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CDAC 2018 Angaroni optimal control

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Presentation at the 2018 Workshop and School on Cancer Development and Complexity (CDAC 2018)
http://cdac2018.lakecomoschool.org

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CDAC 2018 Angaroni optimal control

  1. 1. Optimal Control of chronic myeloid leukemia treatment F.Angaroni May 24, 2018 F.Angaroni (U. Bicocca) Optimal Control of the chronic myeloid leukemia treatment 23 May 2018 1 / 36
  2. 2. 1 The disease Differential equations model 2 Optimal control Analytic solution: the Pontryagins maximum principle Numerical solution: the Pontryagins maximum principle Optimal control: the CML therapy examples Optimal control: the CML therapy 3 Future works F.Angaroni (U. Bicocca) Optimal Control of the chronic myeloid leukemia treatment 23 May 2018 1 / 36
  3. 3. The disease: Mathematical hypothesis This disease is driven by the BRC-ABL oncogene. Since BCR-ABL mutation is present in all leukemic cells the ratio of cancer cells respect to healthy cells is a ”simple” measurements. About 2000 follow up could be found in literature. Their are an example of measurements of the in vivo kinetics (F.Michor) Exponential decay of cancer cells, and exponential blast after the stop of the therapy, tell us that N(t) should be an exponential (solution of first order differential equation) or described by a Poisson process. F.Angaroni (U. Bicocca) Optimal Control of the chronic myeloid leukemia treatment 23 May 2018 2 / 36
  4. 4. The disease: Mathematical hypothesis We divide the cells of the tiusse under study in ne non intersecting ensambles, every ensamble represents a certain stage of cell differentiation. F.Angaroni (U. Bicocca) Optimal Control of the chronic myeloid leukemia treatment 23 May 2018 3 / 36
  5. 5. Compartments Model In this case, we divide the cells in four ensambles: 1 Stem cells (SC) 2 Precursors Cells (PC) 3 Differentiated cells (PC) 4 Terminally differentiated cells (TD) Every collection of these ensembles is a branch. F.Angaroni (U. Bicocca) Optimal Control of the chronic myeloid leukemia treatment 23 May 2018 4 / 36
  6. 6. Differential equations model Since BCR-ABL mutation is present in all leukemic cells we can distinguish between Healthy branch Leukemic branch F.Angaroni (U. Bicocca) Optimal Control of the chronic myeloid leukemia treatment 23 May 2018 5 / 36
  7. 7. Differential equations model We attach a function ci,l (t) to every ensemble, it represents the number of cells in one ensemble. We will study transition rate between classes. Given the following parameters: pi,k the division rate of the cells di,k the death rate of the cells ai,k ∈ [0, 1] the probability of self-renewal λ the probability for unit of time for a SC to develop the mutation s(t) = 1 1+k( 4 i=1 h j=l ci,j (t)) biochemical signal that regulate the cells proliferation, it depends only on the number of mature cells F.Angaroni (U. Bicocca) Optimal Control of the chronic myeloid leukemia treatment 23 May 2018 6 / 36
  8. 8. Differential equations model Considering only symmetric differentiation, the transition rate for the ci,k ensamble are: +2pi,kai,kci (t)s(t) a in-going flux caused by the replication, this contribute is absent for the TD ensembles, F.Angaroni (U. Bicocca) Optimal Control of the chronic myeloid leukemia treatment 23 May 2018 7 / 36
  9. 9. Differential equations model Considering only symmetric differentiation, the transition rate for the ci,k ensamble are: +s(t)pi−1,k(1 − ai−1,k)ci−1,k(t) a in-going flux due to the differentiation in the previous ensemble of the branch, it is not present for the SC (i = 1) F.Angaroni (U. Bicocca) Optimal Control of the chronic myeloid leukemia treatment 23 May 2018 8 / 36
  10. 10. Differential equations model Considering only symmetric differentiation, the transition rate for the ci,k ensamble are: : −di,kci,k out-going flux due to the death of cells, F.Angaroni (U. Bicocca) Optimal Control of the chronic myeloid leukemia treatment 23 May 2018 9 / 36
  11. 11. Differential equations model Considering only symmetric differentiation, the transition rate for the ci,k ensamble are: −s(t)pi ci (t) out-going flux due to the differentiation, F.Angaroni (U. Bicocca) Optimal Control of the chronic myeloid leukemia treatment 23 May 2018 10 / 36
  12. 12. Differential equations model Considering only symmetric differentiation, the transition rate are give by: λc1,k in-going flux for c1,k+1 and out-going flux for c1,k. It represents the generation of the tumor SC from the healthy SC. F.Angaroni (U. Bicocca) Optimal Control of the chronic myeloid leukemia treatment 23 May 2018 11 / 36
  13. 13. Differential equations model From the diagram it is easy to obtain a system of 8 first order differential equations: dc1,h(t) dt = [(2a1,h − 1)p1,hs(t) − d1,h − λ]c1,h(t), dc2,h(t) dt = 2(1 − a1,h)p1,hs(t)c1,h(t) + [−d2,h + (2a2,h − 1)p2,hs(t)]c2,h(t), dc3,h(t) dt = 2(1 − a2,h)p2,hs(t)c2,h(t) + [−d3,h + (2a3,h − 1)p3,hs(t)]c3,h(t), (1) dc4,h(t) dt = 2(1 − a3,h)p3,hs(t)c4,h(t) − d4,hc4,h(t), dc1,l (t) dt = [(2a1,l − 1)p1,l s(t) − d1,l ]c1,l (t) + λc1,h(t), dc2,l (t) dt = 2(1 − a1,l )p1,l s(t)c1,l (t) + [−d2,l + (2a2,l − 1)p2,l s(t)]c2,l (t), dc3,l (t) dt = 2(1 − a2,l )p2,l s(t)c2,l (t) + [−d3,l + (2a3,l − 1)p3,l s(t)]c3,l (t), dc4,l (t) dt = 2(1 − a3,l )p3,l s(t)c4,l (t) − d4,l c4,l (t), F.Angaroni (U. Bicocca) Optimal Control of the chronic myeloid leukemia treatment 23 May 2018 12 / 36
  14. 14. Differential equations model This tumor has a molecularly targeted therapy: Imatinib an inhibitor of BCR-ABL gene. The therapy is simulate as a decrease of 3 order of magnitude of division rate of tumoral cells. Figure: From:Dynamics of chronic myeloid leukemia F.Michor et al. F.Angaroni (U. Bicocca) Optimal Control of the chronic myeloid leukemia treatment 23 May 2018 13 / 36
  15. 15. Differential equations model The model is in a good agreement with the experimental data, defining a figure of merit (FOM) T(t) = 4 i=1 ci,l (t) 4 i=1 h k=l ci,k(t) . (2) F.Angaroni (U. Bicocca) Optimal Control of the chronic myeloid leukemia treatment 23 May 2018 14 / 36
  16. 16. Differential equations model The model presents a steady state where the tumoral stem cells are constant: life-long disease. c∗ 1,l = λc∗ 1,h (2a1,h − 1)p1,hs∗ − d1,l , (3) F.Angaroni (U. Bicocca) Optimal Control of the chronic myeloid leukemia treatment 23 May 2018 15 / 36
  17. 17. Differential equations model The steady state is present only if: d1,l c∗ 1,l = [(2a1,l − 1)p1,l s∗ )]c∗ 1,l + λc∗ 1,h (4) An early estimation of parameters could help in clinical management F.Angaroni (U. Bicocca) Optimal Control of the chronic myeloid leukemia treatment 23 May 2018 16 / 36
  18. 18. Optimal control: motivation The therapy has several drawbacks: 1 Fails in eradicating disease Figure: From:Dynamics of chronic myeloid leukemia F.Michor et al. F.Angaroni (U. Bicocca) Optimal Control of the chronic myeloid leukemia treatment 23 May 2018 17 / 36
  19. 19. Optimal control: motivation The therapy has several drawbacks: 1 Fails in eradicating disease 2 Too expensive to be effective in a epidemiology contest Figure: From:The price of drugs for chronic myeloid leukemia (CML) is a reflection of the unsustainable prices of cancer drugs: from the perspective of a large group of CML experts F.Angaroni (U. Bicocca) Optimal Control of the chronic myeloid leukemia treatment 23 May 2018 18 / 36
  20. 20. Optimal control: motivation The therapy has several drawbacks: 1 Fails in eradicating disease 2 Too expensive to be effective in a epidemiology contest 3 Is a life-long therapy (?) Figure: From:Early molecular response and female sex strongly predict stable undetectable BCR-ABL1, the criteria for imatinib discontinuation in patients with CML F.Angaroni (U. Bicocca) Optimal Control of the chronic myeloid leukemia treatment 23 May 2018 19 / 36
  21. 21. Optimal control: motivation The therapy has several drawbacks: 1 Fails in eradicating disease 2 Too expensive to be effective in a epidemiology contest 3 Is a life-long therapy (?) 4 The therapy fails in approximately 15 − 25% of patients due to the presence of resistant subclones, but Pharmacologic inhibitors for imatinib-resistant CML exist and dose escalation can improve the response in a subset of patients with resistance to standard dose imatinib F.Angaroni (U. Bicocca) Optimal Control of the chronic myeloid leukemia treatment 23 May 2018 20 / 36
  22. 22. Control Theory Control Theory deals with systems that can be controlled, i.e. whose evolution can be influenced by some external agent described by u ∈ U ⊆ Rn, e.g.: dx dt = f (x, u) (5) There are two classes of control: 1 Open loop. Choose u as function of time t, 2 Closed loop or Feedback. Choose u as function of space variable x Optimal control belongs to the first class. F.Angaroni (U. Bicocca) Optimal Control of the chronic myeloid leukemia treatment 23 May 2018 21 / 36
  23. 23. Open loop control: the problem Given the dynamics: dx dt = f (x, u) (6) T is the ensemble of final possible state x(T) at time t = T, if x(0) = x0 L(x(t), u)dt is the running cost φ(x(T)) is the pay-off the optimal control problem is to find u∗(t) such that: min[φ(x(T)) + t 0 dtL(x(t), u(t))] x(0) = x0 x(T) ∈ T (7) One approach to the optimal problem is the maximum principle : F.Angaroni (U. Bicocca) Optimal Control of the chronic myeloid leukemia treatment 23 May 2018 22 / 36
  24. 24. Optimal control: Pontryagins Maximum Principle The cornerstone of optimal control is the Pontryagins maximum principle. If u∗(t) and x∗(t) are optimal optimal solution and H(t, x(t), u(t), λ(t)) is an Hamiltonian defined as follows: H(t, x(t), u(t), λ(t)) = λf (t, u(t), x(t)) + L(u(t), x(t)) (8) then there exists a piecewise differentiable adjoint variable λ(t) such that: dλ(t) dt = − ∂H(t, x(t), u(t), λ(t)) dx (9) λ(T) = φ(x(T)) (10) H(t, x∗ (t), u(t), λ(t)) ≤ H(t, x∗ (t), u∗ (t), λ(t)) (11) with ∂H ∂u u=u∗ = 0 (12) F.Angaroni (U. Bicocca) Optimal Control of the chronic myeloid leukemia treatment 23 May 2018 23 / 36
  25. 25. Optimal control: Numerical solution Consider an optimal control system: 1) Cost max φ(x(T)) + T t0 L(t, x(t), u(t))dt (13) 2) Dynamics x(t0) = a dx(t) dt = f (t, x(t), u(t)) (14) 3) The dynamics of the adjoint λ(T) = φ(x(T)) dλ dt = − ∂H ∂x (15) 4) Characterization of the optimal control ∂H ∂u u=u∗ = 0 (16) F.Angaroni (U. Bicocca) Optimal Control of the chronic myeloid leukemia treatment 23 May 2018 24 / 36
  26. 26. Numerical solution: Forward-Backward Sweep Method A rough outline of the algorithm to solve the optimal control problem is the following: 1 Make an initial guess for u over the interval. 2 Solve Forward in time the Dynamics: x(t0) = a dx(t) dt = f (t, x(t), u(t)) (17) 3 Solve Backward in time the dynamics of the adjoint: λ(T) = φ(x(T)) dλ dt = − ∂H ∂x (18) 4 Update u using λ(t) and x(t) into the characterization of the optimal control ∂H ∂u u=u∗ = 0 (19) 5 Check convergence. If values of the variables in this iteration and the last iteration are negligibly close, output the current values as solutions. If values are not close, return to Step 2. F.Angaroni (U. Bicocca) Optimal Control of the chronic myeloid leukemia treatment 23 May 2018 25 / 36
  27. 27. Optimal control: the CML therapy The dynamics of the system under treatment is given by: dc1,h(t) dt = [(2a1,h − 1)p1,hs(t) − d1,h − λ]c1,h(t), dc2,h(t) dt = 2(1 − a1,h)p1,hs(t)c1,h(t) + [−d2,h + (2a2,h − 1)p2,hs(t)]c2,h(t), dc3,h(t) dt = 2(1 − a2,h)p2,hs(t)c2,h(t) + [−d3,h + (2a3,h − 1)p3,hs(t)]c3,h(t), (20) dc4,h(t) dt = 2(1 − a3,h)p3,hs(t)c4,h(t) − d4,hc4,h(t), dc1,l (t) dt = [(2a1,l − 1) p1,l u(t) s(t) − d1,l ]c1,l (t) + λc1,h(t), dc2,l (t) dt = 2(1 − a1,l ) p1,l u(t) s(t)c1,l (t) + [−d2,l + (2a2,l − 1) p2,l u(t) s(t)]c2,l (t), dc3,l (t) dt = 2(1 − a2,l ) p2,l u(t) s(t)c2,l (t) + [−d3,l + (2a3,l − 1)p3,l s(t)]c3,l (t), dc4,l (t) dt = 2(1 − a3,l )p3,l s(t)c4,l (t) − d4,l c4,l (t), F.Angaroni (U. Bicocca) Optimal Control of the chronic myeloid leukemia treatment 23 May 2018 26 / 36
  28. 28. Optimal control: the CML therapy Where the control is given by: U = {u(t)|∀t, 1 ≤ u(t)≤103}; (21) We use the following cost: C(t, x(t), u(t)) = T 0 dt{Au2 (t) + B 4 i=1 c2 i,l (t)} (22) where: A economic cost per dose u(t) is the control, B 4 i=1 c2 i,l represents a running cost that depends on the size of the tumor F.Angaroni (U. Bicocca) Optimal Control of the chronic myeloid leukemia treatment 23 May 2018 27 / 36
  29. 29. Optimal control: the CML therapy We define an Hamiltonian: H = Au(t)2 + B 4 i=1 c2 i,l (t) + 4 i=1 h k=l λi,k dc1,k(t) dt (23) We have 8 adjoint equation: dλi,k dt = − ∂H ∂ci,h i = 1, 2, 3, 4 k = l, h and the characterization of the optimal control: 0 = −2Au(t) + log(u(t))(2a1,l − 1)p1,l s(t)c1,l + + log(u(t))(2(1 − a1,l )p1,l c1,l (t) + (2a2,l − 1)p2,l s(t)c2,l (t)))+ + log(u(t))2(1 − a2,l )p2,l s(t)c2,l (24) F.Angaroni (U. Bicocca) Optimal Control of the chronic myeloid leukemia treatment 23 May 2018 28 / 36
  30. 30. Optimal control: the CML therapy The Greedy controller (A = 100, B = 0 =⇒ u(t) = 1) 150 200 250 300 -4 -3 -2 -1 0 1 2 t (days) Log[T]+2 150 200 250 300 0 2000 4000 6000 8000 10000 t (days) cost The Careful controller (A = 0, B = 1 =⇒ u(t) = 103) 150 200 250 300 -4 -3 -2 -1 0 1 2 t (days) Log[T]+2 150 200 250 300 0 5.0×1014 1.0×1015 1.5×1015 t (days) cost F.Angaroni (U. Bicocca) Optimal Control of the chronic myeloid leukemia treatment 23 May 2018 29 / 36
  31. 31. Optimal control: the CML therapy Problems: the optimal solution exists and it is unique only for certain costs Optimal control solutions implies continuous measurements and action on the system: Optimal control is very complicated to apply to a real system, Since the functional space is very rich compared to a “discrete” space the optimal control gives the upper bond of a decision process and could be used to evaluate the sustainability of a therapy F.Angaroni (U. Bicocca) Optimal Control of the chronic myeloid leukemia treatment 23 May 2018 30 / 36
  32. 32. Therapy control: future works Using a random optimizer: 1 Consider the real pharmacodynamics and pharmacokinetics Figure: From:Pharmacokinetics and pharmacodynamics of dasatinib in the chronic phase of newly diagnosed chronic myeloid leukemia F.Angaroni (U. Bicocca) Optimal Control of the chronic myeloid leukemia treatment 23 May 2018 31 / 36
  33. 33. Therapy control: future works Using a random optimizer: 1 Consider the real pharcodynamics 2 Consider a more realistic cost, not quadratic C(t, x(t), u(t)) = T 0 dt{Ad(u) + Bd(u) + D 4 i=1 ci,l (t)} (25) where: d(u) the function that represent the dosage of the molecular therapy, A economic cost per dose, B quality of life cost per dose (toxicity) D 4 i=1 ci,l (t) represents a cost that depends on the number of leukemic cells F.Angaroni (U. Bicocca) Optimal Control of the chronic myeloid leukemia treatment 23 May 2018 32 / 36
  34. 34. Therapy control: future works Using a random optimizer: 1 Consider the real pharcodynamics 2 Consider a more realistic cost 3 Consider a resistant branch and then manage the switch of the therapy F.Angaroni (U. Bicocca) Optimal Control of the chronic myeloid leukemia treatment 23 May 2018 33 / 36
  35. 35. Therapy control: future works Using a random optimizer: 1 Consider the real pharcodynamics 2 Consider a more realistic cost 3 Consider a resistant branch and then manage the switch of the therapy 4 Consider a discrete follow up F.Angaroni (U. Bicocca) Optimal Control of the chronic myeloid leukemia treatment 23 May 2018 34 / 36
  36. 36. Therapy control: future works Using a random optimizer: 1 Consider the real pharcodynamics 2 Consider a more realistic cost 3 Consider a discrete follow up 4 Consider resistant branch and then the switch of the therapy 5 Consider a hybrid therapy with “conventional“ radiotherapy or chemotherapy with the aim of eradicate residual cancer stem cells F.Angaroni (U. Bicocca) Optimal Control of the chronic myeloid leukemia treatment 23 May 2018 35 / 36
  37. 37. Thank you F.Angaroni (U. Bicocca) Optimal Control of the chronic myeloid leukemia treatment 23 May 2018 36 / 36

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