Optimisation of time-scheduled regimen for anti-cancer drug infusion

Abstract

Abstract. The chronotherapy concept takes advantage of the circadian rhythm of cells physiology in maximising a treatment efficacy on its target while minimising its toxicity on healthy organs. The object of the present paper is to investigate mathematically and numerically optimal strategies in cancer chronotherapy. To this end a mathematical model describing the time evolution of efficiency and toxicity of an oxaliplatin anti-tumour treatment has been derived. We then applied an optimal control technique to search for the best drug infusion laws. The mathematical model is a set of six coupled differential equations governing the time evolution of both the tumour cell population (cells of Glasgow osteosarcoma, a mouse tumour) and the mature jejunal enterocyte population, to be shielded from unwanted side effects during a treatment by oxaliplatin. Starting from known tumour and villi populations, and a time dependent free platinum Pt (the active drug) infusion law being given, the mathematical model allows to compute the time evolution of both tumour and villi populations. The tumour population growth is based on Gompertz law and the Pt anti-tumour efficacy takes into account the circadian rhythm. Similarly the enterocyte population is subject to a circadian toxicity rhythm. The model has been derived using, as far as possible, experimental data. We examine two different optimisation problems. The eradication problem consists in finding the drug infusion law able to minimise the number of tumour cells while preserving a minimal level for the villi population. On the other hand, the containment problem searches for a quasi periodic treatment able to maintain the tumour population at the lowest possible level, while preserving the villi cells. The originality of these approaches is that the objective and constraint functions we use are L ∞ criteria. We are able to derive their gradients with respect to the infusion rate and then to implement efficient optimisation algorithms.Mathematics Subject Classification. 49M29, 92B05, 92C50.

Optimisation of Cancer Drug Treatments Using Cell Population Dynamics

“…The Gompertz model was initially developed in the context of insurance [62] and was first used in the nineties to fit experimental data of tumour growth [81]. A lot of studies on drug control are based on these models [14,15,35,96,97,98,106,107,108]. For instance, Murray [106,107] considered a two-population Gompertz growth model with a loss term to model the effect of the cytotoxic drug.…”

“…More recently, one of us and his co-workers [15,35] investigated the effects of oxaliplatin on tumour cells and healthy cells. To model tumour growth, they used a Gompertz model modified by a "therapeutic efficacy term" as a death term depending on the drug concentration.…”

“…In [35], the work presented in [15] is extended by coupling this model of tumour growth to a model of healthy cell growth and to a three-compartment model describing the time evolution of the concentration of oxaliplatin in plasma, healthy tissue and tumour tissue. This work was done on the basis of experimental data related to oxaliplatin PK-PD and tumour growth curves with or without drug injection to determine the model parameters and compared time-scheduled infusion schemes with constant ones.…”

Modelling Physiological and Pharmacological Control on Cell Proliferation to Optimise Cancer Treatments

“…A L 2 or L 1 average is not very satisfying to consider because it does not exclude hectic behaviour (peaks and troughs) of the tumour cell population that is seldom in accordance with clinical observations. I proposed with C. Basdevant, see [20], to use minimisation of either the absolute minimal tumour cell number (if zero can be reached, then the tumour is eradicated); or the maximal tumour cell number within a given time interval, in the perspective of renewed chemotherapy courses. In the latter case, if this maximal number can be kept below a clinically tolerable threshold, compatible with survival with a good quality of life even if the tumour has not been eradicated, then the treatment may also be considered as successful.…”

“…For instance, in a feasibility study reported in [20,53], a jejunal toxicity limit has been chosen, for it had been shown that in mice (the model parameters had been identified in mice) the jejunum was the most important toxicity target for the drug considered there, oxaliplatin. But it is more frequent to study toxicity effects on the haemotologic bone marrow, as has been done in a few studies with clinical applications [130,131,216].…”