Analytical derivation of elasticity in breast phantoms for deformation tracking
Abstract
Purpose
Patientspecific biomedical modeling of the breast is of interest for medical applications such as image registration, image guided procedures and the alignment for biopsy or surgery purposes. The computation of elastic properties is essential to simulate deformations in a realistic way. This study presents an innovative analytical method to compute the elastic modulus and evaluate the elasticity of a breast using magnetic resonance (MRI) images of breast phantoms.
Methods
An analytical method for elasticity computation was developed and subsequently validated on a series of geometric shapes, and on four physical breast phantoms that are supported by a planar frame. This method can compute the elasticity of a shape directly from a set of MRI scans. For comparison, elasticity values were also computed numerically using two different simulation software packages.
Results
Application of the different methods on the geometric shapes shows that the analytically derived elongation differs from simulated elongation by less than 9% for cylindrical shapes, and up to 18% for other shapes that are also substantially vertically supported by a planar base. For the four physical breast phantoms, the analytically derived elasticity differs from numeric elasticity by 18% on average, which is in accordance with the difference in elongation estimation for the geometric shapes. The analytic method has shown to be multiple orders of magnitude faster than the numerical methods.
Conclusion
It can be concluded that the analytical elasticity computation method has good potential to supplement or replace numerical elasticity simulations in gravityinduced deformations, for shapes that are substantially supported by a planar base perpendicular to the gravitational field. The error is manageable, while the calculation procedure takes less than one second as opposed to multiple minutes with numerical methods. The results will be used in the MRI and Ultrasound Robotic Assisted Biopsy (MURAB) project.
Keywords
Biopsy Magnetic resonance imaging Elastic calibration BreastIntroduction
Screening and staging of breast cancer for diagnosis and subsequent treatment is based on medical images acquired on different acquisition modalities and includes mammography (Xray), ultrasound (US) and MRI.
After image acquisition, proper localization of the tumor is essential for biopsy procedures to take tissue samples or to remove the tumor during surgery. To take full benefit from the previously acquired medical images, the location of the tumor should be aligned from the preoperative imaging into the operating room. The position of the patient can vary from prone during MRI scanning to supine position required for breast surgery for example. During ultrasound scanning and ultrasoundguided biopsy, the patient is returned on her back and additional compression is induced by the ultrasound probe. The computation of the elastic properties will serve as input for realtime adjustments of realistic deformations between preoperative and intraoperative images. For effective deformation models, the elasticity of the model needs to be known with good accuracy, i.e., the difference between computed and actual elasticity must be small. In this study, we aim for a maximum difference in the order of 10%, or at most two times the elasticity variation among FEMsimulated elasticity values. Image registration techniques based on image intensities could be used for small deformations [24], but do not work in cases with large deformations such as the alignment from prone to supine configurations [4].
Deformation of the breast occurs due to body movements. Various physicsbased numerical procedures have been presented for biomechanical modeling and soft tissue deformation. The most common computational schemes are based on linear or nonlinear biomechanical models including massspring methods (MSM) [2, 7, 20, 23], the masstensor method [10, 22], the boundary element method [13, 17] and conventional finite element modeling (FEM) [3, 25, 26].
In an MSM system, an object is modeled by a collection of point masses linked together with massless springs.
A patientspecific biomechanical model [11] was presented before to provide an initial deformation of the breast before registration between prone and supine MRI images. A zerogravity reference state for both prone and supine configurations was estimated. The patientspecific unloaded configuration was obtained [12]. The biomechanical methods serve in most cases for the initialization of intensitybased image registration techniques, as in [9] or [18]. The sliding motion of the breast on the chest wall was observed [6], but usually a fixed muscle surface is applied during the FEM simulations [14, 18, 21].
This study introduces a method to analytically derive the elastic modulus of the breast from a pair of MRI scans, taking local differences in tissue density and elasticity into account. The two MRI scans differ by the direction of the gravitational field, which are opposite to each other. Contrary to FEMbased numerical simulations, it is not needed to convert the MRI scan into a volumetric mesh, so mechanical properties on voxel scale are preserved. Also, only one iteration over all voxels is necessary, which makes the method relatively fast
The proposed analytical method requires the breast to be vertically supported by a rigid planar base. As the rib cage is approximately cylindrical, a human breast would need to be supported by a patientmounted flat plate with a hole for the breast. In an MRI scanner, the breast coil could serve this purpose.
Materials and methods
Four breast phantoms were constructed (Fig. 1, right), consisting of a rigid base with three fiducials, stiff superficial tissue, soft deep tissue and 3–4 lesions.
Figure 2 shows the outline of a breast phantom in a neutral reference state. Depending on the orientation (prone or supine), it is deformed by the gravitational field and tip is displaced toward the anterior or posterior direction. The magnitude of these deformations is related to the elasticity, and the approach of the research is to reconstruct the elasticity from these deformations using different methods.
The base represents a rigid inertial frame, which must be planar and normal to the gravitational direction. While a patient’s rib cage provides a rigid supportive base, it is not planar but approximately cylindrical. An external structure such as a breast coil (Fig. 2) may be required to provide this planar support.
The scanner was previously calibrated using a custom 3D calibration grid (Fig. 3, left) from which a fifthorder correction polynomial correction function was constructed. The ideal, distorted and corrected grid patterns are shown in Fig. 3. The measured residual error is \(0.2\, \hbox {mm}\), so subpixel resolution is feasible.
The distortioncorrected MRI scans (Fig. 4, left) were segmented by intensity thresholding and automatically aligned with a rigid transformation using the three fiducials, in which the rootmeansquare registration error was found to be 0.2–0.3 mm. From these data, surface and volumetric meshes in different levels of detail were constructed.
Figure 4 right shows two configurations of phantom I, overlaid on each other, after segmentation and registration. A significant displacement of the tip resulting from the change in gravity field direction can be observed.
Elasticity estimation
Preamble
Analytical derivation of elasticity
Figure 5 schematically shows the forces and pressures acting on a shape with inhomogeneous density and elasticity, hanging from a planar, rigid attachment on the top. At a given height h, the crosssectional area is A(h), the mass of the body below it is denoted as m(h) and the gravitational force acting on it F(h). We now derive expressions for the vertical stress \(\sigma (h)\) and elongation \(\epsilon (h)\) for every height, leading to a formula for the displacement D of the lower extremity of the body.
Analyzing the prone and supine scans of a phantom, we have \(\beta _\mathrm{p}\) and \(\beta _\mathrm{s}\) for prone and supine, respectively. In general, \(\beta _\mathrm{p} \ne \beta _\mathrm{s}\), because the shapes are significantly different: The total volume and crosssectional area at the base are approximately equal, but due to difference in height the crosssectional shape is more squeezed in prone position than in the supine one.
The phantom height H is illdefined due to possible irregularities at the tip, but the difference \(\Delta H = H_\mathrm{p}H_\mathrm{s}\) can be accurately determined by comparing point clouds around the tip using, e.g., the iterative closest point algorithm [5], and optimizing \(\Delta H\) such that the total point distance is minimal, or alternatively by comparing the centroids of the point clouds.
Numerical simulation of deformations
The purpose of FEM simulations is to determine the elasticity E of the different phantoms, based on the segmented models. The general strategy is to apply a gravitational field to the FEM model of a phantom in a specific direction. This deformed model is then compared to a reference phantom which was scanned in a different orientation, providing information about the elasticity parameter.
In the following subsections, we present two strategies to find the Young’s modulus by simulation, of which one strategy is performed by two different simulation software packages.
Estimating the \(\beta \) values by simulation in SOFA
In “Analytical derivation of elasticity” section, we have introduced a method to derive the values of \(\beta \) for the four phantoms in different orientations directly from a DICOM scan. In this section, we find \(\beta \) by simulation in SOFA at five different mesh resolutions [1]. For each mesh resolution, we have run a simulation with the phantom’s Young’s modulus set to \(E=6000\,\hbox {Pa}\) and gravity \(g=2.0\, \hbox {m/s}^{2}\). After 100 iterations, the simulation has stabilized and the vertices of the mesh in this configuration were extracted and analyzed. The displacement from the initial position follows by comparing the point clouds around the tip. The value of \(\beta \) then follows from Eq. (10). This procedure is repeated for each resolution of the mesh and for both prone and supine orientations, then the mean \(\beta _\mathrm{s}\) and \(\beta _\mathrm{p}\) values were computed. From the \(\beta _\mathrm{s}\), \(\beta _\mathrm{p}\) and \(\Delta H\), and assuming linearity of the displacement to g / E ratio, the Young’s modulus E can be derived using Eqs. (11) and (12).
Supine–prone and prone–supine simulation and matching in SOFA and Febio
Taking a phantom scanned in supine configuration, the base of the phantom is immobilized and a force field sized two times the gravity (\(19.62\,\hbox {m/s}^{2}\)) in anterior direction is applied to the phantom. After stabilization in simulation, the final state is extracted and compared to the phantom in prone position, which serves as the reference phantom.
The error value, \(\epsilon \), is defined as the distance between the simulated and reference phantoms in the area around the tip of the breast and can be positive or negative. The actual value is dependent on the elasticity parameter E of the phantom, which is optimized to bring \(\epsilon \) to zero.
The minimization is performed using the Newton’s method computed over E and the distance error, corrected by an adaptive step approach (when the FEM analysis software diverges). When procedure ends, i.e., when the method achieves a predefined error or when it reaches a maximum number of iterations, the estimated E parameter is returned with its associated error.
Results
Validation of analytical stress calculation on geometric shapes
Nine homogeneous geometric shapes were generated and analyzed: two cylinders with different aspect ratios, a cone, a Tpiece in normal and upsidedown orientation, a half sphere, a sphere, an hourglass and a snakelike shape.
Figure 6 shows the stress distribution along the vertical midway plane for all nine shapes. The first row uses the analytical computation method. The assumption that the stress distribution is constant in a crosssectional area parallel to the base, is reflected in having constant colors in horizontal direction. The second row shows the tensile stress from numerical simulations using the SOFA software package under the same conditions.
Table 1 lists the calculated and simulated \(\beta \) values for the same geometric shapes.

For cylinder, cubic and prismlike shapes that have a constant crosssectional area (a and b), the numerically derived stress distribution matches the analytically derived one quite well. The \(\beta \) values derived by both methods are well comparable (deviation under 9%).

For shapes that do not have a constant crosssectional area, but are substantially vertically supportive (ch), the analytically calculated and SOFAsimulated \(\beta \) values are still comparable (deviation up to 18%) although the stress distribution is different.

For shapes in which the lower extremity is not vertically supported by the base, i.e., no vertical line of maximum height can be drawn that entirely lies within the model (i), both the analytically calculated \(\beta \) value and the stress distribution are inconsistent with simulations.
Calculated and simulated \(\beta \) values for the nine geometric shapes
Geometric shape  Calculated \(\beta \)  Simulated \(\beta \) 

a  2375  2169 
b  2373  2229 
c  772  724 
d  1638  1581 
e  4500  4979 
f  213  205 
g  1932  2276 
h  3802  3797 
i  4942  26,499 
Analytical derivation of elasticity of phantoms
Each of the four phantoms was scanned in prone and supine position, and from the resulting DICOM scans, the \(\beta _\mathrm{p}\) and \(\beta _\mathrm{s}\) values are computed using Eq. 9 and assuming a homogeneous density and elasticity distribution. From these values plus the observed vertical displacements, the E parameters are computed using Eq. 12 and the results are listed in Table 2. It can be observed that phantom IV has the highest \(\beta \) and E values, making it the stiffest phantom, while phantom II is the softest one. In general, the \(\beta \) values are higher in prone position, which is as expected.
Simulation of \(\beta \) in SOFA
Analytically derived properties of four phantoms, under the assumption of constant tensile stress in each cross section
Phantom  \(\beta _\mathrm{s}\)  \(\beta _\mathrm{p}\)  \(\Delta H\)  E 

I  1215  1298  3.28  7514 
II  1129  1269  4.73  4972 
III  1356  1444  3.58  7673 
IV  1420  1471  2.93  9677 
Properties of four phantoms, derived by numerical simulation in SOFA in five different resolution scales and then averaged
Phantom  \(\beta _\mathrm{s}\)  \(\beta _\mathrm{p}\)  \(\Delta H\)  E 

I  \(1007 \pm 58\)  \(1134 \pm 38\)  3.28  \(6403 \pm 207\) 
II  \(947 \pm 41\)  \(1125 \pm 36\)  4.73  \(4297 \pm 113\) 
III  \(1131 \pm 48\)  \(1259 \pm 61\)  3.58  \(6549 \pm 213\) 
IV  \(1170 \pm 43\)  \(1383 \pm 34\)  2.93  \(8548 \pm 184\) 
Numerical simulation by supine–prone and prone–supine matching in SOFA
Table 4 lists the elasticities obtained by numerical simulation from supine to prone position and vice versa, in SOFA. As opposed to the \(\beta \) computation method, the prone–supine simulation method also takes nonlinearities into account which theoretically results in a more accurate estimate of the E value.
Elasticity values found by numerical simulations from supinetoprone (\(E_\mathrm{sp}\)) and pronetosupine (\(E_\mathrm{ps}\)) in four different resolution scales and then averaged, using SOFA
Phantom  \(E_\mathrm{sp}\)  \(E_\mathrm{ps}\)  Mean E 

I  \(5047 \pm 374\)  \(6459 \pm 373\)  \(5688 \pm 272\) 
II  \(3395 \pm 189\)  \(4513 \pm 272\)  \(3895 \pm 159\) 
III  \(5381 \pm 376\)  \(6828 \pm 438\)  \(6040 \pm 288\) 
IV  \(6245 \pm 433\)  \(7445 \pm 322\)  \(6805 \pm 283\) 
Elasticity values found by simulating from supinetoprone (\(E_\mathrm{sp}\)) and pronetosupine (\(E_\mathrm{ps}\)) in four different resolution scales and then averaged, using FEBio as software package
Phantom  \(E_\mathrm{sp}\)  \(E_\mathrm{ps}\)  Mean E 

I  \(5046 \pm 272\)  \(5252 \pm 307\)  \(5145 \pm 254\) 
II  \(4290 \pm 351\)  \(4298 \pm 273\)  \(4291 \pm 276\) 
III  \(5291.52 \pm 383\)  \(5639 \pm 456\)  \(5459 \pm 400\) 
IV  \(7916 \pm 1165\)  \(7564 \pm 957\)  \(7731 \pm 1016\) 
Numerical simulation by supine–prone and prone–supine matching in FEBio
Comparison of different elasticity measurement methods
Figure 8 graphically shows the elasticity of the four phantoms, derived using the different methods, while Table 6 lists the overall phantom elasticities, averaged from the four different methods.
Mean elasticity values for each phantom, taken as the average of the separate values derived by the four different methods
Phantom  Mean E 

I  \(6188 \pm 886\) 
II  \(4364 \pm 387\) 
III  \(6430 \pm 815\) 
IV  \(8190 \pm 1057\) 
The numerical results from FEBio simulations are in accordance with SOFA matching simulations, which is an indication that the simulations are consistent.
Discussion
We have presented a new method to analytically evaluate the elasticity of breast phantoms, from a pair of MRI scans in prone and supine position. The values found from analyzing the gravityinduced deformations are comparable to the elasticities derived from FEM simulations using FEBio and SOFA, with deviations of up to 18%. A study on nine geometric shapes has shown that the method is not only applicable to breast shapes, but also to other bodies and geometric objects as long as it is substantially supported by a planar rigid base.
The advantages of the analytical method are that the elasticity calculation is very fast (\(< 1\) s) and takes each individually scanned voxel into account, without need for mesh generation. As the voxel intensity in a scan gives certain information about tissue type, density and/or elasticity (depending on scanning protocol), tissue inhomogeneities can be directly incorporated in the analytical computations. The main limitations are that the method is only suitable for deformations in the linear range, and that the shapes must be substantially supported by a planar base perpendicular to the gravitational field.
The fact that a human breast is relatively flexible and the chest wall is not planar but cylindrically shaped, makes clinical application difficult. An artificial planar support base could be constructed by using a patientmounted breast coil, ideally in combination with a patient rotation system. The presented methods may also have applications in different domains, wherever deformation of bodies is involved in situations that meet the aforementioned boundary conditions The conclusion is that under specific conditions, the elasticity of a deformable object such as a human breast can be quickly computed from a pair of volumetric scans with sufficient accuracy, without need for FEM simulations. This promising result opens the door to new applications which can benefit from this complementary and nearrealtime elasticity computation method.
Notes
Compliance with ethical standards
Conflict of interest
The authors declare that they have no conflict of interest.
Human and animal rights
This article does not contain any studies with human participants or animals performed by any of the authors. This articles does not contain patient data.
References
 1.Allard J, Cotin S, Faure F, Bensoussan PJ, Poyer F, Duriez C, Delingette H, Grisoni L (2007) Sofaan open source framework for medical simulation. In: MMVR 15medicine meets virtual reality, vol 125, pp 13–18. IOP PressGoogle Scholar
 2.Altomonte M, Zerbato D, Botturi D, Fiorini P (2008) Simulation of deformable environment with haptic feedback on GPU. In: IEEE/RSJ international conference on intelligent robots and systems, 2008. IROS 2008, pp 3959–3964. IEEEGoogle Scholar
 3.Azar FS, Metaxas DN, Schnall MD (2001) A deformable finite element model of the breast for predicting mechanical deformations under external perturbations. Acad Radiol 8(10):965–975CrossRefPubMedGoogle Scholar
 4.Behrenbruch C, Marias K, Armitage P, Moore N, Clarke J, Brady J (2001) Pronesupine breast MRI registration for surgical visualisation. In: Medical image understanding and analysisGoogle Scholar
 5.Besl PJ, McKay ND (1992) A method for registration of 3D shapes. IEEE Trans Pattern Anal Mach Intell 14(2):239–256CrossRefGoogle Scholar
 6.Carter T, Tanner C, BeecheyNewman N, Barratt D, Hawkes D (2008) MR navigated breast surgery: method and initial clinical experience. In: Medical image computing and computerassisted interventionMICCAI 2008, pp 356–363CrossRefGoogle Scholar
 7.Chang YH, Chen YT, Chang CW, Lin CL (2010) Development scheme of hapticbased system for interactive deformable simulation. Comput Aided Des 42(5):414–424CrossRefGoogle Scholar
 8.Chevrier MC, David J, Khoury ME, Lalonde L, Labelle M, Trop I (2016) Breast biopsies under magnetic resonance imaging guidance: challenges of an essential but imperfect technique. Curr Probl Diagn Radiol 45(3):193–204. https://doi.org/10.1067/j.cpradiol.2015.07.002 CrossRefPubMedGoogle Scholar
 9.Conley RH, Meszoely IM, Weis JA, Pheiffer TS, Arlinghaus LR, Yankeelov TE, Miga MI (2015) Realization of a biomechanical modelassisted image guidance system for breast cancer surgery using supine MRI. Int J Comput Assist Radiol Surg 10(12):1985–1996CrossRefPubMedPubMedCentralGoogle Scholar
 10.Cotin S, Delingette H, Ayache N (2000) A hybrid elastic model for realtime cutting, deformations, and force feedback for surgery training and simulation. Vis Comput 16(8):437–452CrossRefGoogle Scholar
 11.Eiben B, Han L, Hipwell J, Mertzanidou T, Kabus S, Bülow T, Lorenz C, Newstead G, Abe H, Keshtgar M, Ourselin S, Hawkes DJ (2013) Biomechanically guided pronetosupine image registration of breast MRI using an estimated reference state. In: 2013 IEEE 10th international symposium on biomedical imaging (ISBI), pp 214–217. IEEEGoogle Scholar
 12.Eiben B, Vavourakis V, Hipwell JH, Kabus S, Lorenz C, Buelow T, Hawkes DJ (2014) Breast deformation modeling: comparison of methods to obtain a patient specific unloaded configuration. In: Proceedings of SPIE, vol 9036, pp 903615–903618Google Scholar
 13.Greminger MA, Nelson BJ (2003) Deformable object tracking using the boundary element method. In: 2003 IEEE computer society conference on computer vision and pattern recognition, 2003. Proceedings, vol 1, pp I–I. IEEEGoogle Scholar
 14.Han L, Hipwell J, Mertzanidou T, Carter T, Modat M, Ourselin S, Hawkes D (2011) A hybrid fembased method for aligning prone and supine images for image guided breast surgery. In: 2011 IEEE international symposium on biomedical imaging: from nano to macro, pp 1239–1242. IEEEGoogle Scholar
 15.Han L, Hipwell JH, Eiben B, Barratt D, Modat M, Ourselin S, Hawkes DJ (2014) A nonlinear biomechanical model based registration method for aligning prone and supine MR breast images. IEEE Trans Med Imaging 33(3):682–694CrossRefPubMedGoogle Scholar
 16.Han L, Hipwell JH, Tanner C, Taylor Z, Mertzanidou T, Cardoso J, Ourselin S, Hawkes DJ (2011) Development of patientspecific biomechanical models for predicting large breast deformation. Phys Med Biol 57(2):455CrossRefPubMedGoogle Scholar
 17.James DL, Pai DK (2005) A unified treatment of elastostatic contact simulation for real time haptics. In: ACM SIGGRAPH 2005 courses, p 141. ACMGoogle Scholar
 18.Lee A, Schnabel J, Rajagopal V, Nielsen P, Nash M (2010) Breast image registration by combining finite elements and freeform deformations. In: Digital mammography, pp 736–743CrossRefGoogle Scholar
 19.Liu GR, Quek SS (2013) The finite element method: a practical course. ButterworthHeinemann, BostonGoogle Scholar
 20.Maciel A, Boulic R, Thalmann D (2003) Deformable tissue parameterized by properties of real biological tissue. In: Surgery simulation and soft tissue modeling, pp 74–87. SpringerGoogle Scholar
 21.Pathmanathan P, Gavaghan DJ, Whiteley JP, Chapman SJ, Brady JM (2008) Predicting tumor location by modeling the deformation of the breast. IEEE Trans Biomed Eng 55(10):2471–2480CrossRefPubMedGoogle Scholar
 22.Picinbono G, Delingette H, Ayache N (2003) Nonlinear anisotropic elasticity for realtime surgery simulation. Graph Models 65(5):305–321CrossRefGoogle Scholar
 23.Roose L, De Maerteleire W, Mollemans W, Suetens P (2005) Validation of different soft tissue simulation methods for breast augmentation. In: International congress series, vol 1281, pp 485–490. ElsevierGoogle Scholar
 24.Rueckert D, Sonoda LI, Hayes C, Hill DL, Leach MO, Hawkes DJ (1999) Nonrigid registration using freeform deformations: application to breast MR images. IEEE Trans Med Imaging 18(8):712–21CrossRefGoogle Scholar
 25.Samani A, Bishop J, Yaffe MJ, Plewes DB (2001) Biomechanical 3D finite element modeling of the human breast using MRI data. IEEE Trans Med Imaging 20(4):271–279CrossRefPubMedGoogle Scholar
 26.Schnabel JA, Tanner C, CastellanoSmith AD, Degenhard A, Leach MO, Hose DR, Hill DL, Hawkes DJ (2003) Validation of nonrigid image registration using finiteelement methods: application to breast MR images. IEEE Trans Med Imaging 22(2):238–247CrossRefPubMedGoogle Scholar
 27.Whelan B, Liney GP, Dowling JA, Rai R, Holloway L, McGarvie L, Feain I, Barton M, Berry M, Wilkins R, Keall P (2017) An MRIcompatible patient rotation system design, construction, and first organ deformation results. Med Phys 44(2):581–588CrossRefPubMedGoogle Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.