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We discuss a novel diagnostic method for predicting the early recurrence of liver cancer with high accuracy for personalized medicine. The difficulty with cancer treatment is that even if the types of cancer are the same, the cancers vary depending on the patient. Thus, remarkable attention has been paid to personalized medicine. Unfortunately, although the Tokyo Score, the Modified JIS, and the TNM classification have been proposed as liver scoring systems, none of these scoring systems have met the needs of clinical practice. In this paper, we convert continuous and discrete data to categorical data and keep the natively categorical data as is. Then, we propose a discrete Bayes decision rule that can deal with the categorical data. This may lead to its use with various types of laboratory data. Experimental results show that the proposed method produced a sensitivity of 0.86 and a specificity of 0.49 for the test samples. This suggests that our method may be superior to the well-known Tokyo Score, the Modified JIS, and the TNM classification in terms of sensitivity. Additional comparative study shows that if the numbers of test samples in two classes are the same, this method works well in terms of the

Liver cancer is one of the refractory cancers, and overcoming it is of national concern. Despite complete surgical resection, the problem with refractoriness lies in the high percentage of recurrences of liver cancer [

The difficulty with cancer treatment is that even if the types of cancer are the same, the cancers vary depending on the patient. Even a wide variety of tests such as blood tests or CT scans reveal only certain aspects of cancer, and there is no test that can perfectly detect cancer. Nevertheless, by using various combinations of laboratory data (also called markers), the Tokyo Score [

The diagnosis of leukemia was made possible using levels of gene expression, and this triggered the diagnosis of cancer by machine learning [

Generally, with laboratory data, quantitative data (metric data or continuous data), which are represented by numerical numbers, intermingle with qualitative data (nonmetric data or categorical data). Unfortunately, as with the machine learning described above, the conventional Bayes classifier, which is particularly popular in statistical pattern recognition [

As previously mentioned, the discrete Bayes classifier is characterized by the fact that it can handle discrete data. Given

Assuming that, in general, the events in which the discretized laboratory data belong to any of the divisions are mutually independent, the class-conditional probability

The posterior probability

Cases of the relationship between the divisions and the number of patients are shown in Table

Divisions and numbers of patients with each marker.

Division of markers | | |
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We explain the discrete Bayes decision rule by using concrete cases. Assume that in the case of

Division and number of training samples in each class.

Division of markers | Number of samples | |
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29 | 89 | |

Recurrence within 1 year | Nonrecurrence within 1 year | |

| 15 | 60 |

| 14 | 29 |

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| 6 | 47 |

| 11 | 29 |

| 12 | 13 |

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| 10 | 18 |

| 19 | 71 |

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| 14 | 44 |

| 15 | 45 |

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| 9 | 16 |

| 20 | 73 |

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| 16 | 70 |

| 13 | 19 |

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| 18 | 67 |

| 11 | 22 |

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| 17 | 61 |

| 12 | 28 |

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| 25 | 79 |

| 4 | 10 |

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| 15 | 60 |

| 14 | 29 |

Arrangement of training samples of the recurrence class for markers

Now, let us compare the discrete Bayes classifier to the conventional Bayes classifier, both of which classify patterns based on the posterior probability. In the conventional Bayes classifier, a pattern is represented as a multidimensional vector that consists of quantitative data, and the statistical information of the pattern distribution is in the mean vector and covariance matrix. In general, the number of patients is small and the number of dimensions is large. In small sample size situations, the discrete Bayes decision rule is also influenced by the number of samples, as with the conventional Bayes decision rule, and the discrimination ability deteriorates. In the conventional Bayes decision rule, an inverse of a covariance matrix may not exist. At this time, although the conventional Bayes classifier cannot be designed, the discrete Bayes classifier can be designed, irrespective of an inverse matrix. The first advantage of the discrete Bayes decision rule is that it does not need the inverse of a covariance matrix. Second, from the viewpoint of computational cost, in the conventional Bayes classifier, which deals with quantitative data alone, the computational cost increases sharply with an increasing number of dimensions (number of markers). Meanwhile, in any of the discrete Bayes classifiers, which deal with discrete data alone, computation is only scalar computation. Even if a dimension is high, the discrete Bayes classifier can be easily calculated, indicating that the classifier is practical. To simplify the discussion, we have so far dealt with a two-class problem, but this decision rule can easily be extended to multiclass problems.

Knowing which markers are used for the discrete Bayes classifier is essential in discrimination. This is a problem of feature selection in the statistical pattern recognition fields [

Selection of optimal markers.

First, training samples are randomly divided into virtual training samples and virtual test samples. Then, the number of markers at the start of searching is determined, and one combination of markers of interest is used. Second, based on the discrete Bayes decision rule with the markers of interest, the class-conditional probability

Data were obtained from patients whose liver cancers were entirely excised during surgery at Yamaguchi University School of Medicine. Of these patients, 57 experienced a recurrence of liver cancer within one year and 177 experienced no recurrence. Liver cancer is classified as type C liver cancer, type B liver cancer, and others depending on the infecting virus type. The virus types of liver cancer used are shown in Table

Classification of samples by virus type.

Breakdown of training samples

Virus type | Recurrence within 1 year | Nonrecurrence within 1 year |
---|---|---|

B | 6 | 16 |

C | 18 | 56 |

Samples that are neither B type nor C type | 5 | 17 |

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Total number of samples | 29 | 89 |

Breakdown of test samples

Virus type | Recurrence within 1 year | Nonrecurrence within 1 year |
---|---|---|

B | 6 | 16 |

C | 17 | 55 |

Samples that are neither B type nor C type | 5 | 17 |

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Total number of samples | 28 | 88 |

Among candidate markers such as ALB, tumor number × tumor size [

We explain the details of Table

The holdout method [

Flow of classifier design and evaluation.

Next, the optimal combination of markers was fixed, and the number of training samples is discussed. Generally, an increase in the number of training samples results in an improvement of the discrimination ability of the classifier. Therefore, a designer is interested in how many training samples are needed for classifier design. Then, as a subset of the increased number of training samples, we assume that a series of 6 training subsets, as shown in Figure

Relationships between the training sample subsets.

Then, a comparison of the performance of the classifiers with the existing scoring formulae was conducted. For comparison, we adopted accuracy, sensitivity, specificity, the Youden index [

Here, we explain the score formulae as targets for comparison. For the Tokyo Score [

Finally, as described previously, the discrete Bayes classifier uses scalar function alone for discrimination, and, thus, it has a computational advantage. To clarify this advantage, we prepared 1,160,000 artificial test samples that were obtained from 116 actual test samples copied 10000 times. Then, using combinations of the markers shown in Table

Optimal combinations of markers per number of markers and their discrimination performances obtained using training samples.

Number of markers | Sensitivity | Specificity | Youden index | Combination of markers | |||||
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3 | 0.79 | 0.50 | 0.29 | Tumor number × tumor size | vp | Liver damage | |||

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4 | 0.80 | 0.50 | 0.30 | Tumor number × tumor size | vp | ICG | Liver damage | ||

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5 | 0.75 | 0.50 | 0.25 | ALB | Tumor number × tumor size | vv | Degree of differentiation | Liver damage | |

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6 | 0.74 | 0.51 | 0.24 | ALB | Tumor number × tumor size | vp | ICG | Degree of differentiation | Liver damage |

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7 | — | — | — | ||||||

8 | — | — | — | ||||||

9 | — | — | — |

Table

The relationship between the number of training samples and sensitivity is shown in Figure

Relationship between the number of training samples and sensitivity.

Discrimination performances were compared between the discrete Bayes classifier and existing liver scoring systems. The classifier was evaluated by well-known indices such as

Results from the proposed method and existing liver scoring systems.

Results using 28 recurrence test samples and 88 nonrecurrence test samples

Index | Proposed method | Modified JIS with 3 | TNM classification with 2 | Tokyo score with 2 |
---|---|---|---|---|

Accuracy | 0.58 | 0.77 | 0.34 | 0.49 |

Sensitivity, recall | 0.86 | 0.57 | 0.96 | 0.71 |

Specificity | 0.49 | 0.83 | 0.14 | 0.42 |

| 0.49 | 0.54 | 0.41 | 0.40 |

Youden index | 0.35 | 0.40 | 0.10 | 0.13 |

Diagnostic odds ratio | 5.73 | 6.49 | 4.26 | 1.81 |

Results using 28 test samples/class obtained by resampling

Index | Proposed method | Modified JIS with 3 | TNM classification with 2 | Tokyo score with 2 |
---|---|---|---|---|

Accuracy | 0.67 [0.66, 0.69] | 0.70 [0.69, 0.71] | 0.55 [0.54, 0.56] | 0.57 [0.55, 0.58] |

Sensitivity, recall | 0.86 | 0.57 | 0.96 | 0.71 |

Specificity | 0.49 [0.46, 0.52] | 0.83 [0.81, 0.85] | 0.14 [0.12, 0.16] | 0.42 [0.39, 0.44] |

| 0.73 [0.72, 0.73] | 0.66 [0.65, 0.67] | 0.68 [0.68, 0.69] | 0.62 [0.62, 0.63] |

Youden index | 0.35 [0.32, 0.37] | 0.40 [0.38, 0.42] | 0.10 [0.08, 0.12] | 0.13 [0.11, 0.16] |

Diagnostic odds ratio | 6.03 [5.39, 6.67] | 7.88 [6.22, 9.53] | 4.62 [3.86, 5.38] | 1.95 [1.74, 2.15] |

ROC curve.

The relationship between the number of markers and CPU time as the number of markers is changed one by one from 3 to 6 is shown in Figure

Relation between the number of markers and discrimination time.

Based on the results of this experiment, it was revealed that a combination of 4 markers (tumor number

Meanwhile, in terms of accuracy and the

In addition, we point out an advantage of the discrete Bayes classifier over the existing scoring systems. Because the existing scoring systems require the use of specific markers, they cannot be used when the data of the markers are insufficient. However, the proposed technique can be used by selecting an optimal combination of markers from the laboratory data that a patient already has. Moreover, the physician is presented with the markers that should be added to improve discrimination performance for each patient. In this way, on the basis of the technique proposed here, the best personalized medicine can be expected.

In this paper, a discrete Bayes decision rule that predicts early recurrence of highly refractory liver cancer with a high degree of accuracy was proposed. This discrete Bayes decision rule can deal not only with qualitative data but also with quantitative data by discretization. Discrimination experiments enabled us to predict early recurrence of liver cancer with higher sensitivity than that of the Tokyo Score, the Modified JIS, and the TNM classification, which are existing representative scoring systems. Realization of personalized medicine via this discrete Bayes decision rule may be expected.

One of the main limitations of this paper was the use of a small amount of sample data from a single institution. This limited the evaluation of the performance. Further study is needed to evaluate the proposed method using data collected from other institutions. In addition, this method uses the hypothesis of independence without discussion to simplify the calculations. Although verification of independence is difficult, a study on independence such as the adoption of Bayesian networks will be a future challenge. Also, because the discrimination ability of this method depends on the cutoff value used when marker values are discretized, optimization will also be an interesting challenge.

The authors declare that there are no competing interests regarding the publication of this paper.

This work was supported by JSPS KAKENHI Grant no. JP15K00238.