
MBW:Optimal Chemotherapy StrategiesFrom MathBioArticle review by Sigmund Stoerset, March 2010. Article: Optimizing Drug Regimens in Cancer Chemotherapy by an EfficacyToxicity Mathematical Model, Illiadis A., Barbolosi D. Computers and Biomedical Research 2000; 33: 211226.
ContentsSummaryThis article looks at common clinically used protocols for chemotherapeutics for cancer. These are usually overruled by constraints on maximum drug concentration in plasma and total amount of drug introduced to a patient's body. By instead introducing constraints concerning total white blood cell counts in the patient's body, the authors are capable of optimizing a simulation resulting in largely reduced tumor sizes at the end of treatment, without putting the patient at risk. This is accomplished by introducing small infusion intervals, and optimizing drug infusion rate at each interval such that the tumor is minimized in size, without violating any of the constraints on white blood cell counts, which give a general indicator of the patients overall health or side effect response by the treatment. By the simulations in this study, the authors show that it is possible to monitor chemotherapeutic side effects by dynamically observing white blood cell counts, instead of statically observing plasma concentrations and total drug amount. This shows promising results as a new way of achieving optimized drug delivery protocols for minimizing tumor size. Biological backgroundWhen a chemotherapeutic agent (a drug) is introduced into the blood stream, both cancer cells and normal, healthy cells are affected and killed. Because of this it is important to monitor the delivery of the drug to the patient, since overexposure could lead to dramatic side effects and death of the patient. Since the drug is highly present in the blood stream of the patient, monitoring the white blood cell concentration in plasma as a function of time, drug concentration and drug infusion rate could give a good idea of at which level of risk the patient is at each given time during the treatment. Also, by using this approach, a drug delivery regimen can be optimized such that cancer cell count is minimized at the end of treatment without putting the patient at risk during the cure. Model characterizationThis article studies the effect of various chemotherapy administration strategies on cancerous tumors, with an emphasis on patient safety. It proposes a new approach to chemotherapy treatment, monitoring white blood cell count in addition to total drug levels in the body. To support this strategy, the authors develop a staged cellularscale model in which drug concentration is described by modified massaction and delay differential equations. HistoryChemotherapy of cancer is a field which have been studied extensively over the past decades. Cancer is a very complex group of diseases which strikes numerous people around the world each year, and the wish to find a good and effective cure has driven research forward in this field. Chemotherapy of cancer is a way of killing cancer cells by introducing poison to the body. This poison mainly effect proliferating cells, which the bulk of most cancer tumors consists of. The poisons introduced can attack the cancer cells in many different ways. One method is to induce DNAdamage to activate the p53pathway and induce apoptosis in the cells^{[1]}. Other pathways can interact with the cell cycle and result in cell cycle arrest^{[2]}. Common for all chemotherapeutic schemes is that they do not influence cancer cells alone. The chemotherapeutic poison is often introduced intravenously leading to a poison distribution throughout the body. This gives cell death of healthy cells around the body leading to severe side effects ^{[3]} often leading to great suffering of the patient. Because of this, research within chemotherapy have also been focused on how to distribute the chemotherapeutic in a best possible way such that side effects are minimized and tumor regression is maximized. This has been done using several constraints such as maximum drug exposure, maximum injection rate etc. Here, the constraints investigated are instead the white blood cell count per unit volume as a function of time, drug concentration and drug infusion rate. Mathematical modelOutlineThe model is designed to give information of the chemotherapy's effect on both tumor cells and white blood cells. To do this, a compartment model is used to illustrate the stages the drug can be in in the body (figure 1). The model consists of four compartments. On the side of pharmacokinetics, the plasma drug concentration, c_{1}(t), and the active drug concentration, c_{2}(t) ares present, with volumes of distribution V_{1} and V_{2}. The active drug concentration is the portion of the drug which can affect the tumor cells, while the drug in plasma will affect the white blood cells. The two other compartments in the model are present on the pharmacodynamic side of the model, and are the white blood cell count per unit volume, w(t), and the tumor cell count per unit volume, n(t). These compartments constitute the phamacodynamics of the model. Differential equationsThe plasma concentration of the drug is modeled by
ConstraintsSeveral constraints are put on the model to insure that the patient can handle the toxic side effects of the drug. Firstly, the maximum plasma drug concentration must never exceed a given value:
Drug deliveryThe optimization problem considers how to best infuse the drug over a period of time, t_{N}, divided into N time treatment intervals, such that the tumor size is minimized without violating any of the given constraints. The infusion rates can vary between each of the time intervals such that
The model and resultsParameter valuesThere are quite many parameters involved with this model, and to get accurate results, it is important to chose values as close to reality as possible. The values used in this simulation are mainly taken from literature, but assumptions on some of the parameter values are done as well. Table 1 includes all parameter values with comments on each value.
Clinical protocolsUsing the model described it was possible to simulate both tumor cell count an white blood cell count using conventional clinical protocols for chemotherapeutic drug infusion. The protocols simulated were
(2) 2 – h infusions of 100 mgm^{2}day^{1} for the first 5 days of cycle. (3) 1 – h infusions of 150 mgm^{2}day^{1} for days 1, 3, and 5. (4) Continuous infusion for 3 day at the rate of 125 mgm^{2}day^{1}. (5) Continuous infusion over 21 days at the rate of 25 mgm^{2}day^{1}.
The simulation results for cancer cell count, n(t), and white blood cell count, w(t), using these 5 protocols in the described model are presented in figure 2a and 2b respectively. Considering white blood cell counts, the results show that w(t) never drops below the lower limit W_{D} for any of the protocols. All protocols also hold for constraint (8), a(T) <= A_{MAX}. Cancer cell counts decrease for all five protocols, and decreases from an initial cancer mass of 30 g to values from 25.3 g to 28.1 g by the end of treatment. The simulation shows quite different results for protocol 1 through 4 and protocol 5. The four first protocols all involve high infusion rates early in the treatment cycle and a long recovery step, while protocol 5 involves low continuous infusion over the whole treatment period. The result show that for protocol 1 through 4, both white blood cell and cancer cell count fall quickly during the beginning of treatment and then rise slowly. By the end of treatment, the white blood cell count had rose to almost initial values. For protocol 5, both counts fall slowly until end of treatment, and then slowly rise again. In this protocol, the white blood cell count did not reach initial values by the end of treatment. Optimized protocolsThe model was next used to optimize infusion protocol to minimize tumor size without violating the constraints of the model. Two different modeling problems were designed:
(P1) Given the drug infusion function, (11), determine the amount of d_{i} at each interval which minimizes the minimal tumor size over the time interval 0 < t > T while satisfying the constraints (6), (9) and (10).
(P2) Given the drug infusion function, (11), determine the amount of d_{i} at each interval which minimizes the minimal tumor size over the time interval 0 < t > T while satisfying the constraints (6) and (8).
By this, problem 1 concerns the constraints on white blood cell counts, while problem 2 concerns the constraint on total drug exposure. Both problems involve constraint (6) which limits the maximum plasma drug concentration allowed. The two problems both seek to minimize the minimum tumor size, but have different constraints overruling the simulations. The optimized result for problem 1 is given in figure 3a and 3b. Figure 3a shows the optimized drug infusion rates, u^{*}(t), the white blood cell count (shifted τ backwards), w(tτ), and the plasma drug concentration, c_{1}(t). Figure 3b shows the optimized drug infusion rates, u^{*}(t), the cancer cell count, n(t), and the active drug concentration, c_{2}(t). AnalysisInspection of problem 1Inspection of the optimized infusion rates show an initial high infusion rate leading to a fast increase of c_{1}(t) until it reaches the maximum allowed value, C_{MAX}. The white blood cell count drops in the same region until it reaches its minimum value, W_{D}. To avoid violation of the constraints, u^{*}(t) is lowered in the following time intervals, and finally put equal to zero, to rise the white blood cell count above W_{U} to avoid violating constraint (10). The infusion until this break is the first phase of the treatment. The break continuous for two days, before the second phase of treatment starts, where infusion rates in general are lower than in phase one. In phase two, both white blood cell count and plasma drug concentration are kept at levels above and below their limit values. Moving our attention to figure 3b we see that the cancer cell count first increases slightly until the drug reaches the active compartment, and after that decreases until after the end of treatment. The minimum cancer cell count is found at t^{*} = 23.6 days and has the value of n(t^{*}) = 5.1 g. The minimum tumor size is actually achieved after the end of treatment. The total amount of drug infused over the treatment period was 4.39 g. Inspection of problem 2Optimized infusion rates for problem 2 were found be high rates for the first 6 days and then no infusion until the end of the treatment period. For this optimization, the minimum tumor size was found to be n(t^{*}) = 16.6 g and t^{*} = 9.0 days. The total amount of drug infused over the treatment period was 1.5 g. DiscussionWe see immediately that the minimum tumor size is achieved with the procedure of problem 1, and this result is dramatically better then the result achieved in problem 2 (4.39 g versus 16.6 g minimal tumor mass). The total amount of drug infused is also allowed to be elevated in problem 1 compared to problem 2. This is because of the different means of measuring maximal allowed toxicity of the drug in the different procedures. The area under the curve approach of problem 2 is a static approach for measuring maximal toxicity. The infusion has to stop when a(t) reaches A_{MAX}. Problem 1 on the other hand solves this issue by dynamically measuring the response in the white blood cell count and putting constraints on which white blood cell levels that are allowed. By this no constraint is put on the maximum amount of infused drug, and the result is a larger amount of drug infused, and a smaller minimum tumor size. The simulation does indeed show promising results of designing drug regimens using the dynamics of white blood cell count as limiting constraints instead of the static area under the curve, AUC, approach. To arrive at the optimized protocol, a combination of the clinical protocols were used. It starts with high infusion doses and lower the doses to low constant doses later in the treatment. By this, both the initial fast drop in tumor size, n(t), observed using clinical protocols 1 through 4 and the slow decay observed using protocol 5 is achieved, without violating the constraints on white blood cell counts. The results in an exponentiallike decay of tumor size. These findings will hopefully contribute to the consideration of drug toxicity surveillance using dynamic models such as the one presented here, instead of solely relying on static data. Recent ExtensionIn a 2003 paper, Mechanistic models for myelosuppression,^{[5]} Lena E. Friberg and Mats O. Karlsson cite the Illiadis paper in a review of chemotherapeutic treatment optimization models. Their paper analyzes five different models and analyzes their strengths and shortcomings. They argue that mechanistic models, based on physiological realities rather than simply empirical data, are more helpful in determining optimal treatments because they more reliably allow for extrapolation. However, they note that empirical data, such as that obtained through monitoring blood levels, is still necessary for the development of such models. Friberg's citation of the Illiadis paper references it as a "first approach to integrate both pharmacokineticpharmacodynamic relations of effect and toxicity in the same model to optimize therapy.' However, they criticize the great extent to which the Illiadis paper bases its model on empirical data, and propose that the model would be more robust if it had a stronger foundation in the underlying mechanisms of drugtumor interactions. The Friberg paper supports the use of delay equations in describing drug effects, and indicates that drug administration schedules are a valuable topic of research in combatting cancer. They recognize the Illiadis paper is an early contribution to this area. External links
