# A Regression Model for Predicting Shape Deformation after Breast Conserving Surgery

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Dataset for Learning Breast Healing Deformations

#### 2.1. Breast Anatomy

#### 2.2. Wound Healing Simulator

#### 2.2.1. Finite Element Model

#### 2.2.2. Pose Transformation

#### 2.2.3. Tumor Definition

#### 2.3. Dataset Construction

^{®}) [43] - can be considered for weighting the material property values described in [24], and represent several breast densities. This reporting system identifies 4 categories of breast density (A, B, C and D), which are described in Table 1. Following this strategy, the fibroglandular/fat ratios: A—$10/90$; B—$35/65$; C—$60/40$; D—$85/15$, were used to average material properties of each category, as detailed in Table 2.

^{®}reporting system ($4\times 6=24$ cases), then different quadrants for the tumor location ($4\times 24=96$ cases) and, finally, 3 different tumor sizes for each location ($3\times 96=288$ cases). In the end, the dataset sums up to a total of 288 cases representing all the possible combinations of the most prominent clinical factors reported to affect breast shape after BCS.

## 3. Methodology

#### 3.1. Features

#### 3.1.1. Breast Characteristics

#### 3.1.2. Tumor Characteristics

#### 3.1.3. Feature Engineering

#### 3.2. Regression Models

#### 3.2.1. Random Forests

#### 3.2.2. Gradient Boosting Regression

#### 3.2.3. Multi-Output Regression

#### 3.3. Summary

## 4. Results

#### 4.1. Random Forests

#### 4.1.1. PCL Sampling

#### 4.1.2. Assigning Weights

#### 4.1.3. Feature Importance

#### 4.2. Gradient Boosting Regression

#### 4.3. Multi-Output Regression

#### 4.4. Summary

## 5. Discussion

## 6. Conclusions and Future Work

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Ethics

## Abbreviations

2D | 2 Dimensional |

3D | 3 Dimensional |

ACR | American College of Radiologists |

BCS | Breast Conserving Surgery |

BI-RADS | Breast Imaging Reporting and Data System |

FE | Finite Elements |

FEM | Finite Element Model |

GBR | Gradient Boosting Regression |

LIQ | Lower-Inner Quadrant |

LOPO | Leave One Patient Out |

LOQ | Lower-Outer or inferolateral Quadrant |

MIMO | Multiple Input Multiple Output |

MISO | Multiple Input Single Output |

MOR | Multi-Output Regression |

MRI | Magnetic Resonance Image |

OF | Objective Function |

PCL | Pointcloud |

RF | Random Forests |

UIQ | Upper-Inner Quadrant |

UOQ | Upper-Outer Quadrant |

## References

- American Cancer Society. Breast Cancer Detailed Guide; Technical Report; American Cancer Society: Atlanta, GA, USA, 2014. [Google Scholar]
- Gomes, N.S.; Silva, S.R.d. Avaliação da autoestima de mulheres submetidas à cirurgia oncológica mamária. Text Context Nurs.
**2013**, 22, 509–516. [Google Scholar] [CrossRef] - Sakorafas, G.H. Breast cancer surgery—Historical evolution, current status and future perspectives. Acta Oncol.
**2001**, 40, 5–18. [Google Scholar] [CrossRef] [PubMed] - Cardoso, M.J.; Oliveira, H.P.; Cardoso, J.S. Assessing cosmetic results after breast conserving surgery. J. Surg. Oncol.
**2014**, 110, 37–44. [Google Scholar] [CrossRef] [PubMed] - Hill-Kayser, C.E.; Vachani, C.; Hampshire, M.K.; Di Lullo, G.A.; Metz, J.M. Cosmetic outcomes and complications reported by patients having undergone breast-conserving treatment. Int. J. Radiat. Oncol. Biol. Phys.
**2012**, 83, 839–844. [Google Scholar] [CrossRef] [PubMed] - Aerts, L.; Christiaens, M.; Enzlin, P.; Neven, P.; Amant, F. Sexual functioning in women after mastectomy versus breast conserving therapy for early-stage breast cancer: A prospective controlled study. Breast
**2014**, 23, 629–636. [Google Scholar] [CrossRef] [PubMed] - Kim, M.K.; Kim, T.; Moon, H.G.; Jin, U.S.; Kim, K.; Kim, J.; Lee, J.W.; Kim, J.; Lee, E.; Yoo, T.K.; et al. Effect of cosmetic outcome on quality of life after breast cancer surgery. Eur. J. Surg. Oncol.
**2015**, 41, 426–432. [Google Scholar] [CrossRef] [PubMed] - Tőkés, T.; Torgyík, L.; Szentmártoni, G.; Somlai, K.; Tóth, A.; Kulka, J.; Dank, M. Primary systemic therapy for breast cancer: Does the patient’s involvement in decision-making create a new future? Patient Educ. Couns.
**2015**, 98, 695–703. [Google Scholar] [CrossRef] [PubMed] - Oliveira, H.P.; Cardoso, J.S.; Magalhães, A.; Cardoso, M.J. Methods for the Aesthetic Evaluation of Breast Cancer conservation treatment: A technological review. Curr. Med. Imaging Rev.
**2013**, 9, 32–46. [Google Scholar] [CrossRef] - Garbey, M.; Thanoon, D.; Bass, B. Multi-Scale modeling in computational surgery: Application to Breast conservative therapy. J. Serbian Soc. Comput. Mech.
**2011**, 5, 81–89. [Google Scholar] - De Heras Ciechomski, P.; Constantinescu, M.; Garcia, J.; Olariu, R.; Dindoyal, I.; Le Huu, S.; Reyes, M. Development and implementation of a web-enabled 3D consultation tool for Breast augmentation surgery based on 3D-Image reconstruction of 2D pictures. J. Med. Int. Res.
**2012**, 14, e21. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Eiben, B.; Han, L.; Hipwell, J.; Mertzanidou, T.; Kabus, S.; Buelow, T.; Lorenz, C.; Newstead, G.M.; Abe, H.; Keshtgar, M.; et al. Biomechanically guided prone-to-supine image registration of breast MRI using an estimated reference state. In Proceedings of the IEEE 10th International Symposium on Biomedical Imaging, San Francisco, CA, USA, 7–11 April 2013; pp. 214–217. [Google Scholar]
- Han, L.; Hipwell, J.H.; Eiben, B.; Barratt, D.; Modat, M.; Ourselin, S.; Hawkes, D.J. A nonlinear biomechanical model based registration method for aligning prone and supine mr breast images. IEEE Trans. Med. Imaging
**2014**, 33, 682–694. [Google Scholar] [PubMed] - Rajagopal, V. Modelling Breast Tissue Mechanics under Gravity Loading. Ph.D. Thesis, University of Auckland, Auckland, New Zealand, 2007. [Google Scholar]
- Hipwell, J.H.; Vavourakis, V.; Han, L.; Mertzanidou, T.; Eiben, B.; Hawkes, D.J. A review of biomechanically informed breast image registration. Phys. Med. Biol.
**2016**, 61, R1–R31. [Google Scholar] [CrossRef] [PubMed] - Mertzanidou, T.; Hipwell, J.; Johnsen, S.; Han, L.; Eiben, B.; Taylor, Z.; Ourselin, S.; Huisman, H.; Mann, R.; Bick, U.; et al. MRI to X-ray mammography intensity-based registration with simultaneous optimisation of pose and biomechanical transformation parameters. Med. Image Anal.
**2014**, 18, 674–683. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hopp, T.; Dietzel, M.; Baltzer, P.A.; Kreisel, P.; Kaiser, W.A.; Gemmeke, H.; Ruiter, N.V. Automatic multimodal 2D/3D breast image registration using biomechanical FEM models and intensity-based optimization. Med. Image Anal.
**2013**, 17, 209–218. [Google Scholar] [CrossRef] [PubMed] - Shih, T.C.; Chen, J.H.; Liu, D.; Nie, K.; Sun, L.; Lin, M.; Chang, D.; Nalcioglu, O.; Su, M.Y. Computational simulation of breast compression based on segmented breast and fibroglandular tissues on magnetic resonance images. Phys. Med. Biol.
**2010**, 55, 4153–4168. [Google Scholar] [CrossRef] [PubMed] - Sturgeon, G.M.; Kiarashi, N.; Lo, J.Y.; Samei, E.; Segars, W.P. Finite-element modeling of compression and gravity on a population of breast phantoms for multimodality imaging simulation. Med. Phys.
**2016**, 43, 2207–2217. [Google Scholar] [CrossRef] [PubMed] - Azar, F.S.; Metaxas, D.N.; Schnall, M.D. A deformable finite element model of the breast for predicting mechanical deformations under external perturbations. Acad. Radiol.
**2001**, 8, 965–975. [Google Scholar] [CrossRef] - Carter, T.J. Biomechanical Modelling of the Breast for Image-Guided Surgery. Ph.D. Thesis, University of London, London, UK, 2009. [Google Scholar]
- Carter, T.J.; Tanner, C.; Crum, W.; Beechey-Newman, N.; Hawkes, D. A framework for image-guided breast surgery. In International Workshop on Medical Imaging and Virtual Reality; Springer: Berlin/Heidelberg, Germany, 2006; Volume 4091, pp. 203–210. [Google Scholar]
- Garbey, M.; Bass, B.L.; Berceli, S. Multiscale mechanobiology modeling for surgery assessment. Acta Mech. Sin.
**2012**, 28, 1186–1202. [Google Scholar] [CrossRef] - Vavourakis, V.; Eiben, B.; Hipwell, J.H.; Williams, N.R.; Keshtgar, M.; Hawkes, D.J. Multiscale Mechano-Biological Finite Element Modelling of Oncoplastic Breast Surgery—Numerical Study towards Surgical Planning and Cosmetic Outcome Prediction. PLoS ONE
**2016**, 11, e0159766. [Google Scholar] [CrossRef] [PubMed] - Rajagopal, V.; Nielsen, P.M.F.; Nash, M.P. Modeling breast biomechanics for multi-modal image analysis-successes and challenges. Wiley Interdiscip. Rev. Syst. Biol. Med.
**2010**, 2, 293–304. [Google Scholar] [CrossRef] [PubMed] - Bardinet, E.; Cohen, L.D.; Ayache, N. A Parametric Deformable Model to Fit Unstructured 3D Data. Comput. Vis. Image Underst.
**1998**, 71, 39–54. [Google Scholar] [CrossRef] - Rueckert, D.; Sonoda, L.I.; Hayes, C.; Hill, D.L.G.; Leach, M.O.; Hawkes, D.J. Nonrigid Registration using free-form deformations: Application to breast MR images. IEEE Trans. Med. Imaging
**1999**, 18, 712–721. [Google Scholar] [CrossRef] [PubMed] - Gallo, G.; Guarnera, G.C.; Catanuto, G.; Pane, F. Parametric representation of human breast shapes. In Proceedings of the IEEE International Workshop on Medical Measurements and Applications, MeMeA 2009, Cetraro, Italy, 29–30 May 2009; pp. 26–30. [Google Scholar]
- Lee, A.W.C.; Schnabel, J.A.; Rajagopal, V.; Nielsen, P.M.F.; Nash, M.P. Breast image registration by combining finite elements and free-form deformations. In Digital Mammography; Springer: Berlin/Heidelberg, Germany, 2010; pp. 736–743. [Google Scholar]
- Pernes, D.; Cardoso, J.S.; Oliveira, H.P. Fitting of superquadrics for breast modelling by geometric distance minimization. In Proceedings of the 2014 IEEE International Conference on Bioinformatics and Biomedicine, Belfast, UK, 2–5 November 2014; pp. 293–296. [Google Scholar]
- Chen, D.T.; Kakadiaris, I.a.; Miller, M.J.; Loftin, R.B.; Patrick, C. Modeling for Plastic and Reconstructive Breast Surgery. In Medical Image Computing and Computer-Assisted Intervention—MICCAI 2000; Springer: Berlin/Heidelberg, Germany, 2000; pp. 1040–1050. [Google Scholar]
- Seo, H.; Cordier, F.; Hong, K. A breast modeler based on analysis of breast scan. Comput. Animat. Virtual Worlds
**2007**, 18, 141–151. [Google Scholar] [CrossRef] - Gallo, G.; Guarnera, G.C.; Catanuto, G. Human Breast Shape Analysis Using PCA. In Proceedings of the Third International Conference on Bio-inspired Systems and Signal Processing, Valencia, Spain, 20–23 January 2010; pp. 163–167. [Google Scholar]
- Kim, Y.; Lee, K.; Kim, W. 3D virtual simulator for breast plastic surgery. Comput. Animat. Virtual Worlds
**2008**, 19, 515–526. [Google Scholar] [CrossRef] - Bessa, S.; Zolfagharnasab, H.; Pereira, E.; Oliveira, H.P. Prediction of Breast Deformities: A Step Forward for Planning Aesthetic Results After Breast Surgery. In Pattern Recognition and Image Analysis; Springer: Cham, Switzerland, 2017; Volume 4478, pp. 267–276. [Google Scholar]
- Ramião, N.G.; Martins, P.S.; Rynkevic, R.; Fernandes, A.A.; Barroso, M.; Santos, D.C. Biomechanical properties of breast tissue, a state-of-the-art review. Biomech. Model. Mechanobiol.
**2016**, 15, 1307–1323. [Google Scholar] [CrossRef] [PubMed] - Thanoon, D.; Garbey, M.; Bass, B.L. Computational Modeling of Breast Conserving Surgery (BCS) Starting from MRI Imaging. In Computational Surgery and Dual Training; Garbey, M., Bass, B.L., Berceli, S., Collet, C., Cerveri, P., Eds.; Springer New York: New York, NY, USA, 2014; pp. 67–86. [Google Scholar]
- Jennifer, A.; Harvey, M.D.; Viktor, E.B. Quantitative Assessment of Mammographic Breast Density: Relationship with Breast Cancer Risk. Radiology
**2004**, 230, 29–41. [Google Scholar] - Bernardini, F.; Mittleman, J.; Rushmeier, H.; Silva, C.; Taubin, G. The ball-pivoting algorithm for surface reconstruction. IEEE Trans. Vis. Comput. Graph.
**1999**, 5, 349–359. [Google Scholar] [CrossRef] - Cignoni, P.; Callieri, M.; Corsini, M.; Dellepiane, M.; Ganovelli, F.; Ranzuglia, G. MeshLab: An Open-Source Mesh Processing Tool; Scarano, V., Chiara, R.D., Erra, U., Eds.; The Eurographics Association: Geneva, Switzerland, 2008. [Google Scholar]
- Geuzaine, C.; Remacle, J.F. Gmsh : A 3-D finite element mesh generator with built-in pre- and post-processing facilities. Int. J. Numer. Methods Eng.
**2009**, 79, 1309–1331. [Google Scholar] [CrossRef] - Del Palomar, A.P.; Calvo, B.; Herrero, J.; López, J.; Doblaré, M. A finite element model to accurately predict real deformations of the breast. Med. Eng. Phys.
**2008**, 30, 1089–1097. [Google Scholar] [CrossRef] [PubMed] - D’Orsi, C.J.; Sickles, E.A.; Mendelson, E.B.; Morris, E.A. ACR BI-RADS Atlas: Breast Imaging Reporting and Data System; American College of Radiology: Reston, VA, USA, 2013. [Google Scholar]
- Hansen, J.; Netter, F. Netter’s Clinical Anatomy; Saunders/Elsevier: Philadelphia, PA, USA, 2014. [Google Scholar]
- Clough, K.B.; Kaufman, G.J.; Nos, C.; Buccimazza, I.; Sarfati, I.M. Improving Breast Cancer Surgery: A Classification and Quadrant per Quadrant Atlas for Oncoplastic Surgery. Ann. Surg. Oncol.
**2010**, 17, 1375–1391. [Google Scholar] [CrossRef] [PubMed] - Tsai, C.F.; Wu, J.W. Using neural network ensembles for bankruptcy prediction and credit scoring. Expert Syst. Appl.
**2008**, 34, 2639–2649. [Google Scholar] [CrossRef] - Ho, T.K. A Data Complexity Analysis of Comparative Advantages of Decision Forest Constructors. Pattern Anal. Appl.
**2002**, 5, 102–112. [Google Scholar] [CrossRef] - Breiman, L. Prediction Games and Arcing Algorithms. Neural Comput.
**1999**, 11, 1493–1517. [Google Scholar] [CrossRef] [PubMed] - Mason, L.; Baxter, J.; Bartlett, P.; Frean, M. Boosting algorithms as gradient descent. In Proceedings of the 12th International Conference on Neural Information Processing Systems; MIT Press: Cambridge, MA, USA, 1999; pp. 512–518. [Google Scholar]
- Chen, T.; Guestrin, C. XGBoost: A Scalable Tree Boosting System. CoRR
**2016**, abs/1603.02754. [Google Scholar] [CrossRef] - Friedman, J.H. Greedy function approximation: A gradient boosting machine. Ann. Stat.
**2001**, 29, 1189–1232. [Google Scholar] [CrossRef] - Borchani, H.; Varando, G.; Bielza, C.; Larrañaga, P. A survey on multi-output regression. Wiley Interdiscip. Rev. Data Min. Knowl. Discov.
**2015**, 5, 216–233. [Google Scholar] [CrossRef] - Pedregosa, F.; Varoquaux, G.; Gramfort, A.; Michel, V.; Thirion, B.; Grisel, O.; Blondel, M.; Prettenhofer, P.; Weiss, R.; Dubourg, V.; et al. Scikit-learn: Machine Learning in Python. J. Mach. Learn. Res.
**2011**, 12, 2825–2830. [Google Scholar] - Liaw, A.; Wiener, M. Classification and Regression by randomForest. R News
**2002**, 2, 18–22. [Google Scholar] - Chan, J.C.W.; Paelinckx, D. Evaluation of Random Forest and Adaboost tree-based ensemble classification and spectral band selection for ecotope mapping using airborne hyperspectral imagery. Remote Sens. Environ.
**2008**, 112, 2999–3011. [Google Scholar] [CrossRef]

**Figure 1.**Breast anatomy: fibroglandular and fat tissues, which have distinct mechanical properties, compose most part of the breast - adapted from (https://commons.wikimedia.org/).

**Figure 2.**BCS simulation followed pipeline, as proposed by Vavourakis et al. [24].

**Figure 3.**Pose transformation followed pipeline, as proposed by Vavourakis et al. [24].

**Figure 4.**Breast quadrants definition: Upper-Outer (UOQ), Upper-Inner (UIQ), Lower-Outer (LOQ), Lower-Inner (LIQ) quadrants - adapted from (https://commons.wikimedia.org).

**Figure 5.**Definition of breast quadrants planes in supine position. (

**a**) Pectoral plane; (

**b**) Superior-Inferior plane (red); (

**c**) Lateral-Medial plane (green).

**Figure 6.**Alignment of a predefined cylinder (

**a**) through the direction given by pectoral normal; in relation to the tumor location (

**b**).

**Figure 7.**Definition of the cylinder to excise and the correspondent points labelled as damaged: small (

**a**,

**d**), medium (

**b**,

**e**) and large tumor sizes (

**c**,

**f**).

**Figure 9.**Influence of breast density on deformations: PCLs are shown for the same patient with variable breast density and fixed tumor position (UOQ) and size (small). Pre and post-surgical data on black and red, respectively. Blue arrows indicate displacement direction and magnitude. (

**a**) BI-RADS A; (

**b**) BI-RADS B; (

**c**) BI-RADS C; (

**d**) BI-RADS D.

**Figure 10.**Influence of breast shape/laterality: PCLs are shown for two patients with BI-RADS A and the largest tumor positioned on UOQ. Pre- and post-surgical data on black and red, respectively. Blue arrows indicate displacement direction and magnitude. (

**a**) Patient’s right breast; (

**b**) Patient’s left breast.

**Figure 11.**Influence of tumor position on deformations: PCLs are shown for the same patient with the tumor positioned in different breast quadrants/regions. Largest tumors are shown for BI-RADS A. Pre- and post-surgical data on black and red, respectively. Blue arrows indicate displacement direction and magnitude. (

**a**) UIQ; (

**b**) UOQ; (

**c**) LIQ; (

**d**) LOQ.

**Figure 12.**Influence of tumor size: PCLs are shown for the same patient with BI-RADS A, and tumor on region UOQ. Pre- and post-surgical data on black and red, respectively. Blue arrows indicate displacement direction and magnitude. (

**a**) Small; (

**b**) Medium; (

**c**) Large.

**Figure 14.**Impact of sampling breast PCL on the average (

**a**) and Hausdorff (

**b**) with respect to the pair-wise distance.

**Figure 16.**Visual evaluation: Three PCLs from dataset with different prediction distances. The post-surgery PCL is shown with red points while the predicted PCL is visualized with black points. The displacements between corresponding points are colored in blue. (

**a**) Poor prediction, ${D}_{pred\to post}=1.624$ mm; (

**b**) Fair prediction, ${D}_{pred\to post}=1.044$ mm; (

**c**) Good prediction, ${D}_{pred\to post}=0.827$ mm.

Density Categories | % Fibroglandular Tissue | Description |
---|---|---|

A | $<25\%$ | Almost entirely fatty breast |

B | 25–50% | Scattered areas of fibroglandular density |

C | 50–75% | Heterogeneously dense breast |

D | $>75\%$ | Extremely dense breast |

**Table 2.**Combination of tissues biomechanical properties for each density category. ${c}_{1}$ and ${c}_{2}$ are parameters of the Mooney-Rivlin biomechanical model, and ${\rho}_{0}$ is the material density. The reference values from Vavourakis et al. [24] were used.

BI-RADS^{®} | Biomechanical Properties | ||
---|---|---|---|

${\mathit{c}}_{1}$ (Pa) | ${\mathit{c}}_{2}$ (Pa) | ${\mathit{\rho}}_{0}$ (kg·m${}^{-3}$) | |

A | 0.1 × 120 + 0.9 × 80 = 84 | 0 | 0.1 × 1020 + 0.9 × 910 = 921 |

B | 0.35 × 120 + 0.65 × 80 = 94 | 0 | 0.35 × 1020 + 0.65 × 910 = 948.5 |

C | 0.6 × 120 + 0.4 × 80 = 104 | 0 | 0.6 × 1020 + 0.4 × 910 = 976 |

D | 0.85 × 120 + 0.15 × 80 = 114 | 0 | 0.85 × 1020 + 0.15 × 910 = 1003.5 |

**Table 3.**Characterization of breast MRI data used for data augmentation: size, volume and laterality.

Breast Characteristics | Patient #1 | Patient #2 | Patient #3 | Patient #4 | Patient #5 | Patient #6 |
---|---|---|---|---|---|---|

Size | Small | Medium | Large | Medium | Large | Small |

Volume (mm${}^{3}$) | 495,948 | 802,661 | 1,314,990 | 1,052,500 | 1,202,500 | 559,677 |

Laterality | Left | Left | Right | Left | Right | Right |

**Table 4.**Description of features used in the regression models to predict breast deformations after BCS.

Features | ID | Type | Space | Description | |||
---|---|---|---|---|---|---|---|

Point’s coordinate | ${p}_{x}$ | Quantitative Continuous | $\in {\mathrm{I}\phantom{\rule{-0.166667em}{0ex}}\mathrm{R}}^{3}$ | Breast points coordinates in 3D space. The center of geometry of PCLs should be translated to the origin. | |||

${p}_{y}$ | |||||||

${p}_{z}$ | |||||||

Coordinate difference to the excised cylinder | $dis{p}_{x}$ | Quantitative Continuous | $\in {\mathrm{I}\phantom{\rule{-0.166667em}{0ex}}\mathrm{R}}^{3}$ | Difference of each healthy point (signed) of pre-surgery PCL to the excised cylinder | |||

$dis{p}_{y}$ | |||||||

$dis{p}_{z}$ | |||||||

Distance to the excised cylinder | ${d}_{x,y,z}$ | Quantitative Continuous | $\in \mathrm{I}\phantom{\rule{-0.166667em}{0ex}}\mathrm{R}$ | Euclidean distance of each point of pre-surgery PCL to the excised cylinder | |||

Polar distance to the excised cylinder | $\rho $ | Quantitative Continuous | $\in {\mathrm{I}\phantom{\rule{-0.166667em}{0ex}}\mathrm{R}}^{3}$ | Polar difference of each healthy point (signed) of pre-surgery PCL to the excised cylinder | |||

$\varphi $ | |||||||

z | |||||||

Tumor Size | ${s}_{1}$ | Categorical Ordinal | 100 | Defines the size of tumor ($5\%$, $7.5\%$, or $10\%$ of total breast volume) | |||

${s}_{2}$ | 010 | ||||||

${s}_{3}$ | 001 | ||||||

Breast Laterality | R | Categorical Nominal | 1 | Indicates the laterality of breast (right or left) | |||

L | 0 | ||||||

Breast Density | A | Categorical Ordinal | 1000 | Determines breast density level (A, B, C, or D) | |||

B | 0100 | ||||||

C | 0010 | ||||||

D | 0001 | ||||||

Tumor Region | ${R}_{1}$ | Categorical Nominal | 1000 | Specifies the region of breast with the tumor (UOQ,UIQ, LOQ, or LIQ) | |||

${R}_{2}$ | 0100 | ||||||

${R}_{3}$ | 0010 | ||||||

${R}_{4}$ | 0001 |

**Table 5.**Numerical evaluation: Pair-wise distance (in mm) based on the both average and Hausdorff OFs for training set sampled with $65\%$, and $75\%$, respectively. Also, the last column denotes the evaluation of dummy method.

Average-Based OF | Hausdorff-Based OF | Baseline Evaluation | |
---|---|---|---|

${\mathit{D}}^{\mathit{p}\mathbf{2}\mathit{p}}$ | ${\mathit{D}}^{\mathit{p}\mathbf{2}\mathit{p}}$ | ${\mathit{D}}^{\mathit{p}\mathbf{2}\mathit{p}}$ | |

$\mu $ | 1.052 | 1.146 | 2.206 |

$\sigma $ | 0.920 | 0.934 | 1.920 |

$Max$ | 5.210 | 4.101 | 8.410 |

**Table 6.**Numerical evaluation: Global distance (in mm) based on both average and Hausdorff OFs for training set sampled with $65\%$, and $75\%$, respectively. The last two columns denote the evaluation of the dummy method.

Average-Based OF | Hausdorff-Based OF | Baseline Evaluation | ||||
---|---|---|---|---|---|---|

${\mathit{D}}_{\mathit{pred}\to \mathit{post}}^{\phantom{\rule{3.33333pt}{0ex}}\mathit{global}}$ | ${\mathit{D}}_{\mathit{post}\to \mathit{pred}}^{\phantom{\rule{3.33333pt}{0ex}}\mathit{global}}$ | ${\mathit{D}}_{\mathit{pred}\to \mathit{post}}^{\phantom{\rule{3.33333pt}{0ex}}\mathit{global}}$ | ${\mathit{D}}_{\mathit{post}\to \mathit{pred}}^{\phantom{\rule{3.33333pt}{0ex}}\mathit{global}}$ | ${\mathit{D}}_{\mathit{pre}\to \mathit{post}}^{\phantom{\rule{3.33333pt}{0ex}}\mathit{global}}$ | ${\mathit{D}}_{\mathit{post}\to \mathit{pre}}^{\phantom{\rule{3.33333pt}{0ex}}\mathit{global}}$ | |

$\mu $ | 0.973 | 0.970 | 1.091 | 1.071 | 1.758 | 1.731 |

$\sigma $ | 0.787 | 0.748 | 0.825 | 0.761 | 1.333 | 1.277 |

Max | 4.926 | 4.814 | 4.025 | 4.010 | 6.512 | 6.317 |

**Table 7.**Numerical evaluation: Pair-wise distance (in mm) between predicted PCLs and post-surgery models using RF with adaptive weights. The training set is sampled with rate of $65\%$ and $75\%$ due to average and Hausdorff OFs, respectively.

Average-Based OF | Hausdorff-Based OF | Baseline Evaluation | |
---|---|---|---|

${\mathit{D}}^{\mathit{p}\mathbf{2}\mathit{p}}$ | ${\mathit{D}}^{\mathit{p}\mathbf{2}\mathit{p}}$ | ${\mathit{D}}^{\mathit{p}\mathbf{2}\mathit{p}}$ | |

$\mu $ | 1.048 | 1.189 | 2.206 |

$\sigma $ | 0.905 | 0.981 | 1.920 |

$Max$ | 5.240 | 4.083 | 8.410 |

**Table 8.**Numerical evaluation: Global distance (in mm) between predicted PCLs and post-surgery models using RF with adaptive weights. The training set is sampled with rate of $65\%$ and $75\%$ due to average and Hausdorff OFs, respectively.

Average-Based OF | Hausdorff-Based OF | Baseline Evaluation | ||||
---|---|---|---|---|---|---|

${\mathit{D}}_{\mathit{pred}\to \mathit{post}}^{\phantom{\rule{3.33333pt}{0ex}}\mathit{global}}$ | ${\mathit{D}}_{\mathit{post}\to \mathit{pred}}^{\phantom{\rule{3.33333pt}{0ex}}\mathit{global}}$ | ${\mathit{D}}_{\mathit{pred}\to \mathit{post}}^{\phantom{\rule{3.33333pt}{0ex}}\mathit{global}}$ | ${\mathit{D}}_{\mathit{post}\to \mathit{pred}}^{\phantom{\rule{3.33333pt}{0ex}}\mathit{global}}$ | ${\mathit{D}}_{\mathit{pre}\to \mathit{post}}^{\phantom{\rule{3.33333pt}{0ex}}\mathit{global}}$ | ${\mathit{D}}_{\mathit{post}\to \mathit{pre}}^{\phantom{\rule{3.33333pt}{0ex}}\mathit{global}}$ | |

$\mu $ | 0.961 | 0.951 | 1.124 | 1.094 | 1.758 | 1.731 |

$\sigma $ | 0.951 | 0.861 | 0.958 | 0.937 | 1.333 | 1.277 |

Max | 5.182 | 5.178 | 4.022 | 3.980 | 6.512 | 6.317 |

**Table 9.**Features’ importance (both individual and grouped) measured in % for RF with adaptive weights.

Features | ID | Trained Model for X | Trained Model for Y | Trained Model for Z | |||
---|---|---|---|---|---|---|---|

Individual $(\%)$ | Group $(\%)$ | Individual $(\%)$ | Group $(\%)$ | Individual $(\%)$ | Group $(\%)$ | ||

Points’ coordinate | ${\mathit{p}}_{\mathit{x}}$ | 8.39 | 7.03 | 6.94 | 7.67 | 3.75 | 6.25 |

${p}_{y}$ | 7.46 | 8.37 | 6.93 | ||||

${p}_{z}$ | 5.25 | 7.68 | 8.08 | ||||

Coordinate difference to the excised cylinder | $dis{p}_{x}$ | 1.17 | 1.73 | 0.90 | 1.19 | 1.37 | 1.75 |

$dis{p}_{y}$ | 1.46 | 1.25 | 1.57 | ||||

$dis{p}_{z}$ | 2.57 | 1.39 | 2.32 | ||||

Distance to the excised cylinder | ${D}_{x,y,z}$ | 8.23 | 8.23 | 8.09 | 8.09 | 8.00 | 8.00 |

Polar distance the the excised cylinder | $\rho $ | 0.11 | 0.42 | 0.04 | 0.47 | 0.29 | 0.70 |

$\psi $ | 0.48 | 0.66 | 0.82 | ||||

z | 0.68 | 0.70 | 1.00 | ||||

Tumor Size | ${s}_{1}$ | 2.79 | 2.99 | 1.93 | 3.31 | 2.67 | 3.19 |

${s}_{2}$ | 3.03 | 2.21 | 3.21 | ||||

${s}_{3}$ | 3.16 | 5.79 | 3.69 | ||||

Breast Laterality | R | 4.21 | 3.81 | 3.72 | 3.75 | 4.12 | 4.16 |

L | 3.42 | 3.79 | 4.19 | ||||

Breast Density | ${B}_{1}$ | 7.13 | 6.08 | 6.29 | 6.74 | 5.44 | 6.23 |

${B}_{2}$ | 7.00 | 7.30 | 6.35 | ||||

${B}_{3}$ | 5.10 | 6.51 | 6.44 | ||||

${B}_{4}$ | 5.11 | 6.87 | 6.69 | ||||

Tumor Region | ${R}_{1}$ | 4.77 | 5.95 | 4.44 | 5.81 | 5.97 | 5.76 |

${R}_{2}$ | 6.77 | 4.53 | 5.51 | ||||

${R}_{3}$ | 6.80 | 5.06 | 5.72 | ||||

${R}_{4}$ | 4.94 | 5.50 | 5.85 |

**Table 10.**Individual evaluation of the clinical features (breast density, tumor size, and tumor region) with respect to the adaptive weight RF in scope of the average OF.

Breast Density | Tumor Size | Tumor Region | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

A | B | C | D | S | M | L | UOQ | UIQ | LOQ | LIQ | |

$\mu $ | 1.052 | 1.050 | 1.048 | 1.044 | 1.045 | 1.049 | 1.051 | 1.052 | 1.049 | 1.051 | 1.042 |

$\sigma $ | 0.931 | 0.910 | 0.890 | 0.891 | 0.900 | 0.905 | 0.911 | 0.920 | 0.901 | 0.910 | 0.891 |

$Max$ | 5.257 | 5.256 | 5.218 | 5.230 | 5.178 | 5.233 | 5.310 | 5.291 | 5.206 | 5.273 | 5.191 |

**Table 11.**Numerical evaluation: Pair-wise distance (in mm) between predicted PCLs and post-surgery models using GBR. The training set is sampled with rate of $65\%$ and $75\%$ due to average and Hausdorff OFs, respectively.

Average-Based OF | Hausdorff-Based OF | Baseline Evaluation | |
---|---|---|---|

${\mathit{D}}^{\mathit{p}\mathbf{2}\mathit{p}}$ | ${\mathit{D}}^{\mathit{p}\mathbf{2}\mathit{p}}$ | ${\mathit{D}}^{\mathit{p}\mathbf{2}\mathit{p}}$ | |

$\mu $ | 1.326 | 1.631 | 2.206 |

$\sigma $ | 0.943 | 1.026 | 1.920 |

$Max$ | 5.933 | 5.564 | 8.410 |

**Table 12.**Numerical evaluation: Global distances (in mm) between predicted PCLs and post-surgery models using GBR. The training set is sampled with rate of $65\%$ and $75\%$ due to average and Hausdorff OFs, respectively.

Average-Based OF | Hausdorff-Based OF | Baseline Evaluation | ||||
---|---|---|---|---|---|---|

${\mathit{D}}_{\mathit{pred}\to \mathit{post}}^{\phantom{\rule{3.33333pt}{0ex}}\mathit{global}}$ | ${\mathit{D}}_{\mathit{post}\to \mathit{pred}}^{\phantom{\rule{3.33333pt}{0ex}}\mathit{global}}$ | ${\mathit{D}}_{\mathit{pred}\to \mathit{post}}^{\phantom{\rule{3.33333pt}{0ex}}\mathit{global}}$ | ${\mathit{D}}_{\mathit{post}\to \mathit{pred}}^{\phantom{\rule{3.33333pt}{0ex}}\mathit{global}}$ | ${\mathit{D}}_{\mathit{pre}\to \mathit{post}}^{\phantom{\rule{3.33333pt}{0ex}}\mathit{global}}$ | ${\mathit{D}}_{\mathit{post}\to \mathit{pre}}^{\phantom{\rule{3.33333pt}{0ex}}\mathit{global}}$ | |

$\mu $ | 1.287 | 1.269 | 1.604 | 1.590 | 1.758 | 1.731 |

$\sigma $ | 0.928 | 0.906 | 0.985 | 0.972 | 1.333 | 1.277 |

Max | 5.735 | 5.701 | 5.439 | 5.394 | 6.512 | 6.317 |

**Table 13.**Numerical evaluation: Pair-wise distance (in mm) between predicted PCLs and post-surgery models using MOR. The training set is sampled with rate of $65\%$ and $75\%$ due to average and Hausdorff OFs, respectively.

Average-Based OF | Hausdorff-Based OF | Baseline Evaluation | |
---|---|---|---|

${\mathit{D}}^{\mathit{p}\mathbf{2}\mathit{p}}$ | ${\mathit{D}}^{\mathit{p}\mathbf{2}\mathit{p}}$ | ${\mathit{D}}^{\mathit{p}\mathbf{2}\mathit{p}}$ | |

$\mu $ | 1.173 | 1.330 | 2.206 |

$\sigma $ | 0.993 | 1.022 | 1.920 |

$Max$ | 5.746 | 4.738 | 8.410 |

**Table 14.**Numerical evaluation: Global distances (in mm) between predicted PCLs and post-surgery models using MOR. The training set is sampled with rate of $65\%$ and $75\%$ due to average and Hausdorff OFs, respectively.

Average-Based OF | Hausdorff-Based OF | Baseline Evaluation | ||||
---|---|---|---|---|---|---|

${\mathit{D}}_{\mathit{pred}\to \mathit{post}}^{\phantom{\rule{3.33333pt}{0ex}}\mathit{global}}$ | ${\mathit{D}}_{\mathit{post}\to \mathit{pred}}^{\phantom{\rule{3.33333pt}{0ex}}\mathit{global}}$ | ${\mathit{D}}_{\mathit{pred}\to \mathit{post}}^{\phantom{\rule{3.33333pt}{0ex}}\mathit{global}}$ | ${\mathit{D}}_{\mathit{post}\to \mathit{pred}}^{\phantom{\rule{3.33333pt}{0ex}}\mathit{global}}$ | ${\mathit{D}}_{\mathit{pre}\to \mathit{post}}^{\phantom{\rule{3.33333pt}{0ex}}\mathit{global}}$ | ${\mathit{D}}_{\mathit{post}\to \mathit{pre}}^{\phantom{\rule{3.33333pt}{0ex}}\mathit{global}}$ | |

$\mu $ | 1.130 | 1.114 | 1.301 | 1.286 | 1.758 | 1.731 |

$\sigma $ | 0.935 | 0.903 | 0.983 | 0.966 | 1.333 | 1.277 |

Max | 5.695 | 5.623 | 4.682 | 4.621 | 6.512 | 6.317 |

**Table 15.**Numerical evaluation: Point-wise and global distance (in mm) between predicted and post-surgery PCLs using heuristic model. The training set is sampled with rate of $65\%$.

Average OF | |||
---|---|---|---|

${\mathit{D}}^{\mathit{p}\mathbf{2}\mathit{p}}$ | ${\mathit{D}}_{\mathit{pred}\to \mathit{post}}^{\phantom{\rule{3.33333pt}{0ex}}\mathit{global}}$ | ${\mathit{D}}_{\mathit{post}\to \mathit{pred}}^{\phantom{\rule{3.33333pt}{0ex}}\mathit{global}}$ | |

$\mu $ | 1.634 | 1.522 | 1.503 |

$\sigma $ | 1.496 | 1.193 | 1.129 |

$Max$ | 5.763 | 5.326 | 5.136 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zolfagharnasab, H.; Bessa, S.; Oliveira, S.P.; Faria, P.; Teixeira, J.F.; Cardoso, J.S.; Oliveira, H.P.
A Regression Model for Predicting Shape Deformation after Breast Conserving Surgery. *Sensors* **2018**, *18*, 167.
https://doi.org/10.3390/s18010167

**AMA Style**

Zolfagharnasab H, Bessa S, Oliveira SP, Faria P, Teixeira JF, Cardoso JS, Oliveira HP.
A Regression Model for Predicting Shape Deformation after Breast Conserving Surgery. *Sensors*. 2018; 18(1):167.
https://doi.org/10.3390/s18010167

**Chicago/Turabian Style**

Zolfagharnasab, Hooshiar, Sílvia Bessa, Sara P. Oliveira, Pedro Faria, João F. Teixeira, Jaime S. Cardoso, and Hélder P. Oliveira.
2018. "A Regression Model for Predicting Shape Deformation after Breast Conserving Surgery" *Sensors* 18, no. 1: 167.
https://doi.org/10.3390/s18010167