# A Mathematical Description of the Bone Marrow Dynamics during CAR T-Cell Therapy in B-Cell Childhood Acute Lymphoblastic Leukemia

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## Abstract

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## 1. Introduction

^{+}relapse, on the other hand, is related to a lack of expansion or persistence of the product [3]. Clinical trials and associated research have found conflicting evidence regarding the relationship between dose, leukemia burden, and response. It has been pointed out that the reasons for treatment failure may be drug-intrinsic, probably related to characteristics of the patient’s T-cells before extraction and manufacturing [18,19].

^{+}cells, and therefore linked to the level of CAR expansion and leukemia burden. Neurotoxicity is another, less elucidated side effect. Finally, B-cell aplasia (BCA) is a kind of on-target off-tumor toxicity due to the fact that healthy B cells also express CD19 [20,21]. In fact, BCA in peripheral blood is used as a surrogate marker for CAR T persistence, and loss of BCA is associated with a higher probability of disease recurrence [22,23]. This is because new B cells are continuously being generated in the bone marrow, providing an endless source of stimulation for the CAR T-cell population and acting as an endogenous vaccine [24]. In this regard, it is important to note that most data about the outcome of the therapy, such as absolute leukocyte count, cytokine levels, or B cell and CAR T number, come from peripheral blood samples. However, leukemia is a disease of the bone marrow, where blood cells are produced. From that perspective, peripheral blood can be thought of only as a surrogate marker, since what is observed in blood has probably occurred before or is linked to what has previously occurred in bone marrow. Obtaining aspirates or bone marrow biopsies are more invasive procedures and are not carried out as often as peripheral blood extraction. Thus, data on relevanT-cellular components in bone marrow dynamics are scarce. Recent studies related to clinical trials have nonetheless recognized the importance of tracking bone-marrow dynamics [24].

## 2. Mathematical Model

#### 2.1. Main Elements of the Model

^{+}cells. They proliferate on encountering them and differentiate into effector cells, inducing the death of antigen-bearing cells and dying shortly afterwards. Part of the expanded population differentiates into long-lived memory cells that retain the ability to proliferate again upon repeated exposure to the antigen [39].

#### 2.2. Hematopoietic and Leukemic Compartments

#### 2.3. CAR T-Cell Compartment

^{+}compartments, namely ${B}_{1}\left(t\right)$, ${B}_{2}\left(t\right)$, ${B}_{3}\left(t\right)$, and $L\left(t\right)$, include an additional mass action term representing elimination by activated CAR T-cells ${C}_{A}\left(t\right)$, with killing capacity $\alpha $ [49].

#### 2.4. Parameter Estimation

^{+}cellular compartments (${B}_{1}$, ${B}_{2}$, and ${B}_{3}$) was compared to clinical and literature data to obtain characteristic parameter values. We followed the reasoning in that work to provide values for the Pro-B compartment and for the leukemic cell population. A more detailed explanation and references to experimental works can be found in the Supplementary Information. For the CAR T-cell compartment, typical parameter values can be obtained from studies of Tisagenlecleucel kinetics, the FDA-approved CAR T product for B-cell ALL [23,24]. There, the authors fitted patient data to a mixed-effects model and provided values for proliferation rate ${\rho}_{C}$, memory transition rate ${\gamma}_{AM}$, and both activated and memory characteristic lifetimes ${\tau}_{A}$ and ${\tau}_{M}$. The estimation of the rest of parameters is detailed in the Supplementary Information. All parameter values and their meanings are listed in Table 1.

#### 2.5. Computational Details

`ode45`, which uses an explicit Runge–Kutta formula of 4–5 order (Dormand–Prince) and adaptive step size. Plots were produced in the same software and exported using an

`export_fig`package.

## 3. Results

#### 3.1. The Mathematical Model without B-Cell Development Reproduces Clinical Data

#### 3.2. Effector and Memory CAR T-Cells Are Able to Control the Disease

^{+}cells, healthy and malignant. Pro-B cells increased steadily in the meantime, in response to the shortage of mature B cells. From day 30 onwards, healthy B cells started to recover. Around day 50, CAR T-cells responded to this increase: memory cells reactivated and CAR T expanded again. Cycles of CAR T expansion and B-cell reduction ensued, and, after six months, all cell types had reached a steady state and the disease had been controlled. This simulation explains the role of B cells as an endogenous vaccine. Instead of the expected number of activated cells in peripheral blood, we observed their reactivation due to CD19 antigen recovery. The steady although significantly lower level of self-renewing B cells kept CAR T-cells in a state of engagement, explaining their persistence.

^{−}Pro-B cell compartment, and the remaining compartments were ordered according to their maturation stage. In normal conditions, the Pre-BII compartment is the most abundant, followed by the immature compartment [38,40]. Reports of successful CAR T therapy in B-cell malignancies indicate BCA in peripheral blood as a marker of therapy response [14,15,16]. While this seems to be in contradiction with the results presented here, where we observe a recovery of B cells, this is actually due to the sensitivity of clinical detection. Indeed, here the immature compartment in the steady state after therapy is composed of approximately ${10}^{6}$ cells, barely $0.0002\%$ of the total bone marrow capacity. This means that the small number of B cells in blood would go undetected. Thus, BCA in peripheral blood would be compatible with the attempts at B cell recovery in the bone marrow.

#### 3.3. Initial State Does Not Affect CAR T Expansion and Outcome

#### 3.4. CAR T Product Characteristics Determine Therapy Success or Failure

^{+}. In order to simulate the former, we would need to describe CD19 expression with a continuous variable or include a CD19${}^{-}$ leukemic clone compartment. We therefore restricted the analysis to CD19

^{+}relapses. Clinical trials agree that non-responding patients show limited and slower CAR T expansion and shorter persistence, being unable to remove the cancer completely [14,15,16].

^{+}B-cell compartments ${B}_{1}$, ${B}_{2}$, and ${B}_{3}$ are plotted together. For each heatmap, we show an example of a responding (R) and a non-responding (NR) patient. Responding patients were defined by the absence of leukemic cells at day 30 (${L}_{30}<1$). Non-responding patients are defined as those with the presence of leukemic cells on that same day (${L}_{30}\ge 1$). The interesting feature of this figure is in analyzing the variety of possible responses. The three responding patients displayed here showed similar behavior, comparable to that of Figure 3: Expansion followed by decay and oscillations. We found that the disease could be controlled for a range of fold expansions. The relative proportion of activated to memory cell was constant, since we were not modifying transition rates ${\gamma}_{AM}$ and ${\gamma}_{MA}$. The level of steady state CAR T-cells did change and was related to the magnitude of the initial expansion. For the non-responding patients, we observed more variety. First of all, disease recurrence can occur for different magnitudes of expansion, from one to three orders of magnitude (bottom and middle subfigures, respectively). Secondly, this recurrence is detectable from the first to the third month after infusion (top and middle subfigures, respectively). The level of B cells increased at the time of relapse, while responding patients (left column) showed no increase. This supports the reported association between BCA and therapy response (B-cell recovery indicates a possible relapse). Leukemic cells and CAR T-cells were also subject to the oscillations that were observed in responding patients, and correspond to cycles of reactivation and decay. Although this model is not reducible to a Lotka–Volterra model, due to the existence of an activation function, these cycles can be interpreted in the same way: an increase in leukemic cells occurred after the decay of activated CAR T-cells, which then regrew in response to the recurring clone. Interestingly, in some cases, the therapy was able to control the disease after relapse. This behavior has been observed in simulations of simpler models. In previous work [36], we already remarked that this scenario would not be observed in the clinical context due to the prompt actions taken after any signs of disease recurrence. Therefore, while from the dynamical point of view one could describe some cases as ‘Delayed Response’, from the clinical point of view, they would still be considered as ‘No Response.’ Finally, leukemic cells and CAR T could also coexist in equilibrium (top subfigure, NR column).

#### 3.5. Second Infusion in Non-Responding Patients May Improve the Therapy Outcome

## 4. Discussion

^{+}relapses, the model was not able to capture long-term relapse, occurring from month three onwards. We have observed here that, from this date, the system typically reaches a steady state, in coexistence with either healthy B cells (in responding patients) or both leukemic clone and B cells (in non-responding patients). A steady state of this kind is consistent with patients in whom BCA has been ongoing for long periods of time, but not those who either relapse or recover B cells in peripheral blood in the meantime. In addition, differences in expansion and time to maximum expansion between responding and non-responding patients should be larger. This may be due to our arbitrary selection of parameter values in their respective ranges, or to the fact that we did not take into account correlations between them. Nonetheless, we cannot discard the possibility that other processes are at work.

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CAR | Chimeric Antigen Receptor |

ALL | Acute Lymphoblastic Leukemia |

CD | Cluster of Differentiation |

BCA | B-Cell Aplasia |

FDA | Federal Drug Administration |

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**Figure 1.**B cells progressively express CD19 as they become more mature. Four maturation stages were considered, namely Pro B, Pre BI, Pre BII, and immature B cells. These cell types were assumed to proliferate with rate ${\rho}_{i}$ ($i=0,\dots ,2$) and progress with rates ${\gamma}_{j}$ ($j=0,\dots ,3$), except for immature B cells that no longer proliferate, leaving bone marrow with an exit rate ${\gamma}_{3}$. Leukemic cells originate from a B-cell precursor and proliferate with rate ${\rho}_{L}$, invading the bone marrow and eventually migrating to peripheral blood with an exit rate ${\gamma}_{L}$. CAR T-cells are assumed to travel to bone marrow from blood and proliferate with rate ${\rho}_{C}$ upon encounters with CD19-expressing cells. They carry out effector functions on CD19 expressing cells with killing capacity $\alpha $. Pro-B cells are not affected by this action since they do not express CD19. Part of the activated CAR T pool becomes long-lived memory cells with rate ${\gamma}_{AM}$. These cells can regain effector function upon repeated exposure to CD19, with reactivation rate ${\gamma}_{MA}$.

**Figure 2.**Dynamics of leukemic cells and CAR T-cells without new B-cell generation. Simulation of the evolution over time of leukemic cells (solid gray line), total CAR T-cells (solid orange line), and memory CAR T-cells (dotted orange line). In this simulation, we used only Equations (12)–(14), so we observed no recovery of B cells. The initial state was $L\left(0\right)=2\times {10}^{10}$ cells, ${C}_{A}\left(0\right)=5\times {10}^{6}$ cells, and ${C}_{M}\left(0\right)=0$ cells. Parameters were as in Table 1 with $\alpha =3\times {10}^{-10}$ day${}^{-1}$· cell${}^{-1}$ and $h=5\times {10}^{8}$ cells. (

**A**) dynamics for the first month of therapy; (

**B**) dynamics for the first nine months of therapy.

**Figure 3.**Successful control of the disease in bone marrow. (

**A**) dynamics of Equations (8)–(14). Simulation now includes B cells: Pro-B cells (solid blue line), Pre-BI cells (dashed blue line), Pre-BII cells (dashed-dotted blue line), and Immature B cells (dotted blue line). Initial state is ${B}_{0}\left(0\right)={10}^{6}$ cells, ${B}_{1}\left(0\right)=3\times {10}^{7}$ cells, ${B}_{2}\left(0\right)=2\times {10}^{8}$ cells, ${B}_{3}\left(0\right)=9\times {10}^{7}$ cells, $L\left(0\right)=2\times {10}^{11}$ cells, ${C}_{A}\left(0\right)=5\times {10}^{7}$ cells and ${C}_{M}\left(0\right)=0$ cells. Parameters are those of Table 1 with $\alpha =3\times {10}^{-10}$ day${}^{-1}$· cell${}^{-1}$ and $h={10}^{9}$ cells; (

**B**) proportions of B cell subsets in bone marrow, relative to the total B-cell population, under normal conditions (light blue) and after successful CAR T therapy (dark blue). Density plots are a continuous representation of the proportions, obtained with shape-preserving piecewise cubic interpolation using Matlab’s function

`interp1`.

**Figure 4.**Influence of initial leukemia load, B-cell level, and CAR T dose on therapy outcome. Evolution with time of leukemic cells (gray) and total CAR T-cells (orange) for different initial configurations (In decreasing order: solid, dashed, dotted lines). When unchanged, initial values are ${B}_{0}\left(0\right)={10}^{6}$ cells, ${B}_{1}\left(0\right)=3\times {10}^{7}$ cells, ${B}_{2}\left(0\right)=2\times {10}^{8}$ cells, ${B}_{3}\left(0\right)=9\times {10}^{7}$ cells, $L\left(0\right)=2\times {10}^{11}$ cells, ${C}_{A}\left(0\right)=5\times {10}^{7}$ cells and ${C}_{M}\left(0\right)=0$ cells. Parameters are those of Table 1 with $\alpha =3\times {10}^{-10}$ day${}^{-1}$· cell${}^{-1}$ and $h={10}^{9}$ cells. (

**A**) influence of leukemia burden; (

**B**) influence of initial B-cell population, defined as the sum of all B-cell compartments; (

**C**) influence of CAR T dose.

**Figure 5.**Influence of CAR T product characteristics. Evolution in time of leukemic cells (gray) and total CAR T-cells (orange) for different CAR T product attributes (In decreasing order: solid, dashed, dotted lines). Initial state is ${B}_{0}\left(0\right)={10}^{6}$ cells, ${B}_{1}\left(0\right)=3\times {10}^{7}$ cells, ${B}_{2}\left(0\right)=2\times {10}^{8}$ cells, ${B}_{3}\left(0\right)=9\times {10}^{7}$ cells, $L\left(0\right)=2\times {10}^{11}$ cells, ${C}_{A}\left(0\right)=5\times {10}^{7}$ cells and ${C}_{M}\left(0\right)=0$ cells. The remaining parameter values are those of Table 1 with $\alpha =3\times {10}^{-10}$ day${}^{-1}$· cell${}^{-1}$ and $h=5\times {10}^{8}$ cells. (

**A**) influence of killing capacity; (

**B**) influence of proliferation rate; (

**C**) influence of activation threshold.

**Figure 6.**Exploration of parameter ranges for the dynamics of Responding (R) and Non–Responding (NR) patients. (

**A**) number of leukemic cells at day $+30$, in logarithmic scale, for different regions of the parameter space. The unchanged product attribute is displayed on the y-axis. The remaining parameter values are those from Table 1; (

**B**) examples of responding and non-responding patients for the first six months of therapy. Parameter values are marked in subfigure (

**A**). Represented are activated and memory CAR T-cells (solid and dotted orange line, respectively), leukemic cells (solid gray line) and CD19${}^{-1}$ and CD19

^{+}B cells (solid and dotted blue lines, respectively). The initial state of the simulations in this figure is ${B}_{0}\left(0\right)={10}^{6}$ cells, ${B}_{1}\left(0\right)=3\times {10}^{7}$ cells, ${B}_{2}\left(0\right)=2\times {10}^{8}$ cells, ${B}_{3}\left(0\right)=9\times {10}^{7}$ cells, $L\left(0\right)=2\times {10}^{11}$ cells, ${C}_{A}\left(0\right)=5\times {10}^{7}$ cells and ${C}_{M}\left(0\right)=0$ cells.

**Figure 7.**Characteristics of product expansion for Responding (R) and Non-Responding (NR) patients. (

**A**) graphical representation, on a typical simulation, of the characteristics of the product expansion; (

**B**) differences in time to peak and fold expansion for 4000 simulated patients with parameters from Table 1 and $\alpha $ in the range $5\times {10}^{-11}$–$5\times {10}^{-9}$ day${}^{-1}$· cell${}^{-1}$, h in the range $5\times {10}^{9}$–$5\times {10}^{11}$ cells and ${\rho}_{C}$ in the range 0.4–1 day${}^{-1}$. Initial state was set to ${B}_{0}\left(0\right)={10}^{6}$ cells, ${B}_{1}\left(0\right)=3\times {10}^{7}$ cells, ${B}_{2}\left(0\right)=2\times {10}^{8}$ cells, ${B}_{3}\left(0\right)=9\times {10}^{7}$ cells, $L\left(0\right)=2\times {10}^{11}$ cells, ${C}_{A}\left(0\right)=5\times {10}^{7}$ cells and ${C}_{M}\left(0\right)=0$ cells. The boxplot shows median and first and third quartiles.

**Figure 8.**Effect of a second infusion of CAR T-cells in non-responding patients. Dynamics of leukemic cells, B cells and CAR T-cells for the three non-responding patients in Figure 6. (

**A**) standard second dose of $5\times {10}^{8}$ CAR T-cells; (

**B**) increased second dose of $5\times {10}^{9}$ CAR T-cells; (

**C**) second dose of $5\times {10}^{8}$ CAR T-cells of a newly-manufactured product with improved attributes. For the first patient, the new attributes were $h=5\times {10}^{8}$ and ${\rho}_{C}=0.9$. For the second patient, the new attributes were $h=5\times {10}^{8}$ and $\alpha =6\times {10}^{-10}$. For the third patient, the new parameters were $\alpha =5\times {10}^{-8}$ and ${\rho}_{C}=0.7$. We show activated and memory CAR T-cells (solid and dotted orange line, respectively), leukemic cells (solid gray line) and CD19${}^{-}$ and CD19

^{+}B cells (solid and dotted blue lines, respectively). The red vertical line represents the time of the second infusion. The initial state for the simulations and the other parameter values were as in Figure 6.

Parameter | Meaning | Value | Units |
---|---|---|---|

${\rho}_{0}$ | Pro-B proliferation rate | $ln\left(2\right)$/8 | day${}^{-1}$ |

${\rho}_{1}$ | Pre-BI proliferation rate | $ln\left(2\right)$/1 | day${}^{-1}$ |

${\rho}_{2}$ | Pre-BII proliferation rate | $ln\left(2\right)$/1.5 | day${}^{-1}$ |

${\gamma}_{0}$ | Transition rate: Pro-B to Pre-BI | $0.02$ | day${}^{-1}$ |

${\gamma}_{1}$ | Transition rate: Pre-BI to Pre-BII | $0.168$ | day${}^{-1}$ |

${\gamma}_{2}$ | Transition rate: Pre-BII to Immature | $0.144$ | day${}^{-1}$ |

${\gamma}_{3}$ | Blood exit rate | $0.288$ | dayt${}^{-1}$ |

k | Signal intensity | ${10}^{-10}$ | cell${}^{-1}$ |

${\rho}_{L}$ | Leukemic cell proliferation rate | ${\rho}_{1}$ | day${}^{-1}$ |

${L}_{max}$ | Leukemic cell carrying capacity | ${10}^{12}$ | cell |

${\gamma}_{L}$ | Leukemic cell blood exit rate | $0.001\xb7{\gamma}_{3}$ | day${}^{-1}$ |

$\alpha $ | Activated CAR T killing capacity | $3\times {10}^{-9}$–$3\times {10}^{-11}$ | day${}^{-1}$· cell${}^{-1}$ |

${\rho}_{C}$ | Activated CAR T proliferation rate | $0.9$ | day${}^{-1}$ |

${\tau}_{A}$ | Activated CAR T mean lifetime | $6.5$ | day |

${\gamma}_{AM}$ | Activated to memory transition rate | $0.001$ | day${}^{-1}$ |

${\gamma}_{MA}$ | Memory to activated transition rate | $0.33$ | day${}^{-1}$ |

${\tau}_{M}$ | Memory CAR T mean lifetime | 300 | day |

h | CAR T activation threshold | ${10}^{8}$–${10}^{11}$ | cell |

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Martínez-Rubio, Á.; Chulián, S.; Blázquez Goñi, C.; Ramírez Orellana, M.; Pérez Martínez, A.; Navarro-Zapata, A.; Ferreras, C.; Pérez-García, V.M.; Rosa, M.
A Mathematical Description of the Bone Marrow Dynamics during CAR T-Cell Therapy in B-Cell Childhood Acute Lymphoblastic Leukemia. *Int. J. Mol. Sci.* **2021**, *22*, 6371.
https://doi.org/10.3390/ijms22126371

**AMA Style**

Martínez-Rubio Á, Chulián S, Blázquez Goñi C, Ramírez Orellana M, Pérez Martínez A, Navarro-Zapata A, Ferreras C, Pérez-García VM, Rosa M.
A Mathematical Description of the Bone Marrow Dynamics during CAR T-Cell Therapy in B-Cell Childhood Acute Lymphoblastic Leukemia. *International Journal of Molecular Sciences*. 2021; 22(12):6371.
https://doi.org/10.3390/ijms22126371

**Chicago/Turabian Style**

Martínez-Rubio, Álvaro, Salvador Chulián, Cristina Blázquez Goñi, Manuel Ramírez Orellana, Antonio Pérez Martínez, Alfonso Navarro-Zapata, Cristina Ferreras, Victor M. Pérez-García, and María Rosa.
2021. "A Mathematical Description of the Bone Marrow Dynamics during CAR T-Cell Therapy in B-Cell Childhood Acute Lymphoblastic Leukemia" *International Journal of Molecular Sciences* 22, no. 12: 6371.
https://doi.org/10.3390/ijms22126371