Model-based control is currently considered the state-of-the-art in the field of process control and is the technique focused on solely within this dissertation. A generic representation of this technique is shown in Equations (1.1)-(1.4):
min
u(t) F (x(tf), x(t), u(t)) (1.1)
s.t. ˙x(t) = g(x(t), u(t)) (1.2)
A1 ≥ u(t) ≥ A2 (1.3)
B1 ≥ x(t) ≥ B2 (1.4)
This control technique uses an explicit representation of the system as a model (Equa-tions (1.2)) to make decisions on how to change the manipulated variable (u(t)) in order to obtain a desired outcome (based upon the formulation of the objective function, F (x(tf), x(t), u(t))). Objective function terms typically include some combination of penalties (weights) on the state values (x(t)), input magnitude or energy (u(t)), and final time state values (x(tf)) of the system over the optimization horizon. Input changes are accomplished while satisfying both input (Equation (1.3)) and state (Equation (1.4)) constraints, which may be either strict equality or inequality relationships and are specifically
chosen for the system. This technique has been applied across a range of fields, and a comprehensive list of areas where model-based control has been applied is beyond the scope of this dissertation. Instead, discussion will focus on the implementation of this technique specifically for the cancer treatment problem
Cancer treatment design has been investigated using a variety of control techniques, including optimal control theory [65, 75, 117, 118, 119, 120, 121], control vector param-eterization [122, 123], nonlinear model predictive control (NMPC) [79, 80], and mixed-integer linear/nonlinear programming (MILP/MINLP) [64]. Lumped tumor models (i.e., the exponential and Gompertz) coupled with a bilinear kill term have been investigated using optimal control theory in order to minimize a tumor volume at a final time allowing for the continuous administration of a therapeutic. Such dose scheduling approaches, however, have failed to account for the constraints commonly encountered in a clinical setting. First, intravenous (IV) chemotherapeutic administration often occurs during clinical hours, and drugs taken orally at home should be scheduled during the day (e.g., the patient will not routinely wake up in the middle of the night to take a pill). Furthermore, many chemotherapeutics administered IV will not be administered continuously for extended periods of time due to drug toxicity. For pills taken on regular intervals (i.e., daily or weekly administration), the model predictive control framework can be used to develop the dosing schedule [79, 80]. Evaluation of drug schedules over non-regular intervals (i.e., stochastic weekday decisions), however, may require more complex controller algorithms such as MILP/MINLP [64].
The implemented objective function must also be chosen carefully to ensure the existence of solutions which are both nontrivial and clinically relevant. For an objective function based on the final-time tumor volume and an underlying exponential or Gompertz tumor growth model with bilinear drug effect, the solutions returned by the controller will be clinically indistinguishable but numerically different (final tumor volume differences of <1%) [124].
Also, for the above listed objective function, the algorithm predicts a characteristic treatment profile: maximum initial drug delivery, followed by a non-dosing period, with the remainder of the drug delivered at the end of the treatment window [63]. Ethically, however, a doctor cannot allow a tumor to grow untreated, thereby invalidating the controller formulation.
In addition, bulk dosing at the end of the cycle, instead of at the beginning, prohibits dosing immediately after the treatment design window due to patient toxicity. Dose schedule development, therefore, requires either an alternative objective function, PD effect, clinical constraints, or model structure to obtain useful scheduling results. A more suitable model structure for dose schedule development would include the cell-cycle [75, 117, 118, 125], which more accurately captures the phase specificity of drug action, and/or a toxicity model [3, 75, 111, 113] to ensure dosing safety and provide an additional measure of treatment efficacy.
Dose schedule development has been investigated using a number of tumor growth model structures, including lumped models [63, 67], cell-cycle models with cycle-specific treatment effects [75, 80, 125], and tumor growth models with subpopulations of susceptible and resistant cells [117,118]. The inclusion of a toxicity model within the controller formulation, not just input bounds or limits on the total drug administered over a cycle, however, is limited to a few studies from the literature. Fister and Panetta investigated the case of both healthy and cancerous cells proliferating at different growth rates while subjected to a cycle-specific chemotherapeutic [65]. Using an optimal control algorithm with an output objective of maximizing both total drug delivered and the final quantity of healthy cells remaining, the results demonstrated that the therapeutic should be administered preferentially towards the end of the simulation, again neglecting ethical issues and downstream toxicity. Afenya used an optimal control formulation to determine the minimum time necessary to switch between administration of the drug (e.g., drug delivery was either “on” or “off”) in the presence of two concurrent cell populations: (i) healthy; and (ii) cancerous proliferating cells [113].
Results demonstrated that by shortening the time between dose administration, overall cycle time could be reduced by 1-2 weeks without adverse toxicity effects, a result that is termed
“dose-dense” therapy currently practiced clinically in breast cancer treatment [70]. Finally, Harrold et al. combined a nonlinear PK model for the therapeutic, 9-nitrocamptothecin (9NC), with a switched exponential model for tumor progression and bilinear PD effect on tumor regression [124]. A toxicity model altered the body weight of the mouse in response to 9NC administration, and the resulting objective function was set to minimize the final tumor volume subject to output constraints on the body weight of the mouse throughout
the treatment. The model of body weight response to therapy was inexact, not to mention non-ideal as a toxicity dosing metric (body weight alterations would manifest after intestinal damage had already taken effect), and this dramatically impacted the dosing schedule design, illustrating the importance of accurate toxicity models [124].
The investigation of closed-loop solutions to dose schedule development may often be too specific to the problem being modeled (i.e., highly sensitive to model parameters, narrow dosing windows, etc.). Furthermore, the solutions obtained from such formulations are highly dependent on the objective function and system constraints, and the true “optimal” solution may be numerically indistinguishable from a number of other more clinically implementable solutions. Given all possible degrees of freedom in clinical implementation of a control algorithm, it may be useful to begin dose schedule evaluation in an open-loop framework.
Open-loop studies based on cultured tumor cell dynamics have also been investigated, focusing on the ideal treatment interval to maximize cell death [84, 126], ensure statistical elimination of the tumor [127], or prevent the development of resistance [85]. Using more involved PBE representations for tumor proliferation, all of these models were consistent in predicting drug delivery with increased frequency compared to normal clinical treatment (approximately cycles of 12 hours). Lacking within these studies were evaluations on how uncertainty in parameter mean values may alter treatment, in vivo validation of the proposed cell fraction distribution oscillations exploited during treatment, or an accompanying toxicity model to assess whether the increased frequency of doses were clinically acceptable. Other open-loop studies by Dyson et al. [84] and Sherer et al. [86] used concurrent models for the proliferation of cancerous and healthy proliferating cells exposed to the same drug concentration. Rather than investigating dose level alteration, these studies focused on the delivery intervals of chemotherapeutics to maximize tumor elimination, again, attempting to exploit oscillations in cell-cycle fractions induced following treatment with cycle-specific therapeutics. Results from both of these studies concluded that the window of delivery for improving anti-tumor effect while sparing healthy proliferating cells was too narrow to exploit in a clinical setting, though no validation of the healthy proliferating cells within an in vivo setting was provided.