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Optimal chemo-immunotherapy scheduling: A hybrid QPSO-SQP approach with Michaelis-Menten pharmacodynamics

Bereket Sitotaw Kidane, Md Samiul Haque Motayed, Shuo Wang

arXiv:2609.11213v1math.OCcs.SYeess.SY

TL;DR

Many chemo-immunotherapy control models use control-affine pharmacodynamics, which generally favor boundary-driven schedules and do not structurally enforce interior dose modulation. This paper incorporates Michaelis–Menten saturation, characterizes the resulting control regimes, and computes schedules with solutions showing PMP-consistent transitions between boundary and interior regimes.

  • Problem

    Many chemo-immunotherapy control formulations use control-affine pharmacodynamics, which favor boundary-driven schedules and do not structurally enforce interior modulation.

    Source: §1 · p. 2

  • Method

    The paper embeds saturating Michaelis–Menten effects in tumor–immune therapy dynamics, characterizes control regimes with PMP, and computes trajectories using hybrid QPSO–SQP optimization.

    Source: §1 · pp. 2–3, §2.1 · p. 3

  • Results

    Across Cases A–C and diagnostic studies, computed solutions satisfy the hard terminal tumor constraint and show PMP-consistent transitions between boundary and interior regimes.

    Source: §6 · p. 20

  • Takeaways & Limitations

    Exposure regularization combined with saturating pharmacodynamics is sufficient to produce continuously modulated dosing arcs as direct consequences of PMP.

    Source: §6 · p. 20

    The deliberately minimal model omits explicit pharmacokinetics, treatment delays, state-path toxicity constraints, and tumor heterogeneity or resistance mechanisms.

    Source: §6 · p. 20

Abstract

from arXiv · show

Optimal scheduling of combined chemo-immunotherapy is often formulated as a control-affine optimal control problem, which generically yields boundary-selected (bang-bang-type) protocols unless singular arcs occur. This structure, while convenient, neglects saturating pharmacodynamics at high dose rates. We incorporate Michaelis-Menten saturation directly into the therapy channels, making the dynamics non-control-affine and the Hamiltonian nonlinear in each control. With strictly positive exposure penalties, the Hamiltonian admits an explicit three-regime pointwise minimization law: each input is chosen at the lower bound, the upper bound, or as a unique interior minimizer. On interior intervals the strict Legendre condition holds, so continuously modulated dosing arises as a regular interior extremal rather than a singular-arc or smoothing artifact. The resulting transcription is nonconvex; we therefore use a hybrid Quantum Particle Swarm Optimization (QPSO)-Sequential Quadratic Programming (SQP) pipeline, where QPSO provides a constraint-aware warm start and SQP enforces feasibility and Karush-Kuhn-Tucker (KKT) optimality on a collocation grid. Costate reconstruction corroborates the Pontryagin Minimum Principle (PMP) structure and the predicted boundary/interior regime transitions.

1. Introduction.

The paper embeds saturating Michaelis–Menten pharmacodynamics in a chemo–immunotherapy optimal-control model, yielding regular interior dosing arcs under positive exposure penalties. It pairs this structural analysis with a hybrid QPSO–SQP method and evaluates the formulation in controlled cases.

Introduction

  • Combination therapy must balance chemotherapy’s short-term cytotoxic effects against immune competence needed for durable control, since cytotoxic agents can also suppress immunity.Earlier linear dose–response formulations typically yield boundary-selected protocols unless singular-arc conditions or explicit smoothing are introduced [15, 16].Source: §1 · p. 2
  • Positive exposure penalties with saturating pharmacodynamics produce regular interior dosing minimizers, so continuous titration is neither a singular-control phenomenon nor a smoothing artifact.The Hamiltonian becomes nonlinear in the controls, and interior minimizers occur when optimal inputs lie strictly within their bounds.Source: §1 · pp. 2–3
  • The hybrid QPSO–SQP pipeline explores the nonconvex transcription before refining it to tight feasibility and KKT tolerances, mitigating initialization sensitivity and poor local minima.QPSO locates a favorable basin; SQP refines the collocation solution.Source: §1 · p. 3
  • The model isolates pharmacodynamic structure in a homogeneous tumor population, targeting a prescribed terminal tumor threshold with minimal exposure and treatment duration rather than acquired resistance.The authors identify resistant subpopulations and evolutionary selection as possible extensions.Source: §1 · p. 3
  • Controlled numerical cases—balanced, asymmetric ablation, and minimum-time limit—test how exposure regularization selects boundary versus interior arcs while enforcing a hard terminal tumor threshold.The cases are designed to isolate the causal role of exposure regularization in arc structure.Source: §1 · p. 3

2. Mathematical model and optimal control problem formulation.

The section formulates a three-state tumor–immune model with saturating chemo-immunotherapy and a bounded optimal-control problem for reaching a prescribed tumor target while balancing exposure, treatment duration, and state penalties. It also states the trajectory and optimal-control existence conditions and introduces the PMP conditions used to characterize schedules.

2.1. Tumor–immune dynamics with saturation kinetics.

  • The three-state model uses normalized Michaelis–Menten/Emax therapy maps instead of linear exposure, reflecting capacity-limited pharmacology and dose-response models considered more realistic.Source: §2.1 · pp. 3–4
  • The saturating response functions are bounded and strictly increasing on the admissible control set, with normalized half-saturation constants Ku = Kv = 1.Source: §2.1 · p. 4
  • Tumor burden follows logistic growth with mass-action immune killing and a saturable chemotherapy cytotoxic term whose asymptotic maximum rate is α.Source: §2.1 · pp. 4–5
  • Effector dynamics include antigen-driven proliferation, natural influx and decay, chemotherapy toxicity, and immunotherapy stimulation that acts on existing effectors through a saturating signal.Source: §2.1 · p. 5
  • Tumor-associated antigen increases through intrinsic shedding and chemotherapy-induced release, then decays at rate μ.Source: §2.1 · p. 5

2.2. Optimal control problem formulation.

  • The objective drives tumor burden to terminal target xf while minimizing cumulative drug exposure and treatment duration, with optional penalties on immune effector density and antigen burden.Source: §2.2 · p. 5
  • Controls are normalized to [0, 1], and measurable bounded controls yield unique absolutely continuous trajectories that remain bounded on the treatment horizon.Source: §2.2 · p. 6
  • Compact control bounds and continuity support existence of an optimal relaxed control, while PMP couples state and costate trajectories through adjoint equations and pointwise Hamiltonian minimization.Source: §2.2 · pp. 6–7

3. Optimality conditions and control characterization.

The section specifies PMP endpoint conditions and shows that saturating pharmacodynamics with positive exposure penalties yield a three-regime pointwise control law, including regular interior controls.

3. Optimality conditions and control characterization.

  • For fixed x(T)=xf with free terminal y(T) and z(T), λ1(T) is free and determined by the free-final-time condition H(T)=0.Source: §3 · p. 7

3.1. Characterization of optimal controls.

  • With A,B>0, each therapy channel has a three-regime pointwise minimizer: the lower bound, a unique interior minimizer, or the upper bound.For u, u*=0 when Φu≥−A, u*=umax when Φu≤−A(1+umax)^2, and otherwise the unique interior root; the v-channel uses (B, Φv, vmax).Source: §3.1 · pp. 8–9
  • Saturating pharmacodynamics make the Hamiltonian nonlinear in each control, allowing regular interior dosing without quadratic smoothing or singular-control phenomena.On interior intervals, the strict Legendre condition holds, and the stationary control is a unique local minimizer.Source: §3.1 · pp. 7–8

4. Hybrid numerical optimization framework.

The paper uses a hybrid QPSO–SQP framework to solve the nonconvex NLP produced by transcribing the coupled tumor–immune dynamics and saturating therapy channels. QPSO explores for a promising basin, while SQP refines the solution under the original constraints.

QPSO global search

  • QPSO searches control trajectories on 101 nodes and simulates candidates with ode45, producing a coarse warm start near feasibility.The best particle's terminal penalty is used only to steer warm-start generation, not in the final constrained solve.Source: §4 · p. 10

SQP refinement

  • SQP refines a smoothed QPSO trajectory by trapezoidal collocation, treating final time as a decision variable and enforcing terminal equality and nonnegative states.The Gaussian smoothing window is 10 nodes; collocation defects and terminal constraints are driven below specified feasibility and optimality tolerances of 10^-6.Source: §4 · p. 10

Hybrid framework

  • The hybrid strategy addresses the nonconvex problem: uninformed SQP often reached suboptimal basins, while QPSO explores globally and SQP refines feasibility and optimality.Preliminary failures included static monotherapy profiles and premature boundary switching.Source: §4 · p. 9, §5 · p. 11

5. Numerical results and discussion.

The numerical cases show that Michaelis–Menten saturation combined with exposure penalties produces regular interior dosing, while removing those penalties recovers boundary-selected control. Simulations enforce the terminal tumor target as a hard equality and test balanced, asymmetric, minimum-time, and sensitivity regimes.

5.1. Balanced objectives and emergence of interior modulation (Case A).

  • In balanced Case A, chemotherapy transitions from an initial upper-bound arc through interior titration to shutoff, while immunotherapy has a broad interior pulse; strict Legendre checks support regular interior arcs.Diminishing returns from Michaelis–Menten saturation compete with linear exposure cost to produce continuous titration.Source: §5.1 · pp. 12–14, Figure 1 · p. 12, Figure 2 · p. 13

5.2. Mechanism Isolation via asymmetric regularization (Case B).

  • In asymmetric Case B, chemotherapy retains an interior titration segment under exposure regularization, whereas unregularized immunotherapy is boundary-selected at its upper bound.PMP diagnostics show chemotherapy stationarity on its interior interval and a sign condition selecting the immunotherapy boundary action.Source: §5.2 · p. 14, Figure 3 · p. 15

5.3. Validation via the minimum-time limiting case.

  • With A = B = 0, Case C has no interior stationary minimizers; chemotherapy switches bang–bang at a transversal Φu zero near t ≈ 3.4, ruling out a sustained singular interval.The minimum-time limit is a negative control: removing exposure penalties eliminates the interior arcs found in Cases A–B.Source: §5.3 · pp. 15–17, Figure 4 · p. 16

5.4. Sensitivity to admissible dose bounds and local parameter robustness.

  • Across umax = vmax ∈ {1, 1.5, 2}, normalized Case A schedules retain similar shapes without monotonically increasing upper-bound time; under ±20% re-optimization, costs are less sensitive to immune than tumor-growth parameters.The dose-bound sweep uses fixed normalized half-saturation scaling, so it is not an invariant bound-independence claim; moderate parameter mismatch does not qualitatively destabilize the schedules.Source: §5.4 · pp. 17–19, Figure 5 · p. 18, Figure 6 · p. 18

5.5. Comparative synthesis.

  • Across three cases, exposure regularization with saturating pharmacodynamics yields mixed or channel-specific interior dosing, while removing exposure penalties recovers boundary-selected bang–bang-type control.The comparative synthesis describes these as trade-offs among cytotoxic exposure, treatment speed, and immune-mediated control; Table 3 summarizes comparative performance, but its cell values are not supplied here.Source: §5.3 · p. 17, §5.5 · p. 19, Table 3 · p. 19

6. Conclusion.

Across Cases A–C and diagnostic studies, hybrid QPSO–SQP solutions satisfy the hard terminal tumor constraint and exhibit arc transitions consistent with PMP. Within the model’s limited scope, saturation coupled with exposure regularization produces continuously modulated dosing arcs.

6. Conclusion.

  • Across Cases A–C and diagnostic studies, hybrid QPSO–SQP solutions satisfy the hard terminal tumor constraint and show PMP-consistent boundary/interior transitions.QPSO explores for a high-quality basin of attraction, while SQP enforces feasibility and first-order optimality on a refined collocation grid for the nonconvex transcription.Source: §6 · pp. 19–20
  • Exposure regularization coupled with saturation suffices to generate interpretable, continuously modulated dosing arcs as PMP consequences within the paper’s scope.Source: §6 · p. 20
  • The minimal model omits explicit pharmacokinetics, treatment delays, state-path toxicity constraints, and heterogeneity/resistance mechanisms; therapy coefficients are illustrative under normalized dosing.These omissions and assumptions are identified as important extensions for patient-specific translation.Source: §6 · p. 20

A. Detailed derivation of the adjoint system. · Appendix

The appendix states that its derived expressions match the summary in Section 3 and gives the Hamiltonian as a linear combination of the state variables, controls, and a constant term.

Appendix

A.1. Adjoint equation for tumor volume (λ1).

The tumor-volume adjoint equation is obtained by differentiating the Hamiltonian with respect to x.

A.1. Adjoint equation for tumor volume (λ1).

  • The λ1 adjoint equation is derived by differentiating H with respect to x.Source: §A.1 · p. 20

A.2. Adjoint equation for immune density (λ2).

The supplied passages give the expression for the adjoint equation for immune density (λ2).

A.2. Adjoint equation for immune density (λ2).

  • The λ2 adjoint-equation expression is a(1 −bx)z −δ − κu 1 + u + νv 1 + v.The same expression appears in both supplied passages.Source: §A.2 · p. 21

A.3. Adjoint equation for antigen concentration (λ3).

The subsection obtains the adjoint equation for antigen concentration λ3 by differentiating the Hamiltonian with respect to z.

A.3. Adjoint equation for antigen concentration (λ3).

  • The adjoint equation for antigen concentration λ3 is obtained by differentiating the Hamiltonian with respect to z.Source: §A.3 · p. 21

Appendix B. Analytic resolution of interior control arcs. · B.1. Chemotherapy control u(t). · B.2. Immunotherapy control v(t).

Interior control candidates are found by setting the Hamiltonian derivatives with respect to each control to zero; outside the interior regimes, minimizers occur at admissible bounds. For positive exposure coefficients, real interior stationary points require negative switching potentials.

Appendix B. Analytic resolution of interior control arcs.

  • On intervals where either control lies strictly inside its bounds, its Hamiltonian derivative must vanish; otherwise its minimizer is attained at an admissible bound.The interior conditions are 0 < u < umax and 0 < v < vmax.Source: Appendix B · p. 21

B.1. Chemotherapy control u(t).

  • For chemotherapy, stationarity gives A + Φu(t)(1 + u)^2 = 0, or (1 + u)^2 = −Φu(t).Source: §B.1 · p. 21
  • With A > 0, a real interior chemotherapy stationary point requires Φu(t) < 0.Source: §B.1 · p. 21

B.2. Immunotherapy control v(t).

  • For immunotherapy, stationarity gives B + Φv(t)(1 + v)^2 = 0, or (1 + v)^2 = −Φv(t).Source: §B.2 · p. 21
  • With B > 0, a real interior immunotherapy stationary point requires Φv(t) < 0.Source: §B.2 · p. 21
  • The implementation addresses near-zero negative switching potentials that can cause spurious complex square-root values before projecting controls onto their admissible intervals.The note gives Φu ≈ −10^-16 as an example of the finite-precision issue.Source: §B.2 · pp. 21–22
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