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

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.

Modelling the effects of different fracture geometries and healing stages on ultrasound signal loss across a long bone fracture.

The effect on the signal amplitude of ultrasonic waves propagating along cortical bone plates was modelled using a 2D Finite Difference code. Different healing stages, represented by modified fracture geometries were introduced to the plate model. A simple transverse and oblique fracture filled with water was introduced to simulate the inflammatory stage. Subsequently, a symmetric external callus surrounding a transverse fracture was modelled to represent an advanced stage of healing. In comparison to the baseline (intact plate) data, a large net loss in signal amplitude was produced for the simple transverse and oblique cases. Changing the geometry to an external callus with different mechanical properties caused the net loss in signal amplitude to reduce significantly. This relative change in signal amplitude as the geometry and mechanical properties of the fracture site change could potentially be used to monitor the healing process.

Empirical angle-dependent Biot and MBA models for acoustic anisotropy in cancellous bone.

The Biot and the modified Biot-Attenborough (MBA) models have been found useful to understand ultrasonic wave propagation in cancellous bone. However, neither of the models, as previously applied to cancellous bone, allows for the angular dependence of acoustic properties with direction. The present study aims to account for the acoustic anisotropy in cancellous bone, by introducing empirical angle-dependent input parameters, as defined for a highly oriented structure, into the Biot and the MBA models. The anisotropy of the angle-dependent Biot model is attributed to the variation in the elastic moduli of the skeletal frame with respect to the trabecular alignment. The angle-dependent MBA model employs a simple empirical way of using the parametric fit for the fast and the slow wave speeds. The angle-dependent models were used to predict both the fast and slow wave velocities as a function of propagation angle with respect to the trabecular alignment of cancellous bone. The predictions were compared with those of the Schoenberg model for anisotropy in cancellous bone and in vitro experimental measurements from the literature. The angle-dependent models successfully predicted the angular dependence of phase velocity of the fast wave with direction. The root-mean-square errors of the measured versus predicted fast wave velocities were 79.2 m s(-1) (angle-dependent Biot model) and 36.1 m s(-1) (angle-dependent MBA model). They also predicted the fact that the slow wave is nearly independent of propagation angle for angles about 50 degrees , but consistently underestimated the slow wave velocity with the root-mean-square errors of 187.2 m s(-1) (angle-dependent Biot model) and 240.8 m s(-1) (angle-dependent MBA model). The study indicates that the angle-dependent models reasonably replicate the acoustic anisotropy in cancellous bone.

An in vitro study of ultrasound signal loss across simple fractures in cortical bone mimics and bovine cortical bone samples.

Measurements have been performed on Sawbones and bovine cortical bone samples at 200 kHz using an axial transmission technique to investigate the factors that determine how ultrasonic waves propagate across a simulated fracture. The peak amplitude of the first arrival signal (FAS) was studied. Results taken from intact specimens were compared with those produced when a simple transverse fracture was introduced. These fracture simulation experiments were found to be consistent with Finite Difference modelling of the experimental conditions. The peak amplitude showed a characteristic variation across the fracture caused by interference between reradiated and scattered/diffracted waves at the fracture site and a net Fracture Transmission Loss (FTL). For small fracture gaps, the change in amplitude was sensitive to the presence of the fracture. This sensitivity suggests that this parameter could be a good quantitative indicator for the fracture healing process assuming the relative change in this parameter brought about by healing is measurable.

Ultrasonic propagation in cortical bone mimics.

Understanding the velocity and attenuation characteristics of ultrasonic waves in cortical bone and bone mimics is important for studies of osteoporosis and fractures. Three complementary approaches have been used to help understand the ultrasound propagation in cortical bone and bone mimics immersed in water, which is used to simulate the surrounding tissue in vivo. The approaches used were Lamb wave propagation analysis, experimental measurement and two-dimensional (2D) finite difference modelling. First, the water loading effects on the free plate Lamb modes in acrylic and human cortical bone plates were examined. This theoretical study revealed that both the S0 and S1 mode velocity curves are significantly changed in acrylic: mode jumping occurs between the S0 and S1 dispersion curves. However, in human cortical bone plates, only the S1 mode curve is significantly altered by water loading, with the S0 mode exhibiting a small deviation from the unloaded curve. The Lamb wave theory predictions for velocity and attenuation were then tested experimentally on acrylic plates using an axial transmission technique. Finally, 2D finite difference numerical simulations of the experimental measurements were performed. The predictions from Lamb wave theory do not correspond to the measured and simulated first arrival signal (FAS) velocity and attenuation results for acrylic and human cortical bone plates obtained using the axial transmission technique, except in very thin plates.

Energy and phase velocity considerations required for attenuation and velocity measurements of anisotropic composites.

Viscoelastic fibre-reinforced composite materials have a number of possible advantages for use in underwater acoustic applications. In order to exploit these materials it is important to be able to measure their complex stiffness matrix in order to determine their acoustic response. Ultrasonic transmission measurements on parallel-sided samples, employing broadband pulsed transducers at 2.25 MHz and an immersion method, have been used to determine the viscoelastic properties of a glass-reinforced composite with uniaxially aligned fibres. The composite measured was constructed from Cytecfiberite's CYCOM 919 E-glass. The theory of acoustic propagation in anisotropic materials shows that the direction of energy propagation is, in general, different from that given by Snell's Law. At 15 degrees incidence, Snell's Law implies a refracted angle of 40 +/- 2 degrees, whereas the energy direction is observed to be 70 +/- 2 degrees. Despite this, the experimental data indicates that the position of the receiving transducer has relatively little effect on the apparent phase velocity measured. The phase velocities measured at positions determined from the refracted angle and energy direction are 3647 and 3652 +/- 50 m s(-1), respectively. However, the amplitude of the received signal, and hence estimate of attenuation, is highly sensitive to the receiver position. This indicates that the acoustic Poynting vector must be considered in order to precisely determine the correct position of the receiving transducer for attenuation measurements. The beam displacement for a 17.6 mm sample at 15 degrees incidence is 9.5 and 40 mm by Snell's Law and Poynting's Theorem, respectively. Measured beam displacements have been compared with predictions derived from material stiffness coefficients. These considerations are important in recovering the complex stiffness matrix.

Comparison of finite element and heated disc models of tissue heating by ultrasound.

This paper compares different techniques used to model the heating caused by ultrasound (US) in a phantom containing a layer of bone mimic covered by agar gel. Results from finite element (FE) models are compared with those from two techniques based on the point-source solution to the bioheat transfer equation (BHTE): one in which the bone mimic is considered to be an absorbing disc of infinitesimal thickness and the other in which the region through which the US travels is considered to be a volume heat source. The FE results are also compared with experimental measurements. The results from the models differed by up to 40% compared with those from the FE model. Furthermore, for the intensity distribution considered, which corresponds to that in the focal zone of a single-element transducer, the top hat distribution predicts a temperature rise 1.8 times greater than that for a more realistic one based on measured values.

Nonlinear propagation in ultrasonic fields: measurements, modelling and harmonic imaging.

In high amplitude ultrasonic fields, such as those used in medical ultrasound, nonlinear propagation can result in waveform distortion and the generation of harmonics of the initial frequency. In the nearfield of a transducer this process is complicated by diffraction effects associated with the source. The results of a programme to study the nonlinear propagation in the fields of circular, focused and rectangular transducers are described, and comparisons made with numerical predictions obtained using a finite difference solution to the Khokhlov-Zabolotskaya-Kuznetsov (or KZK) equation. These results are extended to consider nonlinear propagation in tissue-like media and the implications for ultrasonic measurements and ultrasonic heating are discussed. The narrower beamwidths and reduced side-lobe levels of the harmonic beams are illustrated and the use of harmonics to form diagnostic images with improved resolution is described.

Finite difference modelling of the temperature rise in non-linear medical ultrasound fields.

Non-linear propagation of ultrasound can lead to increased heat generation in medical diagnostic imaging due to the preferential absorption of harmonics of the original frequency. A numerical model has been developed and tested that is capable of predicting the temperature rise due to a high amplitude ultrasound field. The acoustic field is modelled using a numerical solution to the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation, known as the Bergen Code, which is implemented in cylindrical symmetric form. A finite difference representation of the thermal equations is used to calculate the resulting temperature rises. The model allows for the inclusion of a number of layers of tissue with different acoustic and thermal properties and accounts for the effects of non-linear propagation, direct heating by the transducer, thermal diffusion and perfusion in different tissues. The effect of temperature-dependent skin perfusion and variation in background temperature between the skin and deeper layers of the body are included. The model has been tested against analytic solutions for simple configurations and then used to estimate temperature rises in realistic obstetric situations. A pulsed 3 MHz transducer operating with an average acoustic power of 200 mW leads to a maximum steady state temperature rise inside the foetus of 1.25 degrees C compared with a 0.6 degree C rise for the same transmitted power under linear propagation conditions. The largest temperature rise occurs at the skin surface, with the temperature rise at the foetus limited to less than 2 degrees C for the range of conditions considered.

A theoretical investigation of the effect of nonlinear propagation on measurements of mechanical index.

Safety parameters for diagnostic ultrasound scanners are calculated from measurements made in water, which are derated to account for the attenuation of tissues. Sound is attenuated less by water than by tissue, and so the effects of nonlinear propagation are greater in water. This study compares mechanical index (MI) and derated intensity with the analogous quantities in idealised soft tissue, for simplified models of scanners with source amplitudes up to 2.5 MPa. As expected, MI is much smaller than implied by linear extrapolation from low-amplitude measurements but, in a system with moderate gain, the reduction in tissue is commensurate with that in water, MI and derated intensity underestimating the values in tissue by at most 20%. Determining MI at the location of peak negative pressure halves the error. In high gain systems, however, MI can be less than 60% of the value in tissue.

In vitro heating of human fetal vertebra by pulsed diagnostic ultrasound.

The temperature rise generated at the surface of unperfused human fetal vertebrae in vitro by an ultrasound beam with characteristics typical of those used in pulsed Doppler examinations has been measured. The bone samples were from fetuses that ranged in age from 14 to 39 weeks, dating from the last menstrual period. The samples were embedded in agar gel and the temperature rise at their surface was measured using a 50-microm diameter K-type thermocouple. The power in the ultrasound beam was 50 +/- 2 mW and the -6 dB diameter was 2.9 mm. The temperature rise at 295 s ranged from 0.6 degrees C in the youngest sample to 1.8 degrees C in the oldest. Approximately 70% of the temperature rise occurred in the first min.

Nonlinear propagation applied to the improvement of resolution in diagnostic medical ultrasound.

Medical B-mode scanners operating under conditions typically encountered during clinical work produce ultrasonic wave fields that undergo nonlinear distortion. In general, the resulting harmonic beams are narrower and have lower sidelobe levels than the fundamental beam, making them ideal for imaging purposes. This work demonstrates the feasibility of nonlinear harmonic imaging in medical scanners using a simple broadband imaging arrangement in water. The ultrasonic system comprises a 2.25-MHz circular transducer with a diameter of 38 mm, a membrane hydrophone, also with a diameter of 38 mm, and a polymer lens with a focal length of 262 mm. These components are arranged coaxially giving an imaging geometry similar to that used in many commercial B-scanners, but with a receiver bandwidth sufficient to record the first four harmonics. A series of continuous wave and pulse-echo measurements are performed on a wire phantom to give 1-D transverse pressure profiles and 2-D B-mode images, respectively. The reflected beamwidths wn decrease as wn/W1 = 1/n0.78, where n is the harmonic number, and the reflected sidelobe levels fall off quickly with increasing n. In imaging terms, these effects correspond to a large improvement in lateral resolution and signal-to-clutter ratio for the higher harmonics.

Forces acting in the direction of propagation in pulsed ultrasound fields.

This paper considers some non-thermal effects resulting from absorption of acoustic energy from an ultrasound beam. An experimental investigation of the location of the 'source pump', responsible for the generation of streaming in high amplitude diagnostic fields in water, is reported. Acoustically transparent membranes were inserted in the ultrasound field in order to restrict the streaming volume. It is shown that the major contribution to an acoustic stream is generated in the region near to the focus of a transducer where the intensity in the beam and the degree of non-linear distortion are both high. In the second part of the paper a simple model of non-linear propagation is used to predict the magnitude of the maximum pressure gradient induced in a medium by the absorption of acoustic energy from a beam. Propagation in water, in tissue and in amniotic fluid are considered. Within the limitations of this model it is shown that the pressure gradients induced in pulsed acoustic fields do not result in the ultimate shear stress of tissue being exceeded.

An experimental investigation of streaming in pulsed diagnostic ultrasound beams.

Streaming is shown to occur in water in the focused beams produced by a number of medical pulse-echo devices. The use of hot film anemometry to measure the streaming velocity is described and velocities measured in water using commercial equipment are quoted. The highest velocities occur in pulsed Doppler mode with a maximum velocity of 14 cm s-1 being observed. An experimental set-up was used to investigate the parameters affecting streaming and it was found that the harmonic content of the pulse waveform had a major effect on the streaming velocity. The time taken for a stream to become established at the focus of the acoustic beams studied was typically approximately 0.5 s.

The development of harmonic distortion in pulsed finite-amplitude ultrasound passing through liver.

The progressive development of finite-amplitude distortion of ultrasonic pulses has been investigated in excised bovine liver using pulsed focused ultrasonic beams at nominal frequencies of 2.5 and 3.5 MHz. Both the transducers and the powers used were those which may be encountered with clinical imaging equipment. Significant distortion of the waveform was observed to occur, particularly at higher powers. For example, at 2.5 MHz, with a mean input pressure (p0) of 0.58 MPa, the second harmonic in the pulse spectrum showed a maximum value of 10.5 dB below the fundamental and the highest third harmonic component was 19 dB below the fundamental. These particular observations illustrate that finite-amplitude distortion may be of considerable significance in the transmission through tissue of ultrasonic pulses during diagnostic scanning.

Evidence for ultrasonic finite-amplitude distortion in muscle using medical equipment.

Finite-amplitude distortion of ultrasonic waves from medical equipment has been observed to occur following transmission through calf muscle in human volunteers. Measurements were made using both dynamic pulse-echo imaging equipment and physiotherapy equipment. In both cases irradiation was carried out under operating conditions commonly used clinically. Pressure waveforms were measured at the skin surface using a broadband polyvinylidene difluoride membrane hydrophone. Using a pulsed, weakly focused 2.5-MHz beam with input peak pressure of 0.8 MPa and a pressure gain of 5.3 at the focus, the mean second harmonic peak magnitude (16 measurements) was 17 dB below the fundamental peak. A 1.1-MHz continuous wave therapy set with input peak pressure of 0.5 MPa showed mean second harmonic magnitude 23 dB below the fundamental.