3D noninvasive ultrasound Joule heat tomography based on acousto-electric effect using unipolar pulses: a simulation study

Abstract

Electrical properties of biological tissues are highly sensitive to their physiological and pathological status. Thus it is of importance to image electrical properties of biological tissues. However, spatial resolution of conventional electrical impedance tomography (EIT) is generally poor. Recently, hybrid imaging modalities combining electric conductivity contrast and ultrasonic resolution based on acouto-electric effect has attracted considerable attention. In this study, we propose a novel three-dimensional (3D) noninvasive ultrasound Joule heat tomography (UJHT) approach based on acouto-electric effect using unipolar ultrasound pulses. As the Joule heat density distribution is highly dependent on the conductivity distribution, an accurate and high resolution mapping of the Joule heat density distribution is expected to give important information that is closely related to the conductivity contrast. The advantages of the proposed ultrasound Joule heat tomography using unipolar pulses include its simple inverse solution, better performance than UJHT using common bipolar pulses and its independence of any priori knowledge of the conductivity distribution of the imaging object. Computer simulation results show that using the proposed method, it is feasible to perform a high spatial resolution Joule heat imaging in an inhomogeneous conductive media. Application of this technique on tumor scanning is also investigated by a series of computer simulations.

1. Introduction

Noninvasive imaging of electrical properties of biological tissues has drawn much attention due to its sensitivity to physiological and pathological conditions of living system (Geddes and Baker 1967). Conventional electrical impedance tomography (EIT) technique uses current injection and noninvasive surface voltage measurements to reconstruct tissue impedance images. EIT has the merits of low cost, real-time speed and safety. On the other hand its major limitations include low spatial resolution and degraded sensitivity in the center of imaging object (Wen 1999). Recently imaging techniques based on acousto-electric (AE) effect (Fox et al 1946, Jossinet et al 1998, 1999, Lavandier et al 2000a, 2000b) has attracted considerable interest (Zhang and Wang 2004, Witte et al 2007, Olafsson et al 2009a, Sumi 2009, Yang et al 2011) because of its potential to image tissue's electric property or electrophysiological functioning with high resolution. Such kind of high resolution impedance or current density imaging is made possible through the AE effect which describes the local electrical impedance changes in a conductive volume occupied by an ultrasonic wave packet. Previous studies have shown the potential of the AE based imaging methods. For example, acousto-electric tomography (AET) has been proposed for high resolution electrical impedance imaging of breast tissue (Zhang and Wang 2004). Hybrid current density imaging techniques based on AE effect have also been pursued extensively (Witte et al 2007, Olafsson et al 2009a). The potential of AE imaging with sub-millimeter spatial resolution and good sensitivity to detect small current densities (2 – 4mA/ cm²) was demonstrated by experiment under controlled conditions (Witte et al 2006, 2007, Olafsson et al 2006). Utilizing the acousto-electric effect to image current flow in tissues with physiologically realistic current densities was also reported (Witte et al 2006, 2007, Olafsson et al 2006). Based on these works, ultrasound current source density imaging (UCSDI) was proposed by Olafsson et al (2008) and its potential application in cardiac activation mapping has been well demonstrated in a live rabbit heart model (Olasson et al 2009a). Recently, a three-dimensional (3D) inverse UCSDI solution using unipolar ultrasound pulses has also been proposed (Yang et al 2011). However, for current density reconstruction in UCSDI, a priori knowledge of the conductivity distribution and lead fields of measurement electrodes in the imaging object are required. Although in practice, such kind of knowledge may be obtained through another high resolution imaging modality such as MRI and known bench top impedance measurement of different tissue types, it poses uncertainty and possible imaging errors to the reconstructed current density using UCSDI approach. To avoid these errors, mapping the internal Joule heat consumption using acousto-electric effect was proposed (Sumi 2009). Such kind of ultrasound Joule heat imaging method does not need any priori knowledge of the conductivity distribution and the corresponding lead fields therefore is expected to be more reliable than UCSDI in practice. Although the Joule heat is an indirect measurement of the electrical conductivity of the imaging object, an accurate, high resolution mapping of Joule heat is still expected to give important contrast information related to the conductivity distribution and may provide valuable information for diagnosis in possible clinical applications. However, to our best knowledge, effective 3D solution to ultrasound Joule heat tomography has not yet been developed previously. One of the challenges in deriving the 3D solution comes from the use of common bipolar ultrasound pulses which make the AE signal almost zero when the ultrasound wave packet travels in a spatially slowly varying Joule heat field.

In the present study, a novel 3D ultrasound Joule heat tomography (UJHT) approach using unipolar ultrasound pulses is proposed for imaging biological tissues. Three-dimensional forward equation and inverse solution of the proposed method was derived. Computer simulations were conducted to demonstrate the feasibility of the proposed method and were used to assess its performance for possible applications in tumor scanning. Utilizing specially designed unipolar ultrasound pulses (Holé and Lewiner 1996, 1998) and by confining the Joule heat density perturbation to the ultrasound focus, simulation results show that we are able to obtain accurate 3D ultrasound Joule heat density image. Comparison of the 3D ultrasound Joule heat tomography method using unipolar ultrasound pulses with that using bipolar ultrasound pulses was also conducted in our computer simulations. In the investigation of its performance for possible tumor imaging, several practical imaging parameters are explored including ultrasound beam diameter, ultrasound pulse duration and measurement noise level. In addition, the imaging performance was also assessed in numerical phantoms with tumors of different size, different conductivity contrast and at different position. Our simulation results indicate that 3D tumor scanning with high spatial resolution is feasible by using the proposed method.

2. Methods

2.1. Theory

2.1.1. Overview of 3D ultrasound Joule heat tomography

The diagram of the proposed 3D ultrasound Joule heat tomography approach is illustrated in figure 1. A low-frequency alternating current is injected into the object through two electrodes on the object surface. The injected current flowing in the object volume will also establish a Joule heat distribution. The ultrasound pulse sent by a transducer at the bottom causes a local electrical resistivity change in the ultrasound wave packet when it travels in the object due to the acousto-electric effect. This change in the local electrical resistivity generates a perturbation in voltage that could be measured by surface electrodes. This voltage perturbation, named AE signal, can be measured by the same two current injection electrodes and separated from the low-frequency current signal by a high-pass or band-pass filter. A 3D ultrasound scan is performed in such a way that the ultrasound transducer's focusing point scans over the whole object volume while the corresponding AE signals related to the periodic ultrasound pulse are collected at each scanning location. The 3D Joule heat distribution can then be calculated using the collected AE signals at the time point when the ultrasound pulse packet passes its focusing point. Because the Joule heat distribution mainly depends on the conductivity distribution of the imaging object, a Joule heat mapping contains important contrast information related to electrical conductivity.

Figure 1.

Diagram of the 3D ultrasound Joule heat tomography approach based on acousto-electric effect.

2.1.2. Acousto-electric (AE) effect

Acousto-electric (AE) effect describes the phenomenon of resistivity modulation by ultrasound (Fox et al 1946, Jossinet et al 1998, 1999, Lavandier et al 2000a, 2000b). For an electric and acoustic media, using the AE effect, the temporal change Δρ(x, y, z, t) in resistivity ρ(x, y, z) due to an ultrasound pressure wave P(x, y, z, t) can be written as

$$\Delta \rho (x,y,z,t)=K{\rho }_0(x,y,z)P(x,y,z,t)$$ (1)

where K is an interaction constant on the order of 10^–9 Pa^–1 in a 0.9% NaCl solution (Jossinet et al 1998, 1999, Lavandier et al 2000a, 2000b).

2.1.3. AE signal and internal Joule heat imaging

According to the lead field theory, a pair of electrodes is called a lead. A lead's sensitivity distribution is called its lead field. The measured voltage of lead i with lead field ${\mathbf{J}}_i^{\mathrm{L}}={\mathbf{J}}_i^{\mathrm{L}}(x,y,z)$ due to a current density distribution J^I(x, y, z) can be written as (Malmivuo and Plonsey 1995, He 2004)

$$V^i={\int \int \int }_V\rho {\mathbf{J}}_i^{\mathrm{L}}•{\mathbf{J}}^{\mathrm{I}}\mathrm{d}v.$$ (2)

When the voltage is measured by the same lead injecting current, ${\mathbf{J}}_i^{\mathrm{L}}={\mathbf{J}}^{\mathrm{I}}(x,y,z)∕I$, where I is the injected current. By combining equation (1) and (2) we can obtain equation (3) on the surface voltage measurement with existing AE effect:

$$V={\int \int \int }_V{\rho }_0{\mid \mathbf{J}(x,y,z)\mid }^2∕I\mathrm{d}x\mathrm{d}y\mathrm{d}z-{\int \int \int }_{V}KP{\rho }_0{\mid \mathbf{J}(x,y,z)\mid }^2∕I\mathrm{d}x\mathrm{d}y\mathrm{d}z.$$ (3)

As shown in equation (3), the voltage V consists of two terms. One is called the base current field voltage $V_{\mathrm{B}}={\int \int \int }_V{\rho }_0{\mid \mathbf{J}(x,y,z)\mid }^2\mathrm{d}x\mathrm{d}y\mathrm{d}z$. The other term, generated due to the local resistivity change with the ultrasound pulse wave packet traveling in the imaging object (AE effect), is called AE signal

$$V_{\mathrm{AE}}={\int \int \int }_{V}KP{\rho }_0(x,y,z){\mid \mathbf{J}(x,y,z)\mid }^2∕I\mathrm{d}x\mathrm{d}y\mathrm{d}z$$ (4)

Since V_B is a low frequency signal that has the same frequency with the injected current, and V_AE is a high frequency signal generated due to the local resistivity change by ultrasound (AE effect) , they can be easily separated using band-pass filter (Witte et al 2007, Olafsson et al 2008, 2009a).

The ultrasound pressure as a spatial and temporal function can be expressed as (Olafsson et al 2008)

$$P(x,y,z,t)=P_{0}b(x,y,z)a\left(t\right)$$ (5)

where P₀ is the amplitude of ultrasound pressure wave, b(x, y, z) is the beam pattern, and a(t) describes the ultrasound waveform. For a focused spherical acoustic source, the expression of beam pattern can be found in the literatures (Lucas et al 1982, Chen et al 1993). The -6dB beam diameter can be estimated as B_D =1.02Fc /(fD), where F is the focal length, c is the sound velocity in the media, f represents the central frequency and D is the diameter of the transducer element. Commonly, beam diameter of B_D = 0.4–3mm, ultrasound pulse duration time of T_d =0.1–1us are used. Substituting equation (5) into equation (4), the measured AE signal can be further expressed as

$$V_{AE}={\int \int \int }_{V_p}K{P}_{0}b(x,y,z)a\left(t\right)w(x,y,z)∕Idxdydz$$ (6)

where w(x, y, z) =ρ₀(x, y, z)|J(x, y, z)|² is the internal Joule heat density distribution in the imaging object. V_p represents the volume of the ultrasound wave packet which travels inside the imaging object volume and is determined by the focus beam profile and pulse duration. Generally in the process of ultrasound transmission, strong acoustic pressure only exists in the pulse wave packet V_p and the pressure level outside this region is very weak. Therefore V_AE mainly depends on the local phenomena within the ultrasound wave packet and the AE effect outside this region can be neglected. As shown in equation (6), with the setup of UJHT, the internal Joule heat distribution is time/spatially encoded in the collected AE signal. Another fact we can notice from equation (6) is that, as the AE signal is a volume integration, if a(t) is an oscillatory bipolar pulse, the integration and therefore the AE signal will be very small in a uniform media. If the local Joule heat density w(x, y, z) is near constant inside the ultrasonic pulse wave packet, the acquired AE signal would be close to zero. In order to enhance AE signal, it is therefore desirable to use unipolar ultrasonic pulse for the ultrasound Joule heat tomography. On the other hand, researchers have shown that it is possible to generate unipolar pressure pulse in the transducer's focal zone using piezoelectric transducer (van der Pauw 1966, Brown and Weight 1974, Foster and Hunt 1978, Eisenmenger and Haardt 1982, Maeda et al 1991, Holé and Lewiner 1996, 1998). The merit of unipolar ultrasound pulse was also previously introduced by Lavandier et al (2000a) to quantitatively assess ultrasound induced resistance change in saline solution and will be used in the present 3D UJHT approach.

In order to apply the unipolar pressure pulse to all the spatial locations inside the imaging object, we can do a 3D raster scanning, i.e. move the transducer to make its focusing point scan over the imaging object point by point and collect the AE signal V_AE generated around time t′ that corresponds to the transducer's focusing point where the unipolar pressure pulse maintains (Holé and Lewiner 1996, 1998, Lavandier et al 2000a). By doing this we can also reduce the influence of propagation procedure, beam strength variance and attenuation to the Joule heat imaging results. Assuming the ultrasound pulse packet arrivals at position z = 0 at t = 0, so z=ct. When the focusing point of the transducer moves to (x′, y′, z′) the corresponding AE signal can be further expanded as

$$V_{AE}(x^{\prime },y^{\prime },z^{\prime },t^{\prime })={\int }_{c{t}^{\prime }-c{T}_{\mathrm{d}}∕2}^{c{t}^{\prime }+c{T}_{\mathrm{d}}∕2}\iint KP(x-x^{\prime },y-y^{\prime },z-z^{\prime },t^{\prime })w(x,y,z)∕I\mathrm{d}x\mathrm{d}y\mathrm{d}z$$ (7)

In addition, we can assume w(x, y, z) to be constant in the small focusing region V_f to simplify the calculation because it is very small as compared to the object size. We can then rewrite equation (7) as following

$$V_{AE}(x^{\prime },y^{\prime },z^{\prime },t^{\prime })\approx I^{-1}Aw(x^{\prime },y^{\prime },z^{\prime })$$ (8)

where $A={\int \int \int }_{V_f}KP(x-x^{\prime },y-y^{\prime },z-z^{\prime },t^{\prime })dxdydz$. The internal Joule heat density distribution can then be estimated by

$$w(x^{\prime },y^{\prime },z^{\prime })\approx I{V}_{AE}(x^{\prime },y^{\prime },z^{\prime },t^{\prime })∕A$$ (9)

From the imaging perspective, equations (8) and (9) describe the forward problem and inverse problem of the 3D ultrasound Joule heat imaging method using unipolar ultrasound pulses, respectively. These relationships are simple and clear in physics. In addition, the inverse solution as shown in equation (9) can be considered as a voxel/pixel based inversion and complicated matrix inversion as in UCSDI can be avoided. Furthermore, we should note that the total Joule heat consumption is actually I[V_B +(–V_AE)], and IV_AE(x′, y′, z′, t′) represents the local Joule heat perturbation caused by the ultrasound pulse packet. Another advantage of the proposed UJHT method as compared to UCSDI is that it doesn't need a priori knowledge of the resistivity distribution of the imaging object and the corresponding lead field, which is not trivial to obtain for individual imaging object.

2.2. Computer Simulation

Computer simulations were conducted to demonstrate the feasibility of the proposed imaging approach and to evaluate its performance for possible tumor scan application. Comparison of ultrasound Joule heat tomography using unipolar ultrasound pulses with that using bipolar ultrasound pulses is also shown.

2.2.1. Simulation of electric fields and the induced Joule heat distributions

In our simulations, all applied electric fields and established Joule heat distributions were computed by mean of the finite element method (FEM) using COMSOL software (Comsol AB, Stockholm, Sweden). Equation (4) was used to simulate the measured AE signals. The AE interaction constant K, whose value is on the order of 10^-9Pa^-1, changes very little among NaCl solutions of different concentrations and has the same order in cardiac tissues (Li et al 2010). In addition, the influence caused by the variation of this constant in tissues on UCSDI imaging can generally be neglected (Olafsson et al 2009a and 2009b, Wang et al 2010). We therefore assume the AE interaction coefficient is uniform and K = 1 × 10 ^–9 Pa^-1 in all the present simulation studies.

2.2.2. Simulation of ultrasound field

In the present study, we adopted the method provided by Holé and Lewiner (1996, 1998) to simulate the unipolar ultrasound pulse. The simulated unipolar pulse waveform is shown in figure 2(a). For comparison, a typical oscillatory bipolar ultrasound pulse used in the simulation is illustrated in figure 2(b) as well (frequency 2MHz, bandwidth1.176 MHz, duration 0.85us) (Cobbold 2006). For the beam pattern of a focused spherical acoustic source, a numerically convergent solution with error smaller than 0.01 was provided by Chen et al (1993). This method was adopted to calculate the beam pattern in our simulations. Figure 2(c) illustrated the beam pattern calculated for a focused acoustic source with frequency of 2MHz, diameter of the active element D=3 cm, the radius of curvature R₀=15cm, beam diameter B_D = 0.4 cm.

Figure 2.

(a) Normalized unipolar ultrasound pulse waveform. (b) Bipolar ultrasound pulse used in the simulation for comparison. (c) The beam pattern generated by a focused spherical transducer with central frequency of 2MHz, diameter of the active element to be 3cm and the radius of curvature of 15cm.

2.2.3. UJHT Simulation models

As illustrated in figures 3(a) and 5(a), 3D models of 10×10×10 cm³ are built as the imaging object in the present simulation study. In not noted, sinusoidal current with 200 Hz frequency and 20mA peak amplitude was injected by two disk-shaped electrodes with radii of 1.5 cm, thickness of 0.02cm attached on the two opposite surfaces of the imaging object. The origin of the coordinate system was set at the center of the object. The ultrasound transducer emitted ultrasound pulse from the bottom of the object. Nonconductive deionized water was assumed to be the coupling media between the ultrasound transducer and the object. As many previous studies found out that cancerous tissues exhibit higher conductivity in comparison to surrounding normal tissues (Surowiec et al 1988, Silva et al 2000, Hu et al 2011), we use small balls of elevated conductivity to simulate tumors in these 3D models.

Figure 3.

(a) Diagram of the numerical phantom of a piece of homogeneous tissue with embedded tumors; (b) the target Joule heat density distribution calculated using FEM (normalized for each slice in order to visualize the detailed contrast); (c) reconstructed Joule heat density distribution using the proposed method. The three slices locate at x=-3.0, 0.0 and 3.0 cm, respectively. (d) The target Joule heat density distribution at x=0.0 cm as a 2D image; (e) the true Joule heat density distribution (FEM simulated) at x=0.0 cm without tumors inside; (f) reconstructed Joule heat density distribution at x=0.0 cm using the proposed UJHT method with unipolar pulses; (g) Joule heat density imaging by UJHT using bipolar ultrasound pulses; (h) line profiles along the white dotted lines in images (d)–(g).

Figure 5.

Simulated phantom for testing imaging tumors in tissue with layered structure (a); (d) the target Joule heat distribution in a slice at x=0.0cm established in the phantom; (e) the Joule heat distribution without tumors inside; (f) imaging result by the proposed ultrasound Joule heat tomography using unipolar ultrasound pulses; (b) imaging result using bipolar ultrasound pulses; (c) line profiles of these images (d), (e), (b) and (f) (line profile of imaging result using bipolar ultrasound pulses has been enlarged and shifted for comparison).

(1) Tumor model with homogeneous background tissue

As shown in figure 3(a), the first tumor model has a homogeneous background tissue which has conductivity value of 1.76 S/m. This homogeneous background part is the simplest model of the normal tissue surrounding those tumors. Seven small balls with elevated conductivity value of 7.04 S/m were built in the model to simulate tumors of different size (Surowiec et al 1988, Silva et al 2000, Gao and He 2008). These simulated tumors are labeled with number 1 to 7 and have radii of 0.4, 0.5, 0.6, 0.7, 0.8, 0.5, 0.6 cm respectively.

(2) Tumor model with inhomogeneous background tissue

In order to assess if the proposed tomography technique can be used to scan tumors embedded in inhomogeneous background tissue, we built another model with the background tissue having a conductivity distribution of 0.199 + 100(x² + y² + z²)^0.5 S/m. In this model, the background tissue conductivity varies from 0.199 to 8.859 S/m. The peak amplitude of the injected sinusoidal current is 45mA here. Other setups in this model are the same as that in the first model.

(3) Tumor model with layered structure in the background tissue

As illustrated in figure 5(a), to simulate an even more complicated surrounding background tissue, this model contains several layered structures which are commonly seen in biomedical tissues. The four simulated tumors have same radius of 0.06 cm. These tumors are labeled with number 1 to 4 and have different conductivity values of 0.6, 0.9, 1.2 and 1.5 S/m, respectively. Among those layered structures, there are two pairs of thick slabs and two pairs of thin slabs. The two pairs of thick slabs have conductivity value of 0.15 S/m, while the two pairs of thin slabs have conductivity value of 0.6 S/m. The conductivity value of the background media is 0.3 S/m. The peak value of the injected sinusoidal current is 12mA for this investigation.

2.2.4. Impact of the ultrasound beam diameter, pulse duration and noise level

In order to systematically estimate the imaging performance under different imaging parameters such as the ultrasound beam diameter, pulse duration and different noise level, we built a model similar to the first model as shown in figure 3(a) but with only five tumors inside. The four tumors in the outer loop have the same radius of 0.5 cm, located at (0,-3,0), (0,0,3) (0,3,0) and (0,0,-3)cm, respectively. The radius of the central tumor is 1 cm. We first fixed T_d = 0.125us and tested the imaging performance with the ultrasound beam diameter changing from 0.04 to 0.6cm under noise free condition. The imaging performance as T_d increases from 0.135 to 2.703 us, with fixed B_D = 0.071cm was also tested. After that Gaussian noise with zero mean was added to the simulated AE signals to investigate the impact of noise. In the present study, the standard deviation of the AE signal was referred as signal strength. The ratio of the standard deviation of the added Gaussian noise over signal strength was denoted as noise index I_n.

2.2.5. Imaging performance with tumors of different size, conductivity contrast and at different position

In order to test how the imaging performance is influenced by the tumor parameters, we conducted simulation studies with tumor models that have different tumor sizes, electrical conductivity contrast (tumor to surrounding tissue), and at different position using the aforementioned model under noise free condition. Here we first changed radius of the outer four tumors from 0.1 to 1cm to test the imaging spatial resolution for tumor scan application under electrical conductivity contrast of 4 (the ratio between the conductivity of the tumor tissue and that of the surrounding tissue). In the second test, the conductivity value of the tumors was changed from 1.5 to 6 times larger than the surrounding tissue to test the imaging performance under different conductivity contrast. After that, position dependency of the proposed imaging method was tested by changing positions of those tumors in x axis from 0 to 3.0 cm with electrical conductivity contrast of 4 and radius of the outer four tumors of 0.5 cm. In all these tests, we fixed B_D = 0.1cm , T_d = 0.125us and no measurement noise was considered. The radius of the central tumor is twice of the outer four.

2.2.6. Quantitative evaluation of the imaging performance

In the present study, the relative error (RE) and correlation coefficient (CC) between the imaging result and the target Joule heat distribution, defined as equation (10) and (11), respectively, are used to quantitatively assess the performance of the proposed ultrasound Joule heat tomography technique.

$$CC=\frac{\sum _n\sum _m(H_{mn}^{\widehat{}}-\overline{H^{\widehat{}}})(H_{mn}^0-\overline{H^0})}{\sqrt{\left[\sum _n\sum _m{(H_{mn}^{\widehat{}}-\overline{H^{\widehat{}}})}^2\right]\left[\sum _n\sum _m{(H_{mn}^0-\overline{H^0})}^2\right]}}$$ (10)

$$RE=\frac{\Vert H^{\widehat{}}-H^0\Vert }{\Vert H_0\Vert }=\frac{\sqrt{\sum _n\sum _m{(H_{mn}^{\widehat{}}-H_{mn}^0)}^2}}{\sqrt{\sum _n\sum _m{\left(H_{mn}^0\right)}^2}}$$ (11)

where H⁰, H^{^} are the target and reconstructed Joule heat distribution, respectively.

2.2.7. Quantitative evaluation of the imaging method for tumor detection

In order to estimate the potential of the proposed method for possible tumor imaging applications in a more quantitative way, we define a Joule heat contrast (JHC) as in equation (12). As in equation (12), the JHC is defined as the ratio of the mean value of Joule heat density inside the tumor (S1) to the mean value in the surrounding circular area whose outer radius is 1.5 times of the tumor's radius (S2), as illustrated in figure 3(d).

$$JHC=\frac{average\{H_{mn}\mid H_{mn}\in S1\}}{average\{H_{mn}\mid H_{mn}\in S2\}}$$ (12)

Joule heat contrast without tumors in that location (JHC^-), which is calculated by assuming the tumors don't exist at that location as illustrated in figure 3(e), is used as a reference value for comparison. Joule heat contrast of the true Joule heat distribution (FEM simulated) established when performing the proposed imaging (JHC*) is used to assess the theoretical contrast of the technique for tumor scan application. Joule heat contrast of the imaging result by the proposed tomography (JHC⁺) is used to assess its contrast to scan tumor in tissue.

For convenience of narrative, CC⁺ denotes correlation coefficient (CC) between the imaging result with tumors inside and the true Joule heat distribution with tumors inside (FEM simulated), while CC^- represents CC between the imaging result with tumors inside and the true Joule heat distribution without tumors (FEM simulated). The difference CC⁺ – CC^– and JHC⁺-JHC^- are used to evaluate the discrimination of the proposed ultrasound Joule heat tomography for tumor scan application. The greater the two difference values CC⁺ – CC^– and JHC⁺-JHC^- are, the better the ability to distinguish tumor.

3. Results

3.1. Imaging tumors in homogeneous tissue

Figure 3 shows the simulation results of imaging tumors in homogeneous background tissue using the proposed ultrasound Joule heat tomography technique with B_D = 0.2cm, T_d = 0.125us under noise free condition. Figures 3(b) and (d) display the target Joule heat distribution, while figures 3(c) and (f) show the imaging result by the proposed method. Figure 3(e) illustrates the true Joule heat distribution without tumors inside. Comparing the imaging result and the target Joule heat density, it is clear that the imaging result is consistent with the target Joule heat distribution. By comparing the imaging result, which displays the tumors clearly as in figure 3(f), with the map of true Joule heat distribution without tumors inside, as in figure 3(e), the tumors buried in normal tissue can be clearly identified by the proposed imaging method. For comparison, the imaging result using bipolar ultrasound pulses is shown in figure 3(g). It is obvious that the proposed method gives better imaging result than that using bipolar ultrasound pulses under the same conditions. Figure 3(h) shows the line profiles of these mappings for a direct comparison. As summarized in table 1, using the proposed method, correlation coefficient (CC) of 0.9814 and relative error (RE) of 0.0682 for Joule heat imaging can be obtained in this simulation investigation. Good tumor detection using Joule heat contrast can be achieved with CC⁺ – CC^– = 0.2714, JHC⁺ – JHC^– = 0.9167, as shown in this simulation study.

Table 1.

Performance evaluations of the ultrasound Joule heat tomography in the three examples.

	CC	RE	CC+-CC-	JHC+-JHC-
example 1	0.9814	0.0682	0.2714	0.9167
example 2	0.9945	0.0403	0.6907	0.5332
example 3	0.9801	0.1084	0.1014	0.6329

3.2. Imaging tumors in inhomogeneous tissue

Figure 4 depicts the simulation results of imaging tumors in inhomogeneous background tissue using the proposed technique with B_D = 0.2cm, T_d = 0.125us under noise free condition. By comparing the target Joule heat distribution, shown in figure 4(a), with imaging result by the proposed method, shown in figure 4(c), we can see that the imaging result agrees well with the target Joule heat distribution. From the imaging result, the tumors can also be seen clearly in this case. Comparison among the target Joule heat distribution with tumors inside as in figure 4(a), the true Joule heat distribution without tumors inside as in figure 4(b), and the imaging result as in figure 4(c), we can see that the Joule heat images reconstructed using the proposed technique can provide valuable information for tumor detection even in a inhomogeneous tissue background. For a direct comparison, figure 4(d) displays the line profiles of these images. Correlation coefficient (CC) of 0.9945 and relative error (RE) of 0.0403 for Joule heat density imaging can be achieved as listed in table 1. CC⁺ – CC^– = 0.6907, JHC⁺ – JHC^– = 0.5332 in this case.

Figure 4.

(a) The target Joule heat density distribution in a slice at x=0.0cm generated in the model of a piece of inhomogeneous tissue with tumors embedded; (b) the Joule heat density distribution without tumors inside; (c) imaging result by the proposed method; (d) line profiles of these images along the white dotted lines.

3.3. Imaging tumors in tissue with layered structure

Figure 5 shows the simulation results of imaging tumors in tissue with layered structure using the proposed technique with B_D = 0.2cm, T_d = 0.125us under noise free condition. Comparison among the target Joule heat distribution as in figure 5(d), the true Joule heat distribution without tumors inside as in figure 5(e), and the imaging result by the proposed method as in figure 5(f), shows that the reconstructed Joule heat image is consistent with the target Joule heat distribution and may be used for tumor detection in tissue environment even with complicated layered structure. Comparing the imaging result using bipolar ultrasound pulses, displayed in figure 5(b), with imaging result using unipolar ultrasound pulses, shown in figure 5(f), it is obvious that ultrasound Joule heat tomography using unipolar ultrasound pulses gives better imaging result than that using bipolar ultrasound pulses. The reconstructed Joule heat distribution using bipolar ultrasound pulses is quite flat in most uniform regions and with noisy spikes at the boundaries between different conductivity regions due to its coupling with bipolar ultrasound wave. Figure 5(c) displays the line profiles of these images (d), (e), (b) and (f) for a direct comparison. As seen in table 1, correlation coefficient (CC) of 0.9801 and relative error (RE) of 0.1084 for Joule heat density imaging can be obtained. CC⁺ – CC^– = 0.1014, JHC⁺ – JHC^– = 0.6329 can be achieved in this case.

3.4. Impact of ultrasound parameters and noise level

Figure 6 depicts the investigation results of the impact of beam diameter to the proposed ultrasound Joule heat tomography in which no measurement noise was considered. As shown in figures 6(a) and (b), as beam diameter increases the correlation coefficient decreases slowly and the relative error increases slowly. However, from figures 6(c) and (d), we can see that desirable Joule heat density imaging with CC>0.92 and RE<0.14 can still be achieved even when the beam diameter B_D increases to 0.6cm. These tumors can still be seen clearly from the mapping with B_D = 0.6cm.

Figure 6.

The correlation coefficient (CC) (a) and relative error (RE) (b) between the imaging results and the true Joule heat distribution versus ultrasound beam diameter, (c) the imaging result by the proposed method for a slice at x=0.0cm when beam diameter reaches 0.6cm, (d) the target Joule heat distribution and imaging result along the line cross the center of the phantom as marked by the white dotted line in (c).

Figure 7 shows the investigation results of the impact of ultrasound pulse duration to the proposed technique under noise free condition. The curves of CC and RE between the imaging result and the target Joule heat density as a function of the pulse duration are displayed in figures 7(a) and (b), respectively. With the increase of pulse duration, the correlation coefficient has a slight decrease and the relative error has a slight increase. However, as shown in figures 7(c) and (d), excellent imaging result with CC>0.96 and RE<0.09 can still be obtained even pulse duration increased to 2.7us.

Figure 7.

The correlation coefficient (a) and relative error (b) between the imaging results and the target Joule heat distribution versus ultrasound pulse duration, (c) the imaging result by the proposed method for a slice at x=0.0cm when pulse duration reaches 2.703us, (d) the line profiles of target Joule heat distribution and imaging result along the line as marked by the white dotted line in (c).

Figure 8 depicts the investigation results of the impact of measurement noise to the proposed technique with B_D = 0.071cm, T_d = 0.125us. As displayed in figures 8(a) and (b), with the noise index increases, the correlation coefficient decreases gradually and the relative error increases slowly. However, as illustrated in figures 8(c) and (d), good Joule heat density imaging with CC>0.92 and RE<0.13 can still be achieved even noise index increased to 0.4. These tumors can still be seen clearly from the imaging with the maximum noise index. The proposed imaging technique is not vulnerable to noise due to the fact that it is almost a direct determined mapping between collected AE signals and the Joule heat distribution, as shown in equation (9).

Figure 8.

The correlation coefficient (a) and relative error (b) between the imaging results and the true Joule heat distribution versus noise index, (c) the imaging result by the proposed method for a slice at x=0.0cm when noise index reaches 0.4, (d) the target Joule heat distribution and imaging result along the line cross the center of the phantom as marked by the white dotted line in (c).

3.5. Performance under tumors of different size, conductivity contrast and at different position

Figure 9 shows the investigation results of spatial resolution of the proposed imaging method to scan tumors. As displayed in figures 9(a) and (b), respectively, CCs (CC⁺s) are all greater than 0.98 and REs are all smaller than 0.09 for various tumors size varies from 0.1 to 1.0cm. From figure 9(c), we can see that the Joule heat contrast without tumors inside (JHC^-) changes very little and is almost equal to 1 under different tumor sizes. The Joule heat contrast of the true Joule heat distribution (FEM simulated) established when performing the imaging (JHC*) increases near linearly with the increase of tumor size. The Joule heat contrast of the imaging result by the proposed tomography (JHC⁺) agrees well with JHC* for different tumor sizes. As shown in figures 9(a) and (c), as tumor size increases, CC⁺-CC^- increases gradually and reaches maximum at tumor size of 0.75cm and then decreases slowly, while JHC⁺-JHC^- increases near linearly. As shown in figures 9(d)-(i), the tumors of different sizes can be seen clearly from these images. The small tumors of radius 0.1cm can also be detected accurately by the proposed imaging method.

Figure 9.

Performances of the proposed imaging method to scan tumors of different sizes. (a) CC^-, CC⁺ and CC⁺-CC^- (in black, blue and red color, respectively) for different tumor size. (b) The relative error versus tumor size. (c) shows Joule heat contrast without tumors inside JHC^-(black), Joule heat contrast of established target Joule heat distribution with tumors inside JHC*(blue), Joule heat contrast of imaging result by the proposed method JHC⁺(red) and JHC⁺-JHC^- (green) under different tumor size. (d)-(f) show imaging results for the outer tumors of radii 0.1, 0.55 and 1.0cm, respectively (the center tumor of radii 0.2, 1.1 and 2.0cm, respectively). (g)-(i) illustrate corresponding line profiles of the imaging results and target Joule heat distributions along the line cross the center of the phantom (marked by the white dotted lines in (d), (e) and (f), respectively).

Figure 10 depicts the investigation results of performance of the proposed ultrasound Joule heat tomography for scanning tumors of different electrical conductivity contrast. As shown in figures 10(a) and (b), CCs (CC⁺s) keep high values greater than 0.98 and REs keep low values smaller than 0.07 under a range of electrical conductivity contrast 1.5-6.0. From figure 10(c), we can see that JHC^- is almost equal to 1. JHC* increases gradually with the increase of conductivity contrast, and then reaches a stable value close to 2. JHC⁺ is close to JHC* under different electrical conductivity contrast. From figures 10(a) and (c), it can be seen that as conductivity contrast increases CC⁺-CC^- increases near linearly, while JHC⁺-JHC^- increases gradually and then reaches a stable value close to 1 after conductivity contrast is greater than 4. As illustrated in figures 10(d)-(i), the tumors can be identified clearly from these imaging results under the range of electrical conductivity contrast 1.5-6.0 even as low as 1.5. Excellent ultrasound Joule heat density imaging with good performance to distinguish tumor can be obtained under such range of electrical conductivity contrast.

Figure 10.

Performances of the proposed ultrasound Joule heat tomography under different electrical conductivity contrast (tumor to surrounding normal tissue). (a) CC^-, CC⁺ and CC⁺-CC^- versus electrical conductivity contrast. (b) The relative error (RE) versus electrical conductivity contrast. (c) shows Joule heat contrast without tumor inside JHC^-, Joule heat contrast of established target Joule heat distribution JHC*, Joule heat contrast of the imaging result JHC⁺ and JHC⁺-JHC^- versus electrical conductivity contrast. (d)-(f) show imaging results for tumors of electrical conductivity contrast 1.5, 3.5 and 6.0, respectively. (g)-(i) show corresponding line profiles of the imaging results and established target Joule heat distributions along the line cross the center of the phantom, marked by the white dotted line in (d)-(f), respectively.

Figure 11 shows the investigation results of position dependency of the proposed technique for tumor scan application. For different positions, CCs (CC⁺s) are all greater than 0.98, as displayed in figure 11(a), and REs are all smaller than 0.07, as displayed in figure 11(b). As shown in figure 11(c), JHC^- is equal to 1 at each location. JHC* decreases gradually with the position becomes more and more close to the electrode along x axis. At each location, JHC⁺ is consistent with JHC*. As seen in figures 11(a) and (c), as position of the tumors becomes more and more close to the electrode, both CC⁺-CC^- and JHC⁺-JHC^- decrease gradually. Figures 11(d)-(i) illustrate the imaging results for tumors located at different positions. The tumors can be seen clearly from these imaging results even located at x=3.0cm. Theoretically, the position dependency is caused by lead filed distribution and it can be improved by increasing the area of the two flat electrodes, as illustrated in figures 11(j)-(l), or moving the two electrodes to make their projection cover the tumor.

Figure 11.

The investigation results of the position dependency of the proposed imaging method for tumor scan application. (a) CC^-, CC⁺ and CC⁺-CC^- for tumors located at different position along x axis. (b) The relative error (RE) for tumors located at different position. (c) shows Joule heat contrast without tumor inside JHC^-, Joule heat contrast of established target Joule heat distribution JHC*, Joule heat contrast of imaging result JHC⁺ and JHC⁺-JHC^- for tumors located at different position. (d)-(f) show imaging results for tumors located at x=0.02, 1.6 and 3.0cm, respectively. (g)-(i) show corresponding line profiles of the imaging results and established target Joule heat distributions along the line as marked by the white dot line in (d)-(f), respectively. (j) and (k) display the imaging results for tumors located at x=3.0cm with disk-shaped electrodes of radius 2.0 and 2.6 cm, respectively, and (l) shows their line profiles along the white dotted lines as comparison to the target Joule heat distributions.

These investigation results indicate that the proposed imaging technique has good spatial resolution, desirable contrast and the ability to detect small simulated tumors in the deep regions for tumor scan application.

4. Discussion

It is of importance to image electrical properties of biological tissues. Recently hybrid imaging modality combining electric conductivity contrast and ultrasonic resolution through the acousto-electric (AE) effect has attracted considerable interest (Zhang and Wang 2004, Witte et al 2007, Olafsson et al 2009a, Sumi 2009, Yang et al 2011). Acousto-electric tomography (AET) has been proposed first for high resolution bioimpedance imaging (Zhang and Wang 2004). However, 3D electrical impedance imaging based on AE effect is difficult to develop since the AE signal depends on applied current density distribution and measurement lead field, whereas the established current density distribution and measurement lead field depend on the unknown resistivity distribution of imaging object in turn. Next, ultrasound current source density imaging (UCSDI) was proposed for mapping small biological current and confirmed by recent experiment studies (Witte et al 2007, Olafsson et al 2008, 2009a and 2009b, Wang et al 2011). However, for current density reconstruction in UCSDI, assumption of the conductivity distribution and measurement lead fields is required (Olafsson et al 2008, Yang et al 2011), which limits its applications. In the present study, we propose a novel 3D ultrasound Joule heat tomography (UJHT) approach using unipolar ultrasound pulses. This technique doesn’t require priori knowledge of conductivity distribution, shape of the whole body of the imaging object and calculation of lead field. Therefore it is more flexible and convenient. It is worthy to notice that the frequency bands of testing current and inherent biological current should not be overlapped with that of the AE signal for easy separating the AE signal from testing current and inherent biological current. Fortunately, the frequency band of AE signal is about 0.01-several MHz, i.e., a high frequency signal, while the biological signal is generally below KHz, i.e., a low frequency signal (Olafsson et al 2008, 2009a). So it is generally easy to extract the AE signal by using a band pass filter. From the theory of the AE signal generation, we can see that temperature is not directly related to the AE voltage measurement which is used to estimate the Joule heat density distribution. However, we should note that the increased tissue temperature caused by ultrasound exposure may affect the tissue's electrical conductivity (Miyakawa and Bolomey 1995). For noninvasive or nondestructive evaluation, the increased temperature is an error source in the mapping of the internal Joule heat distribution. However, for living tissues, a perfusion may mitigate such a problem (Sumi 2009). Besides, although the acoustic properties of soft tissues and liquid are similar and their influences to AE effect based imaging can be neglected, the impact of the reflecting and shielding effect at the interface between soft tissue and bone will be considerable (Cobbold 2006, Olafsson et al 2009a). The shielding effect will block the propagation of ultrasound pulse and the reflecting effect would make the imaging have a dim non-superimposed mirror image. So performing the ultrasound Joule heat scanning should avoid shielding effect of bone. Unipolar ultrasound pulses is used instead of common bipolar ultrasound pulses to enhance AE signal and obtain a direct mapping of internal Joule heat distribution due to bipolar oscillations tend to average localized phenomena leading to a loss of AE signal (Holé and Lewiner 1996, 1998, Lavandier et al 2000a, Yang et al 2011). As shown in figures 3 and 5, it is obvious that the proposed method gives better imaging result than that using bipolar ultrasound pulses under the same conditions. As special equipment for ultrasound Joule heat tomography system, only ultrasound equipment, a simple electric circuit, filter and a synchronizing circuit for data acquisition are used, the cost of the technique should be low compared with MREIT (Gao and He 2008).

By comparing the imaging result, which displays the tumors clearly, with the standard map of Joule heat distribution without tumors inside, doctor can easily identify the tumors buried in normal tissue from the ultrasound Joule heat tomography. We should note that the proposed ultrasound Joule heat tomography is not a linear mapping of conductivity distribution. In this proof of concept study, for simplicity, we use CC⁺-CC^- and JHC⁺-JHC^- to quantitatively assess the performance of the proposed imaging method for possible tumor scan application, better evaluation method is still needed. How to use the Joule heat tomography for tumor detection in clinic still needs further studies, but it is beyond the scope of the present study. In future specific application, the design and arrangement of electrodes would need carefully design for improving imaging performance. It is worthy to note that previous experimental studies indicated AE effect has good sensitivity to detect safe small current (2 – 4mA/ cm²) injected into tissue (Witte et al 2006, 2007, Olafsson et al 2006, 2009a). So measurable AE signal would be available for ultrasound Joule heat tomography with safe current injection. Of course, how to fit the technique for specific application still needs further research, but it is beyond the scope of the present study.

In summary, we have proposed a novel 3D ultrasound Joule heat tomography (UJHT) method using unipolar ultrasonic pulses. Utilizing local Joule heat perturbation to specially designed unipolar ultrasound pulses, we are able to obtain 3D Joule heat imaging with electrical conductivity contrast. Comparing to ultrasound current source density imaging, this tomography doesn't require priori knowledge of conductivity distribution and lead fields. Then the capability and performance of the proposed imaging method with simulation settings for possible tumor scan applications is investigated by a series of computer simulations, with consideration of ultrasound beam diameter, pulse duration, measurement noise level, size of tumors, electrical conductivity contrast and position dependency. Computer simulation results show that 3D Joule heat imaging can be obtained using the proposed method. Tumors can be seen clearly from the imaging results utilizing Joule heat contrast. Our simulation results also show that the proposed method has desirable performance and capability under a certain systems parameter setups and application environment. Further experimental study will be needed in order to fully test the capability and performance of the proposed 3D ultrasound Joule heat tomography.