## 3 Development of a Mixed Prehabilitation Program Model
### Mathematical Formula of the Mixed Prehabilitation Program  
The main objective of developing a mathematical model is to track and provide feedback on patient activities during the prehabilitation program. The unique feature of this model is the real-time feedback on the specific prehabilitation activities undertaken by the patient and the quantification of exercise intensity and duration. This may be made available to the various parties of interest through the cloud-based system. In addition, the prehabilitation mathematical formulas described in this paper were developed from the various research findings that were discussed in previous sections. Table 2 shows a summary of these features. Prescribed physical activity can be performed on any number of days during the week and all the physical efforts will be accumulated as a credit gain. One credit point will equate to the completion of a physical activity performed at the prescribed duration and intensity on any given day. Any further efforts beyond the prescribe exercises will be logged as additional credit points. Theoretically, the system continues calculating and accumulating the credit gain for patients performing more exercise than originally prescribed by the clinician. In this model, two points are the maximum number of credits gained per day, which is equivalent to approximately 60 min of moderate exercise based on Equation (1) below. \[P=W_{e}*I*\left(\frac{D_{e}}{T}\right) \tag{1}\]  
where:  
\(P\) is the credit gained for the prehabilitation program independent of the total program duration. Common prehabilitation programs have a length of four or six weeks. In this study, we used a six-week prehabilitation program that is represented as \(P_{b}\). A four-week prehabilitation program is represented by the symbol \(P_{a}\). \(I\) indicates the exercise intensity level. Here 0.75 is allocated for light intensity, 1 for moderate intensity, and 2 for vigorous intensity. The weighting of the different intensities is based the relative effect of each intensity on improvements in health and fitness [20; 22; 28; 29] described in Section 2.4. \(T\) is the minimum threshold time. This threshold is 10 min of physical activity at a moderate or vigorous intensity [15]. \(D_{e}\) is the duration of exercise, which is proportional to the standard unit time \(T\). Currently, the system considers and calculates each effort done by the patient regardless of meeting threshold time target (10 min at moderate intensity). This is because it is still unclear whether very short bursts exercise (less than 10 min) at a moderate or vigorous intensity should be discarded or not. \(W_{e}\) represents the modality or type of prescribed physical exercise. At this stage of the model, all nine types of exercise are initially given the same value. The value for type of exercise is based on previous studies [8; 18; 30; 31] where the patient must perform 150 min of moderate intensity exercise (e.g., 30 min of moderate physical activity five times per week) or equivalent per week, for a period of six weeks. In this example, the total volume of exercise over the six weeks (\(P\)) equates to a total of 30 credit points. The exercise intensity (\(I\)) would be 1 (moderate intensity), the time per session (\(De\)) would be 30 (30 min), and the minimal threshold for time (\(T\)) would be 10 min. Given that \(P\), \(We\), \(De\) and \(T\) are known, Equation (1) can be re-arranged to calculate \(W_{e}\) (see formula below). This would result in \(W_{e}\) being 0.335. \[P=W_{e}*I*\left(\frac{D_{e}}{T}\right),\;1=W_{e}*1*\left(\frac{30}{10}\right) \text{ then the value of }W_{e}=0.335\]  
Future work may differentiate among activities when it comes to weight assignment. The total credits of each program can be varied based on the program length, prescribed physical exercises, and the duration and frequency of each exercise per week. As mentioned previously, the six-week prehabilitation program \(P_{b}\) was selected as a case for the implementation of the mathematical formula.