Improved Curvature Estimation for Computer-aided Detection of Colonic Polyps in CT Colonography

Rationale and Objectives

Current schemes for computer-aided detection (CAD) of colon polyps usually use kernel methods to perform curvature-based shape analysis. However, kernel methods may deliver spurious curvature estimations if the kernel contains two surfaces, because of the vanished gradient magnitudes. The aim of this study was to use the Knutsson mapping method to deal with the difficulty of providing better curvature estimations and to assess the impact of improved curvature estimation on the performance of CAD schemes.

Materials and Methods

The new method was compared to two widely used kernel methods in terms of the performance of two stages of CAD: initial detection and true-positive and false-positive classification. The evaluation was conducted on a database of 130 computed tomographic scans from 67 patients. In these patient scans, there were 104 clinically significant polyps and masses >5 mm.

Results

In the initial detection stage, the detection sensitivity of the three methods was comparable. In the classification stage, at a 90% sensitivity level on the basis of the input of this step, the new technique yielded 3.15 false-positive results per scan, demonstrating reductions in false-positive findings of 30.2% ( P < .01) and 27.9% ( P < .01) compared to the two kernel methods.

Conclusions

The new method can benefit CAD schemes with reduced false-positive rates, without sacrificing detection sensitivity.

According to recent statistics from the American Cancer Society , colorectal cancer is the third most common cause of both cancer deaths and new cancer cases in both men and women in the United States. Most colon cancers start from benign polyps, and the transformation from benignity to malignance often takes 5 to 10 years. Therefore, early detection and removal of colonic polyps prior to their malignant transformation can effectively decrease the incidence of colon cancer . Colon polyps are not commonly associated with any symptoms. Therefore, adequate time interval screening is recommended for people aged > 50 years by the American Cancer Society . As a new minimally invasive screening technique, computed tomographic colonography (CTC) or computed tomography–based virtual colonoscopy has shown several advantages over the traditional optical colonoscopy . To improve the performance of CTC in detecting polyps, computer-aided detection (CAD) has shown the potential to assist physicians in finding and analyzing polyps in the colon .

In most currently available CAD systems in CTC, principal curvatures, estimated through kernel methods , and principal curvature–related measures, such as mean curvatures, Gaussian curvatures, sphericity ratio, shape index, and curvedness, are widely used to characterize the shapes of colon polyps . However, when a kernel contains two surfaces (as happens for thin slab objects such as colonic folds and spherical objects such as polyps), spurious estimations of curvatures are frequently observed, indicating false high curvature . This is due to the discontinuity at thin structures, where the curvatures are discontinuous because of the diminished gradient magnitude. Fortunately, this discontinuity problem can be solved using Knutsson mapping , which maps a discontinuous orientation field into a continuous one. In this study, we adapted the Knutsson mapping technique to accurately estimate principal curvatures and explored the possible benefits of such improved curvature estimation for CAD performance in CTC. Evaluation of the new technique using both phantom experiments and patient studies demonstrated a noticeable reduction in the false-positive (FP) rate for CAD of colonic polyps compared to two current kernel methods.

Materials and methods

Methods

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Kernel methods for estimation of principal curvatures

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kT=k×N, k

T

=

k

×

N

,

where k is the curvature vector of the concerned curve at that point, and N is the unit normal at that point on the surface. Nonsingular curves that have the same tangent vector T will have the same curvature kT . Among all possible tangent vectors, the maximum and minimum values of kT at that point are called the principal curvatures, k 1 and k 2 , and the directions of the corresponding tangent vectors are called principal directions.

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kT=−(TtΩHT)/∥g∥, k

T

=

−

(

T

t

Ω

H

T

)

/

‖

g

‖

,

where Ω H is the Hessian matrix of the volume image, and g is the gradient vector (the normal vector). The resulted principal curvatures and directions were expressed as functions of the first and second derivatives of the image. In the kernel method developed by Thirion and Gourdon , denoted as KM 2 in the following text, equation (1) was rewritten as

k1,2=H±Δ−−√withΔ=H2−K, k

1

,

2

=

H

±

Δ

with

Δ

=

H

2

−

K

,

where H = ( k 1 + k 2 ) and K = k 1 k 2 are the mean and Gaussian curvatures. They can be computed using the following formulas:

H=12∥g∥3[I2x(Iyy+Izz)−2IyIzIyz+I2y(Ixx+Izz)−2IxIzIxz+I2z(Ixx+Iyy)−2IxIyIxy, H

=

1

2

‖

g

‖

3

[

I

x

2

(

I

y

y

+

I

z

z

)

−

2

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y

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z

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+

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2

(

I

x

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+

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z

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−

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x

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+

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2

(

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+

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)

−

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K=1∥g∥4[I2x(IyyIzz+I2yz)−2IyIz(IxzIxy−IxxIyz)+I2y(IxxIzz+I2xz)−2IxIz(IyzIxy−IyyIxz)+I2z(IxxIyy+I2xy)−2IxIy(IxzIyz−IzzIxy). K

=

1

‖

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4

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)

.

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Knutsson mapping

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kT=∥∇TN∥. k

T

=

‖

∇

T

N

‖

.

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|k1,2|=12√∥∇e2,3M(e1)∥=12√∥∥∑3n=1∂Mij∂xnen2,3∥∥, |

k

1

,

2

|

=

1

2

‖

∇

e

2

,

3

M

(

e

1

)

‖

=

1

2

‖

∑

n

=

1

3

∂

M

i

j

∂

x

n

e

2

,

3

n

‖

,

where M ij is the element of the second-order tensor M ( e 1 ), and x n a coordinate. The norm is the Fröbenius norm ∥M∥2=∑i,jm2ij ‖

M

‖

2

=

∑

i

,

j

m

i

j

2 . To be clearer, equation 7 can be laid out as

|k1,2|=12√∥∥∑3n=1∂Mij∂xnen2,3∥∥=12√∥∥∂Mij∂xex2,3+∂Mij∂yey2,3+∂Mij∂zez2,3∥∥, |

k

1

,

2

|

=

1

2

‖

∑

n

=

1

3

∂

M

i

j

∂

x

n

e

2

,

3

n

‖

=

1

2

‖

∂

M

i

j

∂

x

e

2

,

3

x

+

∂

M

i

j

∂

y

e

2

,

3

y

+

∂

M

i

j

∂

z

e

2

,

3

z

‖

,

where ex2,3 e

2

,

3

x , ey2,3 e

2

,

3

y , and ez2,3 e

2

,

3

z represent the coordinates of the vectors e 2 and e 3 . Another scale, σ k , is involved for evaluating first-order derivative ∇Mij=(∂Mij∂x,∂Mij∂y,∂Mij∂z)=Mij×∇G(σk) ∇

M

i

j

=

(

∂

M

i

j

∂

x

,

∂

M

i

j

∂

y

,

∂

M

i

j

∂

z

)

=

M

i

j

×

∇

G

(

σ

k

) through the Gaussian derivative ∇G(σk) ∇

G

(

σ

k

) . The sign of the curvatures is the negative of the sign of Tt Ω H T , where Ω H represents the Hessian matrix.

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Experimental Design

CAD pipeline

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Method parameter configuration

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Table 1

Parameters Involved in Stage 2 of the Computer-Aided Diagnosis (CAD) Pipeline

Method Parameters Kernel method 1 α 1 , α 2 Kernel method 2 σ 1 , σ 2 Knutsson mapping kernel method σ g , σ T , σ k

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Table 2

Means and Standard Deviations of Regions of Interest in Colon Lumen and Tissue Adjacent to Colon Wall

Mean Standard Deviation Lumen (Hounsfield units) 41.54 23.64 Adjacent tissue (Hounsfield units) 937.81 38.77

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Table 3

Size Distribution of the Polyps

Size (mm) n (%) 6–9 118 (56.7) 10–30 82 (39.4) >30 8 (3.8)

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CTC Database

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Table 4

Shape Distribution of the Polyps

Shape n (%) Sessile 103 (49.5) Pedunculated 92 (44.2) Flat 13 (6.3)

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Performance evaluation for patient studies

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Results

Phantom Study

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![Figure 3, Plots of error (vertical axis) (ie,

k 1,2

− k theory ) according to objects with various sizes ( r = 5, 15, 35, and 45 from top to bottom rows) and various values of σ T and σ k (two horizontal axes) with the Knutsson mapping kernel method. The left and right columns indicate the errors of k 1 and k 2 .](https://storage.googleapis.com/dl.dentistrykey.com/clinical/ImprovedCurvatureEstimationforComputeraidedDetectionofColonicPolypsinCTColonography/2_1s20S1076633211002066.jpg)

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Table 5

Statistics of | k 1 | and | k 2 | Resulting From the Three Methods

KM 1 KM 2 KM M Mean Variance Mean Variance Mean Variance_r_ = 14 (| k 1 | = | k 2 | ≈ 0.0714) k 1 | 2.24 1.01 1.88 0.86 0.0684 0.000210 k 2 | 3.15 1.15 2.87 0.97 0.0672 0.00722r = 15 (| k 1 | = | k 2 | ≈ 0.0667) k 1 | 23.04 302 21.53 402 0.0641 0.000204 k 2 | 31.21 411 27.31 433 0.0640 0.00249r = 16 (| k 1 | = | k 2 | ≈ 0.0625) k 1 | 1.94 0.58 1.77 0.77 0.0620 0.000332 k 2 | 2.58 0.89 2.33 0.95 0.0614 0.00745

KM M , Knutsson mapping kernel method; KM 1 , kernel method 1; KM 2 , kernel method 2.

Theoretical values are listed in the first column.

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Patient Studies

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Results of IPC extraction

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Table 6

Performance of Initial Polyp Candidate Extraction According to the Three Methods

Variable KM 1 KM 2 KM M Sensitivity (%) 93.8 94.2 94.2 False-positive rate 24.3 23.8 22.5 Number of false-negative findings 13 12 12

KM M , Knutsson mapping kernel method; KM 1 , kernel method 1; KM 2 , kernel method 2.

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Results of TP and FP classification

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Table 7

False-positive Rates at 90% Detection Sensitivity on the Basis of the Input of the True-positive and False-positive Classification Stage

IPCs Features KM 1 KM 2 KM M KM 1 4.51 (4.21–4.81) 5.07 (4.72–5.42) 3.28 (3.07–3.49) KM 2 5.01 (4.64–5.38) 4.37 (4.02–4.72) 3.38 (3.14–3.62) KM M 4.61 (4.34–4.88) 4.57 (4.31–4.83) 3.15 (2.95–3.35)

IPC, initial polyp candidate; KM M , Knutsson mapping kernel method; KM 1 , kernel method 1; KM 2 , kernel method 2.

In each row, the IPC set is from the same curvature estimation method, but the characterizing features are from various methods.

Table 8

P Values of the Operating Points in Figure 5

IPCs Features KM 1 vs KM 2 KM 1 vs KM M KM 2 vs KM M KM 1 0.14 0.00317 0.000211 KM 2 0.21 0.000531 0.00353 KM M 5.87 0.000342 0.000355

IPC, initial polyp candidate; KM M , Knutsson mapping kernel method; KM 1 , kernel method 1; KM 2 , kernel method 2.

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Discussion

Phantom Study

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Patient Study

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Conclusion

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Acknowledgment

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