A feasibility study of magnetic resonance electrical impedance tomography for prostate cancer detection

Abstract

Magnetic resonance electrical impedance tomography (MREIT) is an imaging technique that reconstructs the conductivity distribution inside the subject using magnetic flux density or current density measurements acquired by a magnetic resonance imaging (MRI) system. Since the primary prostate cancer diagnostic method, prostate biopsy, has limited accuracy in cancer diagnosis and malignant tissues have shown significantly different electrical properties from normal or benign tissues, MREIT has potential application in prostate cancer detection. The feasibility of utilizing MREIT in detecting prostate cancer was evaluated via a series of well-designed computer simulations in the present study. MREIT techniques with three different electrode configurations (external, trans-rectal, and trans-urethral electrode arrays) and two different reconstruction algorithms (J-substitution algorithm and harmonic B_z algorithm) were successfully developed. The performance of different MREIT techniques were evaluated and compared based on the imaging accuracy of the reconstructed conductivity distribution in the prostate. Without the presence of noise, the external MREIT achieves a better imaging accuracy than the two endo-MREIT (trans-rectal and trans-urethral) techniques, while the trans-urethral MREIT achieves the best imaging accuracy in noisy environments. We also found that the J-substitution reconstruction algorithm consistently offered better imaging accuracy than the harmonic B_z algorithm. When Gaussian distributed random noise with a standard deviation of 0.25 nT was added, the relative errors (RE) between the reconstructed and target conductivity distributions inside the prostate were observed to be 14.18% and 17.35% by the trans-urethral MREIT with the J-substitution and harmonic B_z algorithms respectively. The lower REs of 9.64% and 11.17% were achieved respectively when the standard deviation of noise was reduced to 0.05 nT. The simulation results demonstrate the feasibility of applying MREIT for prostate cancer detection.

1. Introduction

Statistical data from the American Cancer Society suggest that prostate cancer is by far the most commonly diagnosed cancer among American men and remains the second leading cause of cancer death in men (Siegel et al 2011). Prostate-specific antigens (PSAs), a primary tool for prostate cancer screening, unfortunately lack sufficient specificity to definitively diagnose the disease and instead act as surrogate markers for clinical testing (Borsic et al 2009a, 2010). Currently, ultrasound-guided biopsy is considered as the ‘gold standard’ for definitive diagnosis of the prostate cancer. While these biopsies yield very accurate information regarding the area they sample, they are performed at discrete points and provide no information on the adjacent tissues (Shini et al 2011).

A number of studies have examined the differences between normal and cancerous prostatic tissues and suggested that electric properties, conductivity and permittivity, were very valuable for the early detection and subsequent treatment of prostate cancer (Halter et al 2007, 2008, 2009 a, b). Electrical impedance tomography (EIT), which has been commonly used to reconstruct cross-sectional images of resistivity distribution inside human body (Abascal et al 2011, Solà et al 2011), could be possibly utilized for prostate cancer detection. Considering the fact that the prostate is located deep within the pelvis and surrounded by complicated tissues, EIT techniques with endo-electrode arrays have been developed specifically for prostate imaging in order to make the injected current more easily reach the imaging target (Jossinet et al 2002, 2004, 2006, Borsic et al 2009b, 2010 and Wan et al 2010). A probe printed with an electrode array is inserted into the human body through the urethra or rectum in order to reach as close to the prostate as possible. This configuration presents an open-domain problem, where the current density used for imaging concentrates mainly near the probe surface and decays rapidly with increasing distance from the probe. The particular challenge associated with the open-domain problem together with the ill-posed characteristics of the inverse problem in the EIT make current EIT techniques suffer from poor spatial resolution and accuracy in prostate cancer detection.

In magnetic resonance electrical impedance tomography (MREIT), magnetic flux density distributions in the 3-dimentional (3D) space of the subject are captured with a magnetic resonance imaging (MRI) system and used to reconstruct the conductivity images (Oh et al 2003, Nam et al 2008, Liu et al 2009, Seo et al 2011). In this case, the ill-posedness of traditional EIT is eliminated, since a huge number of measurements can be acquired not only over the surface but also inside the subject. Experimental MREIT research has been conducted extensively on biological tissue phantoms (Oh et al 2005), postmortem subjects (Kim et al 2007), in vivo animal models (Muftuler et al 2006, Kim et al 2008, 2011) and in vivo human models (Kim et al 2009, Meng et al 2012). Kim et al (2010) have applied MREIT to image the conductivity distribution of a male canine pelvis and the reconstructed conductivity images effectively distinguish different organs of the pelvis including the prostate, sacrum, rectum, and surrounding muscles. The feasibility of utilizing MREIT for the purpose of prostate cancer detection, however, has not been examined.

In the present study, the feasibility of utilizing MREIT to detect prostate cancer was assessed via well-designed computer simulations. The performance of MREIT with different electrode configurations and different reconstruction algorithms was evaluated and the results were compared in terms of imaging accuracy in prostate cancer detection.

2. Method

2.1. Forward problem

Let Ω be a bounded and electrically conductive domain in R³ with boundary Γ. The conductivity distribution inside Ω is σ and assumed to be positive. When small currents are injected into Ω through electrodes in an isotropic, linear, and conductive medium, the electromagnetic field satisfies the following Poisson’s equation and Neumann boundary conditions:

$$\{\begin{array}{cc}\hfill \nabla ·(\sigma \nabla u)=0\hfill & \hfill \text{in}\mathrm{\Omega }\hfill \\ \hfill \sigma \frac{\partial u}{\partial \mathbf{n}}=j_I\hfill & \hfill \text{on}\mathrm{\Gamma }\hfill \end{array}$$ (1)

where u is the voltage due to the injection current, n is the unit outward normal vector, and j_I is the normal component of the boundary current density induced by the injection current. The electric field E and the current density J can be calculated as:

$$\mathbf{E}=-\nabla u$$ (2)

$$\mathbf{J}=\sigma \mathbf{E}$$ (3)

The relation between current density J and magnetic flux density B is given by the B iot-Savart law as,

$$\mathbf{B}(\mathbf{r})=\frac{{\mu }_0}{4\pi }\underset{{\mathrm{R}}^3}{\int }\mathbf{J}(\mathrm{r}\text{'})\times \frac{\mathbf{r}-\mathbf{r}\text{'}}{|\mathbf{r}-\mathbf{r}\text{'}{|}^3}dr\text{'}$$ (4)

For the complicated human body, we cannot analytically solve the boundary value problem (1); a numerical calculation method is needed instead. In the present study, the finite element (FE) method was adopted to solve the forward problem of MREIT.

2.2. Inverse problem

Given the measured magnetic flux density or current density, the calculation of the inner conductivity distribution is referred to as the inverse problem of MREIT. It has been verified that, in order to achieve the unique solution for the MREIT inverse problem, at least two currents need to be injected into the subject (Kwon et al 2002, Ïder et al 2003). Let J⁰ and V⁰ represent the measured current density and voltage difference between the two current injection electrodes induced by the injected current I⁰. J¹ and V¹ are the corresponding measurements induced by another injected current I¹. The J-substitution algorithm (Kwon et al 2002, Liu et al 2009) and harmonic B_z algorithm (Oh et al 2003), two widely used algorithms for solving the MREIT inverse problem based on current density and magnetic flux density measurements respectively, may then be applied. The steps below were adopted in the J-substitution algorithm to reconstruct the conductivity distribution inside the subject iteratively:

• Step 1. Let the iteration number m=0 and assume an initial homogeneous conductivity σ⁰.

• Step 2. For given conductivity σ^2m+q (q=0, 1 corresponding two injection currents), compute $u_m^q$ by solving the following forward problem given by

  $$\{\begin{array}{cc}\hfill \nabla ·({\sigma }^{2m+q}\nabla u_m^q)=0\hfill & \hfill \text{in}\mathrm{\Omega }\hfill \\ \hfill {\sigma }^{2m+q}\frac{\partial u_m^q}{\partial \mathbf{n}}=j_{I^q}\hfill & \hfill \text{on}\mathrm{\Gamma }\hfill \end{array}$$ (5)
• Step 3. Update the conductivity based on the following strategy:

  $${\sigma }^{2m+q+1}=\frac{\left|{\mathbf{J}}^q\right|}{|\nabla u_m^q|}\frac{V_{{\sigma }^{2m+q}}^q}{V^q}$$ (6)

  where $V_{{\sigma }^{2m+q}}^q$ is the voltage difference between the two current injection electrodes when the conductivity distribution is σ^2m+q.

• Step 4. If |σ^2m+q+1 −σ^2m+q|<ε (given tolerance), stop; otherwise, go back to Step 2 with q=q+1 when q=0 or m=m+1 and q=0 when q=1.

The harmonic B_z algorithm was described as follows:

• Step 1. Let the iteration number m=0 and assume an initial homogeneous conductivity σ₀.

• Step 2. Solve the forward problem below, for q=0 and 1.

  $$\{\begin{array}{cc}\hfill \nabla ·({\sigma }^m\nabla u_{m+1}^q)=0\hfill & \hfill \text{in}\mathrm{\Omega }\hfill \\ \hfill {\sigma }^m\frac{\partial u_{m+1}^q}{\partial \mathbf{n}}=j_{I^q}\hfill & \hfill \text{on}\mathrm{\Gamma }\hfill \end{array}$$ (7)
• Step 3. Calculate the gradient of conductivity $\left[\begin{array}{c}\frac{\partial {\sigma }^{m+1}}{\partial x}\hfill \\ \frac{\partial {\sigma }^{m+1}}{\partial y}\hfill \end{array}\right]$ equation (8).

  $$\left[\begin{array}{cc}\frac{\partial u_{m+1}^0}{\partial y}\hfill & -\frac{\partial u_{m+1}^0}{\partial x}\hfill \\ \frac{\partial u_{m+1}^1}{\partial y}\hfill & -\frac{\partial u_{m+1}^1}{\partial x}\hfill \end{array}\right]\times \left[\begin{array}{c}\frac{\partial {\sigma }^{m+1}}{\partial x}\hfill \\ \frac{\partial {\sigma }^{m+1}}{\partial y}\hfill \end{array}\right]=\frac{1}{{\mu }_0}\left[\begin{array}{c}{\nabla }^2{B}_z^0\hfill \\ {\nabla }^2{B}_z^1\hfill \end{array}\right]$$ (8)
• Step 4. Apply the layer potential equation below to each imaging slice S and compute conductivity distribution.

  $${\sigma }^{m+1}(\mathbf{r})=-\underset{s}{\int }{\nabla }_{\mathbf{r}\text{'}}\varphi (\mathbf{r}-\mathbf{r}\text{'})·\nabla {\sigma }^{m+1}(\mathbf{r}\text{'})d\mathbf{r}\text{'}+\underset{\partial s}{\int }{n}_{\mathbf{r}\text{'}}·{\nabla }_{\mathbf{r}\text{'}}\varphi (\mathbf{r}-\mathbf{r}\text{'}){\sigma }^{m+1}(\mathbf{r}\text{'})dl\mathbf{r}\text{'}$$ (9)

  where $\varphi (\mathbf{r}-\mathbf{r}\text{'})=\frac{1}{2\pi }\text{log}|\mathbf{r}-\mathbf{r}\text{'}|$.

• Step 5. If |σ^m+1 −σ^m|<ε (given tolerance), stop; otherwise, go back to Step 2 with m=m+1.

For more details regarding these two reconstruction algorithms, please refer to Kwon et al (2002) and Oh et al (2003).

3. Computer simulation

In the present study, a homogeneous cylinder was used to mimic the pelvis in which an upside-down chestnut was included to mimic the prostate. The cylinder was constructed with a radius of 13 cm and a height of 9 cm. The chestnut with coronal, axial, and sagittal dimensions of 4 cm, 3 cm, and 2 cm, respectively (Jossinet et al 2006), was placed 3cm from the bottom and 2.7 cm off from the axis of the cylindrical pelvis. The conductivity of the pelvis was set to 0.24 S/m (Rush et al 1963, Geddes et al 1967). Two spherical anomalies with radii of 0.5 cm and 0.3 cm respectively were placed 2.2 cm from the bottom of the prostate to mimic the cancerous tissues. The conductivity values of normal and malignant tissues were set at 0.15 S/m and 0.1 S/m, respectively (Borsic et al 2010). The FE meshing of the model and the magnetic flux density calculation were then carried out by ANSYS 13.0.

In order to avoid the “inverse crime” (Lionheart 2004, Wirgin 2004) in the computer simulation, different mesh models with different element sizes were utilized for the forward and inverse simulations involved in this study. The model for the forward simulation was meshed into fine hexahedral elements with equal size of 2×2×2 mm³ which results in 760,500 elements and 789,406 nodes. The model for the inverse simulation was segmented into three parts. The middle part, which contained the prostate, was meshed into fine hexahedral elements with equal sizes of 2×2×2 mm³. The top and the bottom parts of the model, each with a 3 cm height, were meshed into coarse hexahedral elements with equal sizes of 2×2×10 mm³. There were 354,900 elements and 377,542 nodes in total in the inverse simulation model. The part including the prostate was meshed with fine elements in order to achieve high spatial resolution in prostate, while the top and bottom parts were meshed with coarse elements to reduce the computational effort.

In practice, either the magnetic flux density measured by a MRI scanner or the current density calculated from the measured magnetic flux density via Ampere’s law is used to reconstruct the conductivity distribution inside the object. In the present simulation, the magnetic flux density B was first simulated by solving the forward problem with an assumed target conductivity distribution. Various levels of Gaussian distributed random noise were then added to the calculated magnetic flux density to mimic the noise-contaminated magnetic flux density B_n, and the noisy current density J_n was calculated from B_n. Finally, both the noisy magnetic flux density B_n and the noisy current density J_n were used as the ‘measurements’ to reconstruct the conductivity distribution inside the pelvis. The reconstructed and target conductivity distributions inside the prostate were compared to evaluate the performance of the MREIT in prostate cancer detection. Sadleir et al (2005) reported that a carefully designed MREIT study will be able to reduce noise levels below 0.25 and 0.05 nT at main magnetic field strengths of 3 and 11T, respectively. Therefore, Gaussian distributed random noise with a standard deviation of 0.25 and 0.05 nT was added to the simulated magnetic flux density distribution to mimic the noisy environment within a MR scanner.

The correlation coefficient (CC) and relative error (RE) between the reconstructed and target conductivity distributions were calculated to quantitatively assess the performance of the MREIT for prostate cancer detection. The CC and RE are defined as follows:

$$CC[\sigma (i),{\sigma }^{*}(i)]\frac{\sum _{i=1}^N\sigma (i)·{\sigma }^{*}(i)}{{[\sum _{i=1}^N{\sigma }^2(i)]}^{1/2}·{[\sum _{i=1}^N{\sigma }^{*2}(i)]}^{1/2}}$$ (10)

$$RE[\sigma ,{\sigma }^{*}]=\frac{\Vert {\sigma }^{*}-\sigma \Vert }{\Vert {\sigma }^{*}\Vert }\times 100\%$$ (11)

where σ^* and σ are the target and reconstructed conductivity distributions, respectively.

3.1. External MREIT

Four electrodes with equal sizes of 5 cm × 9 cm and thicknesses of 1 cm were uniformly distributed over the surface of the cylinder. The entity model, forward FE model and cross-section view of the inverse FE model for the external MREIT study are shown in figure 1. In the present study, two bipolar rectangular currents (Gao et al 2006, 2008) were injected from E1 to E3 and from E2 to E4, respectively. A number of research results reported that a current density of 2 A/m² is safe for the human body below 3 kHz (Giland et al 2007, Dalziel 1972). The primary frequency component of the bipolar current pulses used in MREIT, meanwhile, is on the order of 10 Hz (Arpinar et al 2012). Following this safety standard, the injection current strength was set as 4.65 mA for external MREIT, resulting in a maximum current density of 2 A/m² in the pelvis.

Figure 1.

(a) Entity model, (b) forward FE model and (c) cross-section view of the inverse FE model for external MREIT.

Figure 2 presents the simulated B_z and the magnitude of J distributions, induced by the injected current via E1 and E3, over the plane crossing through the center of malignant tissues. It can be seen that both B_z and the magnitude of J decrease gradually from the surface of the pelvis to the center. Figure 3 shows the change of CCs and REs with the process of iteration for external MREIT. In this figure, the J and H refer to the J-substitution and harmonic B_z algorithms, respectively, while the following number specifies the standard deviation of the noise in B_z. As we can see, the best reconstruction results were achieved by the J-substitution and harmonic B_z algorithms after 7 and 4 iterations respectively and further iterations do not continue to improve results. It can also be seen from figure 3 that the J-substitution algorithm is superior to the harmonic B_z algorithm in terms of reconstruction accuracy. To better visualize the conductivity reconstruction results in the prostate, the cross-section of the target and reconstructed conductivity distributions achieved by the J-substitution and harmonic B_z algorithms in the prostate are presented in figure 4. We can discern the location and size of the malignant tissues from the reconstructed conductivity mapping when the standard deviation of noise is 0.05 nT, but this becomes difficult to do when the standard deviation of noise increases to 0.25 nT.

Figure 2.

(a)Simulated z direction magnetic flux density distribution B_z and (b) magnitude distribution of current density J for external MREIT.

Figure 3.

Convergence behaviors of J-substitution and harmonic B_z algorithms for external MREIT. (a) CC and (b) RE curves at different noise levels.

Figure 4.

Target and reconstructed conductivity distributions achieved by the external MREIT. (a)Target conductivity distribution; (b1)–(b3) reconstructed conductivity distributions achieved by J-substitution algorithm at different noise levels; (c1)–(c3) reconstructed conductivity distributions achieved by harmonic B_z algorithm at different noise levels.

3.2. Trans-rectal MREIT

The probe for the trans-rectal MREIT was designed to be 9 cm in length, 1.3 cm in radius, and have a spherical tip at the distal end to facilitate rectal insertion (Borsic et al 2009a). Three electrodes with equal sizes of 1.2 cm × 7.7 cm and thicknesses of 0.2 cm were printed over the surface of the probe. The simulated prostate was placed 0.2 cm away from the surface of the electrode to account for the thickness of the rectal wall (Tucker et al 2004, Balthazar et al 1979). The entity model, forward FE model, and cross-section view of the inverse FE model are shown in figure 5. Only the probe, the electrodes, and the prostate, rather than the whole pelvis, are illustrated in the figure for the purpose of clarity. The forward and inverse FE models of the trans-rectal MREIT are identical to those of the external MREIT in terms of the element size, number of elements, and nodes.

Figure 5.

(a) Entity model, (b) forward FE model and (c) cross-section view of the inverse FE model for trans-rectal MREIT.

Bipolar rectangular currents were injected from E1 to E2 and E1 to E3. Following the same safety standard, the magnitude of the injection current was set at 2.66 mA in the trans-rectal MREIT which resulted in a maximum current density of 2 A/m² in the pelvis. Figure 6 shows the distributions of the simulated B_z and the magnitude of J when the current was injected from E1 to E2. Results show that the magnitude of the current density decreases dramatically with the increasing distance from the surface of the probe.

Figure 6.

(a)Simulated z direction magnetic flux density distribution B_z and (b) magnitude distribution of current density J for trans-rectal MREIT.

Figure 7 presents the reconstruction results of the trans-rectal MREIT technique in terms of the CCs and REs achieved by the J-substitution and harmonic B_z algorithms. Much like the external MREIT, the best reconstruction results were achieved by the J-substitution and harmonic Bz algorithms after 7 and 4 iterations, respectively. Figure 8 shows the cross-section of the target and reconstructed conductivity distributions in the prostate achieved by the J-substitution and harmonic B_z algorithms.

Figure 7.

Convergence behaviors of J-substitution and harmonic B_z algorithms for trans-rectal MREIT. (a) CC and (b) RE curves at different noise levels.

Figure 8.

Target and reconstructed conductivity distributions achieved by the trans-rectal MREIT. (a)Target conductivity distribution; (b1)–(b3) reconstructed conductivity distributions achieved by J-substitution algorithm at different noise levels; (c1)–(c3) reconstructed conductivity distribution achieved by harmonic B_z algorithm at different noise levels.

3.3. Trans-urethral MREIT

Since the membranous urethra, the narrowest segment of urethra, may dilate up to 0.8 cm in diameter (Kawashima et al 2004), the trans-urethral MREIT probe was designed to be 7 cm in length and 0.6 cm in diameter. Four electrodes with equal sizes of 0.2 cm × 6.7 cm and thicknesses of 0.1 cm were printed over the surface of the probe. The probe and all of attached electrodes were surrounded by the simulated prostate since the probe would be placed into the human body through the urethra. In order to limit the maximum current density of 2 A/m² in the prostate, two bipolar rectangular currents with magnitudes of 1.57 mA were injected from electrodes E1 to E3 and E2 to E4.

Figure 9 shows the entity model, forward FE model, and cross-section view of the inverse FE model for trans-urethral MREIT. The forward and inverse FE models of the trans-urethral MREIT are identical to those of the external MREIT and trans-rectal MREIT in terms of the element size, number of elements, and nodes. Figure 10 shows the distributions of the simulated B_z and the magnitude of J, while figure 11 compares the reconstruction results of the trans-rectal MREIT technique in terms of the CCs and REs achieved by the J-substitution and harmonic B_z algorithms. Optimal reconstruction results can be achieved by J-substitution and harmonic B_z algorithms after 5 and 3 iterations, respectively. Figure 12 shows the cross-section of the target and reconstructed conductivity distributions in the prostate achieved by the J-substitution and harmonic B_z algorithms.

Figure 9.

(a) Entity model, (b) forward FE model, and (c) cross-section view of the inverse FE model for trans-urethral MREIT.

Figure 10.

Simulated z direction magnetic flux density distribution B_z and (b) magnitude distribution of current density J for trans-urethral MREIT.

Figure 11.

Convergence behaviors of J-substitution and harmonic B_z algorithms for trans-urethral MREIT. (a) CC and (b) RE curves at different noise levels.

Figure 12.

Target and reconstructed conductivity distributions achieved by trans-urethral MREIT. (a)Target conductivity distribution; (b1)–(b3) reconstructed conductivity distribution achieved by J-substitution algorithm at different noise levels; (c1)–(c3) reconstructed conductivity distribution achieved by harmonic B_z algorithm at different noise levels.

3.4 Comparison and analysis of the simulation results

Simulation results show that both external MREIT and endo-MREIT techniques can successfully reveal the internal conductivity distribution inside the prostate in noiseless cases and that external MREIT offers slightly better reconstruction accuracy than the endo-MREIT. However, the performance of either trans-rectal or trans-urethral MREIT techniques is better than that of the external MREIT in noisy environments.

With the same noise level, the trans-urethral MREIT achieves the best reconstruction results among the three MREIT techniques, as is indicated by it having the largest CC and the smallest RE. By comparing the reconstruction results achieved by the J-substitution algorithm and harmonic B_z algorithm, we can see that the J-substitution algorithm offers better reconstruction performance than the harmonic B_z algorithm at identical noise levels.

4. Conclusion and discussion

In the present study, the feasibility of utilizing MREIT to detect cancer in the small and deep-located human prostate was evaluated. Different electrode configurations and reconstruction algorithms for MREIT techniques were also examined and compared in terms of their imaging accuracy to determine the most appropriate MREIT technique for prostate cancer detection. A FE method was employed to solve the forward problem and the J-substitution algorithm and harmonic B_z algorithm were employed to resolve the inverse problem in MREIT.

The computer simulation results suggest that the reconstruction results of the external MREIT are superior to those achieved by either of the two endo-electrode MREIT techniques when there is no noise. When noise with a standard deviation of 0.25 nT was considered in the external MREIT simulation, however, the REs between the reconstructed and target conductivity distributions achieved by the J-substitution and harmonic B_z algorithms were as high as 25.65% and 31.70%, respectively. Therefore, it is necessary to develop an efficient technique to remove noise from the measurements in order to successfully apply an external MREIT to clinical prostate cancer detection. As a comparison, lower REs of 14.18% and 17.35% were achieved by the J-substitution and harmonic B_z algorithms respectively in trans-urethral MREIT when noise with a standard deviation of 0.25 nT was considered. The promising simulation results achieved by the trans-urethral MREIT demonstrate the feasibility of applying MREIT for prostate cancer detection.

Figures 7 and 11 show that the CCs between the reconstructed and target conductivity distributions within the prostate increase monotonously, while the REs decrease monotonously during the iteration process in noisy environments in endo-MREIT. The changing nature of the CCs and REs indicates that the reconstruction accuracy of the endo-MREIT techniques increases as the iteration number increases. However, we also found that the conductivity reconstruction accuracy of the pelvis deteriorated gradually when the iteration number was increased. This phenomenon occurs because the electrodes in the endo-MREIT were placed in the central location of the pelvis, and the current density induced by the injected currents concentrated mainly in the region surrounding the probe and decayed rapidly with the increasing distance (Jossinet et al 2009, Borsic et al 2009b, 2010). This current dissipation can also be seen from figures 5 and 8. The weak measurements of magnetic flux density and the calculated current density in regions of the pelvis that are far from the probe may be submersed by noise, therefore the reconstruction accuracy deteriorates with an increase in the iteration number.

Conversely, the magnetic flux density in the areas close to the probe can be measured with a relatively higher signal-to-noise ratio (SNR) and hence better conductivity reconstruction accuracy can be achieved. In the trans-rectal and trans-urethral MREIT, the probes were placed close to the prostate to achieve better reconstruction accuracy compared to the external MREIT, in which the electrodes were placed over the skin surface of the pelvis. The current-injection pathways determined by the electrode configuration greatly influence the magnetic flux density and current density distributions in the prostate, which in turn affects the reconstruction results. Further investigations are needed to optimize the electrode configuration and the resulting current-injection pathways in order to maximize and uniformize the signal strength within the prostate.

Reconstruction results achieved by using the two reconstruction algorithms were compared in terms of accuracy. Results showed that the J-substitution algorithm consistently gave us better results than the harmonic B_z algorithm with the same electrode configuration and noise level, which is consistent with the results published by other groups (Arpinar et al, 2012 and Eyüboğlu et al, 2010). The reason for this phenomenon is that the second-order differentiation of B_z is required in the harmonic B_z algorithm, while only a first-order differentiation is required in the J-substitution algorithm. However, three components of magnetic flux density are needed in the J-substitution algorithm and the requirement of 3D rotating the subject in the MRI scanner makes this less practical. Since the sensitivity matrix method (SMM) is more robust to noise (Arpinar et al 2012) and only one component of magnetic flux density measurement is needed to reconstruct the conductivity distribution, this method is more practical in the future clinical application.

To date, 9 mA (Kim et al 2009) has been reported as the highest injected current via 8.0 cm×6.0 cm×0.63 cm Carbon-Hydrogel electrodes without producing painful sensations in the human leg and an injection current of 3 mA has been the lowest employed to successfully image the conductivity distribution of the human knee (Jeong et al 2010) in in vivo MREIT studies. In the present study, highly conductive copper electrodes were used in all three MREIT techniques. As the electrode sizes in the three MREIT techniques were significantly different, the maximum current density in the pelvis induced by the injected current, rather than the magnitude of injection current, was adopted as a measurement for safety evaluation. Giland et al (2007) reported a human sensation threshold at the current density of 2.2 A/m² on the surface of the scalp in a human head EIT. IEEE standard requires a safe current density range of 2–15 A/m² for hand grasp area (IEEE 2002) and a range of 2–11 A/m² was specified by Dalziel (1972). The current density of 2 A/m² adopted in the present study is at the lower end of both safety ranges reported above. It is worth noting that the copper electrode used in this study usually has strong edge singularity (Song et al 2011) as can be seen from figure 2. Carbon-Hydrogel electrodes or optimal geometry recessed electrodes (Song et al 2011) which are supposed to produce more uniform current density distributions should be investigated in future studies.

It is possible that current carrying cables in the trans-rectal/urethral MREIT system could generate extra magnetic flux, thereby creating systematic errors in the measurements. In practice, these possible systematic errors can be eliminated by placing the carrying wires parallel to the main magnetic field direction to make the resultant magnetic flux density equal to zero. Another option would be to calculate out the extra magnetic flux density produced by the carrying wires and subtract it from the measurements to eliminate its effects (Birgül et al, 2003). One limitation of the present study is that the computer simulations were conducted based on a simplified pelvic model which consists of a cylindrical model for the pelvis and an upside-down chestnut for the prostate. In future studies, the proposed MREIT techniques will be further tested based on a realistic geometric pelvic model which will be generated from subjects’ high-resolution MR images (Zhang et al 2009a, b, 2010).

In summary, different MREIT techniques with different electrode configurations and different reconstruction algorithms were developed for prostate cancer detection and their performance in terms of conductivity reconstruction accuracy were compared in a well-designed computer simulation. The trans-urethral MREIT offered the best performance in terms of imaging accuracy and stands for the most appropriate MREIT technique for prostate cancer detection. Simulation results demonstrate the feasibility of utilizing the MREIT in detecting cancer in the small and deep-located prostate.