## 2 Materials and Methods
### Applications
#### 2.5.1 Decision Support System for Pulmonary Embolism Diagnosis and Follow-Up
\begin{cases}\text{absent:}&-1,\\ \text{spontaneous:}&0,\\ \text{palpation:}&1,\end{cases}  \tag{19}\]  
- **Difference of calf size**  
Unilateral enlargement of the lower limb is defined by a difference of three centimeters between the two limbs at the calf level. Risk is high if the lower limb is enlarged, moderate if the difference is between one and two centimeters, and low if the difference is less than one centimeter. This clinical sign is represented by a difference parameter of the calf size, which is continuous and expressed in centimeters. The characteristic function \(\mu\Delta CalfSize(H(x,t)\) is depicted in Figure 21.  
Dynamic Attributes  
- **Heart rate**  
A heart rate above 100 beats per minute indicates a high risk of VTE. The risk is moderate when the heart rate is between 75 and 100 beats per minute, and low if the heart rate is below 75 beats per minute Figure 22.  
Figure 21: Fuzzy characteristic function \(\mu\Delta CalfSize(H(x,t)\) risk of PE. **- Low blood pressure**  
When the systolic blood pressure (SBP) is less than 90 mmHg, there is hypotension, or shock, and the diagnosis of pulmonary embolism is withheld until confirmed, or negated, by computed tomographic pulmonary angiography (CTPA), if it is available. The characteristic function of low blood pressure, \(\mu LBP(H(x,t),\) is presented in Figure 23.  
Figure 23: Fuzzy characteristic function low blood pressure \(\mu LBP(H(x,t)\) risk of PE. Figure 22: Fuzzy characteristic function \(\mu CardiacFreq(H(x,t)\) risk of PE. - **Drop of systolic blood pressure**  
A variation in systolic blood pressure \((\Delta SBP=SBP_{t+1}-SBP_{t})\), or a drop in SBP greater than 40 mmHg for at least 15 min is considered hypotension. It has the same diagnostic value as systolic blood pressure below 90 mmHg. This is represented by the fuzzy characteristic function \(\mu\Delta SBP(H(x,t)\) on Figure 24.  
Since either criterion determines the risk associated with a drop in systolic blood pressure, we can apply the OR-inclusive operator usually used in fuzzy logic (Max), which is compatible with FVSOOMM and can be used when OR conditions are necessary on composition relationships in the UML class diagram. It is possible to combine these functions, as in Equation (20):  
\[\mu RTAS(x,t_{n})=Max(\mu RHypoTAS(x,t_{n}),\mu R\Delta tTAS(x,t_{n},t_{n-15})) \tag{20}\]  
A disease diagnosis results from the values of the attributes defined by successive compositions (the objects, signs, syndromes, and clinical pictures). To set up the disease, it is necessary to set up the different attributes and define the result vectors beforehand. The evaluation function for pulmonary embolism (PE) disease is between \([-1,1]\) and is defined by the resultant of the vectors of the different variables. In this example, variables are aggregated according to the three variable categories (static, evolutionary, and dynamic). The resultant of the three vectors constitutes the PE vector and is shown in Figure 25, which illustrates the three usual diagnostic steps for the disease: step 0, history and suspicion of EP; step 1, thromboembolic syndrome; and step 2, cardiac syndrome that can be combined in a specific way in each patient. Figure 24: Fuzzy characteristic function \(\mu\Delta SBP(H(x,t)\) for 15 min. The corresponding FVS of pulmonary embolism \(\overrightarrow{PE(t)}\) is the resultant of the vectors \(\overrightarrow{HIST}\), \(\overrightarrow{TES(t)}\) and \(\overrightarrow{CS(t)}\), as shown on Figure 26. Note that \(\overrightarrow{HIST}\) (history) is static and can just be updated in the case of missing data. As indicated from Table 2, history remains stable, while the thromboembolic syndrome and the cardiac syndrome evolve, and thus, the resultant vector \(\overrightarrow{PE(t)}\), corresponding to the pulmonary embolism indicator, is changing over the time. Figure 22.0 Pulmonary embolism’s vector. Figure 22.1 Pulmonary embolism’s variables evolution from suspicion to diagnosis.