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The Self-Overlap Method for Assessment of Lung Nodule Morphology in Chest CT

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Abstract

Surface morphology is an important indicator of malignant potential for solid-type lung nodules detected at CT, but is difficult to assess subjectively. Automated methods for morphology assessment have previously been described using a common measure of nodule shape, representative of the broad class of existing methods, termed area-to-perimeter-length ratio (APR). APR is static and thus highly susceptible to alterations by random noise and artifacts in image acquisition. We introduce and analyze the self-overlap (SO) method as a dynamic automated morphology detection scheme. SO measures the degree of change of nodule masks upon Gaussian blurring. We hypothesized that this new metric would afford equally high accuracy and superior precision than APR. Application of the two methods to a set of 119 patient lung nodules and a set of simulation nodules showed our approach to be slightly more accurate and on the order of ten times as precise, respectively. The dynamic quality of this new automated metric renders it less sensitive to image noise and artifacts than APR, and as such, SO is a potentially useful measure of cancer risk for solid-type lung nodules detected on CT.

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Correspondence to Joseph N. Stember.

Appendix

Appendix

Appendix 1

We seek a threshold function Θ that turns μ(i) into a binary matrix whose entries are equal to 1 if they exceed the assigned threshold value and are 0 otherwise, i.e.

(8)

where H is the Heaviside step function and T(μ(i)) is the threshold-generating function for blurred nodule i's image matrix μ(i). The latter function is defined by

$$ T\left( {{\mu^{(i) }}} \right)=\frac{1}{2}\left( {\mathrm{att}_{\mathrm{high}}^{(i) }+\mathrm{att}_{\mathrm{low}}^{(i) }} \right), $$
(9)

where \( \mathrm{att}_{\mathrm{low}}^{(i) } \) is the attenuation value below which approximately 10 % of the attenuation values for nodule i lie. In other words, for a frequency distribution \( {f_i}\left( {\mathrm{at}{{\mathrm{t}}^{(i) }}} \right) \) of nodule i's attenuation values, \( \mathrm{att}_{\mathrm{low}}^{(i) } \) satisfies

$$ \int_0^{{\mathrm{att}_{\mathrm{low}}^{(i) }}} {f_i \left( {\mathrm{at}{{\mathrm{t}}^{(i) }}} \right)d\left( {\mathrm{at}{{\mathrm{t}}^{(i) }}} \right)=\frac{1}{10 }} \int_0^{\max } {^{{\left( {\left\{ {\mathrm{att}_j^{(i) }} \right\}_{j=1}^{{{N_{\mathrm{rows}}}\times {N_{\mathrm{cols}}}}}} \right)}}} {f_i}\left( {\mathrm{at}{{\mathrm{t}}^{(i) }}} \right)d\left( {\mathrm{at}{{\mathrm{t}}^{(i) }}} \right), $$
(10)

where \( \left\{ {\mathrm{att}_j^{(i) }} \right\}_{j=1}^{{{N_{\mathrm{rows}}}\times {N_{\mathrm{cols}}}}}=\mathrm{vec}\left( {{\mu^{(i) }}} \right) \) is the set of attenuation values in μ (i). Similarly, \( \mathrm{att}_{\mathrm{high}}^{(i) } \) is the attenuation value above which the highest 10 % of attenuation values for nodule i reside, so that

$$ \int_{{\mathrm{at}{{\mathrm{t}}_{\mathrm{high}}}}}^{\max } {^{{\left( {\left\{ {\mathrm{att}_j^{(i) }} \right\}_j^{{{N_{\mathrm{rows}}}\times {N_{\mathrm{cols}}}}}} \right)}}{f_i}\left( {\mathrm{at}{{\mathrm{t}}^{(i) }}} \right)d\left( {\mathrm{at}{{\mathrm{t}}^{(i) }}} \right)=\frac{1}{10}\int_0^{\max } {^{{\left( {\left\{ {\mathrm{att}_j^{(i) }} \right\}_{j=1}^{{{N_{\mathrm{rows}}}\times {N_{\mathrm{cols}}}}}} \right)}}{f_i}\left( {\mathrm{at}{{\mathrm{t}}^{(i) }}} \right)d\left( {\mathrm{at}{{\mathrm{t}}^{(i) }}} \right).} } $$
(11)

The hope is that by using this 10 % margin, the attenuation value \( T\left( {{\mu^{(i) }}} \right) \) accurately reflects a typical value characterizing the attenuation distribution for nodule i and is not unduly influenced by aberrantly large or small attenuation values, as these can represent for example pieces of bone or air that are caught in the nodule image slice.

Appendix 2

According to the generalized linear model, we can write any expected value for a dependent variable Y in terms of the independent variable X with parameter set β:

$$ E(Y)={g^{-1 }}\left( {X\beta } \right), $$
(12)

where E(Y) is the expected value for Y and g is the link function. In our case, the outcome Y i is smoothness i , a continuous quantity that expresses the likelihood that nodule i is smooth or not smooth, and X is the self-overlap SO. Since we are interested in predicting whether a given nodule is smooth or not, our distribution is binomial, and we will denote the binary counterpart to smoothness to be “smooth” sm. When sm i  = 1, we can say that nodule i is smooth, while sm i  = 0 is a statement that nodule i is not smooth. Thus, the link function is given by

$$ g\left( {E(Y)} \right)=\mathrm{s}{{\mathrm{m}}_i}=X\beta =\ln \left( {\frac{E(Y) }{1-E(Y) }} \right), $$
(13)

with corresponding mean function

$$ E(Y)=\mathrm{smoothnes}{{\mathrm{s}}_i}=\frac{1}{{1-\exp \left( {-X\beta } \right)}}=\frac{1}{{1-\exp \left( {-{\beta_0}+s{o_i}\times {\beta_1}} \right)}}. $$
(14)

The smooth values \( \left\{ {\mathrm{s}{{\mathrm{m}}_i}} \right\}_{i=1}^{{{N_{\mathrm{nods}}}}} \) were assigned by an experienced radiologist (JPK and DN). Together with \( \left\{ {\mathrm{S}{{\mathrm{O}}_i}} \right\}_{i=1}^{{{N_{\mathrm{nods}}}}} \), we can compute \( \left\{ {\beta_0, {\beta_1}} \right\} \) by least squares regression applied to the system of equations

$$ \left\{ {\mathrm{s}{{\mathrm{m}}_i}={\beta_0}+\mathrm{s}{{\mathrm{o}}_i}\times {\beta_1}} \right\}_{i=1}^{{{N_{\mathrm{nods}}}}}, $$
(15)

called in Matlab as glmfit. Then we can compute \( \left\{ {\mathrm{smoothnes}{{\mathrm{s}}_i}} \right\}_{i=1}^{{{N_{\mathrm{nods}}}}} \) via the mean function (Eq. 14). Finally, using a threshold

$$ {T_{\mathrm{sm}}}=\frac{1}{2}\left( {\mathrm{smoothnes}{{\mathrm{s}}_{\min }}+\mathrm{smoothnes}{{\mathrm{s}}_{\max }}} \right), $$
(16)

we can assign a predicted smooth value

$$ \mathrm{sm}_i^{\mathrm{SO}}=H\left( {\mathrm{smoothnes}{{\mathrm{s}}_i}-{T_{\mathrm{sm}}}} \right), $$
(17)

where H is the Heaviside step function.

Using the same fitting technique, we generate smooth values predicted from area-to-perimeter-length ratio \( \left\{ {\mathrm{sm}_i^{\mathrm{APR}}} \right\}_{i=1}^{{{N_{\mathrm{nods}}}}} \) for comparison with the smooth values \( \left\{ {\mathrm{sm}_i^{\mathrm{SO}}} \right\}_{i=1}^{{{N_{\mathrm{nods}}}}} \) predicted from the self-overlap method.

Appendix 3

We test for robustness by permuting the nodule μ (i) into \( {{\widetilde{\mu}}^{(i) }} \), which now has noise on top of the simulation image. We introduce noise by generating N rp  = 200 randomly positioned trial points and subjecting them to the probability distribution \( {p_{\mathrm{flip}}}=\exp \left( {-{d_{\mathrm{bd}}}} \right) \), where p flip is the probability of a trial point's being accepted (thus flipping the sign of the pixel in that position), and d bd is the distance of the trial point from the nearest point on the nodule–air interface/boundary. This probability distribution weights noise toward the boundary, so that the added noise affects predominantly surface morphology. For each value of l, we generate N perm = 1,000 permuted nodules \( \left\{ {\widetilde{\mu}_m^{(l) }} \right\}_{m=1}^{{{N_{\mathrm{perm}}}}} \) by this approach, giving us sets of nodules that are distorted in various places and to various degrees.

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Stember, J.N., Ko, J.P., Naidich, D.P. et al. The Self-Overlap Method for Assessment of Lung Nodule Morphology in Chest CT. J Digit Imaging 26, 239–247 (2013). https://doi.org/10.1007/s10278-012-9536-9

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