Mathematical deconvolution of CAR T-cell proliferation and exhaustion from real-time killing assay data

Chimeric antigen receptor (CAR) T-cell therapy has shown promise in the treatment of hematological cancers and is currently being investigated for solid tumors including high-grade glioma brain tumors. There is a desperate need to quantitatively study the factors that contribute to the efficacy of CAR T-cell therapy in solid tumors. In this work we use a mathematical model of predator-prey dynamics to explore the kinetics of CAR T-cell killing in glioma: the Chimeric Antigen Receptor t-cell treatment Response in GliOma (CARRGO) model. The model includes rates of cancer cell proliferation, CAR T-cell killing, CAR T-cell proliferation and exhaustion, and CAR T-cell persistence. We use patient-derived and engineered cancer cell lines with an in vitro real-time cell analyzer to parameterize the CARRGO model. We observe that CAR T-cell dose correlates inversely with the killing rate and correlates directly with the net rate of proliferation and exhaustion. This suggests that at a lower dose of CAR T-cells, individual T-cells kill more cancer cells but become more exhausted as compared to higher doses. Furthermore, the exhaustion rate was observed to increase significantly with tumor growth rate and was dependent on level of antigen expression. The CARRGO model highlights nonlinear dynamics involved in CAR T-cell therapy and provides novel insights into the kinetics of CAR T-cell killing. The model suggests that CAR T-cell treatment may be tailored to individual tumor characteristics including tumor growth rate and antigen level to maximize therapeutic benefit. Statement of Significance We utilize a mathematical model to deconvolute the nonlinear contributions of CAR T-cell proliferation and exhaustion to predict therapeutic efficacy and dependence on CAR T-cell dose and target antigen levels.


Introduction
Chimeric antigen receptor (CAR) T-cell therapy is a targeted immunotherapy, demonstrating remarkable antitumor efficacy, particularly in the treatment of hematologic cancers (1,2). CAR T-cell therapy is a specific type of immunotherapy where T-cells are genetically modified to recognize a tumor antigen thereby specifically redirecting T cell cytolytic activity. Inspired by the success of CAR T-cell therapy in liquid tumors, there has been a great interest in expanding the use of CAR T-cells for the treatment of solid tumors, such as glioblastoma (GBM), a highly aggressive form of primary brain cancer. Several clinical trials using CAR T-cells to treat GBM have been initiated all over the world (3)(4)(5)(6).
At this early stage of clinical development, CAR T-cells offer much promise in solid tumors. However, the diversity of current clinical trials employing varying types of CARs for different solid tumors, target patient populations, and preconditioning regimes, presents a significant challenge in identifying which aspects of a given CAR T-cell treatment protocol are most critical for its effectiveness. An additional critical challenge for CAR T-cell therapy is the potential for transient-progression, where the cancer appears to progress before eventually responding to the treatment (7,8).
In order to address these challenges in CAR T-cell therapy for solid tumors, we endeavored to study the kinetics of CAR T-cell killing with an in vitro system and a mathematical model. Mathematical models are useful to describe, quantify, and predict multifaceted behavior of complex systems, such as interactions between cells. A mathematical model is a formalized method to hypothesize systems dynamics, and yield solutions that represent the system's behavior under certain initial conditions. Mathematical models can be versatile and tested with clinical data which may be obtained in vivo from non-invasive imaging (9)(10)(11). When additional information about the system becomes available, the model can be refined and adjusted accordingly. Many mathematical models have been developed to understand tumor progression to guide refinement of cancer therapy regimens (12)(13)(14). As CAR T-cell therapy is a newly advanced treatment modality, relatively few studies have utilized computational modelling to understand and improve this cell-based therapy. Recently computational models have been developed to investigate cytokine release syndrome for toxicity management (15)(16)(17), effect of cytokine release syndrome on CAR T-cell proliferation (18), and mechanisms of CAR T-cell activation (19,20). However, it remains an open challenge how to use mathematical modeling to study and ultimately predict dynamics of CAR T-cell mediated cancer cell killing with respect to CAR T-cell dose, cancer cell proliferation, target antigen expression, and how these factors contribute to the overall effectiveness of CAR T-cell therapy.
Based upon our pre-clinical and clinical experience with our well-characterized IL13Rα2targeted CAR T-cell therapy for recurrent glioblastoma (21,22), we have identified several factors which contribute to the effectiveness of CAR T-cells: rates of proliferation, exhaustion, persistence, and target cell killing. To study these various facets of CAR Tcell killing kinetics, we modeled the dynamics between cancer cells and CAR T-cells as a predator prey system with the CARRGO mathematical model: Chimeric Antigen Receptor t-cell treatment Response in GliOma. We used a real-time cell analyzer experimental system to estimate parameters of the mathematical model and then apply the model to in vivo human data with the long-term aim of developing a model which could be used to predict and eventually to optimize response.

Quick guide to equations and assumptions
The CAR T-cell treatment Response to GliOma (CARRGO) mathematical model is a variation on the classic Lotka-Volterra (23,24) predator-prey equations: where represents the density of cancer cells, is the density of CAR T-cells, is the net growth rate of cancer cells, is the cancer cell carrying capacity, 2 is the killing rate of the CAR T-cells, 5 is the net rate of proliferation including exhaustion or death of CAR

Model assumptions
The CARRGO model treats cancer cell-CAR T-cell dynamics in this experimental condition as a closed predator-prey system. The model assumes 1) the populations are well mixed, 2) cancer cell growth is limited by space and nutrients (culture media) in the in vitro culture system and therefore grow logistically, 3) CAR T-cells kill cancer cells when   they interact via the law of mass action, 4) the CAR T-cell killing rate does not explicitly   assume a dependence on antigen density, 5) CAR T-cells may be stimulated to proliferate or to undergo loss of effector function-defined as exhaustion-upon contact with a cancer cell (29), and 6) the CAR T-cell death rate is independent of cancer cell density.
We chose the Logistic growth model for the cancer cell population because the fixed growth rate and carrying capacity parameters were the biological quantities of interest when comparing CAR T-cell killing kinetics across cell lines. Witzel et al compared sigmoidal growth laws including Logistic, Gompertz, and Richards showed that all these models can be fit equally well to this form of experimental data (30). Data supporting our model assumptions are given in supplemental material1 (Fig. S1,S2)

Dynamical system analysis of the CARRGO model
Closed form solutions cannot be obtained for the relatively simple CARRGO model. To study the possible dynamics with the CARRGO model, we perform classical dynamical system analysis. Detailed mathematical analysis of this model can be found in several textbooks in dynamical systems (24,31). In the interest of informing the reader, we briefly summarize the main points here. We begin by 1) scaling (non-dimensionalizing) the variables in the system and then 2) identify stationary points and classify their stability and finally 3) interpret the stationary points and system dynamics in terms of the initial numbers of cancer cells and CAR T-cells.
First, we scale the variables in the CARRGO model to obtain a model without physical units in order to study the intrinsic dynamics of the system. We scale time, the cancer cell and CAR T-cell populations as These variables are substituted into the CARRGO model (Eq.1,2) to obtain the scaled dimensionless system The steady-state solutions   is then a stable sink (Fig.1c). This case results in a transient increase in cancer cells corresponding to tumor progression followed by a decrease in tumor cells corresponding to treatment response in an oscillating manner. The transient and oscillatory nature of these dynamics may be interpreted as a "pseudo"-failure and "pseudo"-response to the therapy. We note that cancer progression and treatment occur on finite and sometimes small timescales and therefore oscillatory dynamics may not be observed in vivo due to insufficient time to observe these changes.

Cell lines
Low-passage primary brain tumor (PBT) lines were derived from GBM patients undergoing tumor resections at City of Hope as previously described (32,33).

Experimental design
Real-time monitoring of cancer cell growth was performed by using xCELLigence cell analyzer system (38). This system utilizes electrical impedance to non-invasively quantify adherent cell density with a dimensionless number referred to as cell-index (CI). The cellindex read-out from the machine is strongly positively correlated with the number of cells in the well (r 2 > 0.9) and can be used as a linear measure of cell number (30). We therefore

CARRGO model fitting to experimental data
The first 24 hours of the time-series describes the process of cell attachment to the bottom of the plate (Fig. 2). The spatial process of cell adhesion and spreading in the well can be modeled as a reaction-diffusion process, described in supplement material (Fig. S4).
Since we are interested in cell growth kinetics, we omitted the data from first 24 hours during the attachment process.

Model/data fitting to in vitro data
A high goodness of fit of the CARRGO model to the xCELLigence data was observed across all cell lines ( 5 = 0.93 ± 0.1, Fig. 3, Fig. S3). To investigate the sensitivity of our model fitting to sampling frequency, we down-sampled the data by taking time intervals of 2 hours, 5 and 10 hours. No significant variation was observed in the model parameters 2 , 5 and to the down-sampled data (repeated measure ANOVA p>0.1) (Fig. S5, S6).
We consistently observed very small values of the CAR T-cell death rate ( < 10 4h ).

Relating , with tumor growth rate and antigen expression
Tumor growth rate varies significantly (p<0.01) among different cell-lines and with antigen expression level (see supplement material2 Fig. S11a). To investigate the relationship between tumor growth rate and CAR T-cell killing 2 and exhaustion 5 , we evaluated cell lines with antigen levels greater than 80% and treated with BBζ CAR Tcells at an effector to target ratio of 1:5. No significant correlation was found between the cancer cell proliferation rate and killing rate ( 2 ) (Fig. S11b). However, the exhaustion rate 5 is significantly correlated with tumor growth rate (Fig. S11c) with Pearson correlation coefficient = −0.9, p<0.001. Similar results were observed for the cells treated with 28ζ IL13Ra2-CARs. Figure 5 shows the density of IL13R 2 level on cancer cell surface and its relation to CAR T-cell killing for cell line HT1080-H and PBT138-H.
We observed that 2 shows a decreasing trend from medium to high antigen level (Fig.5c) suggesting that high levels of antigen expression may not result in faster rates of CAR Tcell killing. The rate constant 5 increases from low to medium antigen expression and plateaus with high levels (Fig.5d). This suggests limited activation of CAR T-cell at lower antigen expression and exhaustion rate from medium to high antigen may not change significantly and may be the result of over-activation of the CAR T-cells. . CAR T-cell killing rate is observed to decrease with increasing ET ratio for all cell lines. This suggests CAR T-cells kill more cancer cells per unit time at a lower concentration as compared to higher ET ratio. In contrast, the CAR T-cell proliferation / exhaustion rate increases with ET ratio. This suggests that the CAR T-cells are stimulated to proliferate and are less exhausted with higher ET ratio as compared to lower. For reference, CAR T-cells are hypoactivated in mock ( 5 < 0). The CAR T-cell death rate, or persistence, is observed to be independent of target cell line and ET ratio. . CAR T-cell killing rate is observed to decrease with increasing ET ratio for all cell lines. This suggests CAR T-cells kill more cancer cells per unit time at a lower concentration as compared to higher ET ratio. In contrast, the CAR T-cell proliferation / exhaustion rate increases with ET ratio. This suggests that the CAR T-cells are stimulated to proliferate and are less exhausted with higher ET ratio as compared to lower. For reference, CAR Tcells are hypoactiveted in mock (κ 2 < 0). The CAR T-cell death rate, or persistence, is observed to be independent of target cell line and ET ratio.

CARRGO model applied to in vivo human data
To translate the in vitro dynamics to of the model to real patient data (22), we fit the CARRGO model to MRI-derived tumor volume data during CAR T-cell treatment (Fig. 6).
The CARRGO model is able to fit the tumor growth dynamics quite accurately for lesions T6, and T7 with the same set of parameters 2 = 6 × 10 4• (day -1 cell -1 ), 5 = 0.3 × 10 42' (day -1 cell -1 ), = 0.1 × 10 4' (day -1 ) and lesion T9 with 2 = 9 × 10 4" (day -1 cell -1 ), 5 = −2 × 10 42h (day -1 cell -1 ), = 5 × 10 4' (day -1 ). In the case of lesion T9, although the CARRGO model is consistent with the overall tumor dynamics, it does not fit the later time points following CAR T-cell treatment well. This is because lesion T9 received radiation treatment between day 200 to 300, which is not included in the CARRGO model. We note the negative correlation between the tumor growth rate ( = 0.06, 0.07 /day and = 0.2 /day for T6, T7 and T9 respectively) with the CAR T-cell exhaustion rate 5 in the patient data, which is consistent with that observed in the experimental data (Fig. S11c). We remark that the parameters 2 and 5 are on the order of (10 42h ), which appear to be very small, however, these parameters are scaled by the caring capacity in units of cells, which is of order (10 • ). Therefore, these parameter values are comparable with the in vitro data when scaled relative to the carrying capacity (Fig. 4)

Discussion
The   Cancer Cell We observed that the cancer cell growth showed no relation with CAR T-cell killing rate and an inverse relationship with 5 . This may explain variations in patient-specific responses even for the same CAR T-cell dose. For a fixed CAR T-dose, 5 is the principle determinant of treatment failure or success as shown in phase plane analysis (Fig.1), which is also observed in patient data (Fig.6). This result, driven by the CARRGO model This prediction may be consistent with the clinical phenomenon of pseudo-progression, in which the cancer is seen to progress during therapy before eventually responding (7,8).

Identifying characteristics of the patient and the CAR T-cells which may result in pseudo-
progression could have a profound effect on interpretation of these dynamics observed in the clinic.
Interestingly, we found 2 decreases and 5 plateaued from medium antigen level to higher level of antigen expression. One of the possible explanations of this behavior could be the antigen density is more heterogeneous in the higher antigen level cell population as compared to medium and low antigen levels (Fig.5a).     Figure 1) and rapid progression for HT1080-H (case 2, Figure 1). . CAR T-cell killing rate is observed to decrease with increasing ET ratio for all cell lines. This suggests CAR T-cells kill more cancer cells per unit time at a lower concentration as compared to higher ET ratio. In contrast, the CAR T-cell proliferation / exhaustion rate increases with ET ratio. This suggests that the CAR T-cells are stimulated to proliferate and are less exhausted with higher ET ratio as compared to lower. For reference, CAR Tcells are hypoactiveted in mock (κ 2 < 0). The CAR T-cell death rate, or persistence, is observed to be independent of target cell line and ET ratio. . κ 1 shows a decreasing trend from medium to high antigen levels (c) suggesting that high levels of antigen expression may not result in faster rates of CAR T-cell killing(d). κ 2 increases from low to medium antigen expression and plateaus with high antigen levels(d). This suggests limited activation of CAR T-cell at lower antigen expression and that exhaustion rates from medium to high antigen may not change significantly. Cancer Cell The right columns shows CARRGO model fits and dynamics based on the tumor volume data. The CARRGO model parameters are the same for lesions T7 and T6, which responded to CAR T-cell treatment. The model predicts that the non-responding lesion T9 had a smaller rate of CAR T-cell killing and increased rates of exhaustion and CAR T-cell death. Lesion T9 was also observed to have a higher cancer cell proliferation rate (ρ =0.2/day) as compared to T6 and T7 which had very similar rates (ρ =0.06, ρ =0.07, respectively).