# Histopathological Imaging–Environment Interactions in Cancer Modeling

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Data

## 3. Methods

**E**= [E

_{1}, ⋯, E

_{J}] as the N × J matrix of clinical/environmental variables, and

**X**= [X

_{1}, ⋯, X

_{K}] as the N × K matrix of imaging features. As represented by the LUAD data, usually clinical/environmental variables are pre-selected and low-dimensional, and imaging features are high-dimensional.

#### 3.1. Marginal Analysis

**E**, and

**X**have been properly centered.

- (a)
- For $j=1,\dots ,J$ and $k=1,\dots ,K$, consider the linear regression model$$Y={\alpha}_{j}{E}_{j}+{\beta}_{k}{X}_{k}+{\gamma}_{jk}{E}_{j}{X}_{k}+\u03f5,$$
- (b)
- As each model has a low dimension, estimates can be obtained using standard likelihood based approaches and existing software. p-values can be obtained accordingly.
- (c)
- Interactions (and main effects) with small p-values are identified as important. When more definitive conclusions are needed, the false discovery rate (FDR) or Bonferroni approach can be applied.

#### 3.2. Joint Analysis

- (a)
- Consider the joint model$$Y=\sum _{j=1}^{J}{\tau}_{j}{E}_{j}+\sum _{k=1}^{K}{\eta}_{k}{X}_{k}+\sum _{j=1}^{J}\sum _{k=1}^{K}{\eta}_{k}{\theta}_{jk}{E}_{j}{X}_{k}+\u03f5,$$
- (b)
- For estimation, consider the Lasso penalization$$\underset{{\eta}_{k},{\theta}_{jk}}{min}\left|\right|Y-{f(\mathbf{E},\mathbf{X})\left|\right|}^{2}+{\lambda}_{1}\sum _{k}|{\eta}_{k}|+{\lambda}_{2}\sum _{j}\sum _{k}\left|{\theta}_{jk}\right|,$$
- (c)
- Interactions (and main effects) with nonzero estimates are identified as being associated with the outcome.

#### 3.3. Accommodating Survival Outcomes

_{i}’s from the smallest to the largest. The Kaplan–Meier weights can be computed as ${w}_{1}={\displaystyle \frac{{\delta}_{1}}{N}}$, ${w}_{i}={\displaystyle \frac{{\delta}_{i}}{N-i+1}}{\displaystyle \prod _{j=1}^{i-1}}{\left({\displaystyle \frac{N-j}{N-j+1}}\right)}^{{\delta}_{j}}$, $i=2,\dots ,N$. Similar to Equation (3), consider the penalized estimation

## 4. Results

#### 4.1. Analysis of FEV1

#### 4.1.1. Marginal Analysis

#### 4.1.2. Joint Analysis

#### 4.2. Analysis of Overall Survival

#### 4.2.1. Marginal Analysis

#### 4.2.2. Joint Analysis

#### 4.3. Simulation

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Flowchart of the I–E interaction analysis of The Cancer Genome Atlas (TCGA) lung adenocarcinoma (LUAD) data.

**Figure 2.**Kaplan–Meier curves of high and low risk groups identified by the approach that accommodates interactions ((

**a**); logrank test p-value 0.007) and the one with main effects only ((

**b**); logrank test p-value 0.320).

**Table 1.**Marginal analysis of the reference value for the pre-bronchodilator forced expiratory volume in one second in percent (FEV1): identified main effects and interactions, with raw p-values P

_{r}.

Feature Group | Feature Name | Estimate | P_{r} | |
---|---|---|---|---|

Geometry | AreaShape_Zernike_2_2 | Main | 0.270 | 0.002 |

Geometry | AreaShape_Zernike_5_3 | Main | −0.319 | 0.001 |

Geometry | Mean_Identifyhemasub2_AreaShape_Zernike_9_9 | Main | −0.259 | 0.004 |

Geometry | Median_Identifyhemasub2_AreaShape_Zernike_7_1 | Main | −0.249 | 0.005 |

Geometry | Median_Identifyhemasub2_AreaShape_Zernike_8_6 | Main | −0.272 | 0.003 |

Texture | StDev_Identifyeosinprimarycytoplasm_Texture_Correlation_maskosingray_3_01 | Main | 0.280 | 0.002 |

Geometry | StDev_Identifyhemasub2_AreaShape_Zernike_8_8 | Main | −0.251 | 0.005 |

Geometry | StDev_Identifyhemasub2_AreaShape_Zernike_9_1 | Main | −0.259 | 0.004 |

Geometry | StDev_Identifyhemasub2_AreaShape_Center_Y | Sex | 0.291 | 0.002 |

Geometry | StDev_Identifyhemasub2_AreaShape_Zernike_8_2 | Sex | 0.304 | 0.001 |

Geometry | StDev_Identifyhemasub2_Location_Center_Y | Sex | 0.294 | 0.002 |

Feature Group | Feature Name | Main | Age | Stage | Smoking | Sex |
---|---|---|---|---|---|---|

−0.049 | −0.052 | −0.002 | 0.006 | |||

Geometry | AreaShape_Zernike_2_2 | 0.163 | 0.040 | −0.014 | −0.185 | |

Geometry | AreaShape_Zernike_5_3 | −0.053 | ||||

Geometry | AreaShape_Zernike_6_0 | −0.034 | ||||

Texture | Granularity_10_ImageAfterMath | 0.137 | 0.110 | −0.020 | 0.064 | |

Geometry | Location_Center_X | 0.002 | ||||

Geometry | Mean_Identifyeosinprimarycytoplasm_Location_Center_X | 0.005 | ||||

Geometry | Median_Identifyhemasub2_AreaShape_Zernike_7_1 | −0.127 | −0.073 | 0.072 | 0.003 | |

Geometry | StDev_Identifyhemasub2_AreaShape_Zernike_8_2 | −0.170 | −0.083 | 0.188 | ||

Texture | StDev_Identifyhemasub2_Granularity_6_ImageAfterMath | −0.029 | ||||

Texture | Texture_AngularSecondMoment_ImageAfterMath_3_00 | −0.044 | ||||

Texture | Texture_AngularSecondMoment_ImageAfterMath_3_03 | −0.010 |

**Table 3.**Marginal analysis of overall survival: identified main effects and interactions, with raw p-values P

_{r}and false discovery rate (FDR) adjusted p-values P

_{a}.

Feature Group | Feature Name | Estimate | P_{r} | P_{a} | |
---|---|---|---|---|---|

Holistic | Threshold_FinalThreshold_Identifyeosinprimarycytoplasm | Main | −0.301 | 0 | 0.095 |

Holistic | Threshold_OrigThreshold_Identifyeosinprimarycytoplasm | Main | −0.301 | 0 | 0.095 |

Holistic | Threshold_WeightedVariance_identifyhemaprimarynuclei | Main | −0.360 | 0 | 0.077 |

Geometry | AreaShape_Area | Smoking | 0.253 | 0.004 | 0.078 |

Geometry | AreaShape_MaximumRadius | Smoking | 0.266 | 0.004 | 0.074 |

Geometry | AreaShape_MeanRadius | Smoking | 0.265 | 0.005 | 0.079 |

Geometry | AreaShape_MedianRadius | Smoking | 0.266 | 0.005 | 0.079 |

Geometry | AreaShape_MinFeretDiameter | Smoking | 0.257 | 0.003 | 0.073 |

Geometry | AreaShape_MinorAxisLength | Smoking | 0.264 | 0.002 | 0.07 |

Geometry | AreaShape_Zernike_4_4 | Smoking | −0.241 | 0.005 | 0.079 |

Geometry | AreaShape_Zernike_7_3 | Smoking | −0.308 | 0 | 0.027 |

Geometry | AreaShape_Zernike_8_4 | Smoking | −0.242 | 0.007 | 0.096 |

Geometry | AreaShape_Zernike_8_6 | Smoking | −0.252 | 0.005 | 0.079 |

Geometry | AreaShape_Zernike_9_1 | Smoking | −0.303 | 0 | 0.027 |

Texture | Granularity_13_ImageAfterMath.1 | Smoking | −0.317 | 0.001 | 0.054 |

Texture | Mean_Identifyeosinprimarycytoplasm_Texture_Correlation_maskosingray_3_03 | Smoking | 0.232 | 0.005 | 0.079 |

Geometry | Mean_Identifyhemasub2_AreaShape_Area | Smoking | 0.297 | 0.001 | 0.049 |

Geometry | Mean_Identifyhemasub2_AreaShape_MaximumRadius | Smoking | 0.318 | 0.001 | 0.049 |

Geometry | Mean_Identifyhemasub2_AreaShape_MeanRadius | Smoking | 0.318 | 0.001 | 0.049 |

Geometry | Mean_Identifyhemasub2_AreaShape_MedianRadius | Smoking | 0.308 | 0.002 | 0.054 |

Geometry | Mean_Identifyhemasub2_AreaShape_MinFeretDiameter | Smoking | 0.299 | 0.001 | 0.049 |

Geometry | Mean_Identifyhemasub2_AreaShape_MinorAxisLength | Smoking | 0.310 | 0.001 | 0.045 |

Geometry | Mean_Identifyhemasub2_AreaShape_Zernike_4_4 | Smoking | −0.263 | 0.003 | 0.07 |

Geometry | Mean_Identifyhemasub2_AreaShape_Zernike_5_1 | Smoking | −0.268 | 0.002 | 0.07 |

Geometry | Mean_Identifyhemasub2_AreaShape_Zernike_8_2 | Smoking | −0.277 | 0.003 | 0.073 |

Geometry | Mean_Identifyhemasub2_AreaShape_Zernike_8_8 | Smoking | −0.290 | 0.003 | 0.073 |

Geometry | Mean_Identifyhemasub2_AreaShape_Zernike_9_1 | Smoking | −0.226 | 0.004 | 0.074 |

Texture | Mean_Identifyhemasub2_Granularity_13_ImageAfterMath | Smoking | −0.325 | 0.001 | 0.054 |

Texture | Mean_Identifyhemasub2_Texture_Correlation_ImageAfterMath_3_01 | Smoking | 0.330 | 0 | 0.039 |

Texture | Mean_Identifyhemasub2_Texture_Correlation_ImageAfterMath_3_02 | Smoking | 0.297 | 0.002 | 0.07 |

Texture | Mean_Identifyhemasub2_Texture_Correlation_ImageAfterMath_3_03 | Smoking | 0.397 | 0 | 0.01 |

Texture | Mean_Identifyhemasub2_Texture_SumVariance_ImageAfterMath_3_02 | Smoking | 0.258 | 0.007 | 0.093 |

Texture | Median_Identifyeosinprimarycytoplasm_Texture_Correlation_maskosingray_3_03 | Smoking | 0.233 | 0.004 | 0.079 |

Geometry | Median_Identifyhemasub2_AreaShape_Area | Smoking | 0.344 | 0 | 0.027 |

Geometry | Median_Identifyhemasub2_AreaShape_MaxFeretDiameter | Smoking | 0.242 | 0.005 | 0.079 |

Geometry | Median_Identifyhemasub2_AreaShape_MaximumRadius | Smoking | 0.323 | 0.001 | 0.049 |

Geometry | Median_Identifyhemasub2_AreaShape_MeanRadius | Smoking | 0.323 | 0.001 | 0.049 |

Geometry | Median_Identifyhemasub2_AreaShape_MedianRadius | Smoking | 0.266 | 0.005 | 0.079 |

Geometry | Median_Identifyhemasub2_AreaShape_MinFeretDiameter | Smoking | 0.346 | 0 | 0.027 |

Geometry | Median_Identifyhemasub2_AreaShape_MinorAxisLength | Smoking | 0.342 | 0 | 0.027 |

Geometry | Median_Identifyhemasub2_AreaShape_Perimeter | Smoking | 0.247 | 0.006 | 0.085 |

Geometry | Median_Identifyhemasub2_AreaShape_Zernike_4_4 | Smoking | −0.242 | 0.002 | 0.059 |

Geometry | Median_Identifyhemasub2_AreaShape_Zernike_5_1 | Smoking | −0.256 | 0.003 | 0.073 |

Texture | Median_Identifyhemasub2_Granularity_13_ImageAfterMath | Smoking | −0.311 | 0.001 | 0.049 |

Texture | Median_Identifyhemasub2_Texture_Correlation_ImageAfterMath_3_01 | Smoking | 0.319 | 0.001 | 0.049 |

Texture | Median_Identifyhemasub2_Texture_Correlation_ImageAfterMath_3_02 | Smoking | 0.274 | 0.005 | 0.081 |

Texture | Median_Identifyhemasub2_Texture_Correlation_ImageAfterMath_3_03 | Smoking | 0.394 | 0 | 0.01 |

Texture | StDev_Identifyeosinprimarycytoplasm_Texture_SumAverage_maskosingray_3_00 | Smoking | 0.272 | 0.003 | 0.073 |

Texture | StDev_Identifyeosinprimarycytoplasm_Texture_SumAverage_maskosingray_3_01 | Smoking | 0.273 | 0.003 | 0.073 |

Texture | StDev_Identifyeosinprimarycytoplasm_Texture_SumAverage_maskosingray_3_02 | Smoking | 0.270 | 0.004 | 0.074 |

Texture | StDev_Identifyeosinprimarycytoplasm_Texture_SumAverage_maskosingray_3_03 | Smoking | 0.275 | 0.003 | 0.073 |

Geometry | StDev_identifyhemaprimarynuclei_Location_Center_Y | Smoking | −0.245 | 0.007 | 0.093 |

Geometry | StDev_Identifyhemasub2_AreaShape_Zernike_8_4 | Smoking | −0.280 | 0.001 | 0.045 |

Geometry | StDev_Identifyhemasub2_AreaShape_Zernike_8_8 | Smoking | −0.236 | 0.007 | 0.094 |

Texture | StDev_Identifyhemasub2_Texture_SumVariance_ImageAfterMath_3_01 | Smoking | 0.266 | 0.007 | 0.096 |

Texture | StDev_Identifyhemasub2_Texture_SumVariance_ImageAfterMath_3_02 | Smoking | 0.283 | 0.005 | 0.079 |

Texture | StDev_Identifyhemasub2_Texture_SumVariance_ImageAfterMath_3_03 | Smoking | 0.283 | 0.006 | 0.084 |

Geometry | StDev_identifytissueregion_Location_Center_Y | Smoking | −0.289 | 0.002 | 0.059 |

Texture | Texture_Correlation_ImageAfterMath_3_01 | Smoking | 0.252 | 0.004 | 0.078 |

Texture | Texture_Correlation_ImageAfterMath_3_03 | Smoking | 0.329 | 0 | 0.027 |

Texture | Texture_Correlation_maskosingray_3_03 | Smoking | 0.237 | 0.004 | 0.074 |

Texture | Texture_Entropy_ImageAfterMath_3_01 | Smoking | 0.220 | 0.007 | 0.093 |

Texture | Texture_Entropy_ImageAfterMath_3_03 | Smoking | 0.233 | 0.004 | 0.074 |

Feature Group | Feature Name | Main | Age | Stage | Smoking | Sex |
---|---|---|---|---|---|---|

−0.024 | −0.317 | −0.038 | −0.088 | |||

Geometry | AreaShape_Zernike_6_0 | −0.038 | ||||

Geometry | AreaShape_Zernike_6_4 | −0.019 | ||||

Geometry | AreaShape_Zernike_6_6 | 0.052 | ||||

Geometry | AreaShape_Zernike_9_3 | 0.027 | ||||

Geometry | AreaShape_Zernike_9_5 | 0.153 | ||||

Texture | Granularity_10_ImageAfterMath.1 | −0.033 | ||||

Texture | Granularity_9_ImageAfterMath | 0.081 | ||||

Geometry | Mean_Identifyhemasub2_AreaShape_Center_X | 0.002 | ||||

Geometry | Mean_Identifyhemasub2_AreaShape_Zernike_5_1 | 0.013 | ||||

Geometry | Mean_Identifyhemasub2_AreaShape_Zernike_6_2 | −0.002 | ||||

Geometry | Mean_Identifyhemasub2_AreaShape_Zernike_6_4 | −0.010 | ||||

Geometry | Mean_Identifyhemasub2_AreaShape_Zernike_9_9 | −0.146 | ||||

Geometry | Mean_Identifyhemasub2_Location_Center_X | 0.002 | ||||

Geometry | Mean_identifytissueregion_Location_Center_X | 0.056 | ||||

Geometry | Median_Identifyeosinprimarycytoplasm_Location_Center_X | −0.071 | ||||

Geometry | Median_Identifyhemasub2_AreaShape_Zernike_4_0 | 0.023 | ||||

Geometry | Median_Identifyhemasub2_AreaShape_Zernike_7_3 | 0.083 | ||||

Geometry | Median_Identifyhemasub2_AreaShape_Zernike_8_4 | −0.120 | ||||

Geometry | Median_Identifyhemasub2_AreaShape_Zernike_8_6 | −0.098 | ||||

Geometry | Median_Identifyhemasub2_AreaShape_Zernike_9_1 | −0.044 | ||||

Geometry | Median_identifytissueregion_Location_Center_Y | −0.063 | ||||

Holistic | Neighbors_SecondClosestDistance_Adjacent | −0.170 | −0.072 | 0.002 | ||

Geometry | StDev_Identifyeosinprimarycytoplasm_Location_Center_Y | 0.095 | ||||

Texture | StDev_Identifyeosinprimarycytoplasm_Texture_DifferenceVariance_maskosingray_3_00 | 0.036 | ||||

Geometry | StDev_Identifyhemasub2_AreaShape_Orientation | −0.159 | ||||

Geometry | StDev_Identifyhemasub2_AreaShape_Zernike_8_8 | −0.146 | ||||

Texture | StDev_Identifyhemasub2_Granularity_12_ImageAfterMath | −0.101 | ||||

Texture | StDev_Identifyhemasub2_Granularity_13_ImageAfterMath | 0.327 | 0.130 | 0.072 | −0.189 | 0.174 |

Texture | StDev_Identifyhemasub2_Granularity_9_ImageAfterMath | 0.003 | ||||

Texture | StDev_Identifyhemasub2_Texture_SumVariance_ImageAfterMath_3_01 | −0.034 | ||||

Geometry | StDev_identifytissueregion_Location_Center_Y | 0.016 |

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## Share and Cite

**MDPI and ACS Style**

Xu, Y.; Zhong, T.; Wu, M.; Ma, S.
Histopathological Imaging–Environment Interactions in Cancer Modeling. *Cancers* **2019**, *11*, 579.
https://doi.org/10.3390/cancers11040579

**AMA Style**

Xu Y, Zhong T, Wu M, Ma S.
Histopathological Imaging–Environment Interactions in Cancer Modeling. *Cancers*. 2019; 11(4):579.
https://doi.org/10.3390/cancers11040579

**Chicago/Turabian Style**

Xu, Yaqing, Tingyan Zhong, Mengyun Wu, and Shuangge Ma.
2019. "Histopathological Imaging–Environment Interactions in Cancer Modeling" *Cancers* 11, no. 4: 579.
https://doi.org/10.3390/cancers11040579