Understanding the role of B-cells in CAR T-cell therapy in leukemia through a mathematical model

TL;DR

Analyzing the oscillatory behavior of the system, the time-dependent dynamics of CAR T cells and leukemic cells can be approximated, shedding light on the impact of initial tumor burden on therapeutic outcomes.

Abstract

Chimeric Antigen Receptor T (CAR-T) cell therapy has been proven to be successful against different leukaemias and lymphomas. This paper makes an analytical and numerical study of a mathematical model describing the competition of CAR-T, leukaemias tumor and B cells. Considering its significance in sustaining anti-CD19 CAR T-cell stimulation, we integrate a B-cell source term into the model. Through stability and bifurcation analyses, we reveal the potential for tumor eradication contingent on the continuous influx of B-cells, uncovering a transcritical bifurcation at a critical B-cell input. Additionally, we identify an almost heteroclinic cycle between equilibrium points, providing a theoretical basis for understanding disease relapse. Analyzing the oscillatory behavior of the system, we approximate the time-dependent dynamics of CAR T-cells and leukemic cells, shedding light on the impact of initial tumor burden on therapeutic outcomes. In conclusion, our study provides insights into CAR T-cell therapy dynamics for acute lymphoblastic leukemias, offering a theoretical foundation for clinical observations and suggesting avenues for future immunotherapy modeling research.

Figures (16)

• Figure 1: Summary of model components and processes. B-cells $B(t)$ exit bone marrow at a rate $I_0/\tau_I$ and die at a rate $1/\tau_B$. Leukemic cells $L(t)$ proliferate with constant rate $\rho_L$. CAR T-cells $C(t)$ proliferate with rate $\rho_C$ upon encounter with antigen-expressing cells, which are removed at a rate $\alpha$. CAR T-cells eventually die at rate $1/\tau_C$.
• Figure 2: Regions determined by the stability of the equilibrium points with $\rho_L=0.2$ day$^{-1}\cdot$ cell$^{-1}$; $\alpha=10^{-11}$ day$^{-1}\cdot$ cell$^{-1}$; $\tau_I=4$ days; $\tau_B=45$ days. Bold text denotes equilibrium points with biological meaning (all components are positive). Green text denotes stable equilibria. Dashed black line represent the standard value of $\tau_C\rho_C=2\cdot 10^{-10}$ cell$^{-1}$. See main text for more details.
• Figure 3: Central panel, value of the components $C_i$, $i=1 \ldots 3$, of the equilibrium points as a function of the B-cell influx $I_0$, for standard values of the parameters ($\rho_C\tau_C=2\cdot 10^{-10}$ cell$^{-1}$; $\rho_L=0.2$ day$^{-1}\cdot$ cell$^{-1}$, $\alpha=10^{-11}$ day$^{-1}\cdot$ cell$^{-1}$, $\tau_I=4$ days, $\tau_B=45$ days). The interval for $I_0$ is broader than the biological one, to incorporate the bifurcations happening at $I_0=0$. Dashed lines indicate that the equilibrium is unstable, while solid lines indicate that the equilibrium is stable. Also, bold lines denote that the equilibrium point is biologically meaningful. The black circle marks the Hopf bifurcation of $P_3$. The two purple circles mark transcritical bifurcations. Panels (a) to (d) show a characteristic phase diagram for several close initial conditions in the different regions of equilibrium ($R_1$ to $R_4$ respectively).
• Figure 4: Simulations from Eqs. \ref{['sys']} displaying focus-like dynamics in Region $R_2$ approaching $P_3$ equilibrium. Parameter values for the orbit shown (blue curve) are $I_0= 0.5 \cdot 10^9$ cells, $\rho_C \tau_C = 2 \cdot 10^{-10}$ cell$^{-1}$, $\rho_L=0.2$ day$^{-1}\cdot$ cell$^{-1}$, $\alpha=10^{-11}$ day$^{-1}\cdot$ cell$^{-1}$, $\tau_I=4$ days, $\tau_B=45$ days. The initial state is given by $B_0=5\cdot10^8$ cells; $L_0=5\cdot10^{10}$ cells and $C_0=5\cdot10^7$ cells. (a) Phase-portrait around $P_3$ highlighting strong and weak stable manifolds. (b,c) Evolution of CAR T-cells with time asymptotically approximated by $x(t)$ (Eqn. (\ref{['approxeq']})) with $K=1/2 \cdot 10^{11}, k_s=k_c=1, d=4.8$.
• Figure 5: Asymptotic behaviour of leukaemic cells approaching equilibrium point $P_3$ as a function of $I_0$, for the same parameter values as in Fig. \ref{['equilibrios']}. (a) Magnitude of the local maxima of $L(t)$ approaching $L_3$ (red curve). (b) Quotient between two consecutive local maxima of $L$. Red curve marks $e^{-(2\pi )\alpha/\omega}$. (c) Difference between the times at which two consecutive local maxima of $L$ occur. Red curve marks $(2\pi )/\omega$. In all three cases, the curves appear in descending order with time.

Theorems & Definitions (5)

• Proposition 1
• Proposition 2
• Proposition 3
• Proposition 4
• Remark 1