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

Abstract

Geometric shape models often help to extract specific contours in digital images (the segmentation process) with major precision. Motivated by this idea, we introduce two models for the representation of prostate shape in the axial plane of magnetic resonance images. In more detail, the models are two parametric closed curves of the plane. The analytic study of the models includes the geometric role of the parameters describing the curves, symmetries, invariants, special cases, elliptic Fourier descriptors, conditions for simple curves and area of the enclosed surfaces. The models were validated for prostate shapes by fitting the curves to prostate contours delineated by a radiologist and measuring the errors with the mean distance, the Hausdorff distance and the Dice similarity coefficient. Validation was also conducted by comparing our models with the deformed superellipse model used in literature. Our models are equivalent in fitting metrics to the deformed superellipse model; however, they have the advantage of a more straightforward formulation and they depend on fewer parameters, implying a reduced computational time for the fitting process. Due to the validation, our models may be applied for developing innovative and performing segmentation methods or improving existing ones.

1. Introduction

Motivated by the possibility to geometrically describe prostate shapes in magnetic resonance (MR) images, this paper presents a study, under several mathematical aspects, of two parametric closed curves of the plane defined by Equations (6) and (7). Making use of these curves, we introduce new 2D models (Equation (8)) for prostate shapes in the axial plane of MR images, which we validate by comparing their shapes with prostate contours (ground truths) delineated by a radiologist with more than 10 years of experience.

Geometric shape models of prostates can be helpful in low-level image analysis, like the segmentation phase, i.e., the process of extraction of prostate contours in digital images like MR, transrectal ultrasound (TRUS) or computed tomography (CT) images. Image segmentation is necessary for prostate volume estimation, which has a crucial role in evaluating and managing prostate disorders, both malignant and benign [1,2]. For example, in benign prostatic hyperplasia (BHP), the knowledge of total prostatic volume, in conjunction with serum prostatic specific antigen, is necessary in the calculation of prostate specific antigen density, a clinically relevant parameter for decision making [3]. Moreover, prostate volume has an essential role in prostate cancer (PCa) (several studies showed a relationship between BHP and incidence of PCa [4,5,6]), and it is also routinely calculated for prognostic factors and treatment purposes. It is crucial for biopsy or surgical planning, focal ablative treatments, radiation therapy or new minimally invasive therapies like high-intensity focused ultrasound and cryotherapy [7,8].

Prostate volume is usually estimated on TRUS, CT or MR images using the ellipsoid formula [9]. However, ellipsoid formula volume estimation is often inaccurate due to the shape variability of the prostate gland (especially in the case of BHP with median lobe enlargement [10]). Manual image segmentation is the best way to obtain a correct prostate volume estimation; however, this method is time-consuming, requires experience and is strongly operator-dependent [11]. Semiautomatic and automatic segmentation techniques in MR, TRUS and CT images have been developed (see, for instance [7,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27]). Recently, deep learning methods for semiautomatic or automatic segmentation of the prostate gland on magnetic resonance images have shown good performance [10,28]. Besides medical applications, deep learning segmentation is employed in various other fields like agriculture [29,30], biology [31,32], robotics [33,34] and autonomous driving [35,36]. Popular methods for deep learning segmentation are based on convolutional neural networks, encoder–decoder and autoencoder models or generative adversarial networks [37]. More generally, artificial intelligence techniques are used in the medical field not only for segmentation but also for classification and prediction (radiomics) [38,39,40,41,42].

The prostate shape in the axial plane of an MR image may change because the prostate consists of more parts or because it is affected by BHP, see [43]. Figure 1 displays examples of the standard shapes (manually extracted by the radiologist).

Curves are often used in computer-aided geometric design to model specific objects (see [32,44,45,46,47]). Furthermore, for the specific case of prostate shape, some models are proposed in the literature. Ladak et al. [16] employed a model made of points which are connected with anatomic data and cubic interpolation. Jeong and Radke [48] resampled the contours by interpolating elliptic Fourier descriptor coefficients. Ellipses have been used in [13,27] for segmentation in TRUS images and in [17,18] for MR images. Makni et al. [19] modeled a zone of the prostate with catenary curves between two points. Catenary curves have equations $y=\frac{a}{2}(e^{\frac{x}{a}}+e^{-\frac{x}{a}})+c$ and have an appearance similar to the parabolas.

Ellipses and catenary curves are not suitable for certain prostate shapes like those shapes with a concavity (see Figure 1). A geometric curve that can assume a concavity is the deformed superellipse, which has been employed to describe prostate shapes and for segmentation in TRUS images by Gong et al. [12] with very good results. Superellipses have found applications in several other fields [49,50,51,52,53,54,55,56,57,58]), and the following equation defines them:

$${\left|\frac{x}{a_x}\right|}^{\frac{2}{ϵ}}+{\left|\frac{y}{a_y}\right|}^{\frac{2}{ϵ}}=1$$

(1)

where $a_x,a_y,ϵ>0$. The parametric form of (1) is

$$\{\begin{array}{c}x\left(t\right)=a_x{|cos\left(t\pi \right)|}^{ϵ}\mathrm{sgn}(cos\left(t\pi \right))\hfill \\ y\left(t\right)=a_y{|sin\left(t\pi \right)|}^{ϵ}\mathrm{sgn}(sin\left(t\pi \right))\hfill & \end{array}0\le t\le 2,$$

(2)

where sgn is the sign function. In Figure 2a, some superellipses with different choices of $ϵ$ are shown. A superellipse can be deformed with the following transformations [59]:

• Linear tapering along y axis

  $$\{\begin{array}{cc}{x}^{\prime }=\left(\frac{T}{a_y}y+1\right)x\hfill & \\ y^{\prime }=y\hfill & \end{array}$$

  (3)

  where $-1\le T\le 1$. The effect of this transformation is shown in Figure 2b.
• Circular bending along the y axis

  $$\{\begin{array}{cc}{x}^{\prime }=\left(\frac{a_y}{\beta }-y\right)sin\left(\frac{x}{\frac{a_y}{\beta }-y}\right)\hfill & \\ y^{\prime }=\frac{a_y}{\beta }-\left(\frac{a_y}{\beta }-y\right)cos\left(\frac{x}{\frac{a_y}{\beta }-y}\right)\hfill & \end{array}$$

  (4)

  where $-1\le \beta \le 1$, $\beta \ne 0$. The effect of this transformation is shown in Figure 2c.
• Rotation and translation

  $$\{\begin{array}{c}{x}^{\prime }=xcos\alpha -ysin\alpha +x_0\hfill \\ y^{\prime }=xsin\alpha +ycos\alpha +y_0\hfill \end{array}$$

  (5)

  where $\alpha \in [0,2\pi [$, $x_0,y_0\in \mathbb{R}$.

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) $ϵ=0.7, 1$ and $1.3$. (b) $T=0, 0.2$ and $0.5$ ($ϵ=1$). (c) $\beta =0.01, 0.6$ and $0.8$ ($ϵ=1$).

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) $ϵ=0.7, 1$ and $1.3$. (b) $T=0, 0.2$ and $0.5$ ($ϵ=1$). (c) $\beta =0.01, 0.6$ and $0.8$ ($ϵ=1$).

The model defined by composing Equations (1) and (3)–(5) is the deformed superellipse [12], and we used it as an object of comparison for the validation of our models. The proof of goodness was obtained by fitting all of our models to the 1227 prostate contours delineated by the radiologist. Moreover, the fitting performance was measured in terms of mean distance, Hausdorff distance and Dice similarity coefficient. The results show that our models are comparable to the deformed superellipse model from the point of view of error fitting. However, the main quality of our models is the simpler formulation; indeed, it can see in Section 2 that the models are defined by few and not complex equations. In addition, our models depend on a smaller number of parameters, and this implies a reduced cost in their implementation.

Other geometric descriptions of prostate shapes can be found in the literature. Saroul et al. [20] considered the model defined by only (1), (3) and (5). Tutar et al. [24] employed superquadrics, which are a 3D version of superellipses. Remaining in the 3D setting, prostate shape was modeled by Tutar [23] with spherical harmonics.

This paper is organized as follows. In Section 2, we present our models based on two parametric closed curves of the real plane (that we call CS and CC curves), and we study their main characteristics. In Section 3, we describe the validation process on our dataset of prostate shapes in the axial plane of MR images. The results of the validation are shown and discussed in Section 4 along with a comparison with the deformed superellipse model. Concluding remarks are in Section 5. Furthermore, in Appendix A, we report more mathematical studies of the curves, like special cases, elliptic Fourier descriptors, conditions for simple curves and area of the enclosed surfaces.

2. Our Models

To describe the contour of the prostate in the axial plane, we propose two types of curves as base models. We then formulate our general models considering any possible rotations and translations in the plane.

The first curve we present is defined in terms of parameters $a,b,c,d\in \mathbb{R}$ with $b,d>0$ as follows:

$$\{\begin{array}{c}x\left(t\right)=(acos(t\pi )+b)cos\left(t\pi \right)\hfill \\ y\left(t\right)=(csin(t\pi )+d)sin\left(t\pi \right)\hfill & \end{array}0\le t\le 2.$$

(6)

The parameters b and d are chosen to be positive because of symmetry reasons (see Section 2.2) and because for $b=0$ or $d=0$, the curve is degenerate. In Figure 3, representative examples of simple curves as defined by (6) are provided. A curve is simple if it does not intersect itself (see also Appendix A.3); we show a picture of simple curves since they are the only cases of interest for the application we deal with in this paper. We named Equation (6) the CS curve (the term CS is an abbreviation to indicate the cosine function in the brackets of the expression of x and the sine function in the brackets of the expression of y).

The second curve is defined for $a,b,c,d\in \mathbb{R}$ and $b,d\ge 0$ as follows:

$$\{\begin{array}{c}x\left(t\right)=(acos(t\pi )+b)cos\left(t\pi \right)\hfill \\ y\left(t\right)=(ccos(t\pi )+d)sin\left(t\pi \right)\hfill & \end{array}0\le t\le 2.$$

(7)

We are actually interested in the cases where none of the conditions $a=b=0$ or $b=c=0$ or $c=d=0$ are satisfied, because each of them corresponds to a degenerate case. In Figure 4, representative examples of simple curves defined by (7) are depicted. With a meaning similar to the previous curve, we thus named Equation (7) the CC curve.

To the best of our knowledge, curves (6) and (7) have not been previously studied. Moreover, they include ellipses in the particular case $a=c=0$ and other known curves, which are listed in Appendix A.1.

The models for prostate shapes which we propose in this paper are defined starting from (6) and (7) and considering any possible rotations and translations. The models are formulated more accurately in the following way:

$$\{\begin{array}{c}\tilde{x}\left(t\right)=x\left(t\right)cos\alpha -y\left(t\right)sin\alpha +x_0\hfill \\ \tilde{y}\left(t\right)=x\left(t\right)sin\alpha +y\left(t\right)cos\alpha +y_0\hfill & \end{array}0\le t\le 2.$$

(8)

where $\alpha \in [0,2\pi [$, $x_0,y_0\in \mathbb{R}$ and $x\left(t\right),y\left(t\right)$ are as in (6) or as in (7). These general models are then obtained by the rotation of angle $\alpha$ followed by a shift of $x_0$ along the x axis and a shift of $y_0$ along the y axis of a CS or a CC curve. They depend on seven parameters (a, b, c, d, $\alpha$, $x_0$, $y_0$). The value of $\alpha$ can be confined in the interval $[0,\pi [$ since, as observed in Section 2.2, by changing the sign of the parameters a and b, one obtains the curve rotated by a straight angle around the origin.

In the next subsections, we describe various aspects of the two types of curves, dedicating a different paragraph to each one. Further analyses are reported in Appendix A.

2.1. Role of the Parameters

We start by discussing geometric meanings of the parameters $a,b,c$ and d of the models. We refer to Figure 5, Figure 6 and Figure 7 for the considerations about CS curves and to Figure 8 and Figure 9 for the considerations about CC curves. The first observations we make are the following particular points of the curves determined by $t=0,\frac{1}{2},1$ and $\frac{3}{2}$.

 

CS curve:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}x\left(0\right)=a+b\hfill \\ y\left(0\right)=0\hfill \end{array}\right.,\left\{\begin{array}{c}x\left(\frac{1}{2}\right)=0\hfill \\ y\left(\frac{1}{2}\right)=c+d\hfill \end{array}\right.,\left\{\begin{array}{c}x\left(1\right)=a-b\hfill \\ y\left(1\right)=0\hfill \end{array}\right.,\left\{\begin{array}{c}x\left(\frac{3}{2}\right)=0\hfill \\ y\left(\frac{3}{2}\right)=c-d\hfill \end{array}\right..\end{array}$$

(9)

 

CC curve:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}x\left(0\right)=a+b\hfill \\ y\left(0\right)=0\hfill \end{array}\right.,\left\{\begin{array}{c}x\left(\frac{1}{2}\right)=0\hfill \\ y\left(\frac{1}{2}\right)=d\hfill \end{array}\right.,\left\{\begin{array}{c}x\left(1\right)=a-b\hfill \\ y\left(1\right)=0\hfill \end{array}\right.,\left\{\begin{array}{c}x\left(\frac{3}{2}\right)=0\hfill \\ y\left(\frac{3}{2}\right)=-d\hfill \end{array}\right..\end{array}$$

(10)

Taking into account (9) and (10), we can state that for every type of curve, b represents the half distance of the points determined by $t=0$ and $t=1$ (i.e., the points B and $B^{\prime }$ in Figure 5 and Figure 8, respectively) and d represents the half distance of the points determined by $t=\frac{1}{2}$ and $t=\frac{3}{2}$ (i.e., the points D and $D^{\prime }$ in Figure 5 and Figure 8, respectively). Hence, b and d play a role as indicators of thickness along the two axes, as it is also shown in Figure 6b,d and Figure 9b,d.

More rigorously, looking to (6) and (7), we note that if b increases, then $x\left(t\right)$ for $0\le t<\frac{1}{2}$ or for $\frac{3}{2}<t\le 2$ increases, and this means that the part of the curve from $D^{\prime }$ to D (taken in an anticlockwise way) moves to the right if b increases. Conversely, $x\left(t\right)$ for $\frac{1}{2}<t<\frac{3}{2}$ decreases if b increases, and this means that the part of the curve from D to $D^{\prime }$ (taken in an anticlockwise way) moves to the left if b increases. A similar argumentation can be applied to the parameter d. Concerning the parameters a and c, the geometric meanings vary between the two types of curves, so we make the related considerations separately below.

2.1.1. CS Curve

From Equation (9) we note that the middle point of the segment $B{B}^{\prime }$, labeled A in Figure 5, has an abscissa equal to a. This suggests that the parameter a regulates the stretching along the x axis (it can also be seen in Figure 6a). For the rigorous proof of the statement, we use Equation (6) and, in particular, that $x\left(t\right)=a{cos}^2\left(t\pi \right)+bcos\left(t\pi \right)$: for a fixed $0\le t\le 2$ except for $t=\frac{1}{2}$ and $t=\frac{3}{2}$, if a increases, then $x\left(t\right)$ increases too. This means that if a increases, the curve stretches to the right of the plane.

A similar observation can be made for the parameter c. The middle point of the segment $D{D}^{\prime }$, labeled C in Figure 5b, has an ordinate equal to c, and furthermore, the parameter c regulates the stretching along the y axis, as shown in Figure 6c.

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 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 $1.5$.

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 $1.5$.

Other parameters of particular interest are $\frac{a}{b}$ and $\frac{c}{d}$. The first value indicates if the curve has a concavity (like in Figure 7a,d) or a ‘flat side’ on the right/left part (like in Figure 7b), while the second value specifies the presence of a concavity (like in Figure 7c) or a ‘flat side’ on the top/bottom part. More precisely, see the following:

• 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 $\frac{c}{d}>\frac{1}{2}$.

Furthermore, the concavity is more accentuated for greater values of $|\frac{a}{b}|$ (respectively, $|\frac{c}{d}|$). The statements above can be proved by studying the signed curvature $k\left(t\right)=\frac{x^{\prime }\left(t\right)y^{\prime \prime }\left(t\right)-x^{\prime \prime }\left(t\right)y^{\prime }\left(t\right)}{{(x^{\prime }{\left(t\right)}^2+y^{\prime }{\left(t\right)}^2)}^{\frac{3}{2}}}$ for $t=0,\frac{1}{2},1,\frac{3}{2}$ [60]. For instance, $k\left(0\right)=\frac{2a+b}{d^2}$, so $k\left(0\right)<0$, i.e., there is a concavity around B if $\frac{a}{b}<-\frac{1}{2}$.

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 $\frac{c}{d}=0.7$.

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

For the parameter a, we can make the same remarks for the previous curve, i.e., a is the abscissa of the middle point of the segment $B{B}^{\prime }$, and moreover, it regulates the stretching along the x axis. The reference figures are Figure 8 and Figure 9a.

The parameter c plays a different role in the case of CS curves. To visualize the effect of this parameter on the curve, we refer to Figure 8 for the points and to Figure 9c. Increasing the value of c, the part of the curve from B to D moves up, the part of the curve from D to $B^{\prime }$ moves down, the part of the curve from $B^{\prime }$ to $D^{\prime }$ moves up and the part of the curve from $D^{\prime }$ to $B^{\prime }$ moves down (each part is taken in an anticlockwise way from the first to the second point excluding the two points).

To prove this statement, we use (7): for example, the part of the curve from B to D (excluding these two points) corresponds to the values of t from 0 and $\frac{1}{2}$; therefore, $cos\left(t\pi \right)>0$ and $sin\left(t\pi \right)>0$. Thus, for a fixed $0<t<\frac{1}{2}$, we have $cos\left(t\pi \right)sin\left(t\pi \right)>0$, so if c increases, then $y\left(t\right)$ increases. For the other parts of the curve, the $cos\left(t\pi \right)$ and $sin\left(t\pi \right)$ may have a different sign, so the argument must be consequently modified.

Moreover, as for CS curves, the parameter $\frac{a}{b}$ is an indicator of concavity:

• 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 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 $1.5$.

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

Here we discuss the actions of some transformations on the parameters and general symmetries of the curves. Parts of the transformations constitute invariants, which can limit the variability of some parameters. The statements below can be easily proved directly from the expressions (6) and (7) and taking into account basic properties of the sine and cosine functions.

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

This section is devoted to the assessment of the models for the representation of prostate shape in the axial plane of MR images. Thanks to the results we obtained (see Section 4), we believe that these models may be useful in specific fields of image analysis (such as segmentation, classification, recognition).

Our dataset is made of 1227 slices of prostate MR images coming from a total of 50 patients. The slices have a resolution of $672\times 672$ and a pixel size of 0.27 mm × 0.27 mm, and the distance between two consecutive slices is 2.5 mm. Figure 10 shows an example of an MR image on the left, a cropped and enlarged version of it in the center and the prostate contour (ground truth) delineated by the radiologist on the right.

As one can see, the prostate contours in Figure 1 are very similar, with eventual rotations, to the examples of model curves in Figure 3 and Figure 4. Of course, visual correspondence is not sufficient to check if a model well represents an object. Therefore, we proceeded to a numerical validation of the model (8). We point out that here we only take care of the validation of the models. Future works will be oriented to their service in a segmentation process.

The validation consisted of three parts: the extraction of the prostate contours in MR images made by a radiologist with more than 10 years of experience, the model fitting and its performance valuation.

First of all, the expert manually extracted the prostate contours in each of the 1227 slice images in our database. Then, the model (8) was validated on the ground truths one time for each of curves (6) and (7). In other words, following the method described below, we found the best CS and CC curves that approximate the given contour. Finally, we selected the best one of the two results for the given contour. Figure 11 shows some ground truths and the relative fittings with CS and CC curves.

In detail, the approximation process was set as follows. We denote by $C_p$ a prostate contour as a set of points $(\tilde{x},\tilde{y})$ in a 2D digital image and by $C_m=C_m(a,b,c,d,\alpha ,x_0,y_0)$ the model curve given by (8) (and (6) or (7), depending on the type of curve). For a given contour $C_p$ and each of the two types of curve (CS and CC), we searched the parameters $a,b,c,d,\alpha ,x_0,y_0$ of the model $C_m=C_m(a,b,c,d,\alpha ,x_0,y_0)$ to find which best fits the prostate contour, in the sense that it minimizes the following metric:

$$\begin{array}{c}\hfill d_m(C_p,C_m(a,b,c,d,\alpha ,x_0,y_0))=\underset{(\tilde{x},\tilde{y})\in C_p}{\mathrm{mean}}\underset{t\in [0,2]}{min}d((\tilde{x},\tilde{y}),(x\left(t\right),y\left(t\right)))\end{array}$$

(11)

that is, the mean distance between the generic point of $C_p$ and the closest point of the model $C_m(a,b,c,d,\alpha ,x_0,y_0)$ [12]. To further evaluate the fitting of the curve found by the minimization above, we also considered the Hausdorff distance [12,61] and the Dice similarity coefficient [62,63]. The Hausdorff distance

$$\begin{array}{c}\hfill d_H(C_p,C_m(a,b,c,d,\alpha ,x_0,y_0))=\underset{(\tilde{x},\tilde{y})\in C_p}{max}\underset{t\in [0,2]}{min}d((\tilde{x},\tilde{y}),(x\left(t\right),y\left(t\right)))\end{array}$$

is the maximum distance between the generic point of $C_p$ and the closest point of the model $C_m(a,b,c,d,\alpha ,x_0,y_0)$. The Dice similarity coefficient is

$$DSC=\frac{2|S_p\cap S_m|}{|S_p|+|S_m|},$$

where $|·|$ denotes the area of a surface, $S_p$ is the prostate surface (i.e., the surface enclosed by the contour $C_p$) and $S_m$ is the surface enclosed by the curve $C_m$.

The local minima for (11) were searched using the gradient descent method, iterating the process until the variation in (11) becomes negligible. In the procedure, all the curves were discretized, with 160 points individuated by uniformly distributed points $\left\{t_i\right\}$ in the interval $[0,2]$. The initialization of the process set the ellipse with axes of axial sizes $l_x$ and $l_y$ of the prostate contour and centered in the center of the prostate (see Figure 12a). In particular, this corresponds to the following parameters: $a=c=0$, $b=\frac{l_x}{2}$ and $d=\frac{l_y}{2}$. When the prostate has a concavity and the process starts with this ellipse, the gradient descent method may find a local minimum which does not give a perfect approximation (for example, this is the case in Figure 12b). For a better result, the method was repeated, and the initial curve taken was one with a concavity if the mean distance was higher than a fixed value at the end of the process (see Figure 12c). This way, a better fitting was usually found, as shown in Figure 12d. More precisely, for the second initialization, we set $a=b=\frac{l_x}{2}$, $c=0$ and $d=\frac{l_y}{2}$ for both the CC and the CS fitting.

Finally, we compared our models with the deformed superellipse model employed in [12], which we described in the Introduction. In particular, to fit the deformed superellipse model to the prostate contours, we followed an approach similar to the one explained for our models. Namely, employing the gradient descent method, for each of the 1227 ground truths, we searched the parameters $a_x$, $a_y$, $ϵ$, T, $\beta$, $\alpha$, $x_0$, $y_0$ of the deformed superellipse that best fit the given contour, i.e., that minimized the mean distance (11). The base curve (1) was implemented by (2) taking, as before, 160 points $\left\{t_i\right\}$ uniformly distributed in the interval $[0,2]$. Then the transformations (3)–(5) were applied. The initial choice of parameters was set corresponding to the initial curve in our models’ validation.

4. Results and Discussion

The results of the validation are reported in

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	$96.5\%\pm 2.7\%$

The models concerning the curves we have introduced are in bold text.

, where in each entry, the first and the second numbers represent the mean and the standard deviation of the corresponding metric over all the 1227 contours, respectively.

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

The models concerning the curves we have introduced are in bold text.

shows the time employed (on an Intel Core i5-9500 CPU running at 3 GHz) for fitting a given contour.

In the first row are the performance metrics when the best model between CS and CC is selected for a given contour. The model selected is the one that gives the lowest mean distance value (11). In Figure 13, two ground truths with the fittings of each model are shown. In particular, we chose two contour examples in which a different model provides the best approximation.

The performance of a single model (CS or CC) is detailed in the second and third rows of Table 1. Finally, the performance metrics of the deformed superellipse model are reported in the last row of Table 1.

The performance metrics of our models are comparable to ones of the deformed superellipse model (a slightly better result is obtained when the best curve between CS and CC is selected as the model). However, concerning the time employed for the fitting process, there is a significant difference. Finding a deformed superellipse that best fits a contour requires about 15 times the cost necessary for fitting a CS or CC model curve.

Furthermore, the deformed superellipse model depends on eight parameters ($a_x$, $a_y$, $ϵ$, T, $\beta$, $\alpha$, $x_0$ and $y_0$); in contrast, each of our models (CS or CC) (8) depends on seven parameters (a, b, c, d, $\alpha$, $x_0$ and $y_0$). Hence, this represents an important benefit of our models because it provides a more straightforward definition and it is also the reason for the reduced computational cost in the fitting. Indeed, to obtain a more precise fitting of deformed superellipses, the method was repeated for different fixed values of b, and this caused a greater time of execution. Moreover, CS and CC curves have a detailed description in Section 2 and in Appendix A. This could help to make specific and quantitative analyses of the prostate shape.

As a limitation, we remark that in some, but few, cases, the prostate has an atypical contour, different to the ones represented in Figure 1, which is not well approximated by the CS and CC curves. One of these outlier cases is shown in Figure 14 with its fittings.

Our goal in this paper was to present and validate two models for prostate shape. Due to the assessment of our models, we believe they can be adopted in prostate segmentation processes. A possible scheme for an automatic segmentation may be the following. A first and approximative prostate contour is searched with elementary image analysis techniques like texture evaluations or morphological operators or with more sophisticated methods like machine/deep learning [28]. Then the CS and CC models are fitted to the first contour to reproduce a more real shape. Alternatively, a second approach may be the definition of an active contour [64] whose shape is restricted to CS and CC curves.

5. Conclusions

We proposed two geometric 2D models for prostate shape in MR images. Each model is defined starting from a parametric curve and depends on a total of seven parameters. We described the parametric curves and validated the models on a database of prostate contours (ground truths) extracted by a radiologist. As part of the validation, we also compared to the deformed superellipse model employed for prostate segmentation in TRUS images in [12]. The results show that our models are comparable with the deformed superellipse model in terms of distance and overlapping errors. We highlight values for the mean distance equal to $0.56\pm 0.32$ mm and $0.61\pm 0.32$ mm and DSC values equal to $96.7\%\pm 3.1\%$ and $96.5\%\pm 2.7\%$ for the best of our models and the deformed superellipse model, respectively. However, our models have a more compact mathematical expression, depend on fewer parameters and require less time for contour fitting.