## 2 Materials and Methods
### Applications
#### 2.5.1 Decision Support System for Pulmonary Embolism Diagnosis and Follow-Up
Static Variables  
-Age  
Age is a predisposing factor in the occurrence of pulmonary embolism. The risk of PE is high when the age of a patient is above 65, moderate for an age between 50 and 65, and low if the age is below 50. The proposed approach in this paper overcomes these thresholds by using a continuous characteristic function when the attribute considered is continuous, which is the case for age (Table 1 and Figure 17). **- Thromboembolic risk factors**  
Three levels (low, moderate, or high) of thromboembolic risk factors are considered in factors that are either present or absent. A multiplicative exponential weight factor is associated with each level. For low risk, \(w_{i}=2\) is used, \(w_{j}=4\) for medium risk, and \(w_{k}=8\) for high risk. There are 9 low-risk factors \(nblr=9\), 18 moderate-risk factors \(nbnr=18\), and 8 high-risk factors \(nbhr=8\). It is possible to add new risk factors and to tune the curve according to local epidemiological data, if necessary. The equation for calculating the score variable is given by Equation (17). \[Score=w_{i}\sum_{i=1}^{nbr=9}Lr_{i}+w_{j}\sum_{j=1}^{nbmr=18}Mr_{j}+w_{k}\sum_{k =1}^{nbr=8}Hr_{k} \tag{17}\]  
According to the previous score, the following fuzzy characteristic function of thromboembolic risk \(\mu ThE(H(x)\) is computed. For example, when two high risk factors are found in the patient, the risk is 16 and \(\mu ThE(H(x))=0.939\) on Figure 18.  
Figure 17: Fuzzy characteristic function \(\mu\)Age(H(x)). - **Cancer risk factors**  
Cancer is classified according to the stage classification of cancers:  
- Stage 0--it is a carcinoma in situ or a precancerous change;  
- Stage 1--the tumor is usually small and has not grown outside the organ in which it originated;  
- Stages 2 and 3--the tumor is large or has grown outside the organ in which it originated into the surrounding tissue;  
- Stage 4--the cancer has spread through the blood or lymphatic system to a distant location (metastatic spread). - Stage (\(-1\))--this is added to signify that there is no cancer. Based on this classification, stages 0 and 1 are considered low risk, stages 2 and 3 are considered moderate risk, and stage 4 is considered high risk for VTE. Stage -1 denotes the absence of a known cancer. This stage attribute x is discrete, and Table 3 shows the values. \(H(e^{x})\) for cancer risk is computed in a similar way as the previous thromboembolism risk, shown in Figure 18. Some attributes are discrete and exponential like the cancer risk function described on Figure 19.  
- **Hospitalization or immobilization after surgery**  
A length of time of one month since surgery, immobilization after a fracture, and hospitalization, are all considered risk factors for pulmonary embolism. Since elapsed time is a continuous variable, the characteristic function \(\mu RiskAfterSurgery(H(x))\) shows that the characteristic function decreases with the time (t is expressed in days) elapsed since the date of fracture or surgery(Figure 20). Evolutionary Variables  
* **Hemoptsis**  
Hemoptsis is defined by its presence or absence. When present, risk is high if the hemoptysis was frank, moderate if it was blood-striped sputum, and none in the absence  
Figure 19: Fuzzy characteristic cancer risk function \(\mu Cancer(H(e^{x})\) of PE. Figure 20: Fuzzy characteristic function \(\mu RiskAfterSurgery(H(x)\) of PE. of hemoptysis. Hemoptysis is a discrete attribute, so the fuzzy characteristic function \(\mu Hemoptysis(x)\) is discrete and expressed by Equation (18)  
\[\mu RHemoptysis(t)= \begin{cases}\text{none:}&-1,\\ \text{striped sputum:}&0,\\ \text{Frank hemoptysis:}&1,\end{cases}  \tag{18}\]  
- **Unilateral pain in the lower limb**  
Unilateral pain in the lower limb is defined by its presence or absence. Risk is high if the pain occurs on deep palpation of the calf, moderate if the pain is spontaneous, and low if the pain is absent. This attribute is discrete, so the characteristic function is expressed in the same way as hemoptysis by Equation (19)  
\[\mu R Pain Limb(t)=