# Flexible Risk Evidence Combination Rules in Breast Cancer Precision Therapy

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Dempster-Shafer Theory

`Blue`,

`Red`and

`Green`, thus $\mathsf{\Omega}=\{B,R,G\}$.

#### 2.2. Evidence Combination Rules

## 3. Results

#### 3.1. Model Adaptation

- The operator ⊗, as defined in Formula (8), is not fully compatible with DST. There is always a dependence between the two receptor status. However, DST in its original form requires independent BBMs. This is obviously not the case for estrogen and progesterone receptors. Our suggestion to absorb this correlation is to replace ⊗ by ${\oplus}_{0.5}$ giving estrogen and progesterone a balanced contribution to both BBM.
- The operator ${\oplus}_{1}$ for combining pieces of evidence coming from gene expression and co-gene expression might be problematic in case of conflicting expression values. In a previous paper [32] we introduced mass limits $\widehat{\alpha}$ and $\widehat{\beta}$ for the BBMs to tackle this issue. We retain these mass limits, but replace ${\oplus}_{1}$ by ${\oplus}_{0.9}$ as an additional reinsurance.
- Combining gene expression evidence with IHC evidence, the operator ${\oplus}_{0}$ will in case of conflict put too much weight into the mass of ignorance, $m\left(\mathsf{\Omega}\right)$. Therefore we suggest slightly increasing $\lambda $ and replacing ${\oplus}_{0}$ by ${\oplus}_{0.1}$. On the lower end of the $\lambda $-range, the influence of $\lambda $ on the ECR is significantly less than on the upper end. As long as there is a profound confidence in the data, particularly in the IHC measurements, replacing ${\oplus}_{0}$ by e.g., ${\oplus}_{0.3}$ is therefore also an option.
- In the past it turned out that the optimal choice for the co-gene of progesterone is mostly estrogen itself. If so, although ${m}_{\mathrm{expr}}^{\mathrm{esr}}$ and ${m}_{\mathrm{co}}^{\mathrm{pgr}}$ are calculated differently and so vary numerically, they are basically generated from the same gene expression data. A preferable assumption in DST is the independence of input data to generate evidence. In contrary to estrogen, progesterone expression data is often diffuse and it might be impossible to find a decent co-gene. This issue can be easily resolved by replacing ${m}_{\mathrm{co}}^{\mathrm{pgr}}$ with the vacuous mass function. Currently, for the sake of consistency, we stick to the current configuration which uses estrogen as co-gene for progesterone.

#### 3.2. Examples

#### 3.3. Analysis

#### 3.4. Decision Making

## 4. Discussion

#### 4.1. Quality of Data

#### 4.2. From Data to Evidence

#### 4.3. The Functionality of $\lambda $ in ${\oplus}_{\lambda}$

#### 4.4. Training of $\lambda $ in ${\oplus}_{\lambda}$ on Real Data

#### 4.5. Enhanced Evidence Combination Rules

#### 4.6. An Evidence Combination Rule with Constant Ignorance

#### 4.7. Modified Frame of Discernment

#### 4.8. Risk Function for Decision Making

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

BBM | basic belief masses, same as basic belief assignment (BBA) |

BC | breast cancer |

DSmT | Dezert-Smarandache theory |

DST | Dempster-Shafer theory of evidence |

ECR | evidence combination rule |

ESR | estrogen |

FOD | frame of discernment |

GEO | Gene Expression Omnibus |

IHC | immunohistochemistry |

PCR | proportional conflict redistribution |

PGR | progesterone |

SLAM | simultaneous localization and mapping |

TBM | transferable belief model |

## Appendix A. Examples of Combining Pieces of Evidence

#### Appendix A.1. Two Rather Consistent Agents

`Red`. While the first one considers

`Blue`as an alternative, the second one’s first alternative is the preferable

`Green`.

**Figure A1.**Adding two rather consistent pieces of evidence with Formula (4). A larger value of $\lambda $ (less conservative) seems to be preferable. Masses for ambiguous outcomes such as $\{R,B\}$ almost vanish. The first and the second rows show different graphical representations of the same situation.

#### Appendix A.2. Two Rather Contradictory Agents

`Red`, while the second one gives most mass to

`Blue`or

`Green`.

**Figure A2.**Adding two rather contradictory pieces of evidence with Formula (4). A smaller value of $\lambda $ (more conservative) seems preferable staying on the safe side. Obviously none of the three possible outcomes receives the necessary support to represent a good choice. Contradicting masses should therefore be mostly allocated to total ignorance, $m\left(7\right)=m\left(\mathsf{\Omega}\right)$ (compare Formulas (A1) and (A2)), displayed by the central grey disk.

## Appendix B. Data Description

## References

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**Figure 1.**Graphical comparison between probability and evidence: (

**a**) a distribution with probability function $Pr\left(a\right)$. (

**b**) an evidence represented by a mass functions $m\left(A\right)$. Note, with three overlapping disks due to the lack of degrees of freedom (5 instead of 6) not all possible constellations can be graphically represented.

**Figure 2.**Closed world vs. open world assumption: (

**a**) In a closed world no mass is given to the empty set, thus no outcome beyond $\mathsf{\Omega}$ is possible. (

**b**) In an open world a basic belief mass is given to the empty set allowing the ability to consider completely unexpected events to the model (e.g., broken coin) or to deal with data of low quality.

**Figure 3.**Belief $\mathrm{Bel}\left(S\right)$ and Plausibility $\mathrm{Pl}\left(S\right)$ illustrated by overlapping disks. The size of arrays represents basic believe masses. The difference between plausibility and believe $\mathrm{Pl}\left(S\right)\backslash \mathrm{Bel}\left(S\right)$ is called uncertainty. (

**a**) $\mathrm{Bel}\left\{R\right\}$ and $\mathrm{Pl}\left\{R\right\}$ of the singleton $\left\{R\right\}$. (

**b**) $\mathrm{Bel}\{B,G\}$ and $\mathrm{Pl}\{B,G\}$ of the set $\{B,G\}$.

**Figure 4.**Intersection of basic belief masses of two pieces of evidence. The seven colors in the mosaic plot represent (in order) the seven sets $\left\{B\right\}$, $\left\{R\right\}$, $\{B,R\}$, $\left\{G\right\}$, $\{B,G\}$, $\{R,G\}$, $\{B,R,G\}$. The 49 rectangles within the two squares are colored according to the intersect.

`White`represents the empty set ∅. For a closed world, the white areas must be redistributed among all others. (

**a**) rather consent pieces of evidence. (

**b**) rather contradicting pieces of evidence.

**Figure 5.**Contradictory data inducing undecidable outcome: (

**a**) model (9) illustrated by a sample with indecisive outcome (sample id = 881). Both IHC measurements ${\mathrm{esr}}_{\mathrm{ihc}}$ and ${\mathrm{pgr}}_{\mathrm{ihc}}$ are receptor negative, but three out of 4 pieces of evidence based on gene expression indicate a receptor positive status. Red areas represent masses $\alpha $ for positive hormone status, blue areas represent masses $\beta $ for negative hormone status, centers represent masses $\theta =1-\alpha -\beta $ for $\mathsf{\Omega}=\{+,-\}$. (

**b**) choosing inappropriate $\lambda $ for the ECRs results in dubious prognosis.

**Figure 6.**Consistent data inducing reliable outcome: (

**a**) model (9) illustrated by a sample with very clear outcome (sample id = 1980). (

**b**) When evidence is highly consistent, the parameter $\lambda $ has practically no influence on the results.

**Figure 7.**Contradicting data inducing uncertainty: (

**a**) model (9) illustrated by a sample with very contradictory input data (sample id = 2365). (

**b**) even when setting all $\lambda =0.5$ in the ECRs no conclusive evidence is generated. Nevertheless, the influence of $\lambda $ can change case by case.

**Figure 8.**Illustration of the shift towards hormone receptor negative outcome by an improved linkage between hormone receptors (Formula (9)): (

**a**) red dots are weights $\alpha $ for receptor positive, blue dots are weights $\beta $ for receptor negative. (

**b**) incorrect favoring of positive hormone receptor status has been revised by using ${\oplus}_{0.5}$ instead of ⊗.

**Figure 9.**Uncertainty vs. flexible risk. (

**a**) Increasing risk decreases uncertainty: The number of uncertain samples depends on the parameter $\lambda $ in combining the hormone receptors with ${m}^{\mathrm{esr}}\otimes {m}^{\mathrm{pgr}}$. The yellow area shows 45 out of 2519 samples (1.8%) which will always be uncertain, independent of the choice of $\lambda $. The red area changes from uncertain to receptor positive with increasing $\lambda $, the blue area changes into receptor negative. (

**b**) fixed mass of ignorance: number of classifications by fixing the weight $m\left(\mathsf{\Omega}\right)=\omega $ as described by Formula (11).

**Table 1.**Clinical decision making vs. flexible risk: (

**a**) change from clinical decision making to Formula (9), $\kappa =0.877$, there is a trend towards receptor negative (upper triangular matrix). (

**b**) an influence of $\lambda $ is only given for numerically problematic samples, $\kappa =0.966$.

(a) | ||||

flexible risk | ||||

pos | unc | neg | ||

clinical | pos | 1287 | 78 | 1 |

unc | 3 | 51 | 86 | |

neg | 0 | 0 | 1013 | |

(b) | ||||

$\lambda =0.5$ constant | ||||

pos | unc | neg | ||

flexible risk | pos | 1268 | 22 | 0 |

unc | 14 | 107 | 8 | |

neg | 0 | 2 | 1098 |

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**MDPI and ACS Style**

Kenn, M.; Karch, R.; Singer, C.F.; Dorffner, G.; Schreiner, W.
Flexible Risk Evidence Combination Rules in Breast Cancer Precision Therapy. *J. Pers. Med.* **2023**, *13*, 119.
https://doi.org/10.3390/jpm13010119

**AMA Style**

Kenn M, Karch R, Singer CF, Dorffner G, Schreiner W.
Flexible Risk Evidence Combination Rules in Breast Cancer Precision Therapy. *Journal of Personalized Medicine*. 2023; 13(1):119.
https://doi.org/10.3390/jpm13010119

**Chicago/Turabian Style**

Kenn, Michael, Rudolf Karch, Christian F. Singer, Georg Dorffner, and Wolfgang Schreiner.
2023. "Flexible Risk Evidence Combination Rules in Breast Cancer Precision Therapy" *Journal of Personalized Medicine* 13, no. 1: 119.
https://doi.org/10.3390/jpm13010119