Deep learning fully convolution network for lumen characterization in diabetic patients using carotid ultrasound: a tool for stroke risk

Abstract

Manual ultrasound (US)-based methods are adapted for lumen diameter (LD) measurement to estimate the risk of stroke but they are tedious, error prone, and subjective causing variability. We propose an automated deep learning (DL)-based system for lumen detection. The system consists of a combination of two DL systems: encoder and decoder for lumen segmentation. The encoder employs a 13-layer convolution neural network model (CNN) for rich feature extraction. The decoder employs three up-sample layers of fully convolution network (FCN) for lumen segmentation. Three sets of manual tracings were used during the training paradigm leading to the design of three DL systems. Cross-validation protocol was implemented for all three DL systems. Using the polyline distance metric, the precision of merit for three DL systems over 407 US scans was 99.61%, 97.75%, and 99.89%, respectively. The Jaccard index and Dice similarity of DL lumen segmented region against three ground truth (GT) regions were 0.94, 0.94, and 0.93 and 0.97, 0.97, and 0.97, respectively. The corresponding AUC for three DL systems was 0.95, 0.91, and 0.93. The experimental results demonstrated superior performance of proposed deep learning system over conventional methods in literature.

Appendices

Appendix 1 Statistical test results

Table 7 Statistical tests

Fig. 14

Wilcoxon box plot for (a) DL1 vs. GT1, (b) DL2 vs. GT2, and (c) DL3 vs. GT3

Fig. 15

Mann-Whitney box plot for (a) DL1 vs. GT1, (b) DL2 vs. GT2, and (c) DL3 vs. GT3

Fig. 16

Paired t test box plot for (a) DL1 vs. GT1, (b) DL2 vs. GT2, and (c) DL3 vs. GT3

Appendix 2 Polyline distance method

1.1 Polyline distance metric

Polyline distance metric (PDM) [40] is used to measure the lumen diameter (LD), LI-far error, and LI-near error. In here, we focus on deriving the PDM given two border contours. Let the first and second contours be denoted as I₁ and I₂. Let the reference point on I₁ be vertex A₁ and the segment in I₂ be defined by vertices A₂ and A₃. Let the distance between A₁ and A₂ be d₁and the distance between A₁ and A₃ be denoted as d₂. Let D(A₁, L) be the polyline distance between vertex A₁ : (x₁, y₁) on I₁ and line segment L formed by two points A₂ : (x₂, y₂)and A₃ : (x₃, y₃). Let phi (φ) be the distance of the reference point, A₁ towards the line segment L. The perpendicular distance between the line segment L and the reference point, A₁, is given by d_P. Then, the polyline distance D(A₁, L) can be defined as:

$$ D\left({A}_1,L\right)=\left\{\begin{array}{c}\mid {d}_P\mid \kern0.5em 0<\varphi <1\\ {}\min \left({d}_1,{d}_2\right)\kern0.5em \varphi <0,\varphi >1\end{array}\right. $$

(B.1)

where,

$$ {d}_1=\sqrt{{\left({x}_1-{x}_2\right)}^2+{\left({y}_1-{y}_2\right)}^2} $$

(B.2)

$$ {d}_2=\sqrt{{\left({x}_1-{x}_3\right)}^2+{\left({y}_1-{y}_3\right)}^2} $$

(B.3)

$$ \varphi =\frac{\left({y}_3-{y}_2\right)\left({y}_1-{y}_2\right)+\left({x}_3-{x}_2\right)\left({x}_1-{x}_2\right)}{{\left({x}_3-{x}_2\right)}^2+{\left({y}_3-{y}_2\right)}^2} $$

(B.4)

and

$$ {d}_P=\frac{\left({y}_3-{y}_2\right)\left({x}_2-{x}_1\right)+\left({x}_3-{x}_2\right)\left({y}_1-{y}_2\right)}{\sqrt{{\left({x}_3-{x}_2\right)}^2+{\left({y}_3-{y}_2\right)}^2}} $$

(B.5)

The process to obtain D(A₁, L) is repeated for the rest of the points of the contour I_j and is given by:

$$ D\left({I}_1,{I}_2\right)={\sum}_{i=1}^ND\left({A}_i,{S}_{I_2}\right) $$

(B.6)

where, N is the total number of points on I₁ and ${S}_{I_2}$ is the segment on contour I₂. This algorithm is repeated in reverse, where I₂ becomes the reference contour and I₁ becomes the segment contour. The reverse is represented as D(I₂, I₁). Finally, by combining both D(I₁, I₂) and D(I₂, I₁), we obtain the PDM which is given by:

$$ {D}_{\mathrm{PDM}}\left({I}_1:{I}_2\right)=\frac{D\left({I}_1,{I}_2\right)+D\left({I}_2,\kern0.5em {I}_1\right)}{\left(\#\mathrm{points}\in {I}_1+\#\mathrm{points}\in {I}_2\right)} $$

(B.7)

Appendix 3 Figure-of-merit and precision-of-merit

1.1 LD and mean LD computation

The LD error is computed as PDM between the ground truth LD (LD_gt) and deep learning LD (LD_dl). The LD for a patient i is computed as a PDM between LI-far ( LI_far(i)) and LI-near ( LI_near(i)) wall of the patient. The ground truth LD (LD_gt(i)) for patient i is given as:

$$ \mathrm{L}{\mathrm{D}}_{\mathrm{gt}(i)}={D}_{\mathrm{PDM}}\left({\mathrm{LI}}_{\mathrm{far}(i)}^{\mathrm{gt}}:{\mathrm{LI}}_{\mathrm{near}(i)}^{\mathrm{gt}}\right) $$

(C.1)

Similarly, deep learning LD (LD_dl(i)) for image i is given as:

$$ \mathrm{L}{\mathrm{D}}_{\mathrm{dl}(i)}={\mathrm{D}}_{\mathrm{PDM}}\left({\mathrm{LI}}_{\mathrm{far}(i)}^{\mathrm{dl}}:{\mathrm{LI}}_{\mathrm{near}(i)}^{\mathrm{dl}}\right) $$

(C.2)

The mean LD can therefore be computed as:

$$ {\overline{\mathrm{LD}}}_{\mathrm{dl}}=\frac{1}{N}{\sum}_{i=1}^N\mathrm{L}{\mathrm{D}}_{\mathrm{dl}(i)} $$

(C.3)

$$ {\overline{\mathrm{LD}}}_{\mathrm{gt}}=\frac{1}{N}{\sum}_{i=1}^N\mathrm{L}{\mathrm{D}}_{\mathrm{gt}(i)} $$

(C.4)

1.2 LD error and mean LD error

The LD error (ε_LD(i)) for an image i is computed as the absolute difference between LD_gt(i) and LD_dl(i) and is mathematically represented as:

$$ {\upvarepsilon}_{\mathrm{LD}(i)}=\left|\mathrm{L}{\mathrm{D}}_{\mathrm{gt}(i)}-\mathrm{L}{\mathrm{D}}_{\mathrm{dl}(i)}\right| $$

(C.5)

If ε_LD(i) represents the LD error for an image i, then the mean LD error (${\overline{\varepsilon}}_{\mathrm{LD}}$) for all N patients is given by:

$$ {\overline{\upvarepsilon}}_{\mathrm{LD}}=\frac{\sum_{i=1}^N{\upvarepsilon}_{\mathrm{LD}(i)}}{N} $$

(C.6)

1.3 Precision-of-merit

Using Eqs. (C.1) and (C.2), one can therefore define mathematically the precision-of-merit (POM) and is given as:

$$ Po{M}_{LD}\left(\%\right)=100-\left(\frac{\sum_{i=1}^N\frac{\left|L{D}_{dl(i)}-L{D}_{gt(i)}\right|}{L{D}_{gt(i)}}}{N}\right)\times 100 $$

(C.7)

1.4 Figure-of-merit

The central tendency of the LD distribution can also be used to tell the difference between the DL-based LD and GT-based LD. Using (C.3) and (C.4), one can therefore compute figure-of-merit (FoM) and can be expressed mathematically as:

$$ \mathrm{Fo}{\mathrm{M}}_{\mathrm{LD}}\left(\%\right)=100-\left[\left(\frac{\left|{\overline{\mathrm{LD}}}_{\mathrm{dl}}-{\overline{\mathrm{LD}}}_{\mathrm{gt}}\right|}{{\overline{\mathrm{LD}}}_{\mathrm{gt}}}\right)\times 100\right] $$

(C.8)

Appendix 4 LI-far and LI-near position errors

1.1 LI-far error

The LI-far error (ε_far(i)) for patient i is computed as the PDM between the GT LI-far wall (${\mathrm{LI}}_{\mathrm{far}(i)}^{\mathrm{gt}}$) and DL LI-far (${\mathrm{LI}}_{\mathrm{far}(i)}^{\mathrm{dl}}$) wall for the patient, which is given by:

$$ {\upvarepsilon}_{\mathrm{far}(i)}={D}_{\mathrm{PDM}}\left({\mathrm{LI}}_{\mathrm{far}(i)}^{\mathrm{gt}}:{\mathrm{LI}}_{\mathrm{far}(i)}^{\mathrm{dl}}\right) $$

(D.1)

If ε_far(i) represents the LI-far error for the patient i, then, the mean LI-far error (${\overline{\varepsilon}}_{\mathrm{far}}$) for all N patients is given by:

$$ {\overline{\upvarepsilon}}_{\mathrm{far}}=\frac{\sum_{i=1}^N{\upvarepsilon}_{\mathrm{far}(i)}}{N} $$

(D.2)

1.2 LI-near error

Similarly, the LI-near error (ε_near(i)) is computed as the PDM between the GT LI-near wall (${\mathrm{LI}}_{\mathrm{near}(i)}^{\mathrm{gt}}$) and DL LI-near (${\mathrm{LI}}_{\mathrm{near}(i)}^{\mathrm{dl}}$) wall for patient i is given by:

$$ {\upvarepsilon}_{\mathrm{near}(i)}={D}_{\mathrm{PDM}}\left({\mathrm{LI}}_{\mathrm{near}(i)}^{\mathrm{gt}}:{\mathrm{LI}}_{\mathrm{near}(i)}^{\mathrm{dl}}\right) $$

(D.3)

The mean LI-near error (${\overline{\varepsilon}}_{\mathrm{near}}$) for all N patients is given by:

$$ {\overline{\upvarepsilon}}_{\mathrm{near}}=\frac{\sum_{i=1}^N{\upvarepsilon}_{\mathrm{near}(i)}}{N} $$

(D.4)

The corresponding symbol table is given in Appendix 5, Table 8.

Appendix 5 Symbol table

Table 8 Symbol table