Mathematical modelling of stem and progenitor cell dynamics during ruxolitinib treatment of patients with myeloproliferative neoplasms

Abstract

Introduction: The Philadelphia chromosome-negative myeloproliferative neoplasms are a group of slowly progressing haematological malignancies primarily characterised by an overproduction of myeloid blood cells. Patients are treated with various drugs, including the JAK1/2 inhibitor ruxolitinib. Mathematical modelling can help propose and test hypotheses of how the treatment works.

Materials and methods: We present an extension of the Cancitis model, which describes the development of myeloproliferative neoplasms and their interactions with inflammation, that explicitly models progenitor cells and can account for treatment with ruxolitinib through effects on the malignant stem cell response to cytokine signalling and the death rate of malignant progenitor cells. The model has been fitted to individual patients' data for the JAK2 V617F variant allele frequency from the COMFORT-II and RESPONSE studies for patients who had substantial reductions (20 percentage points or 90% of the baseline value) in their JAK2 V617F variant allele frequency (n = 24 in total).

Results: The model fits very well to the patient data with an average root mean square error of 0.0249 (2.49%) when allowing ruxolitinib treatment to affect both malignant stem and progenitor cells. This average root mean square error is much lower than if allowing ruxolitinib treatment to affect only malignant stem or only malignant progenitor cells (average root mean square errors of 0.138 (13.8%) and 0.0874 (8.74%), respectively).

Discussion: Systematic simulation studies and fitting of the model to the patient data suggest that an initial reduction of the malignant cell burden followed by a monotonic increase can be recapitulated by the model assuming that ruxolitinib affects only the death rate of malignant progenitor cells. For patients exhibiting a long-term reduction of the malignant cells, the model predicts that ruxolitinib also affects stem cell parameters, such as the malignant stem cells' response to cytokine signalling.

Keywords: JAK2 V617F; blood cancer; mathematical modelling; myeloproliferative neoplasms (MPN); ordinary differential equations; parameter estimation; ruxolitinib; stem cells.

Figure 1

Conceptual compartment diagram of the model. See the text for further description of the model. The lightning symbol represents external factors affecting the cytokines. HSC, Haematopoietic stem cells; HPC, Haematopoietic progenitor cells; MBC, Mature blood cells; mHSC, Malignant haematopoietic stem cells; mHPC, Malignant haematopoietic progenitor cells; mMBC, Malignant mature blood cells.

Figure 2

A simulation of the stem, progenitor and mature cell counts and the JAK2 VAF based on Equations (1) and (2) with the standard parameters from Table 2 . As initial conditions, we choose $x_0(0)=1.0\times {10}^5$ , $x_1(0)=2.5\times {10}^6$ , $x_2(0)=6.4\times {10}^{11}$ , $y_0(0)=1$ , $y_1(0)=0$ , $y_2(0)=0$ , $a(0)=8.1\times {10}^2$ , and $s(0)=1$ . For the plots of cell counts, the green curves represent the number of healthy cells, the solid red curves represent the number of malignant cells, and the dashed black curves represent the sum of healthy and malignant cells. Treatment is initiated after 30 years in the simulation. (A) No effect of treatment. (B). $s_{y_0}$ increased to 6 times its standard value during treatment. (C) $d_{y_1}$ to 6 times its standard value during treatment. (D) $s_{y_0}$ and $d_{y_1}$ to 6 times their standard values during treatment.

Figure 3

Model fit to selected individual patients as described in sections 2.2-2.3. ${\rho }_{s_{y_0}}$ and ${\rho }_{d_{y_1}}$ are the fitted parameters that quantify the strength of the given patient’s response to RUX treatment in terms of the effect on $s_{y_0}$ and $d_{y_{{}_1}}$ , respectively. The solid yellow curves show the optimal fit of the model to the JAK2 VAF data. In the fit, it is assumed that both parameters ${\rho }_{s_{y_0}}$ (response of mHSCs to cytokine signal) and ${\rho }_{d_{y_1}}$ (malignant progenitor cell death) are affected by RUX at the same time. To visualise the impact of each of the two effects (changed response to cytokines and increased progenitor death) on the JAK2 VAF dynamics, the dashed lines show the time evolution of JAK2 VAF if either ${\rho }_{d_{y_1}}$ (blue) or ${\rho }_{s_{y_0}}$ (cyan) is set to 0 and the respective other parameter remains unchanged. The red dots are the data. (A) Patient 1, one of the patients for whom the model fits very well, and for whom the model predicts that RUX affects both $s_{y_0}$ and $d_{y_{{}_1}}$ . (B) Patient 2, a patient for whom the model fits quite well, and the model predicts that RUX affects only $d_{y_{{}_1}}$ ( ${\rho }_{s_{{}_y{}_0}}=0$ ). (C) Patient 3, the patient for whom the model fits worst. (D) Patient 20, a patient from the RESPONSE study for whom the model fits very well, and for whom the model predicts that RUX affects both $s_{y_0}$ and $d_{y_1}$ .

Figure 4

Model fit to individual patients’ data for patients 1-12 as described in sections 2.2-2.3. The solid yellow curves show the optimal fits of the model to the JAK2 VAF data. In the fit it is assumed that both parameters ${\rho }_{s_{y_0}}$ (response of mHSCs to cytokine signal) and ${\rho }_{d_{y_1}}$ (malignant progenitor cell death) are affected by RUX at the same time. To visualise the impact of each of the two effects (changed response to cytokines and increased progenitor death) on the VAF dynamics, the dashed lines show the time evolution of JAK2 VAF if either ${\rho }_{d_{y_1}}$ (blue) or ${\rho }_{s_{y_0}}$ (cyan) is set to 0 and the respective other parameter remains unchanged. The red dots are the data. Patients 1-12 are from the COMFORT-II study.

Figure 5

Continuation of Figure 4 for patients 13-24. Patients 13-18 are from the COMFORT-II study, and patients 19-24 are from the RESPONSE study.

Figure 6

Histogram of RMSE-values for the model fitted to the individual patient JAK2 VAF data as described in sections 2.2-2.3 for all patients (n = 24) as presented in Figures 4 , 5 and in sections S6 and S7 of the supplementary. Yellow: Optimal fits allowing RUX treatment to affect both $s_{y_0}$ and $d_{y_{{}_1}}$ . Blue: Optimal fits allowing only changes in $s_{y_0}$ . Cyan: Optimal fits allowing only changes in $d_{y_{{}_1}}$ . There are no observations with RMSE-values outside the range shown in the plots.