## 2 Materials and Methods
### Development and Evaluation of the MARS-MMA Scoring System  
Our second aim was to investigate whether another scoring system might predict the severe CPR impairment of the ACA and MCA territories better than the PIRAMID scoring system. For this purpose, we propose an optimized new scoring system incorporating the components of the PIRAMID scoring system and additionally the presence of arterial vessel wall contrast enhancement and the Ivy sign on FLAIR images.  
First, we calculated, in a preliminary analysis, the grade of arterial wall contrast enhancement indicating a severe [15O]water PET impairment with the highest sensitivity and specificity. For this purpose, we performed an ROC curve analysis using [15O]water PET as the dependent variable and the arterial vessel wall contrast enhancement as the independent variable. The grade of contrast enhancement with the maximum Youden index was determined as the optimal cut-off to binarize the CVR of each evaluated territory in the further analysis as impaired or not impaired.  
Afterwards, we performed a backward and forward binary logistic regression analysis with [15O]water PET CPR impairment as the dependent variable and all components of the PIRAMID scoring system (prior infarcts, impaired CVR, mSS \(\geq\) II, \(\geq\)2 territories with impaired collaterals), as well as the presence of an arterial wall contrast enhancement and the presence of the Ivy sign, as the independent variables. A new scoring system (MARS-MMA) was created with all factors, revealing the significant prediction of a severe [15O]water PET CPR impairment at \(p<\) 0.05. The relative weighting for each factor was determined by the calculated regression coefficients of the binary logistic regression analysis.  
For clinical use, the MARS-MMA scoring system was divided into three grades, for which two threshold values had to be defined. After the determination of the components of the MARS-MMA score and their weightings, all possible MARS-MMA scores that the territories could achieve by combining the weighted factors were calculated. Grade stratification was performed by calculating the individual grades of each ACA/MCA territory for all possible pairs of threshold values. The optimum pair of threshold values was calculated with ROC curve analysis using [15O]water PET CPR impairment as the dependent variable and the grades of all possible pairs of threshold values as the independent variables.  
The MARS-MMA score and grade were calculated for each ACA/MCA territory. As our aim was to compare the newly developed MARS-MMA scoring system to the PIRAMID scoring system, we calculated a so-called _adapted PIRAMID score_ to avoid the asymmetric overfitting of the new MARS-MMA scoring system. For this purpose, we performed a binary logistic regression analysis with [15O]water PET CPR impairment as the dependent variable and all components of the PIRAMID scoring system (prior infarcts, impaired CVR, mSS \(\geq\) II, \(\geq\)2 territories with impaired collaterals) as the independent variables and optimized the factors' weights based on the regression coefficients to fit our data.  
The comparison between the MARS-MMA scoring system and the PIRAMID scoring system was conducted by performing an ROC curve analysis with [15O]water PET CPR impairment as the dependent variable and the MARS-MMA score and the adapted PIRAMID score as independent variables. The Akaike information criterion (AIC) was calculated to account for different numbers of predictors.