Proefschrift

coefficient values Matrix α 12 α 13 α 21 α 23 α 31 α 32 Λ Stable Point for FBS Volume (2) 0.7 0.8 0.4 0.6 0.5 2.0 0.4 (2082.76, 5206.90, 0.00) 7290 0.4848 (863.45,4436.73,694.82) 5995 (5) 0.6 0.8 0.4 0.7 0.5 2.0 0.39 (2600.00,5099.00,0.00) 7699 (6) 0.7 2.0 0.4 0.5 0.6 1.5 0.4 (0.0, 3764, 4354) 8118 0.35 (1003, 5739,790) 7532 0.31 (2840,5795,0) 8635 Table B.1: All stable points used for further analysis. The matrix label, updated coefficients, abiraterone dose associated with the stable point, the density of each of the subpopulations at the stable point, and the total tumor volume are shown. A total of 6 stable points are available for discussion. B.2 Forwards Backwards Sweep Algorithm The Forward Backward Sweep (FBS) algorithm is a method designed to solve the differential-algebraic system generated by Pontryagin’s Maximum Principle (Pontryagin, 2018; Metz et al., 2016). The Maximum Principle in optimal control theory states that there is a co-state variable Λ i ( t ) where an optimal state x i ( t ) and the optimal control Λ ∗ ( t ) must satisfy the dynamics of the state equations where i ∈ T = { T + , T P , T − } given by ⎧⎪⎪ ⎨ ⎪⎪⎩ ˙ x i = r i x i ⎛ ⎝ 1 − j ∈T α ij x j K i ⎞ ⎠ x i ( t 0 ) = x 0 i Initial conditions are given by the 100 random initial tumor compositions ( x T + ( t 0 ) , x T P ( t 0 ) , x T − ( t 0 )) , shown in Figure 6.2. The two-dimensional projections of these initial tumor compositions are shown in Figure B.4. Figure B.4: Two-dimensional projections for 100 random initial states used by FBS. Projections of the 100 random initial states x ( t 0 ) to x ∗ with Λ ∗ ( · ) used by FBS. ( x T + , x T P )- space in the left panel, ( x T + , x T − )-space in the center panel, and ( x T P , x T − )-space in the right panel. 141