. A comparison between the FTL calculated from numerical modeling data of the signal loss produced by healing stages 1 6 and increasing Young s modulus of the callus material. The change in signal amplitude compared with the baseline data is also expressed as a percentage and given in parentheses. Callus material Healing stage 1 5 GPa 2 10 GPa 3 15 GPa FTL db % % % % % % % % % % % % % % % % % % The effect of increasing Young s modulus E of the callus material on the FTL values produced by the simulations are demonstrated in Figs. 5 a 5 c. Table II summarizes the percentage change in amplitude compared with the baseline data and shows the FTL calculated from these curves for each Young s modulus value used callus materials 1 3, 5 15 GPa. A callus with a Young s modulus of 10 GPa produced a larger drop in FTL between stages 1 and 2 Fig. 5 b, compared to the same fracture geometry with half the stiffness, E=5 GPa Fig. 5 a. Although there exists a continuous fall in FTL from stages 2 to 5, the change is relatively small, in a similar way to that obtained for E =5 GPa. A much larger drop in FTL occurred from stages 5 to 6 than existed for original Young s modulus. When Young s modulus of the callus material was increased further to 15 GPa Fig. 5 c, the same pattern was produced with similar FTL values to those obtained for 10 GPa. C. In-vitro experiments of changing the fracture geometry and mechanical properties of a fracture site Corresponding experimental FAS amplitude results of changing the shape and geometry of the fracture site for a Sawbones bone plate mimic with a 4 mm transverse fracture are shown in Fig. 6 a. For the results presented in Fig. FIG. 5. Numerical modeling results of the SPL versus transducer separation for the six healing stages when Young s modulus value corresponds to callus materials a 1 5 GPa, b 2 10 GPa, and c 3 15 GPa. FTL calculations where preformed using data taken at 120 mm. The first fracture interface is positioned at a transducer separation of 0.07 m. B. Effect of increasing Young s modulus of the callus material FIG. 6. Results of the SPL versus transducer separation measurements for a in-vitro experiments investigating the effect of fracture geometry and mechanical properties on signal amplitude, and b simulations of the invitro experiments. A 6 mm thick Sawbones plate with a 4 mm transverse fracture was used for this work. The first fracture interface is positioned at a transducer separation of 0.07 m. J. Acoust. Soc. Am., Vol. 126, No. 2, August 2009 Gheduzzi et al.: Ultrasound transmission across a healing fracture 891

6 TABLE III. A comparison between the FTL measured from in-vitro experiments and predicted by numerical modeling of the signal loss produced by three different fracture geometries simulated on a Sawbones plate. The change in signal amplitude compared with the baseline data is also expressed as a percentage and given in parentheses. FTL db Gap material Measured Predicted Water % % Bone cement 0.2 3% 0.7 8% Bone cement callus % % 6 the first fracture interface was positioned at a transducer separation of 0.07 m. A large drop in signal amplitude is seen for the inflammatory stage, equivalent to stage 1 in the numerical modeling. When an intermediate material, Palacos R bone cement, was substituted for the water in the gap, the signal loss was reduced significantly, i.e., the signal amplitude at large separation was only slightly lower than the baseline data, giving a small FTL. Adding a callus to the top surface of the bone mimic plate resulted in increased signal loss and hence an increase in FTL. The experimental uncertainty associated with the experimental baseline and callus measurements of amplitude was approximately 10%. D. Modeling of the callus experiment Figure 6 b shows the results of modeling the in-vitro experiment. The model predicted similar changes in the signal amplitude at large separations for the different fracture geometries used in the experiments. When the material in the fracture gap was given the properties of the cement, the signal loss was small compared to the baseline data, but the addition of a callus caused the signal loss to increase and resulted in a larger FTL Table III. One significant difference between the experimental data and the numerical modeling was the shape of characteristic curve for the callus geometry. In the model, a large peak in the reradiated amplitude was produced; this was not present in the experimental data. IV. DISCUSSION Previous studies of the relative change in velocity as a transverse fracture healed 1,2,11 have shown that, after the initial fracture, the velocity tended to increase steadily toward the baseline intact value. The models of a healing transverse fracture used in this study suggest that the relative change in FAS amplitude displays a different variation with the healing stages when compared with the velocity changes. It is also interesting to compare the magnitude of the changes observed in average velocity and transmission amplitude. For example, for a 2 mm fracture gap and 20 mm transducer spacing, the maximum reduction in average velocity reported by Protopappas et al. 2 was of order 20%. For the attenuation measurements reported here for a gap of 2 mm the reductions in transmission are of order 50%. Velocities will in general be easier to measure accurately, but as the transducer separation increases the change in average velocity reduces unlike the FTL. FIG. 7. Numerical simulation snapshots of acoustic pressure for healing stages 1 5. At stage 1 the fracture gap is filled with water, stage 2 represents the initiation of bridging and ossification with a symmetrical noncontinuous callus and cartilage at the fracture site, and stages 3 and 4 represent the continued ossification of callus, the difference being the material filling the fracture gap which is modeled as cartilage in stage 3 and callus in stage 4. The front of the acoustic pulse has just passed the fracture. For a relatively low modulus callus G=5 GPa, the changes in the FAS amplitude were not steady with the healing process stages. The change in geometry, i.e., introduction of a bridging callus stages 1 and 2, and the low stiffness of the callus material did not result in a significant decrease in the FTL. Similarly, the change in mechanical properties, as the fracture geometry remained constant stage 2 5, did not result in significant decreases in the signal loss. This would suggest that the mechanical changes within the callus do not greatly affect the measurements. The most apparent change occurred in the simulated remodeling stage stage 6, where the loss was approximately halved see Table II. This picture changed when the stiffness of the callus material was increased to 10 GPa. Essentially, small increases toward the baseline result decrease in FTL were observed for data taken in stages 2 5, while a marked change was observed between stages 1 and 2 and between stages 5 and 6. The small variation in amplitude for stages 2 5 reinforced the idea that the mechanical changes within the gap do not greatly affect the measurements. A further increase in callus stiffness to 15 GPa did not significantly alter this behavior. The reduction in fracture loss as the callus is added from stages 1 to 2 indicates that the callus helps to guide the acoustic radiation from one side of the fracture to the other, effectively acting as a waveguide. This effect is shown in Fig. 7 which shows snapshots of the numerical simulations for healing stages 1 5 at a time when the front of the pulse has just passed the fracture. It should be noted that from stage 2 the pressure wave spreads out in to callus, with similar pressures in the callus and bone. This effect increases 892 J. Acoust. Soc. Am., Vol. 126, No. 2, August 2009 Gheduzzi et al.: Ultrasound transmission across a healing fracture

7 as the modulus of the callus increases due to the better matching of the bone to the callus. At the same time, however, the callus acts as an additional load on the bone giving rise to an increased re-radiation from the fracture site. This effect then appears to dominate the loss, so changes in the fracture gap material modulus have relatively little effect on the fracture loss stages 2 5. As the density of the callus remains constant, the mass loading will not vary with callus material. As a result, the loss only starts to approach the baseline as the callus reduces in size stages 6 and 7. Some experiments were performed to simulate the effect of changing the mechanical properties and geometry of a fracture site, in this case using flat plates. These experiments provided some useful insights, even though they do not use the same geometries as the numerical modeling. Increasing the stiffness of the simulated healing tissue, i.e., substituting bone cement for water in the fracture gap, resulted in a significant decrease in signal loss higher transmission. The gap filler reduces the mismatch at the two gap interfaces and increases the transmission across the fracture zone. This pattern is also seen in the modeling of these experiments. However, adding a bone cement callus to the top surface across the fracture resulted in an increased signal loss compared to just the gap filler alone. This confirms the concept that the callus itself can give rise to strong scattering/re-radiation and a reduction in signal transmission across the fracture and this may have some clinical relevance depending on the extent of callus formation during healing. In terms of sensitivity of the signal amplitude measurement to changing mechanical and geometrical properties at the fracture site, these models and experiments suggest that only three changes are detectable. The first change from stages 1 to 2 initial inflammatory condition to the formation of a callus is a result of both the geometrical and mechanical properties of the fracture site. However, a stiffer callus material creates a larger relative signal amplitude change at this stage. Mechanical changes within the callus that occur from stages 2 to 5 though potentially clinically significant cannot be easily detected with this experimental set-up. It is possible that the change in signal amplitude from stages 5 to 6 may be proportional to the callus dimensions. Therefore, the next detectable change would occur in the remodeling stage after a significant reduction in the callus size has taken place. The final change occurs when the original geometry of the bone is regained. In these models and experiments, only the transverse case was considered. A number of physical properties need to be investigated to provide a more complete picture of the wave interaction at different healing stages. The thickness of the bone plate at a particular acoustic frequency has been shown to influence FAS velocity and attenuation measurements; therefore, modeling of different geometries, such as oblique fractures, and different plate thicknesses may also provide important insights on the modalities of ultrasound propagation in bone. Other properties, such as cortical bone porosity and material damping, may also have an effect on the relative amount of ultrasound signal loss detected. V. CONCLUSIONS The results presented confirm that the addition of a discontinuity to a cortical bone model, such as a transverse fracture, results in a reduction in the reradiated FAS amplitude beyond the discontinuity. The use of a numerical finite difference model has enabled the effects of the different stages of the fracture healing process on the acoustic transmission to be studied. These show that the changes can be complex, and do not vary steadily with the healing stage. Specifically the largest changes in the transmitted amplitude are observed to occur from stages 1 to 2, the initial inflammatory condition to the formation of a callus, and from stages 5 to 6 in the remodeling stage after a significant reduction in the callus has taken place. The relative significance of these two stages appears to be dependant on Young s modulus selected for the callus, with the former dominating for higher moduli. The progressive changes in the modulus of the material filling the fracture appears to produce less significant changes in the FTL, irrespective of the callus modulus. These results imply that it is important to consider the effect of the callus when considering the changes that occur in acoustic transmission across a fracture through the fracture healing process. The callus itself appears to be a strong scatterer/source of re-radiation, presumably due to its mass loading of the bone plate. These numerical studies have been confirmed by in-vitro experiments performed using flat plates. These demonstrated that substituting bone cement for water in the fracture gap resulted in a large decrease in the FTL, i.e., better signal transmission, compared with the water gap. However, adding a bone cement callus to the top surface across the fracture resulted in an increased signal loss compared to just the gap filler alone. The results indicate that significant changes do occur, both in the re-radiation from the fracture/callus and transmission loss across the fracture site, through the fracture healing sequence. This change in the signal amplitude from an inflammatory healing stage to callus formation stage or to the callus remodeling stage could potentially be used to quantify the healing process. In practice the amplitude values without a fracture would be obtained from measurements of the nonfractured contraleteral bone. The aim of the clinical measurement would then be to evaluate the evolution of the FTL from the fractured bone with time as healing progressed and as it approached, but did not necessarily coincide with, the FTL value for the non-fractured contralateral. For example, a fracture could be deemed sufficiently healed for unsupported weight-bearing if the FTL of the fractured bone was within a certain percentage of that of the contralateral. ACKNOWLEDGMENT The authors are grateful to Action Medical Research for funding this work. 1 V. C. Protopappas, D. A. Baga, D. I. Fotiadis, A. C. Likas, A. Papachristos, and K. N. Malizos, An ultrasound wearable system for the monitoring and acceleration of fracture healing in long bones, IEEE Trans. Biomed. Eng. 52, J. Acoust. Soc. Am., Vol. 126, No. 2, August 2009 Gheduzzi et al.: Ultrasound transmission across a healing fracture 893

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