Eliminating the tumor mass is not the only concern during chemotherapy, as the drugs employed also adversely affect healthy proliferating cells. Common toxicities encountered during the course of chemotherapy include diarrhea, dizziness, bleeding, nausea, vomiting, mucositis, and myelosuppression, with different toxicities manifesting depending on the drug(s) and treatment regimens employed [104]. While maximum dosing limits provide one means of avoiding excessive patient toxicity, patients must still be monitored during a treatment cycle because of inter-patient variability in drug clearance and sensitivity.
Clinicians assess toxicity on a discrete, graded scale from 0-4, with higher grades correlating with worse toxicity [104]. Additional discrete measures, such as discrete events of vomiting and diarrhea introduce another element of complexity in toxicity model development, as well as subjective classification associated with some toxicity measures (such as dizziness or skin irritation).
One approach to toxicity modeling involves identifying those subsets of patients more prone to toxicity and treating them in a less aggressive manner [105, 106]. Others include developing heuristic relationships between a toxicity grade and dose administered, drug plasma concentration area under the curve, or time that plasma drug concentration exceeds a therapeutic value [107, 108, 109]. These relationships provide clinicians with trends and likelihood of toxicity, though the uncertainly on such models precludes them from serving as strict toxicity predictors during treatment. Such models do, however, allow for a continuous formulation of discretely graded variables, facilitating downstream controller design if the toxicity measure is indeed the primary toxicity associated with the drug. Combining both ideas, most toxicity model development has focused on measures that are easily obtainable and already continuous, such as the decrease in platelets, leukocytes, and/or neutrophils following chemotherapeutic treatment [3, 50, 51, 110, 111, 112]. Control algorithms that incorporate models for neutrophil or leukocyte proliferation and drug effect can develop drug schedules that minimize patient toxicity (possibly avoiding other toxicities as well) while simultaneously minimizing overall tumor volume [3, 75, 111, 113]. Secondary toxicities may still manifest in such regimens, but by avoiding the primary toxicity it should be theoretically possible to maintain secondary toxicities within acceptable treatment bounds.
The simplest model for healthy proliferating cells uses the same equations as those used to represent short-term tumor growth or the progression of hematologic cancers. Using either traditional exponential or linear cell-cycle models with adjusted growth rates and appended elimination or apoptotic rates, it is possible to represent leukocyte progression and treatment response over short windows in time. However, these models are either unstable or cannot guarantee convergence back to the original steady state (i.e., will not maintain a steady state cell population in the absence of treatment). Replacing the production rate of cells in the progenitor compartment with a constant (rather than the product of progenitor cells and
a rate) allows for stability of the system and baseline recovery. A typical compartmental neutrophil model without feedback is shown in Figure 1.5a, along with a representative neutrophil time course after administration of a myelosuppressive drug in Figure 1.5b [114, 115]. The structure consists of: (i) a progenitor compartment where drug effect (Edrug) occurs; (ii) intermediate compartments between progenitor and circulating cells representing cell maturation, where the cells are unaffected by the drug; and (iii) a circulating neutrophil compartment that has elimination due to natural cell death but is otherwise unaffected by the drug. Figure 1.5b shows a delayed drop in circulating cell count following drug exposure (bolus at t = 0) as affected cells progress to the circulation; proliferating cells, and eventually the circulating population, return to baseline following treatment. Models of this type, however, fail to account for the internal biological control loop that regulates circulating hematopoietic cell count and demonstrates baseline cell count overshoot following treatment.
An important compound in regulating neutrophil count is granulocyte colony stimulating factor (GCS-F), which promotes the proliferation of hematopoietic stem cells by binding to and activating the GCS-F receptor. In addition, GCS-F binds to circulating neutrophils, which has no effect on neutrophil production. This interaction between circulating neu-trophils and GCS-F results in an equilibrium cell count and systemic GCS-F concentration.
If the baseline cell count of circulating neutrophils falls, such as from a chemotherapeutic treatment, the concentration of unbound GCS-F will increase resulting in an upregulation of hematopoietic stem cell division. Similarly, neutrophil counts above baseline decrease the concentration of bound GCS-F and subsequently downregulate hematopoietic stem cell division. As GCS-F plasma concentrations are not often obtained during treatment, many groups have developed neutrophil feedback models based on the circulating neutrophil count. A low-order model developed by Karlsson and coauthors [3] has model states as described above in addition to a nonlinear feedback mechanism dependent on circulating neutrophil deviation from the homeostatic value. Similar neutrophil models have been developed by Zamboni et al. [50] and Minami et al. [109], though the model output in these cases was percentage decrease in neutrophil count rather the absolute neutrophil count.
More complicated model structures depicting the entire process of human granulopoiesis
incorporate multiple feedback loops, including the effects of granulocyte macrophage colony stimulating factor (GMCS-F), individual drug effects on each subpopulation of precursor cells, and nonlinear transition rates dependent on circulating GCS-F and GMCS-F concentration [51,66]. Such models are able to capture more rapid alterations in circulating neutrophil counts and are useful for describing population or subpopulation behavior during a chemotherapeutic trial; however, these models require more parameters than less complex neutrophil models and are difficult to tailor to individual patients.
A final item is the location of chemotherapeutic effect relative to the measurement of the effect. As previously mentioned, chemotherapeutics target cellular machinery which is upregulated during cell growth and cell division. This allows for preferential selection of tumor cells versus the majority of other cells in the body (under the assumption that tumor cells proliferate more rapidly than healthy cells), though other rapidly proliferating cells, such as hematopoietic stem cells, are simultaneously affected by the treatment. Circulating neutrophils and post-mitotic neutrophils undergoing differentiation, however, do not proceed through the cell-cycle and are not affected by the administration of chemotherapeutics.
Rather, the upstream precursors will undergo apoptosis or growth inhibition which will, in turn, reduce circulating neutrophil count after a period of 3-5 days (the approximate maturation time for neutrophils). Observed fluctuations in circulating neutrophil count, therefore, are a delayed observation of the actual drug effect. More thorough model structures accounting for differentiation states of hematopoietic cells and cell phase can also be found in the literature [51, 66], although the large number of model parameters would hinder the ability to adapt such models for individual patient treatment based on sparse data collection.
Neutrophil toxicity models have been developed for a number of therapeutics including pemetrexed, docetaxel (Doc), paclitaxel, etoposide, vinflunine, irinotecan, deoxy 2’-methylidenecytidine, topotecan, epirubicin, and 5-fluorouracil [3, 50, 110, 111, 116].
Typically, these models were developed based on individual neutrophil counts and were used to construct representative population models of neutrophil progression following treatment. Karlsson and coauthors have updated the model parameters to include patient-specific measures such as α-1 acid glycoprotein (AAG), body surface area (BSA), patient weight, and albumin levels, in order to develop prospective individual neutrophil progression
b)
0 5 10 15 20 25 30
1 1.5 2 2.5 3 3.5 4 4.5 5
Time (days) Neutrophil Count (103 cells/µ L)
a)
Transit Transit Transit
3
1 2 Circ.
Neut.
Neut.
Prol.
ktr ktr ktr ktrkprol(= ktr)
Edrug kcirc(= ktr)
Figure 1.5: A low-order neutrophil model without feedback (a) and representative circulating neutrophil count progression following treatment with a chemotherapeutic (b).
predictions for patients [106]. Results applied to Doc demonstrated that the most important patient parameters included AAG, BSA, and CYP3A enzyme level [106]. Scholz and coauthors used a more involved leukocyte model to investigate the treatment outcomes of a cyclophosphamide/doxorubicin/vincristine/prednisone (CHOP) regimen [66]. These studies focused on defining drug PD parameter profiles representative of patients at high (elderly patients), medium, and low risk for neutrophil toxicity following treatment. Using these groups, subgroup leukocyte progression was simulated and necessary regimen alterations to the cyclophosphamide dose to maintain acceptable toxicity levels were evaluated [66]. To our knowledge, prospective evaluation of alternative dosing regimens have not been investigated using these models in the literature.