# New Parametric 2D Curves for Modeling Prostate Shape in Magnetic Resonance Images

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## Abstract

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## 1. Introduction

- Linear tapering along y axis$$\{\begin{array}{cc}{x}^{\prime}=\left(\frac{T}{{a}_{y}}y+1\right)x\phantom{\rule{1.em}{0ex}}\hfill & \\ {y}^{\prime}=y\hfill & \end{array}$$
- Circular bending along the y axis$$\{\begin{array}{cc}{x}^{\prime}=\left(\frac{{a}_{y}}{\beta}-y\right)sin\left(\frac{x}{\frac{{a}_{y}}{\beta}-y}\right)\phantom{\rule{1.em}{0ex}}\hfill & \\ {y}^{\prime}=\frac{{a}_{y}}{\beta}-\left(\frac{{a}_{y}}{\beta}-y\right)cos\left(\frac{x}{\frac{{a}_{y}}{\beta}-y}\right)\hfill & \end{array}$$
- Rotation and translation$$\{\begin{array}{c}{x}^{\prime}=xcos\alpha -ysin\alpha +{x}_{0}\hfill \\ {y}^{\prime}=xsin\alpha +ycos\alpha +{y}_{0}\hfill \end{array}$$

**Figure 2.**In each subfigure, a parameter changes and ${a}_{x}={a}_{y}=1$. The dotted, dashed and solid curves correspond to the first, second and third value of the variable parameter, respectively. (

**a**) $\u03f5=0.7,1$ and $1.3$. (

**b**) $T=0,0.2$ and $0.5$ ($\u03f5=1$). (

**c**) $\beta =0.01,0.6$ and $0.8$ ($\u03f5=1$).

## 2. Our Models

#### 2.1. Role of the Parameters

- CS curve:$$\begin{array}{c}\hfill \left\{\begin{array}{c}x\left(0\right)=a+b\hfill \\ y\left(0\right)=0\hfill \end{array}\right.\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\left\{\begin{array}{c}x\left(\frac{1}{2}\right)=0\hfill \\ y\left(\frac{1}{2}\right)=c+d\hfill \end{array}\right.\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\left\{\begin{array}{c}x\left(1\right)=a-b\hfill \\ y\left(1\right)=0\hfill \end{array}\right.\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\left\{\begin{array}{c}x\left(\frac{3}{2}\right)=0\hfill \\ y\left(\frac{3}{2}\right)=c-d\hfill \end{array}\right.\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}.\end{array}$$
- CC curve:$$\begin{array}{c}\hfill \left\{\begin{array}{c}x\left(0\right)=a+b\hfill \\ y\left(0\right)=0\hfill \end{array}\right.\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\left\{\begin{array}{c}x\left(\frac{1}{2}\right)=0\hfill \\ y\left(\frac{1}{2}\right)=d\hfill \end{array}\right.\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\left\{\begin{array}{c}x\left(1\right)=a-b\hfill \\ y\left(1\right)=0\hfill \end{array}\right.\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\left\{\begin{array}{c}x\left(\frac{3}{2}\right)=0\hfill \\ y\left(\frac{3}{2}\right)=-d\hfill \end{array}\right.\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}.\end{array}$$

#### 2.1.1. CS Curve

**Figure 5.**Geometric meaning of the parameters $a,b,c$ and d of a CS curve. $B(a+b,0)$ and ${B}^{\prime}(a-b,0)$ are the points determined by $t=0$ and $t=1$, respectively; $D(0,c+d)$ and ${D}^{\prime}(0,c-d)$ are the points determined by $t=\frac{1}{2}$ and $t=\frac{3}{2}$, respectively; $A(a,0)$ is the middle point of the segment $B{B}^{\prime}$; $C(0,c)$ is the middle point of the segment $D{D}^{\prime}$. (

**a**) The length of the segment $B{B}^{\prime}$ is $2b$. The length of the segment $D{D}^{\prime}$ is $2d$. (

**b**) The length of the segment $OA$ is $\left|a\right|$. The length of the segment $OC$ is $\left|c\right|$.

**Figure 6.**Some examples of CS curves. In each subfigure, three parameters are fixed and the remaining one changes. The dotted, dashed and solid curves correspond to the first, second and third value of the variable parameter, respectively. (

**a**) Fixed parameters: $b=1$, $c=0.3$, $d=1$. Variable parameter: $a=0,0.5$ and $1.1$. (

**b**) Fixed parameters: $a=0.5$, $c=0$, $d=1$. Variable parameter: $b=0.6,1$ and 2. (

**c**) Fixed parameters: $a=0.5$, $b=1$, $d=1$. Variable parameter: $c=0,0.5$ and 1. (

**d**) Fixed parameters: $a=1$, $b=0.8$, $c=0.4$. Variable parameter: $d=0.5,1$ and $1.5$.

- The curve has a concavity around B if and only if $\frac{a}{b}<-\frac{1}{2}$;
- The curve has a ‘flat side’ around B if and only if $\frac{a}{b}=-\frac{1}{2}$;
- The curve has a ‘flat side’ around ${B}^{\prime}$ if and only if $\frac{a}{b}=\frac{1}{2}$;
- The curve has a concavity around ${B}^{\prime}$ if and only if $\frac{a}{b}>\frac{1}{2}$;
- The curve has a concavity around D if and only if $\frac{c}{d}<-\frac{1}{2}$;
- The curve has a ‘flat side’ around D if and only if $\frac{c}{d}=-\frac{1}{2}$;
- The curve has a ‘flat side’ around ${D}^{\prime}$ if and only if $\frac{c}{d}=\frac{1}{2}$;
- The curve has a concavity around ${D}^{\prime}$ if and only if $\frac{c}{d}>\frac{1}{2}$.

**Figure 7.**Some examples of CS curves with the values of the indicators $\frac{a}{b}$ and $\frac{c}{d}$ of concavity. (

**a**) $\frac{a}{b}=1.2$ and $\frac{c}{d}=0.3$. (

**b**) $\frac{a}{b}=0.5$ and $\frac{c}{d}=0$. (

**c**) $\frac{a}{b}=-0.3$ and $\frac{c}{d}=2$. (

**d**) $\frac{a}{b}=-0.7$ and $\frac{c}{d}=0.7$.

#### 2.1.2. CC Curve

- The curve has a concavity around B if and only if $\frac{a}{b}<-\frac{1}{2}$;
- The curve has a ‘flat side’ around B if and only if $\frac{a}{b}=-\frac{1}{2}$;
- The curve has a ‘flat side’ around ${B}^{\prime}$ if and only if $\frac{a}{b}=\frac{1}{2}$;
- The curve has a concavity around ${B}^{\prime}$ if and only if $\frac{a}{b}>\frac{1}{2}$.

**Figure 8.**Geometric meaning of the parameters $a,b$ and d of a CC curve. $B(a+b,0)$ and ${B}^{\prime}(a-b,0)$ are the points determined by $t=0$ and $t=1$, respectively; $D(0,d)$ and ${D}^{\prime}(0,-d)$ are the points determined by $t=\frac{1}{2}$ and $t=\frac{3}{2}$, respectively; $A(a,0)$ is the middle point of the segment $B{B}^{\prime}$. (

**a**) The length of the segment $B{B}^{\prime}$ is $2b$. The length of the segment $D{D}^{\prime}$ is $2d$. (

**b**) The length of the segment $OA$ is $\left|a\right|$.

**Figure 9.**Some examples of CC curves. In each subfigure, three parameters are fixed and the remaining one changes. The dotted, dashed and solid curves correspond to the first, second and third value of the variable parameter, respectively. (

**a**) Fixed parameters: $b=0.7$, $c=0.3$, $d=1$. Variable parameter: $a=-1,0$ and 1. (

**b**) Fixed parameters: $a=0.8$, $c=0.5$, $d=1$. Variable parameter: $b=0.6,1$ and 2. (

**c**) Fixed parameters: $a=0.5$, $b=1$, $d=1$. Variable parameter: $c=-1,0$ and 1. (

**d**) Fixed parameters: $a=1$, $b=0.5$, $c=0.5$. Variable parameter: $d=0.5,1$ and $1.5$.

#### 2.2. Symmetries and Invariants

#### 2.2.1. CS Curve

- The transformation $a\mapsto -a$ gives the symmetric curve with respect to the y axis. Thus, if $a=0$, the curve is symmetric with respect to the y axis.
- The transformation $c\mapsto -c$ gives the symmetric curve with respect to the x axis. Thus, if $c=0$, the curve is symmetric with respect to the x axis.
- Each one of the transformations $b\mapsto -b$ and $d\mapsto -d$ leaves the curve invariant. This reason, together with the degenerate cases $b=0$ or $d=0$, justifies our choice to consider b and d as positive numbers made in (6).
- The transformation $(a,c)\mapsto (-a,-c)$ gives the symmetric curve with respect to the origin (or, equivalently, the curve rotated by a straight angle with its center as the origin).
- The transformation $(a,b,c,d)\mapsto (c,d,a,b)$ gives the symmetric curve with respect to the line $y=x$. Thus, if $a=c$ and $b=d$, the curve is symmetric with respect to the line $y=x$.

#### 2.2.2. CC Curve

- The curve is always symmetric with respect to the x axis.
- Each one of the transformations $(a,b)\mapsto (-a,-b)$, $(a,c)\mapsto (-a,-c)$ and $(a,d)\mapsto (-a,-d)$ gives the symmetric curve with respect to the y axis (or, equivalently, the curve rotated by a straight angle with its center at the origin, as in the previous remark).
- Each one of the transformations $(b,c)\mapsto (-b,-c)$, $(b,d)\mapsto (-b,-d)$ and $(c,d)\mapsto (-c,-d)$ leaves the curve invariant. For this reason, we assume that b and d are non-negative numbers in (6).

## 3. Validation of the Models

## 4. Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Further Analysis of CS and CC Curves

#### Appendix A.1. Special Cases

#### Appendix A.1.1. CS Curve

- For $a=c=0$ and $b=d=r$, the curve is the circle centered in the origin with radius r.
- For $a=c=0$, the curve is the ellipse centered in the origin with semiaxes b and d.

#### Appendix A.1.2. CC Curve

- For $a=c=0$ and $b=d=r$, the curve is the circle centered in the origin with radius r.
- For $a=c=0$, the curve is the ellipse centered in the origin with semiaxes b and d.
- For $a=d=0$, the curve is a Lissajous curve.
- For $a=d=0$ and $b=c=1$, the curve is the lemniscate of Gerono.
- For $a=b=c=d$, the curve is a cardioid.
- For $a=c$ and $b=d$, the curve is a limaçon.
- For $a=c=2b=2d$, the curve is a limaçon trisectrix.
- For $a=0$ and $c=d$, the curve is a translation of a piriform quartic.
- For $a=b=-c=d$, the curve is a translation of a deltoid.
- For $a=-c$ and $b=d$, the curve is a translation of a hypotrochoid.
- For $a=1,b=\frac{1}{2},c=-1$ and $d=\frac{1}{2}$, the curve is a translation of a regular trifolium.

#### Appendix A.2. Elliptic Fourier Descriptors

#### Appendix A.2.1. CS Curve

#### Appendix A.2.2. CC Curve

#### Appendix A.3. Simple Curves

#### Appendix A.3.1. CS Curve

**Theorem A1.**

**Proof.**

- If ${a}^{2}{d}^{2}+{b}^{2}{c}^{2}=4{a}^{2}{c}^{2}$ holds, Equation (A5) means $sin\left(\frac{{t}_{1}-{t}_{2}}{2}\pi \right)=\pm 1$. Interchanging, eventually, ${t}_{1}$ with ${t}_{2}$, we can suppose that $sin\left(\frac{{t}_{1}-{t}_{2}}{2}\pi \right)=1$, and then, taking into account that ${t}_{1},{t}_{2}\in [0,2[$, we have ${t}_{1}-{t}_{2}=1$. We write $s:={t}_{1}+{t}_{2}$. So we have that ${t}_{1}=\frac{s+1}{2}$ and ${t}_{2}=\frac{s-1}{2}$. Imposing that ${t}_{1},{t}_{2}\in [0,2[$, we find that$$s\in [-1,3[\phantom{\rule{0.166667em}{0ex}}\cap \phantom{\rule{0.166667em}{0ex}}[1,5[\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}[1,3[.$$For symmetry reasons (see Section 2.2), we can confine ourselves to the case $a>0$ and $c<0$; therefore, by (A4), $s\in \phantom{\rule{0.277778em}{0ex}}]0,1[,$ which is in contradiction to (A6).To summarize, in the case ${a}^{2}{d}^{2}+{b}^{2}{c}^{2}=4{a}^{2}{c}^{2}$, the system (A2) has no solutions, i.e., the curve is simple.

#### Appendix A.3.2. CC Curve

**Theorem A2.**

- (i)
- $c=0$;
- (ii)
- $a=0$ and $d\ge \left|c\right|$;
- (iii)
- $a,b,c\ne 0$, $d\ge \left|c\right|$ and $\frac{b}{2a}\left(\frac{b}{a}-\frac{d}{c}\right)\le 0$;
- (iv)
- $a,b,c\ne 0$, $d\ge \left|c\right|$ and $\frac{b}{2a}\left(\frac{b}{a}-\frac{d}{c}\right)\ge 1$;
- (v)
- $a,b,c\ne 0$, $d\ge \left|c\right|$, $0<\frac{b}{2a}\left(\frac{b}{a}-\frac{d}{c}\right)<1$ and $\frac{b}{2\left|a\right|}\ge \left|\frac{b}{a}-\frac{d}{c}\right|$.

**Proof.**

- $c\ne 0$, $d<\left|c\right|$;
- $a,c\ne 0$, $0<\frac{b}{2a}(\frac{b}{a}-\frac{d}{c})<1$ and $\frac{b}{2\left|a\right|}<\left|\frac{b}{a}-\frac{d}{c}\right|$;
- $b=0$.

**Figure A2.**Some examples of nonsimple CC curves. (

**a**) $a=1$, $b=0.5$, $c=1.5$, $d=1$. (

**b**) $a=0.5$, $b=1.5$, $c=-1.5$, $d=1$. (

**c**) $a=1$, $b=0$, $c=1$, $d=1$. (

**d**) $a=-1.5$, $b=1$, $c=1$, $d=1$. (

**e**) $a=-1.5$, $b=1$, $c=1.5$, $d=1$.

#### Appendix A.4. Area of the Enclosed Surface

- $A=\pi bd$ for CS curves (so it does not depend on the parameters $a,c$ and it is the same as the area of the surface inside an ellipse with semiaxes b and d);
- $A=\pi \phantom{\rule{-0.166667em}{0ex}}\left(\frac{ac}{2}+bd\right)$ for CC curves.

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**Figure 1.**Some contours (in white) of prostates (internal regions) in the axial plane delineated by the radiologist representing the most common shapes (the images are rotated clockwise 90 degrees with respect the usual radiologist’s visualization because of a software setting).

**Figure 10.**From left to right: one of the MR images, the same image cropped and enlarged containing the prostate and finally the contour (in white) extracted by the radiologist.

**Figure 11.**In each row, from the left to the right: a ground truth (in white) and the CS and CC curves (in dashed red) obtained by the fitting process, respectively. For the best view of the figure, please refer to the original version.

**Figure 12.**In each picture, the solid black curve is the ground truth. The dashed red curves are CC curves and represent the following: (

**a**) the first initialization (ellipse); (

**b**) the final curve obtained with the first initialization (${d}_{m}=1.34$ mm, ${d}_{H}=4.69$ mm, $DSC=92.0\%$); (

**c**) the second initialization; (

**d**) the final curve obtained with the second initialization (${d}_{m}=0.48$ mm, ${d}_{H}=2.45$ mm, $DSC=96.8\%$).

**Figure 13.**The solid black curves are ground truths and the dashed red curves are the model curves found by the method (in each figure from the left to the right: CS and CC model). The two comparisons show some cases in which a different model gives a result better than the other one. (

**a**) CS model (${d}_{m}=0.39$ mm, ${d}_{H}=2.45$ mm, $DSC=97.2\%$) and CC model (${d}_{m}=0.66$ mm, ${d}_{H}=4.06$ mm, $DSC=93.3\%$). (

**b**) CS model (${d}_{m}=0.68$ mm, ${d}_{H}=2.92$ mm, $DSC=93.8\%$) and CC model (${d}_{m}=0.26$ mm, ${d}_{H}=1.45$ mm, $DSC=98.0\%$).

**Figure 14.**An example of atypical prostate contour (in solid black curve) and the model fittings (from the left to the right: CS model and CC model) in dashed red curves.

**Table 1.**Mean and standard deviation of the three metrics, mean distance ${d}_{m}$, Hausdorff distance ${d}_{H}$ and Dice similarity coefficient $DSC$, of the fitting of the different models.

Model | ${\mathit{d}}_{\mathit{m}}$ | ${\mathit{d}}_{\mathit{H}}$ | $\mathit{DSC}$ |
---|---|---|---|

Best CS-CC | $0.56\pm 0.32$ mm | $2.36\pm 1.67$ mm | $96.7\%\pm 3.1\%$ |

CS | $0.58\pm 0.34$ mm | $2.42\pm 1.71$ mm | $96.6\%\pm 3.3\%$ |

CC | $0.62\pm 0.35$ mm | $2.51\pm 1.71$ mm | $96.4\%\pm 3.4\%$ |

Deformed superellipse | $0.61\pm 0.32$ mm | $2.44\pm 1.63$ mm | $96.5\%\pm 2.7\%$ |

**Table 2.**The computational cost of the fitting process of our models and the deformed superellipse model.

Model | Time Required by the Fitting Process |
---|---|

CS or CC (a single model) | $2.5\pm 1.8$ s |

Best between CS and CC models | $5.0\pm 3.5$ s |

Deformed superellipse | $36.5\pm 17.3$ s |

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## Share and Cite

**MDPI and ACS Style**

Corso, R.; Comelli, A.; Salvaggio, G.; Tegolo, D.
New Parametric 2D Curves for Modeling Prostate Shape in Magnetic Resonance Images. *Symmetry* **2024**, *16*, 755.
https://doi.org/10.3390/sym16060755

**AMA Style**

Corso R, Comelli A, Salvaggio G, Tegolo D.
New Parametric 2D Curves for Modeling Prostate Shape in Magnetic Resonance Images. *Symmetry*. 2024; 16(6):755.
https://doi.org/10.3390/sym16060755

**Chicago/Turabian Style**

Corso, Rosario, Albert Comelli, Giuseppe Salvaggio, and Domenico Tegolo.
2024. "New Parametric 2D Curves for Modeling Prostate Shape in Magnetic Resonance Images" *Symmetry* 16, no. 6: 755.
https://doi.org/10.3390/sym16060755