Mathematical models for cytarabine-derived myelosuppression in acute myeloid leukaemia

Abstract

We investigate the personalisation and prediction accuracy of mathematical models for white blood cell (WBC) count dynamics during consolidation treatment using intermediate or high-dose cytarabine (Ara-C) in acute myeloid leukaemia (AML). Ara-C is the clinically most relevant cytotoxic agent for AML treatment. We extend a mathematical model of myelosuppression and a pharmacokinetic model of Ara-C with different hypotheses of Ara-C's pharmacodynamic effects. We cross-validate the 12 model variations using dense WBC count measurements from 23 AML patients. Surprisingly, the prediction accuracy remains satisfactory in each of the models despite different modelling hypotheses. Therefore, we compare average clinical and calculated WBC recovery times for different Ara-C schedules as a successful methodology for model discrimination. As a result, a new hypothesis of a secondary pharmacodynamic effect on the proliferation rate seems plausible. Furthermore, we demonstrate the impact of treatment timing on subsequent nadir values based on personalised predictions as a possibility for influencing/controlling myelosuppression.

Fig 1. Schematic model from which all mathematical models were derived.

We assumed clustering of cells and cytarabine (Ara-C) concentrations in compartments with identical properties. White blood cell (WBC) differentiation is represented by a proliferating compartment x_pr, a number n_tr of transit compartments x_tr with different levels of maturation, and a compartment x_ma with mature, circulating WBCs. Cells mature with a maturation rate G. Mature cells x_ma die by apoptosis with a death rate of k_ma. The pharmacodynamic effect of Ara-C is described as a log-linear function E targeting the proliferating cells in the bone marrow. It depends on the concentration x₁ of Ara-C in an assumed central compartment including the circulating blood. The proliferation rate F of x_pr models the replication speed of proliferating progenitor cells. Modelling assumptions were incorporated by choosing different functions F and G (compare Table 1). The estimated model parameters used for personalisation were B, slope, k_tr, γ, and initial conditions.

Fig 2. Simulations of different pharmacokinetic models and Ara-C concentration measurements from Kern et al. [39].

Fig 3. Visualisation of predictive accuracies of personalised mathematical models (PM).

(a) Goodness-of-fit plot for M10. Shown are measured versus calculated white blood cell (WBC) counts. Models were personalised using complete data sets of one to three cycles from 23 patients. The measured counts around the nadir coincide well (RMSE = 0.740) with the calculated WBC counts. (b) As (a), but cross-validated: WBC counts from the last cycle of patients were not used for personalisation, but compared to predictions (RMSE = 0.927). The plot shows cross-validated WBC counts from the last cycle in red, others in blue. The plots are prototypical for M1–M12. (c) PMs based on M10 and either personalisation with WBC counts from one or from all three cycles. 1000 Monte Carlo simulations after personalisation with WBC counts from one cycle were used to indicate the propagated probability density function. (d) As (c), but using WBC counts from the first two cycles for personalisation. More measurements lead to higher prediction accuracy. The uncertainty tube tightens and the predicted trajectory gets closer to the solution that used all available WBC counts.

Fig 4. Comparison of personalised models (PMs) based on M1-M12 and white blood cell (WBC) data.

Patient with three D135 cycles (left) and patient P_D123 with two D123 cycles (right), as indicated on the x-axis. The PMs exemplify reproducability (first row), predictability (second row) and simulation of a different schedule in prediction than estimation (third row). (a) Reproducability: all 12 PMs based on M1–M12 are able to explain the measured WBC counts. (b) As in (a), all PMs explain the measured WBC counts well, particularly around the nadirs. (c) Cross-validated prediction: all PMs explain the WBC counts well, also in the predicted third cycle. (d) As in (c), here with a slightly too slow predicted recovery time in the second cycle for all models. (e) Varied Ara-C schedule: prediction of D123 in the third cycle for a PM based on two D135 cycles shows faster WBC recovery for M9, M10, and M12. (f) Prediction of D135 in the second cycle for a PM based on one D123 cycle shows slower WBC recovery times for M9, M10, and M12.

Fig 5. Analysing the influence of treatment timing on nadir values.

(a) Simulation study in which 20 simulated nadirs were compared with the true nadir of the last CC for the 14 patients who have more than one CC. The simulated nadirs were computed by using the patient’s PM (second row of Table 3) and varying the start of the last CC daily with the maximal starting variation of 10 days earlier or later. (b) Exemplary variation of the CC start for one patient. An earlier (later) start results in a larger (lower) nadir.