Tumor modeling approaches can be grouped into two categories: homogeneous and heterogeneous. The basic homogeneous tumor model is a single, linear ODE characterizing exponential growth with a single parameter governing the doubling time of the tumor.
This model accurately describes blood-born cancer progressions (e.g., leukemia), but over predicts the growth progression of solid tumors except over very short windows (e.g., one doubling) [67]. A slightly more complex nonlinear ODE representation is the Gompertz equation, which has been shown to accurately describe clinical data for a wide range of solid tumors [68, 69]. The two parameter Gompertz equation predicts saturating tumor growth as the total tumor volume approaches a preset “plateau” volume. As an approximation to the Gompertz equation, Simeoni et al. have constructed models that “switch” growth profiles at a predefined tumor volume [49]. This “switch” alters the growth profile from exponential to linear or continues the exponential progression at a slower doubling time. This model structure is useful for describing tumor growth over multiple doubling times where a single exponential model cannot accurately model the data. The anti-tumor PD effect of administered drugs is incorporated as a bilinear term, that is, a product of drug concentration and the total tumor mass present scaled by an estimated drug effect rate [70]. However, assuming an equal drug effect on the entire tumor population may result in an over-estimate of anti-tumor effect, especially as radiation and most chemotherapeutics are more effective during specific phases of the cell-cycle [71]. Incorporating cell phase information within the model structure, therefore, would allow for a more biologically relevant incorporation of treatment effect and increased model accuracy in response to treatment.
Cell-cycle models seek to describe the biological phase progression of cells. There are five identified cell phases, corresponding to growth (G1), DNA replication (S), mitotic preparation (G2), mitosis (M ), and quiescence (Q) as shown in Figure 1.4. Actively proliferating cells progress through the cell division loop (G1to M and back to G1), while non-cycling (i.e., the chemotherapy or radiation insensitive population) reside in the quiescent Q-phase [72]. Cell Q-phases are typically resolved by a combination of proliferation, DNA, or RNA staining; however, it is often difficult to resolve individual phase fractions for all five phases
of the cell-cycle [4]. Therefore, constructing a model with explicit representation of all phases would be overparameterized, thereby leading to high parameter sensitivity and uncertainty.
Grouping cell phases together reduces the amount of information necessary to model the system in this framework while simultaneously reducing overall model complexity. Smith and Martin have reduced the cell-cycle model to its basic component, namely proliferating and quiescent cells [48,73]. This two ODE model better incorporates drug effects targeting proliferating cells, however, most chemotherapeutics target proliferation mechanisms specific to certain phases of the cell-cycle. A number of models have been developed [74,75,76, 77]
to describe either G1−, S−, or M -phase (the latter being a combination of G2- and M -phase cells, hereafter referred to solely as M -phase within this dissertation) specific drug effects with varying degrees of complexity (e.g., additional equations for All of these models have linear transition rates between phases that can only return exponential tumor volume progression at the macroscopic scale when left untreated. The introduction of saturating phase-transition rates, similar in philosophy to the Gompertz model, or the selective accumulation of cells in a non-cycling compartment, Q, overcomes this limitation. Such models maintain cell-cycle specific information while amassing cells in either Q or a lumped G-phase [78,79,80]. These models are often sufficient for describing bulk tumor data observed clinically; however, these models simplify reality by assuming that the tumor is composed of lumped populations of cells. In actuality, these cells have individual transition rates, are exposed to various concentrations of drugs dependent on an erratic vascular network, and may not be actively dividing depending on their overall location within the tumor and nutrient availability.
One class of models that accounts for the heterogeneous properties of tumors is population balance equation (PBE) models. These models incorporate internal cellular properties (e.g., age, mass, or DNA content) and allow for distributed progression through cell phases and drug effect [80, 81, 82, 83, 84,85, 86]. The discussion of this class of tumor models will be explored in greater depth in Chapter 3. An alternative structure, the cell ensemble model, uses equations to define each cell with parameters fixed from a preset distribution [87]. Such models are simpler for a high number of intracellular properties, but are ill-suited for handling cell division, an essential component of tumor progression. Spatial inhomogeneity of the tumor may also be essential for predicting tumor invasiveness and
G 1
k M
G
G M
S
G 2
M k GS k SM
Q
Figure 1.4: Cell cycle model showing phases of the cell-cycle, Q, G1, S, G2, and M . Also shown are experimentally reconcilable phases based on DNA analysis: G (Q + G1), S, and M (G2+ M ).
drug concentration throughout the tumor [88,89,90,91,92,93]. Studies have demonstrated that many chemotherapeutics have difficulty traversing the leaky internal vascular networks of tumors, with a majority of exposure occurring in the outer periphery of a tumor mass.
While this may be ideal from a treatment perspective, as these cells are actively proliferating, failure to account for internal spatial diffusion limitations of chemotherapeutic compounds (not to mention essential nutrients such as oxygen and glucose) can lead to inaccurate model predictions.
Cell automata models and fractal simulations take the cellular tumor model to its penultimate conclusion, representing each cell individually while accounting for invasive migration, predicting metastases, and constructing an appropriate vascular network for the simulated tumor [94,95,96,97,98]. Other factors influencing tumor growth, such as external pressure effects, acquired drug resistance, and gene mutations perpetuating tumor growth provide additional information, allowing for a more physiological representation of tumor progression [87, 99, 100, 101, 102, 103]. While all this information would improve model predictions, data on a per patient basis, or a per animal basis in the preclinical setting, would be difficult, at best, to collect. Distributions for phase transitions and drug effects could be obtained from collected data, but the exact tumor geometry with internal necrosis cannot be determined exactly except by destructive biopsy. An important question to consider, then, is whether control based on a less detailed but identifiable model is sufficient for dose scheduling purposes.