Proefschrift

APPENDIX B. SUPPLEMENT TO CHAPTER 6 It is also required that co-states λ i ( t ) satisfy the transversality conditions given by ⎧⎪⎨ ⎪⎩ ˙ λ i = − ∂H ∂x i λ i ( t f ) = 0 where co-state equations λ i for the Hamiltonian in Equation (6.3) are constructed below. ˙ λ T + = x T + − x ∗ T + φ − λ T + r T + x T + + α 12 x T P + α 13 x T − K T + (Λ) x T P − 1 ... − λ T + r T + x T + K T + (Λ) x T P − α 21 λ T P r T P x T P K T P (Λ) − α 31 λ T − r T − x T − K T − (B.2) ˙ λ T P = x T P − x ∗ T P φ − λ T P r T P α 21 x T + + x T P + α 23 x T − K T P − 1 ... − λ T + r T + x T + α 12 K T + (Λ) x T P − x T + + α 12 x T P + α 13 x T − K T + (Λ) x T P 2 − ... λ T P r T P x T P K T P (Λ) − α 32 λ T − r T − x T − K T − (B.3) ˙ λ T − = x T − − x ∗ T − φ − λ T − r T − x T − + α 31 x T + + α 32 x T P K T − − 1 ... − α 13 λ T + r T + x T + K T + (Λ) x T P − α 23 λ T P r T P x T P K T P (Λ) − λ T − r T − x T − K T − (B.4) where φ = x T + − x ∗ T + 2 + x T P − x ∗ T P 2 + x T − − x ∗ T − 2 (B.5) The first iteration of the FBS algorithm requires as input an initial guess of the control Λ( · ) which for the simulations presented here is set to Λ( t ) = 0 . 5 for all t ∈ [ t 0 , t f ]. For each iteration the initial value problem of the state equation is solved forward in time subject to the initial conditions set by the initial tumor composition ( x T + ( t 0 ) , x T P ( t 0 ) , x T − ( t 0 )). Then, the co-state final value problem is solved backwards in time subject to the transversality conditions λ i ( t f ) = 0. The solution to these are then used to solve for the Hamiltonian given in Equation (6.3). As optimal control attempts to maximize the Hamiltonian given in Equation (6.3), the new guess for the optimal Λ( t ) , denoted here by Λ ◦ ( n +1) , is calculated using the following formula. Λ ◦ ( n +1) = Λ ◦ ( n ) − ∂H ∂ Λ ◦ ( n ) (B.6) where ∂H ∂ Λ = λ T P r T P x T P ( α 21 x T + + x T P + α 23 x T − ) K T P (Λ) 2 + ... λ T + r T + x T + ( x T + + α 12 x T P + α 13 x T − ) 10000 x T P K T P (Λ) 10000 + x T + 2 2 (B.7) 142