Experimental validation of a method for the simulation of backscatter signals

from finite, shaped scatterers

Martin Sälzle1, Jan D'hooge1,2 and Marco M. Voormolen1 1 MI Lab and Department of Circulation and Imaging, Norwegian University of Science and Technology, Trondheim, Norway

E-mail: marco.voormolen@ntnu.no 2 Cardiovascular Imaging and Dynamics, Catholic University of Leuven, Leuven, Belgium

Abstract — An efficient method for the calculation of backscatter signals from finite, shaped scatterers is presented and validated. Good agreement is shown between the proposed method, theoretical results and measurements obtained from single scatterers. It is also shown that the method will breakdown if the dimensions of the scatterer get significantly larger than half the wave length of the applied ultrasonic field. With an average myocyte length of 120μm, the limits of validity were found to be sufficient to make the method suitable for the modeling of myocardial fiber orientation in the echocardiographic frequency range (2-7MHz).

Keywords: ultrasonic backscatter, finite shaped scatterers, myocadial fiber orientation, spatial impulse response

I. INTRODUCTION Existing simulation tools for medical, ultrasonic imaging

use random distributions of infinitely small point scatterers to model tissue. This assumes that human tissue is regionally isotropic [1]. Although this has been a valuable approximation, it is well known that tissue often exhibits a fibrous, anisotropic structure. The myocardium even consists of several layers with different fiber orientations. It has been shown that the anisotropy of tissue has its influence on the properties of the backscattered signal [2]. This has mainly been investigated in relation with tissue characterization. The anisotropy of tissue has however also its influence on B-mode images [3]. The ability to incorporate the anisotropy of tissue in ultrasonic simulated tools would allow for the development of new beamform and image processing strategies using more realistic, virtual B-mode images. As a first step towards this goal we present a method for simulating backscatter signals from finite, shaped scatterers.

To calculate backscatter signals from finite, shaped scatterers a method based on the time-domain first-Born approximation was incorporated in a pulse-echo spatial impulse response scheme. The method was adapted from similar methods, previously presented by Lee and Warhola and Mottley and Miller [4, 5]. In this manuscript the proposed method is theoretically and experimentally validated using pulse-echo signals from single, cylindrical scatterers. In

addition, the limits of validity of the underlying approximations are explored.

Echocardiography is a major application within diagnostic ultrasound and the shape of cardiomyocytes approximates a cylinder with an average length of 120μm and an average diameter of 12μm. These are the reasons why the remainder of this manuscript concentrates on cylindrical shaped scatterers. It should be mentioned that the method is suitable for scatterers of any shape.

II. THEORIE A classic formulation for the calculation of the pulse-echo

response received by a transducer from a point scatterer is given by Jensen and Svendsen [6]:

( ) ( ) ( ) ( )( )2

2 2

0

1, , ,pe pe Tx Rxp r t v t h r t h r t

c t

∂= ∗ ∗

∂

r r r (1)

Where ppe is the received pulse echo signal, c0 the speed of sound, vpe the excitation function including the pulse-echo electromechanical impulse response, hTx the transmit and hRx the receive spatial impulse response of the transducer.

Using the result from Lee and Warhola a similar result can be obtained for a finite, shaped scatterer [4]:

( ) ( ) ( ) ( ) ( )( )2

2 2

0

, , , , ,pe pe Tx Rx

Mp r t v t h r t h r t A t i o

c t

∂= ∗ ∗ ∗

∂

r r r r r (2)

Where M is a scaling parameter that represents the perturbation in material properties of the scatterer from those of the surrounding medium and A is the area function of the scatterer. This result can be obtained by assuming that:

• the scatterer is relatively small, such that the incident field from the transmit aperture can be approximated by a plane wave;

• the time-domain first-Born approximation can be applied;

• the receiving aperture is located in the far field of the backscattered field.

2161978-1-4577-1252-4/11/$26.00 ©2011 IEEE 2011 IEEE International Ultrasonics Symposium Proceedings

10.1109/ULTSYM.2011.0536

The area function A in equation 2 describes the finite shaped scatterer’s cross-sectional area with a plane traveling in the direction of the incidence wave (see figure 1). With i being the direction of incidence and o the direction of the observation, the normal vector n of the intersecting plane equals:

i on

i o

−=

−

r rr

r r (3)

Making use of the superposition principle we can replace the shaped scatterer by a large number of point scatterers. Provided that the assumptions leading to equation 2 are adequate, it then follows from equation 1 and 2 that:

( ) ( ) ( )1

1, ,

N

pe Tx n Rx nv t h r t h r tN

∗ ∗∑r r

(4)

should be equal to:

( ) ( ) ( ) ( )0

2

, , , ,pe Tx c Rx c

c

v t h r t h r t A t i o

V

∗ ∗ ∗r r r r

(5)

Where N is the number of point scatterers, rn is the location of the nth point scatterer, rc is the position of the center of the shaped scatterer and V is the volume of the shaped scatterer. Formula 4 and 5 display the efficiency of the presented method: a quadruple convolution replaces an N times triple convolution.

Figure 1. A cylindrical, shaped scatterer with the parameters that define the area function indicated: i is the direction of incidence, o the direction of the observation and ϕ the angle between them. The cross-sectional area of the

plane (with normal vector n) and cylinder is indicated in grey and represents the amplitude of the area function for this time instance.

III. METHODS The method presented in the previous section was validated

both theoretically and experimentally. The following sections describe the used validation methods.

A. Theoretical validation Our proposed method was theoretically validated by

comparing results from formula 4 and 5. For this purpose pulse-echo signals from a concave transducer with a diameter of 19.1mm (i.e. 0.75inch) and a focal radius of 70mm was used. The finite, cylindrical scatterer had a length of 500μm

and a diameter of 50μm. The scatterer was positioned on the transducer’s symmetry axis at a distance of 50mm and with an angle θ of 30° (the angle between the symmetry axis of the transducer and the scatterer). The cylindrical scatterer was filled with a suitable number of randomly distributed point scatterers. A Gaussian pulse with a fractional bandwidth of 60% was used as input signal vpe.

B. Experimental validation For the experimental validation pulse-echo signals from

single cylindrical scatterers were recorded. The scatterers were made from copper or pencil lead. Figure 2 shows an example from both material types. To avoid interference from supporting structures the scatterers were deposited in a low dose (<1% by mass), liquid agar solution. To validate equation 2 experimentally a balance needed to be found between scatterer size, signal strength and transmit frequency. This was achieved by using a copper scatterer with a length of 2.22mm and a diameter of 0.5mm.

Figure 2. The experimental set-up. The background image shows a close-up of the water tank with the 3.5MHz transducer and a cylindrical, pencil lead scatterer (5.1mm length and 0.55mm diameter). The top-left inset shows an overview of the whole set-up with the 0.3MHz transducer conneced to the robot arm. The bottom-left inset shows microscope images from an pencil

lead and a copper scatterer (both 2.2mm length and 0.7 and 0.5mm diameter respectively).

A custom made concave transducer (Imasonic, Voray sur l’Ognon, France) was used with center frequency of 0.3MHz, a diameter of 55mm and a focal radius of 88mm. The transducer was attached to a robot arm with three orthogonal degrees of freedom. As electrical input signal a Guassian pulse with a fractional bandwidth of 50% and an amplitude of 15V was used. The scatterer was approximately placed in the focal region of the transducer after which the transducer was manipulated to get maximum signal. The scatterer was placed under various insonification angles θ. The exact angulation of the scatterer was determined in retrospect from orthogonal bidirectional photos.

To explore the validity limits of the presented model similar recordings were made using cylindrical scatterers with different sizes and transducers with different center frequencies.

2162 2011 IEEE International Ultrasonics Symposium Proceedings

IV. RESULTS As a compromise between computation time and accuracy

we chose to use 8000 point scatterers. Although the accuracy from the point scatterer simulation was not optimal this already resulted in computation time reduction of 1000 times for the proposed method. Figure 3 shows the result of the theoretical validation. For all three angles the pulse-echo signals and their frequency spectra are practically identical.

Figure 3. Results from the comparison between formula 4 (blue, solid lines) and 5 (red, dashed lines) using a 3.5MHz concave aperture and a cylindrical scatterer of 500μm length and 50μm diameter. Pulse-echo signals and their frequency spectra are shown for insonification angles θ of 0, 45 and 90°

respectively (from top to botom).

A typical frequency interference pattern can be observed for the 0 and 45 degrees cases. The location of these frequency notches can easily be estimated with the following equation:

( )0

2 cos

cf

l θ=

⋅ (6)

Where f is the fundamental frequency of the observed notches, l is the length of the cylindrical scatterer and θ the insonification angle. For figure 3 this results in 1.5MHz (and its multiples) for the 0 degree case and 2.1MHz for the 45 degree case. Equation 6 gives good estimations for cylinders with high aspect ratios, were the diameter of the cylinder has negligible size.

Figure 4 shows the results from primary validation experiments. Only the recorded pulse-echo signals are shown because the transfer function of the transducer was unknown. The unequal phase of the recorded and simulated pulse-echo

signals completely mask their resemblance. From the frequency spectra it can however be seen that the recorded and simulated pulse-echo signals are in close agreement with each other.

Figure 4. Experimetal results from the 0.3MHz transducer and a cylindrical scatterer of 2.2mm length and 0.5mm diameter. Pulse-echo signals and their frequency spectra (blue, solid lines) are shown for insonification angles θ of

12, 44 and 87° respectively (from top to botom). Simulated spectral results are shown in red (dashed lines).

Figure 5 shows a somewhat obvious result from the limits of validity exploration. A rather large scatterer (5.1mm length and 0.55mm diameter) was insonified at 8 degrees with a concave 3.5MHz transducer. The fading internal reverberation can be clearly observed between the stronger front and back echoes. More remarkable is the dense interference pattern in the frequency spectrum. Despite the reverberation, the recorder spectrum still resembles the simulated one.

Figure 5. Experimetal results from the 3.5MHz transducer and a cylindrical scatterer of 5.1mm length and 0.55mm diameter. A pulse-echo signal and its

frequency spectra (blue, solid lines) are shown for insonification angle θ of 8°. Simulated spectral results are shown in red (dashed line).

2163 2011 IEEE International Ultrasonics Symposium Proceedings

The same scatter was used in an experiment with the 0.3MHz transducer. The results are shown in figure 6. The 0 and 90 degrees cases show good agreement between the simulated and recorded pulse-echo spectra. However, at 30 degrees oscillatory behavior can be observed. In the frequency spectrum a clear peak can be seen somewhat below the center frequency of the transducer. Internal oscillations inside the scatterer are ruled out by the sound speed of pencil lead (approximately 6500m/s, estimated from figure 5). Combining that with the appearance of the oscillations at a specific angle, it looks like we have encountered a helical, fluid-born Franz wave [7].

Figure 6. Experimetal results from the 0.3MHz transducer and a cylindrical scatterer of 5.1mm length and 0.55mm diameter. Pulse-echo signals and their

frequency spectra (blue, solid lines) are shown for insonification angles θ of 7, 30 and 85° respectively (from top to botom). Simulated spectral results are

shown in red (dashed lines).

Although the phase of the recorded and simulated pulse-echo signals do not match, it is still possible to compare their maximum amplitude of the envelope (MAE). This was done for the results from figure 4 and 6. The results of figure 6 can still be used for this purpose because the oscillatory signal does not interfere with the amplitude of the direct echo. By using the results from the largest angle to calibrate the other two, the experimental values can be compared with simulations. It can be seen from figure 7 that the experimental MAE’s are in good agreement with the results from the proposed method.

Figure 7. Comparison between the simulated (blue line) and measured (magenta dots) maximum amplitude of the envelope results. The measured

results are extracted from the pulse-echo signals shown in figure 4 and 6. The measured results at the highest insonification angle (red cross) were used to

calibrate the results from other two insonification angles.

V. DISCUSSION AND CONCLUSIONS A method for efficient calculation of backscatter signals

from finite, shaped scatterers was presented and validated. From the results it can be concluded that the proposed method is in close agreement with the theoretical and experimental results. It was also shown that if the size of the finite, shaped scatterer significantly exceeds half the wave length of the applied ultrasonic field the proposed method starts breaking down.

A pulse-echo spatial impulse response method for the simulation of backscatter signals from finite, shaped scatterers was developed and validated. With an average myocyte length of 120μm, the limits of validity were found to be sufficient for the modeling of myocardial fiber orientation in the echocardiographic frequency range (2-7MHz).

REFERENCES [1] J.W. Hunt, et al., “The subtleties of ultrasound images of an ensemble of

cells: Simulation from regular and more random distributions of scatterers,” Ultrasound Med Biol, vol. 21, pp. 329-341, 1995.

[2] C.S. Hall, et al., “Anisotropy of the apparent frequency dependence of backscatter in formalin fixed human myocardium,” J Acoust Soc Am, vol. 101, pp. 563-567, 1997

[3] J. Crosby, et al., “The effect of including myocardial anisotropy in simulated ultrasound images of the heart,” IEEE Trans Ultrason Ferroelectr Freq Control, vol. 56, pp. 326-333, 2009

[4] D.A. Lee and G.T. Warhola, “Time-domain first-Born approximations to backscattering from cylinders,” J Acoust Soc Am, vol. 79, pp. 681-690, 1986

[5] J.G. Mottley and J.G. Miller, “Anisotropy of the ultrasonic backscatter of myocardial tissue: I. Theory and measurements in vitro,” J Acoust Soc Am, vol. 83, pp. 755-761, 1988

[6] J.A. Jensen and N.B. Svendsen, “Calculation of pressure fields from arbitrarily shaped, apodized, and excited ultrasound transducers,” IEEE Trans Ultrason Ferroelectr Freq Control, vol. 39, pp. 226-267, 1992

[7] N.D. Veksler and J.L. Izbicki, “Modal resonances of the Franz waves,” Acta Acustica, vol. 82, pp. 18-26, 1996

2164 2011 IEEE International Ultrasonics Symposium Proceedings