{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,17]],"date-time":"2026-08-17T17:27:01Z","timestamp":1786987621722,"version":"build-2736575974"},"reference-count":39,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2026,8,17]],"date-time":"2026-08-17T00:00:00Z","timestamp":1786924800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We study an optimal control framework for chemotherapy administration in patients with chronic lymphocytic leukemia (CLL) while accounting for immune regulation and treatment-induced allergic reactions. The analysis is based on a previously developed nonlinear delay differential equation model describing the interactions between leukemic cells, immune populations, antigen-presenting cells, and cytokine dynamics, with three distinct biological delays. The chemotherapy infusion rate is introduced as a time-dependent control variable and optimized to reduce leukemic burden, shift the helper T-cell balance toward a Th1-dominant configuration associated with lower hypersensitivity risk, and preserve immune competence. Existence of an optimal control is established for arbitrary delays and horizon, without the commensurability hypothesis required by reductions in delay systems to higher-dimensional delay-free ones; the argument uses only that the control enters the dynamics affinely and the running cost concavely. Necessary optimality conditions are derived via Pontryagin\u2019s Maximum Principle for systems with delays, and the resulting eleven-dimensional adjoint system, which carries advanced arguments generated by the three delays, is written out explicitly. A contraction estimate for the associated sweep operator yields both uniqueness of the optimal control on a short horizon and geometric convergence of the numerical scheme. The optimality system is solved by a forward\u2013backward sweep adapted to the delayed setting, with documented convergence and grid independence and sensitivity analysis over kinetic parameters, delays, initial conditions and objective weights. The optimized schedule is compared not only with the untreated case and a low constant dose, but also with a constant infusion delivering the same cumulative exposure, so that the reported benefit is attributable to the temporal distribution of the dose rather than to its total amount. At equal exposure, the optimal schedule reaches each therapeutic milestone earlier\u2014Th1 dominance 0.9 days sooner and a 90% leukemic reduction 1.6 days sooner\u2014and attains a terminal leukemic burden lower by a factor of 2.25; a constant infusion of the same total dose reaches a comparable configuration later. The benefit of adaptive scheduling in this model is therefore principally one of rate of response at fixed drug exposure. We emphasize that the absolute Th2 population is not reduced by treatment; the reduction in hypersensitivity risk arises from the resulting Th1-dominant relative balance rather than from direct Th2 suppression. To characterize the therapeutic outcome in information-theoretic terms, we describe the two competing goals as distributional balances: an allergy axis, given by the Th1\/Th2\/Treg distribution, and a leukemia axis, given by the immune\/leukemic distribution, each measured by its Shannon entropy and its Kullback\u2013Leibler divergence to a healthy reference profile. These quantities are used in two roles. As diagnostics, they are evaluated along the computed trajectories, and the ordering of dosing strategies is shown to be robust across twenty alternative reference profiles. As an objective, the combined divergence is then taken as the running cost of a second optimal control problem; because it depends on the leukemic population only through a normalized fraction, it prescribes a markedly gentler schedule that administers 37% of the drug and still achieves a 93% leukemic reduction, against 98% for the population-based formulation. These results suggest that adaptive, immune-aware chemotherapy scheduling may accelerate disease control at fixed drug exposure, and that information-theoretic objectives offer a scale-free alternative formulation of the therapeutic goal.<\/jats:p>","DOI":"10.3390\/e28080921","type":"journal-article","created":{"date-parts":[[2026,8,17]],"date-time":"2026-08-17T16:35:08Z","timestamp":1786984508000},"page":"921","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Optimal Chemotherapy Scheduling for Chronic Lymphocytic Leukemia Under Immune and Allergy Constraints"],"prefix":"10.3390","volume":"28","author":[{"given":"Rawan","family":"Abdullah","sequence":"first","affiliation":[{"name":"Department of Arts and Sciences, Rafik Hariri University, P.O. Box 10, Damour-Chouf 2010, Lebanon"},{"name":"Department of Mathematics and Informatics, Faculty of Applied Sciences, National University of Science and Technology POLITEHNICA Bucharest, 060042 Bucharest, Romania"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Andrei","family":"Halanay","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Informatics, Faculty of Applied Sciences, National University of Science and Technology POLITEHNICA Bucharest, 060042 Bucharest, Romania"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0008-0398-8129","authenticated-orcid":false,"given":"Lara","family":"Abou Orm","sequence":"additional","affiliation":[{"name":"Department of Arts and Sciences, Rafik Hariri University, P.O. Box 10, Damour-Chouf 2010, Lebanon"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,8,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Guzev, E., Luboshits, G., Mendrazitsky, S.B., and Firer, M.A. (2021). Experimental Validation of a Mathematical Model to Describe the Drug Cytotoxicity of Leukemic Cells. 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