16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) 2006 Published by Elsevier B.V. 1643 Optimal Delivery of Chemotherapeutic Agents in Cancer Pinky Dua a, Vivek Dua b, Efstratios N. Pistikopoulos a a Centre for Process Systems Engineering, Department of Chemical Engineering, Imperial College London, South Kensington Campus, London SW7 2AZ, United Kingdom b Centre for Process Systems Engineering, Department of Chemical Engineering, University College London, Torrington Place, London WC1E 7JE, United Kingdom Abstract In this paper, derivation of the optimal chemotherapy schedule is formulated and solved as a dynamic optimization problem. For this purpose two models representing the tumour growth kinetics are considered. The dynamic optimization problem for the first model, which is cell cycle non-specific, takes into account multiple time characteristics, drug resistance and toxicity. The discontinuity in the model is formulated by introducing integer variables. For the second model, which is cell cycle specific, the tumour growth is modelled via two compartments: proliferating and resting compartment. Keywords: Drug Delivery Systems, Cancer, Chemotherapy, Optimal Control, Mixed Integer Dynamic Optimization 1. Introduction Cancer is a collective term that describes a group of diseases characterized by uncontrolled and unregulated growth of cells leading to invasion of surrounding tissues and spreading to the parts of the body that are distant from the site of origin. There are around 200 types of cancer and cancers of lungs, breast, bowel and prostrate are the most common ones. There are three main stages in the process of carcinogenesis: initiation, promotion and progression. The normal cell changes to an initiated cell and then to cancer differentiated cell and finally invades and spreads to the surrounding cells. The simplest mathematical model describes the entire cell cycle as a uniform entity, where all the cells contained in a tumour are of the same type. The cell cycle nonspecific models consist of one compartment so that the effect of the anticancer agents is same on all the cells. However these models fail to describe the action of cycle specific drugs due to their over-simplified nature. The more detailed multi-compartment models (cell cycle specific models) are considered for this purpose. Here the cell cycle is divided into compartments depending on the types of cells that are affected by the drug. Chemotherapy is one of the most commonly used treatments for cancer that uses anticancer or cytotoxic drugs to destroy or kill cancer cells. The suitability of chemotherapy and the choice of drugs depend on many factors including the type of cancer, the location of the origin of the cancer in the body, how mature the cancer cells are and whether the cancer cells have spread to the other parts of the body. Chemotherapy targets dividing cells which does not only include cancer cells but any
1644 P. Dua et al. normal cells that are dividing such as the hair producing cells, cells that line mouth and digestive system and those in the bone marrow and the skin. Harrold and Parker (2004) recently proposed a mixed integer linear programming approach for the derivation of optimal chemotherapy schedule. In this paper, optimal chemotherapy schedules are derived with the objective of minimizing the final number of tumour cells at the end of the treatment. The rest of the paper is orgainzed as follows: in Section 2 optimal cancer chemotherapy schedule is derived for a cell cycle non-specific model whereas a cell cycle specific model is considered in Section 3; concluding remarks are presented in Section 4. 2. Optimal Control for Cell Cycle non-specific Model A pharmacokinetic/pharmacodynamic model given in Martin (1992) is used for the derivation of the schedule. The objective is to obtain the drug dosage over a fixed period of time so as to minimize the number of cancer cells at the end of the period subject to constraints on toxicity and resistance of the drugs. The optimal control problem is formulated as follows: min J ( u) = z( T ) u, y s.t. z = λz( + k( υ( υ ) y z(0) = ln θ N 0 υ( = u( γυ( υ(0) = υ0 = 0 0 υ( υ, T υ( s) ds υ 0 max z(21) ln(200) z(42) ln(400) z(63) ln(800) υ( yυth, υ( υ ym, th cum th where u = [u 1,.,u n ] T R n is the vector of the rate of delivery of drug, z( is the nondimensional tumour size, λ is a growth parameter, k is the proportion of the tumour cells killed per unit time per unit drug concentration, υ( is the concentration of the anticancer drug at time t, υ th is the therapeutic concentration of the drug, y is a 0-1 binary variable, θ is the plateau population or the carrying capacity of the tumour, N 0 is the initial tumour cell population, υ max is the maximum amount of drug in the plasma, υ cum is the cumulative toxicity and M is a large positive number. Note that in this formulation the binary variable, y, is introduced to model the discontinuity so that y takes the value 0, if 0 υ( υ th, or the value 1, if υ( υ th and the following transformation is introduced to make the model tractable and tumour size dimensionless (Dua, 2005): z = ln ( θ ) N
Optimal Delivery of Chemotherapeutic Agents in Cancer 1645 where N is the number of tumour cells. The data for the parameters in the model is taken from Martin (1992). This problem is a mixed-integer dynamic optimization problem (Bansal et al., 2003) and is solved using gproms (gproms, 2003) by taking the control interval of one day and a time period of 84 days. The profiles of the optimal chemotherapy and tumour growth are shown in Figures 1 and 2 and are consistent with those reported in the open literature on optimal control strategies for cancer chemotherapy. Rate of Delivery of Drug ([D]/days) 45 40 35 30 25 20 15 10 5 0 0 10 20 30 40 50 60 70 80 Figure 1 Optimal chemotherapy protocol for the cell cycle non specific model 1.2E+10 1.0E+10 Number of Cells 8.0E+09 6.0E+09 4.0E+09 2.0E+09 1.0E+04 0 10 20 30 40 50 60 70 80 90 Figure 2 Predicted tumour growth using the optimal treatment protocol Initially no drug is delivered until the time approaches 21 days, the first interval of the multiple characteristic time constraints, z(21) ln(200), which are introduced to model
1646 P. Dua et al. drug resistance. This can be attributed to the fact that an initial high dose may not necessarily result in an overall decrease in the tumour size at the end of the time period as compared to the minimum tumour size obtained by the optimal chemotherapy protocol, while the constraints on toxicity and drug resistance are satisfied. Moreover, the drugs are given in large doses intermittently rather than in small doses continuously. The tumour follows the Gompertz growth until the first injection of the drug that results in a reduction of 70% of the initial tumour size. High intensity chemotherapy, from time 40 days till the end of the therapy, results in an increase in the rate of tumour reduction. The value of the objective function at the end of the treatment is given by 7.5 10 4 cancer cells. As a conclusion, the optimal way to reduce the tumour size was to apply high intensity therapy towards the end of the chemotherapy period. The optimal chemotherapy protocol shown in Figure 1 produces a 99.9% reduction of the initial tumour size. 3. Optimal Control for Cell Cycle Specific Model The model of Panetta and Adam (1995) describes the administration of the anticancer drug in the case of cell cycle specific chemotherapy. In this model, the effect of the drug depends only on the duration of the injection and not on the amount of the drug that is injected. For this reason, this model is modified so as to relate the effect of the drug with the rate of delivery of the drug (Dua, 2005) and minimization of the final tumour population is formulated as the following optimal control problem: min J ( u) = P( T ) + Q( T ) u( s.t. P = ( a m n) P( + bq( g( P(, Q = mp( bq(, y( y = δy( 1 g( y(, K υ( = u( γυ1(, g( = k1υ 1( ymin y( K, P(0) = P0 Q(0) = Q0 y(0) = y0 υ1(0) = υ0 where P and Q represent the cycling and non-cycling tumour cell mass respectively, a is the cycling cell growth rate, m is the rate at which cycling cells become non-cycling, n is the natural decay of cycling cells, b is the rate at which non-cycling cells become cycling, y is the number of normal cells, δ is the growth rate, K is the carrying capacity of normal tissue, g is the effect of drug on the cells, υ 1 is the concentration of the drug in the body, u is the rate of the delivery of the drug, k 1 is the parameter for the kill rate and γ is the parameter for the decay of the drug. The parameters for this model are taken from Panetta and Adam (1995). The problem was solved by using gproms by considering a time horizon of 60 days and control interval of one day. The optimal chemotherapy schedule derived is shown in Figure 3. Initially, the drug is delivered at a high dose, followed by a low intensity therapy for almost the rest of the treatment period. Finally, a large amount of drug is applied in the last two days. The objective function represents the size of the tumour, including both the proliferating (cycling) and resting (non-cycling) compartments. The
Optimal Delivery of Chemotherapeutic Agents in Cancer 1647 evolution of number of cancer cells during the therapy period is illustrated in Figure 4. After the optimal drug protocol is applied to the patient, the predicted number of cancer cells drops from 10 12 cells to 4 10 11 cells, which corresponds to 60% reduction of the initial tumour size. The final reduction in the number of cancer cells is less when compared to the results obtained for the cell cycle non-specific case. Nevertheless, the cell cycle specific model includes the constraint that places strict limits to the acceptable number of normal cells, i.e. y min = 10 8. As shown in Figure 5, for almost the entire treatment period, the level of normal cells is near to its lower bound. It is obvious that the first high dose of the drug had an enormous effect on normal cells, which did not allow further application of high doses, apart from that in the last two days. Rate of Delivery of Drug ([D]/days) 35 30 25 20 15 10 5 0 0 10 20 30 40 50 60 Figure 3 Optimal chemotherapy protocol for the cell cycle specific model 1.2E+12 1.0E+12 Number of Cells 8.0E+11 6.0E+11 4.0E+11 2.0E+11 0.0E+00 0 10 20 30 40 50 60 Figure 4 Predicted tumour growth using the optimal treatment protocol cycling and non-cycling cells
1648 P. Dua et al. Number of Cells 1.2E+09 1.0E+09 8.0E+08 6.0E+08 4.0E+08 2.0E+08 0.0E+00 0 10 20 30 40 50 60 Figure 5 Predicted growth of the normal cell population using the optimal treatment protocol 4. Concluding Remarks In this paper, two models for the derivation of optimal chemotherapy schedules were considered. The first model involves the use of cell cycle non-specific agents. The tumour is modelled as one compartment and the drug is assumed to affect all the cancer cells in the same way. Toxicity to normal cells is controlled with constraints that limit drug concentration during therapy, as well as the total cumulative toxicity at the end of the treatment. The problem is formulated and solved as a mixed integer dynamic optimization problem. The optimal chemotherapy protocol derived in this case kept the initial administration of the drug at a low level, while most of the drug was delivered towards the end of the therapy. The second model describes the effects of the cell cycle specific drugs. The tumour is divided into two compartments; the cycling and the noncycling compartments. Toxicity is monitored through the differential equation that describes the effect of the drug on normal cell population. The problem is formulated and solved as a dynamic optimization problem. The optimal chemotherapy indicates that the anticancer agent should be administered in high doses at the beginning and at the end of the treatment period. The current work involves formulating these problems such that the objective is not only to minimize the tumour size at the end of a given time period but also to optimize the time period for the therapy. References V. Bansal, V. Sakizlis, R. Ross, J.D. Perkins and E.N. Pistikopoulos, 2003, New algorithms for mixed integer dynamic optimization, Computers and Chemical Engineering, 27, 647-668 P. Dua, 2005, Model based and parametric control for drug delivery systems, PhD Thesis, Imperial College London, University of London gproms, 2003, Introductory User s Guide, Release 2.2, Process Systems Enterprise Limited, London, U. K. J.M. Harrold and R.S. Parker, 2004, An MILP approach to cancer chemotherapy dose regime design, Proc. American Control Conference, 969-974 R.B. Martin, 1992, Optimal control drug scheduling of cancer chemotherapy, Automatica, 28, 1113-1123 J.C. Panetta and J. Adam, 1995, A mathematical model of cycle-specific chemotherapy, Mathematical and Computer Modelling, 22, 67-82