## 2 Methods and Materials
### PK Model Fitting
For a given AIF, we sample the model parameters \(\theta\) and calculate \(C_{tissue}\) according to Equation (1). Sampling is repeated \(10^{4}\) times for each AIF. As a result, there were \(19\times 10^{4}<C(t)\), \(\theta>\) pairs available in each leave-one-out repetition. The calculated \(C_{tissue}\) curves were probed at time steps adjusted to match the temporal resolution of the network input vector. The simulated tracer concentration time curves were optionally corrupted by the Gaussian noise to increase their variability and make them more realistic. Data corruption was performed by adding to each simulated CA concentration a value drawn from the normal distribution, with mean = 0, and standard deviation was adjusted to a current time step, i.e., \(\sigma_{noise}\ =\ s\cdot[C_{tissue}(t)-min(C_{tissue}(t))]\). The scale factor \(s\) was experimentally set to 0.025 to achieve reasonable deflection from the modeled CA concentration time curves. Then, the data set was partitioned into training and test sets in the proportion 7:3. Moreover, 30% of the training data vectors were randomly set apart in each epoch for validation purposes. The last issue which needs to be fixed is the establishing of a proper network architecture. During experiments, it occurred that only one hidden layer cannot accurately encapsulate the non-linear relationship between the PK model's parameters and the variety of CA concentration time-curves. The two-layer structure ensured a significant decrease in the loss function value, while the usage of more than two hidden layers did not offer any further improvement in this respect. Eventually, the number of perceptrons in each layer had to be established. We tested 12 configurations, listed in Table 2. Apart from the various cardinalities of neurons in hidden layers, we also considered inclusion of additional dropout layers between them to overcome the problem of overfitting [30]. For every configuration and each leave-one-out fold, the training was run for a fixed number of 100 epochs. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & **Min** & **Max** & **Mean** & **SD** & **Probability Distribution** \\ \hline \(K^{trans}\) [min\({}^{-1}\)] & 0 & – & 0.25 & 0.1 & Normal \\ \hline \(v_{p}\) [mm\({}^{3}\)] & 0.2 & 0.8 & – & – & Uniform \\ \hline \(\Delta\) [s] & 1 & 3.5 & – & – & Uniform \\ \hline MRT [s] & – & – & 5.5 & 0.7 & Normal \\ \hline \(T_{g}\) [s] & 0.02 & – & – & – & Normal \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter limits and types of probability distributions used for sampling training data set for ANN (artificial neural network). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Configuration No.** & **First Hidden Layer** & **Dropout Rate \({}^{*}\)** & **Second Hidden Layer** & **Output Layer** \\ \hline \#1 & 20 & 0.0 & 10 \\ \#2 & & 0.2 & \\ \hline \#3 & 26 & 0.0 & 13 \\ \#4 & & 0.2 & 15 \\ \hline \#5 & 30 & 0.0 & 15 \\ \#6 & 34 & 0.0 & 17 \\ \#8 & & 0.2 & \\ \hline \#9 & 40 & 0.0 & 20 \\ \#10 & & 0.2 & 25 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Number of neurons in the tested two-layer perceptron structures.