Blasting neuroblastoma using optimal control of chemotherapy

Craig Collins, K. Renee Fister, Bethany Key, Mary Williams
2009 Mathematical Biosciences and Engineering  
A mathematical model is used to investigate the effectiveness of the chemotherapy drug Topotecan against neuroblastoma. Optimal control theory is applied to minimize the tumor volume and the amount of drug utilized. The model incorporates a state constraint that requires the level of circulating neutrophils (white blood cells that form an integral part of the immune system) to remain above an acceptable value. The treatment schedule is designed to simultaneously satisfy this constraint and
more » ... constraint and achieve the best results in fighting the tumor. Existence and uniqueness of the optimality system, which is the state system coupled with the adjoint system, is established. Numerical simulations are given to demonstrate the behavior of the tumor and the immune system components represented in the model. 2000 Mathematics Subject Classification. Primary: 49J15, 49K15; Secondary: 93C15. Control statistics for the year 2003 [8], one quarter of all youth cancer deaths are attributable to the neurological system. One particular drug that has proven successful in the reduction of cancer cells is the drug Topotecan (TPT), which is a novel semisynthetic antiderivative of the anticancer agent Camptothecin [7]. This drug interacts with Topoisomerase I, which is an intranuclear enzyme in the body, to inhibit the replication of DNA and thus result in cell death [5, 23, 36] . Several studies have been conducted using several dosage strategies of TPT in order to kill tumors and reduce side effects [36] . One technique often used to determine appropriate dosage levels is called pharmacokinetic-based (PK-based) dosing [5]. This approach utilizes careful monitoring of drug levels in order to tailor treatments to the particular physiology of each patient [4, 5]. Optimal control theory is a useful mathematical approach for maximizing the results of various treatment strategies. This theory has been applied to biological and mathematical models such as the interaction between tumor cells and chemotherapy [13, 24, 25] . Studies by G. W. Swan helped open the door to using optimal control with biology. In 1980, Swan [34] published a study on optimal control in some chemotherapy problems; two years later, he applied optimal control to diabetes mellitus [35] . Ten years after Swan's paper, J. M. Murray published a more in depth paper on the same subject by adding aspects such as general growth and loss functions [28] . Murray is just one of many authors who have furthered the study of using optimal control with biological models. Control theory has successfully been applied to models that maximize the effect of the chemotherapy while minimizing the damage due to toxicity by Kim et. al [21] , Swan and Vincent [33], and Murray [29] . Optimal control has also been applied to studies for other treatments, such as immunotherapy for cancer (dePillis et. al [11], Ledzewicz et. al [24, 25] ) and HIV by Kirschner et. al [22] . Costa et. al [9, 10] also published work involving optimal chemotherapeutic protocols which include inequality constraints of the control and state variables. We will discuss a system of differential equations which model TPT plasma PK, tumor growth, and absolute neutrophil count (ANC) (Section 2). We then analyze the existence, characterization, and uniqueness of the optimal control in Sections 3 and 4 respectively. In Section 5, numerical analysis and results are then given with explanations relating the numerical results to clinical findings. 2. Model. The model used in this study was developed by a research team at St. Jude Children's Research Hospital [31], using clinical data generated at St. Jude's. The model consists of three components: the TPT plasma PK, the tumor growth model, and the ANC (Absolute Neutrophil Count) model. The state variables are defined as follows:
doi:10.3934/mbe.2009.6.451 pmid:19566120 fatcat:mx5evzfr7rdsdo6iotzoceedoi