Modulus Reconstruction from Prostate Ultrasound Imagesusing Finite Element Modeling

Zhennan Yana, Shaoting Zhanga, S. Kaisar Alamb, Dimitris N. Metaxasa, Brian S. Garrac,Ernest J. Feleppab

aRutgers, the State University of New Jersey, Computer Science Department, NJ, USA;bRiverside Research, New York, NY, USA;

cFood and Drug Administration, Silver Spring, MD, USA

ABSTRACT

In medical diagnosis, use of elastography is becoming increasingly more useful. However, treatments usuallyassume a planar compression applied to tissue surfaces and measure the deformation. The stress distributionis relatively uniform close to the surface when using a large, flat compressor but it diverges gradually alongtissue depth. Generally in prostate elastography, the transrectal probes used for scanning and compression arecylindrical side-fire or rounded end-fire probes, and the force is applied through the rectal wall. These make itvery difficult to detect cancer in prostate, since the rounded contact surfaces exaggerate the non-uniformity ofthe applied stress, especially for the distal, anterior prostate.

We have developed a preliminary 2D Finite Element Model (FEM) to simulate prostate deformation inelastography. The model includes a homogeneous prostate with a stiffer tumor in the proximal, posterior regionof the gland. A force is applied to the rectal wall to deform the prostate, strain and stress distributions canbe computed from the resultant displacements. Then, we assume the displacements as boundary condition andreconstruct the modulus distribution (inverse problem) using linear perturbation method.

FEM simulation shows that strain and strain contrast (of the lesion) decrease very rapidly with increasingdepth and lateral distance. Therefore, lesions would not be clearly visible if located far away from the probe.However, the reconstructed modulus image can better depict relatively stiff lesion wherever the lesion is located.

Keywords: Biopsy Guidance, Elastography, Finite Element Modeling, Inverse Problem, Modulus Reconstruc-tion, Strain, Stress, Prostate Cancer, Ultrasound

1. INTRODUCTION

Prostate cancer (PCa) is a worldwide health crisis. There were about 240,890 new cases and about 33,720 deathsdue to PCa in the United States in 2011.1 And an estimated 241,740 new cases of PCa and 28,170 deaths areexpected in 2012.2 Current trends reveal that about 97% of PCa are diagnosed in men 50 years of age andolder.2 The 5-year relative survival rate for local or regional stages at the time of diagnosis approaches 100%,but only 29% for distant stages.2 This makes early detection, enabling localized treatment, a very high priority.

Due to the elastic properties of soft tissues, physicians routinely use palpation to detect lesions in the breast aswell as in the prostate. This is because that cancerous tissues are often stiffer than surrounding tissue.3 However,manual palpation is of limited use for deep lesions or when elasticity differences are small. Furthermore, it doesn’tprovide quantitative information, and the utility depends on physician skill and experience.

To overcome these limitations, ultrasound-based elasticity-imaging methods4–18 and other elastography meth-ods19 have been developed. While conventional B-mode ultrasonograms convey information regarding acoustic-scattering properties, ultrasonic elasticity images convey information regarding tissue elastic properties, primarilyYoung’s modulus. Ultrasonic elasticity imaging methods can be grouped into two broad categories:17 1) imagingtissue responses to dynamic loading or vibration,9,18 and 2) imaging tissue responses to quasi-static loading or

{zhennany, shaoting, dnm}@cs.rutgers.edu

compression.15 Vibration-based methods typically use Doppler-type methods to sense and depict internal mo-tion. Vibration can be applied more easily and can employ a freehand device. However, the resolution may berelatively poor. The second group of methods (elastography) produces high-resolution elastograms (elastographicimages) that depict local tissue deformation under compression.

Prostate cancers typically arise in the peripheral zone of the prostate, the glandular tissue region, whichis adjacent to the rectal wall and wraps around the sides of the prostate. Although not directly exposed toa transrectal probe, the prostate is separated from it only by a few millimeters of rectal mucosa and smoothmuscle. In addition, the prostate is adjacent to the pubic arch, a convenient ”anvil” for compression for prostateelastography.

We propose to diagnose prostate cancer consisting of the following steps: (1) compress prostate using tran-srectal probes, (2) measure displacement field from the pre- and post-compression ultrasonic signals, and (3)compute the elastic modulus (here the Young’s modulus) by solving an inverse problem numerically.20–22

The strain and displacement fields can be measured from ultrasound images under applied forces. And it ispossible to reconstruct the spatial distribution of relative Young’s modulus of biological tissue, and we observedthat modulus contrast is higher than strain contrast in the prostate.

This work is designed to make significant improvements in the detection of PCa by developing advancedultrasonic elastographic methods for sensing and imaging cancerous foci in the prostate based on their relativestiffness.

2. METHODS

Fig. 1 illustrates our model. We developed a 2D finite element model to simulate prostate deformation inelastography, and used linear perturbation method21 to reconstruct the modulus distribution from the resultantdisplacements. We assumed that tissue is a linear isotropic incompressible elastic material in plane-strain state.

Ultrasound Probe Rectal Wall

Prostate

Probe pushes the rectal wall

Stiff Lesion

Figure 1. Illustration of the simulation of prostate deformation in elastography.

2.1 Simulation modeling

As seen in Fig. 2, we constructed finite element model for prostate and surrounding environment referring to theB-mode image. The model size was 9 cm × 7 cm; prostate was about 3.1 cm × 2.2 cm; and tumor was modeledas a cylinder with a diameter of 0.44 cm. Using constrained Delaunay triangulation method,23 we generatedmore than 1000 points and more than 2000 triangles in the mesh. The displacements of upper boundary werefixed. The initial Young’s modulus for tissues were: rectal wall (30 kPa), background (15 kPa), prostate (30 kPa),tumor (90 kPa). And the poisson’s ratios for all tissues were 0.495.

We applied an external force on rectal wall (here we used displacement boundary conditions moved rectalwall upward 0.3 cm), and used FEM method to simulate the deformation (forward problem).

C010BAC1a10: B−mode (dB)

(a) Prostate B-mode

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Figure 2. 2-D Finite Element model based on a typical ultrasound image. (a) Prostate B-mode image; (b) Prostatefinite-element model, blue mesh is prostate, red mesh is tumor, yellow mesh is rectal wall, green mesh is the background.

2.2 Characterization of tumor position

Since cancerous tissues are generally stiffer than normal tissues,3 lesions should be visible in the strain images.But in endocavity elastography, the compressors (transrectal probes) are small and not flat. So the stress is notuniformly distributed and decreases quickly with increasing depth and lateral distance. If the lesion was locatedat different positions, the contrast of lesion in strain images may be different. To illustrate the relation betweenthe position of lesion and the contrast in strain field, we designed the following characterization simulations.

1. Fix the x-coordinate of tumor’s position (at the horizontal center of prostate), move it vertically, andcompute the contrast of magnitude of strain between inside and outside the tumor. See fig. 3a.

2. Fix the y-coordinate of tumor’s position (where the max contrast appears in simulation 1), move ithorizontally, and compute the contrast of magnitude of strain as well. See fig. 3b.

The contrast of strain is defined as eq. 1, where min strain is minimum absolute value of strain inside thetumor and max strain is maximum absolute value of strain surrounding the tumor (outside the tumor and insidethe prostate).

contrast =max strain−min strain

max strain(1)

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Figure 3. Characterization models: (a) Lesion is moved vertically; (b) Lesion is moved horizontally.

2.3 Modulus reconstruction

We first computed the displacements from the model (fig. 2b). From these displacements, we used the algorithmshown in fig. 4 to reconstruct the elasticity spatial distribution (inverse procedure). Here we set initial Young’s

modulus E0 = 30 kPa and poisson ratio = 0.495 for every element in FEM, error threshold ε = 10−4, andmaximum iteration max iter = 1000. Because for a realistic simulation noise has to be defined for ultrasoundsignals, we will improve the simulation in the future work.

Start

Make guess of initial

modulus distribution 𝐸0,

𝑘 = 0

Compute 𝐸𝑟𝑟𝑜𝑟 =

𝑈 − 𝑈𝑘 2

𝐸𝑟𝑟𝑜𝑟< 𝜖

End

Update 𝐸𝑘

by linear perturbation

method

Update stiffness

matrix in FEM

Solve forward

problem, and get

displacement 𝑈𝑘

Current

𝑘 <𝑚𝑎𝑥_𝑖𝑡𝑒𝑟

Yes Yes

No

No

𝑘 = 𝑘 + 1

Figure 4. Iterative algorithm for inverse problem.

3. RESULTS

Fig. 5 shows the simulation results of the forward procedure using Finite Element Method, including the plotof model after deformation as well as strain and stress distributions. In this case, we can estimate the tumor’sposition from the strain and stress distribution.

X

Y

Figure 5. Simulation results for prostate deformation (forward procedure). Top left is the original model, top center isresult after deformation, top right is maximum stress image; bottom left is strain-X plot, bottom center is strain-Y plot,bottom right is shear-strain image.

3.1 Characterization results

Fig. 6 shows the result of simulation 1 described in section 2.2, in which Y axes represent the Y-coordinate(depth) of the center of tumor. In fig. 6a X axis represents the minimum absolute strain inside the tumor inX and Y directions, in fig. 6b it represents the strain contrast in decibel (dB). Because the Y limit of prostate

is from 1.6 to 3.8, we set the Y limit of center of tumor from 2.13 to 3.24. We observe that the minimumstrain is becoming smaller as the depth of tumor increasing, while the maximum contrast appears at Y = 2.75approximately.

0.026 0.028 0.03 0.032

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Figure 6. With different vertical position, (a) Minimum strain inside the tumor; (b) Strain contrast between inside andoutside of the tumor.

Fig. 7 shows the result of simulation 2 when fixing the depth at 2.75, in which X axes represent the X-coordinate of the center of tumor. Because the X limit of prostate is from 3.12 to 6.24 and it’s smaller alongY = 2.75, we set the X limit of center of tumor from 3.96 to 5.27. The Y axes represent the minimum strain andstrain contrast respectively. We can see that the largest minimum strain appears at the middle of the X limit,and the maximum contrast appears at X = 4.5 approximately. From these two simulations, we conclude thatthe maximum strain contrast is observed when the tumor is at (4.5, 2.75) approximately.

4 4.5 50.0245

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horizontal position of tumor4 4.5 5

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horizontal position of tumor

(b) Strain Contrast

Figure 7. With different horizontal position, (a) Minimum strain inside the tumor; (b) Strain contrast between inside andoutside of the tumor.

3.2 Modulus reconstruction results

In this section, we show the reconstruction results vs. strain images when the tumor is located at (4.5, 2.75)as discussed in last section. Because the inverse problem is ill-posed,20,21 we used regularization tools.24 Infig. 8a, plots show the absolute values of strain-Y along vertical and horizontal lines passing the center of tumor.The strain contrast is good enough to distinguish the tumor from prostate. In fig. 8b, plots show the values ofreconstructed Young’s modulus along the same two lines. The difference between prostate and tumor is moresignificant.

If defining the contrast of tumor (on the two lines and limit in X:3.12 − 6.24, Y:1.6 − 3.8) as maximumdifference between values inside and around tumor divided by maximum value (similar to eq. 1), the contrast

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Figure 8. Strain-Y image vs. reconstructed modulus image. Plots show values along vertical and horizontal lines passingthe center of tumor. (a) Absolute values of strain-Y; (b) Values of reconstructed Young’s modulus.

in strain-Y image is about −12dB, while in modulus image the contrast is about −4dB. It is clear that thedetection of tumor is much easier and more reliable in reconstructed modulus distribution.

To illustrate the advantage of reconstructed result, fig. 9 shows the strain-Y vs. reconstructed modulusdistribution with original Young’s modulus 40 kPa for tumor (closer to the modulus of prostate). In this case,the tumor is clearly visible by modulus reconstruction, but not in the strain image.

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Figure 9. Strain-Y image vs. reconstructed modulus image. Tumor has Young’s modulus 40 kPa.

4. CONCLUSIONS AND FUTURE WORK

In this work, we have developed a 2D finite element model to simulate prostate deformation and to reconstructmodulus spatial distribution by linear perturbation method. This work seeks to advance the prostate elastographymethod for cancer detection. The results indicated that reconstructed modulus images were superior to strainimages for depicting tumors in the low-strain regions. With higher contrast, the reconstructed elasticity couldbe very useful to guide biopsies and improve the disease diagnoses. The framework developed in this work willallow us to evaluate and compare other elasticity imaging modalities in prostate cancer detection, guidance, andmonitoring.

We will include ultrasound simulation (using FIELD II25) with attenuation to make the simulation morerealistic in the next step. Because 2D model cannot represent practical scenario adequately, and we already havesome experience in 3D deformable models such as mass-spring,26 laplacian surface27,28 and meshless model,29

we would like to investigate 3D model for this application. Future work will aim at practical approaches for 3Dprostate elastography, we will acquire 3D scanning data by controlled translation or pivoting of the scan head,

develop transrectal ultrasound probe assemblies to produce ”well-behaved” deformations, and then construct3D stiffness assessments covering the entire gland. If 3D model could generate enough precise results, theimplementation of 3D prostate elastography would be very helpful in clinical manner.

ACKNOWLEDGMENTS

This work was supported in part by the Institutional Research and Development fund of Riverside Research. Weare grateful to Yang Yu(Rutgers) for the help with analyzing improvement of numerical computation.

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