# Using Deep Learning and B-Splines to Model Blood Vessel Lumen from 3D Images

^{*}

## Abstract

**:**

## 1. Introduction

- Three-dimensional segmentation-based quantification;
- Model-fitting-based 2D lumen cross-sections quantification along an approximate centerline.

## 2. Materials and Methods

- Lower-extremities MRA volumes available within the PAVES Grand Challenge [37];
- Computer-synthesized cross-section images generated for purposes of this research.

- These datasets are characterized in the following subsections.

#### 2.1. PAVES MR Dataset

- Vesselness map thresholding to obtain binary lumen representation;
- Binary region thinning to produce a skeleton;
- Skeleton parsing [42] to identify the blood vessel tree branches between bifurcations;
- Approximating the skeleton branches by a differentiable function in 3D (to initialize their centerline).

#### 2.2. CAT08 Coronary CTA Dataset

#### 2.3. Image Formation Model and the Synthetic Dataset Generation

#### 2.4. Least-Squares Model Fitting for the PAVES Dataset

#### 2.5. Quantitative Evaluation of Contours Similarity

#### 2.6. CNN-Based B-Spline Model Parameter Estimation

**d**. The architecture is shown in Figure 6. Three 2D convolution layers are followed by a flattening layer and by four fully connected layers. The three convolutional layers feature 3 × 3 kernels, with a kernel number equal to 8 for each layer, “same” input padding type, and ReLU nonlinear activation function [52]. The flattening layer transforms the multichannel 2D feature maps into vectors. The vectors are then processed by three fully connected layers with 64, 32, and 16 neurons, respectively, and ReLU activation. The output layer estimates the 10 elements of vector

**d**and features a linear activation function. In total, the network has 119,290 trainable parameters and was implemented in keras (version 2.13.1).

#### 2.7. Methods and Tools for Statistical Analysis of the Results

## 3. Results

#### 3.1. PAVES Dataset

#### 3.1.1. LS Model Fitting to PAVES Data

#### 3.1.2. Generation of Synthetic Images for CNN Training

#### 3.1.3. CNN-Estimated B-Spline Model for PAVES Data

#### 3.2. CNN-Based Modeling of Artery Lumen in CAT08 Dataset

#### Analysis of Experimental Results for CAT08 Dataset

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CE | Contrast Enhanced |

CNN | Convolutional Neural Network |

CTA | Computed Tomography Angiography |

DSC | Dice Similarity Coefficient |

HU | Hounsfield units |

LS | Least Squares |

MAE | Mean Absolute Error |

MRA | Magnetic Resonance Angiography |

NURBS | Non-Uniform Rational B-splines |

PSF | Point Spread Function |

RMS | Root-Mean-Square |

SNR | Signal to Noise Ratio |

## Appendix A

#### Appendix A.1. Errors of Lumen Boundary Estimation from Segmented Image

**Figure A1.**($\mathbf{A}$) Lumen boundary crossing the $0x$ axis between sampling points ${x}_{i}$ and ${x}_{i+1}$, $i\in \{-N,\dots ,0,\dots ,N-1\}$, noiseless case. ($\mathbf{B}$) Computer-simulated image intensity profile along the $0x$ axis. Dashed red lines ended with dots: noisy image samples computed with the use of (A6) for $a=0$, $b=1$, ${x}_{e}=0.8{\Delta}_{s}$, $w=1.2{\Delta}_{s}$ and ${\sigma}_{\nu}\approx 0.032$ ($SNR=30$ dB). Continuous green line: image profile (A5) LS-fitted to the noisy samples, computed for optimum parameter values, $a=-0.01$, $b=1.01$, ${x}_{e}=0.84{\Delta}_{s}$ and $w=1.27{\Delta}_{s}$. ($\mathbf{C}$) Root-mean-square values of the model-based edge estimation error ${\u03f5}_{e}^{\left(2\right)}$ computed 500 times for each of the 48 combinations of $w/{\Delta}_{s}$ and SNR.

#### Appendix A.2. Errors of Lumen Boundary Estimation by Model Fitting

**q**be the parameter vector in (A5),

**q**= $(a,b,{x}_{e},w)$. Given the acquired noisy image samples ${I}_{a}\left(i\right)$, one can estimate its elements by minimising the sum of squared differences between the model ${I}_{x}(i{\Delta}_{s};\mathbf{q})$ and the corresponding samples

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**Figure 1.**Example 15 × 15-pixel cross-sections of coronary artery segments in CAT08, pixel size 0.45 × 0.45 mm.

**Figure 2.**Visualization of the PAVES b14 branch of vein–artery lumen based on contours obtained with different LS-identified image formation models, (

**A**) B-spline contours, (

**B**) circular lumen boundaries, (

**C**) overlay of the surfaces in (

**A**,

**B**).

**Figure 3.**(

**A**) Maximum intensity projection for PAVES dataset 5, showing TWIST (subtracted time-resolved acquisition) volume on the axial plane, left volunteer extremity. The arrow indicates a stenosis in the anterior tibial artery. (

**B**) Binary skeleton of the blood vessels, after parsing. The tibial artery branch was assigned code b14. (

**C**) Mosaic of 112 numbered cross-sections of the tibial artery, taken at 0.5-mm intervals along the centerline. An example of a coronal slice MRI of the volunteer’s right leg is shown in Figure S1.

**Figure 4.**Example contours marked by the three observers on coronary artery sections in the CAT08 dataset. Cross-sections were interpolated to $60\times 60$ pixel resolution to make their appearance similar to the example shown in Figure 4 of [38]. The pseudocolor palette was used to enhance the visibility of the intensity variations.

**Figure 5.**(

**A**) Geometry of an eight-parameter B-spline curve normalized to the scale factor R. (

**B**) Ideal lumen region in the cross-section image space. The shaded area represents the constant-intensity lumen region $\Omega $. (

**C**) Low-resolution noiseless image of the lumen on its background.

**Figure 7.**Example sixteen images from the CAT08 data test set. Red line: contours marked by an observer, green line: CNN-predicted contours. (

**A**) Observer 1, (

**B**) Observer 2, (

**C**) Observer 3.

**Figure 8.**Bland–Altman plot for $Are{a}_{Obs1}$ and $Are{a}_{CNN1}$ over the test set. Computed with the ODR option of the blandAltman() function from the $pyCompare$ library [56] to model and remove the multiplicative offset between each assay by orthogonal distance regression.

**Figure 9.**Example contours (light green lines) of the vein–artery lumen obtained via LS model fitting to $15\times 15$-pixel odd-numbered (1, 3, …) sections of branch b14 in the PAVES 05 dataset. Twelve-parameter (a, b, ${d}_{0},\dots ,{d}_{9}$) B-spline lumen model, ${\Delta}_{s}$ = 1.0mm, R = 11, $w/{\Delta}_{s}$ = 0.65. The points marked by a blue “x” symbol indicate the approximate centerline location.

**Figure 10.**(

**A**) B-spline contour example for Section 14 of branch b14 in PAVES 05. (

**B**) B-spline geometry and synthesized image. The control points are marked by numbered red dots. Control point ${\mathbf{C}}_{6}$ is located on the radial line at angle −36° (instead of 144°), which indicates a negative value of ${d}_{6}$. The LS-identified $\mathbf{d}$ vector for this image is (0.5, 0.23, 0.50, 0.25, 0.58, 0.54, −

**0.08**, 0.40, 0.77, 0.57).

**Figure 11.**Histogram of the differences between CNN-estimated and true values of B-spline parameters over the test sets of 1000 images. Training sets size: 8100 images. (

**A**) noiseless case, CNN trained for 154 epochs and (

**B**) noisy case, CNN trained for 204 epochs.

**Figure 12.**Examples of noiseless images of the test set, synthesized with the use of randomly generated B-spline parameters. Contours computed from ground-truth parameters and their CNN-predicted estimates are drawn using turquoise and red lines, respectively. Turquoise lines are practically invisible as they are closely matched by the red ones.

**Figure 13.**Example sections of PAVES b14 branch, taken at random. Turquoise lines: lumen contours obtained via LS model fitting. Red lines: contours computed using B-spline parameters predicted by CNN trained on synthesized noiseless images (

**A**) and noisy images (

**B**).

**Figure 14.**(

**A**) Example Section 27 of PAVES b14 branch with LS-predicted lumen contour. One can note random intensity variations. (

**C**) Sample of GIMP-generated fractal-like texture, visually similar to patterns observed in the PAVES data. (

**B**) The image in (

**C**) added to a noiseless image synthesized using the contour marked by the light green line.

**Figure 15.**Three examples of fitting B-spline curves to observer-marked contours (CAT08 data), to find B-spline parameters for CNN training. Dashed red line: contour marked by an observer, green line: B-spline curve. The numbers in boxes indicate the corresponding $mDist$ value.

**Figure 16.**Plot of the mean values and standard deviations (numbers in brackets) of mean absolute differences between CNN-estimated and Observer 3 contour-related B-spline parameters over the test set of 66 images (CAT08 data). CNN trained on 527 images for 199 epochs.

**Figure 17.**

**Left column**: Comparison of contours marked by the three observers and computed with the corresponding CNN-predicted B-spline parameters for cross-section 11 in the test set (CAT08 dataset 05, segment 07, Section 5). (

**A**) Observer 1, (

**B**) Observer 2, (

**C**) Observer 3.

**Middle column**: Geometric illustration of $mDist$ calculation.

**Right column**: Image regions used to compute the $DSC$.

**Figure 18.**Violin plots of $mDist$ and $DSC$ descriptors of differences between contours marked by different observers on CAT08 images (interobserver differences).

**Figure 19.**Violin plots of $mDist$ and $DSC$ descriptors of differences between contours marked by observers on CAT08 images and computed from CNN predictions (trained on the corresponding observer data) over the test set.

**Table 1.**Descriptors of observer-marked and CNN-predicted contours in Figure 17.

^{†}: Obs1 vs. CNN1,

^{‡}: Obs2 vs. CNN2, *: Obs3 vs. CNN3.

Obs 1 | Obs 2 | Obs3 | |
---|---|---|---|

$mDist$, mm | $0.203{}^{\u2020}$ | $0.209{}^{\u2021}$ | $0.119{}^{*}$ |

$DSC$, − | $0.870{}^{\u2020}$ | $0.871{}^{\u2021}$ | $0.901{}^{*}$ |

$Are{a}_{Obs}$, mm^{2} | 7.81 | 7.61 | 5.39 |

$Are{a}_{CNN}$, mm^{2} | 9.15 | 9.35 | 4.64 |

**Table 2.**Median/mean values of $mDist$ and $DSC$ descriptors of shape differences between contours marked by different observers on the same images in CAT08 over the test set.

Obs1 vs. Obs2 | Obs1 vs. Obs3 | Obs2 vs. Obs3 | |
---|---|---|---|

$mDist$, mm | 0.422/0.462 | 0.440/0.527 | 0.334/0.380 |

$DSC$, − | 0.813/0.771 | 0.809/0.743 | 0.841/0.832 |

**Table 3.**Median/mean values of $mDist$ and $DSC$ descriptors of shape differences between observer-marked and CNN-predicted contours for CAT08 images over the test set.

Obs1 vs. CNN1 | Obs2 vs. CNN2 | Obs3 vs. CNN3 | |
---|---|---|---|

$mDist$, mm | 0.284/0.333 | 0.336/0.405 | 0.298/0.335 |

$DSC$, − | 0.876/0.844 | 0.849/0.837 | 0.878/0.851 |

**Table 4.**Median/mean values for the contour area marked by observers $Are{a}_{Obs}$ and predicted by CNNs $Are{a}_{CNN}$ over the test set drawn from the CAT08 repository.

Obs 1 | Obs 2 | Obs 3 | |
---|---|---|---|

$Are{a}_{Obs}$, mm^{2} | 2.90/4.79 | 2.83/4.71 | 2.23/4.12 |

$Are{a}_{CNN}$, mm^{2} | 3.73/5.83 | 4.18/5.68 | 3.26/4.82 |

**Table 5.**p-values of the paired Wilcoxon test for $Are{a}_{Obs}$ and $Are{a}_{CNN}$ coefficients over the test set drawn from the CAT08 repository.

Observer | CNN | Observer vs. CNN | |
---|---|---|---|

Obs1 vs. Obs2 | 0.775 | 0.751 | − |

Obs1 vs. Obs3 | 0.085 | 0.034 | − |

Obs2 vs. Obs3 | 3.5 × 10^{−5} | 1.9 × 10^{−5} | − |

Obs1 vs. CNN1 | − | − | 1.4 × 10^{−8} |

Obs2 vs. CNN2 | − | − | 1.2 × 10^{−7} |

Obs3 vs. CNN3 | − | − | 9.7 × 10^{−9} |

**Table 6.**Mean values (in mm

^{2}) of the differences between paired $Are{a}_{Obs}$ and $Are{a}_{CNN}$ coefficients over the test set drawn from the CAT08 repository. Columns ‘None’, ‘Linear’, and ‘ODR’ correspond to detrending options used in $pyCompare\left(\right)$ to compare the distributions [56].

‘None’ | ‘Linear’ | ‘ODR’ | |
---|---|---|---|

Obs1 vs. CNN1 | −1.04 | −0.20 | −0.11 |

Obs2 vs. CNN2 | −0.98 | −0.76 | −0.33 |

Obs3 vs. CNN3 | −0.70 | −0.93 | −0.65 |

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**MDPI and ACS Style**

Materka, A.; Jurek, J.
Using Deep Learning and B-Splines to Model Blood Vessel Lumen from 3D Images. *Sensors* **2024**, *24*, 846.
https://doi.org/10.3390/s24030846

**AMA Style**

Materka A, Jurek J.
Using Deep Learning and B-Splines to Model Blood Vessel Lumen from 3D Images. *Sensors*. 2024; 24(3):846.
https://doi.org/10.3390/s24030846

**Chicago/Turabian Style**

Materka, Andrzej, and Jakub Jurek.
2024. "Using Deep Learning and B-Splines to Model Blood Vessel Lumen from 3D Images" *Sensors* 24, no. 3: 846.
https://doi.org/10.3390/s24030846