A framework integrating multiscale in silico modeling and experimental data predicts CAR-NK cell cytotoxicity across target cell types

Significance

Predicting the activity of chimeric antigen receptor (CAR)-natural killer (NK) cells against their targets is challenging due to the diverse NK cell receptor repertoire and the integration of signals that control their activation. We present an integrated framework combining an in silico multiscale model with experimental data that predicts in-vitro and in-vivo CAR-NK cytotoxicity with high accuracy. We apply our model to explore mechanisms of CAR-NK cells which maximize lysis of cancer cells while minimizing lysis of healthy cells. We show that nonmodified expanded NK cells provide better differentiation between the lysis of tumor cells and healthy cells. This framework can be extended to predict CAR-NK cytotoxicity against multiple cancer cell types and to further understand CAR-NK biology.

Abstract

Natural killer (NK) cells may be engineered with chimeric antigen receptors (CARs) to recognize tumor-associated antigens which bolsters their antitumor activity. More so than CAR-T cells, CAR-NK cell responses result from an integration of signals from a wider range of innate activating cytotoxic receptors, inhibitory receptors, and adhesion receptors in addition to the engineered CAR, making computational modeling of CAR-NK cell cytotoxicity more difficult than CAR-T cells. Uncovering mechanisms and predicting tumor cell responses to CAR-NK cytotoxicity is essential for improving therapeutic efficacy. The complexity of these effector–target interactions and the donor-to-donor variations in NK cell receptor (NKR) repertoire preclude the use of predictive models based on a single receptor, requiring function to be determined experimentally for each donor, CAR, and target combination. Computational modeling generates frameworks that allow the relationships of these factors to biologic outcomes to be explored without resource-consuming experiments. Here, we developed a computational mechanistic multiscale model which considers heterogenous expression of CARs, NKRs, adhesion receptors, and their cognate ligands, signal transduction, and NK cell-target cell population kinetics. The model is trained with quantitative flow cytometry and in-vitro cytotoxicity data and accurately predicts the short-term, long-term, and in-vivo cytotoxicity of CAR-NK cells. Furthermore, using Pareto optimization we explored the effect of CAR proportion and NK cell signaling on the differential cytotoxicity of CD33CAR-NK cells to cancer and healthy cells. This model can be extended to predict CAR-NK cytotoxicity across many antigens and tumor targets and serves as a tool to mechanistically explore CAR-NK signaling and biology.

Natural killer (NK) cells are lymphocytes of the innate immune system and interact with target cells using a wide array of activating, inhibitory, and adhesion receptors expressed on the cell surface (1). Integration of opposing signals initiated by diverse receptor–ligand interactions determine cytotoxic or tolerogenic response of NK cells toward their targets (2). Tumor or virally infected cells upregulate ligands cognate to activating NK cell receptors (NKRs) or downregulate ligands cognate to inhibitory NKRs which usually lead to lysis of the tumor cells by interacting NK cells (3). NK cells may be engineered to express chimeric antigen receptors (CARs) that expand their repertoire of recognized antigens (4–6). Compared to CAR-T cell–based therapies, CAR-NK cell therapy potentially minimizes both antigen escape and off-tumor toxicities by avoiding reliance on the expression of a single targeted antigen and utilizing healthy tissue-sparing inhibitory signaling (7).

Understanding and predicting tumor cell responses to CAR-NK cell cytotoxicity is essential for designing approaches to improve therapeutic efficacy. The overall antitumor activity of each manufactured CAR-NK cell product is variable and dependent on many factors including the efficiency of CAR transduction, tumor expression of the targeted antigen, and the innate cytolytic capacity of the nonmodified NK cell starting material. Due to this complexity, the therapeutic efficacy of each manufactured CAR-NK cell product should be assessed individually through time and resource consuming functional assays. Computational modeling offers an avenue to explore the biological parameters which impact CAR-NK cell cytotoxicity and predict real-world outcomes.

Several previous computational models have explored the CAR design space, primarily in the context of CAR-T cells. These have revealed important parameters such as antigen-receptor affinity tuning, dosing protocols, and cellular transition states that predict biological outcomes and, ultimately, patient responses to CAR immunotherapies (8–13). However, computational modeling of CAR-NK cells presents a unique challenge compared to modeling CAR-T cells due to the diverse array of germline-encoded activating and inhibitory receptors that regulate their innate cellular cytotoxicity (2). Since CAR-NK cell effector functions are determined by the integration of signals initiated by both the CAR and the diverse NKRs and adhesion receptors, the cytotoxic response of CAR-NK cells may not follow an intuitive dependence on the availability of the CAR antigens or strength of signaling domains. Accounting for this diversity of receptors is computationally challenging as the decision-making process of NK cell cytotoxicity is still incompletely understood, despite insights from mathematical modeling (14, 15). Indeed, a previous agent-based model of CAR-NK cell cytotoxicity does not take into account the expression of coreceptors at the single-cell level that are responsible for NK cell innate cytotoxicity (16).

Here, we develop a framework combining in-vitro experiments with a multiscale mechanistic in silico model which integrates single-cell level ligand and receptor expressions, single-cell signal transduction, and population kinetics of CAR-NK cells and target cells to quantitatively explore the roles of CAR, activating and inhibitory NKRs, and adhesion receptors in determining cytotoxicity against tumor and healthy cells. Our multiscale in silico model is trained with single-cell abundances of tumor ligands and NKRs measured by quantitative flow cytometry along with in-vitro cytotoxicity data. Our model predicts short-term, long-term, and in-vivo CAR-NK cell cytotoxicity against novel tumor cell lines and healthy cells. Furthermore, we have employed a Pareto optimization approach to determine the optimal CAR expression and NK cell signaling parameters which enables CAR-NK cells to discriminate between tumor cells and healthy cells. This model serves as a proof-of-concept tool for designing, evaluating, and predicting CAR-NK cell cytotoxicity and can be adapted and trained for a wide range of predictive clinical and translational applications.

Results

CAR-NK Cell Cytotoxicity Displays Nonlinearity with Antigen Expression.

It has been shown experimentally that CAR-T cell activation increases monotonically with increasing cognate antigen expression (Fig. 1A) (17, 18). The increased expression of antigens on target cells give rise to increased CAR–antigen complexes, increasing the strength of the CAR signaling, and consequently increasing CAR-T cell effector functions such as cytotoxicity. Prior mathematical models of CAR-T cell cytotoxicity based on this reasoning are consistent with experimental outcomes (9, 10, 19, 20). Similarly, analysis of previously published studies show that CAR-NK cell responses correlate with CAR antigen availability and can be monotonic within the same tumor cell line when all other ligand levels are kept constant (Fig. 1B) (21). However, to determine whether CAR antigen density alone would account for the dynamic variability in CD33CAR-NK cell cytotoxicity across different target cells, we performed a flow cytometry experiment and correlated CD33 expression with CD33CAR-NK cell cytotoxicity. Contrasting with the monotonic relationship in T cells (Fig. 1C) (22), we observed a nonmonotonic relationship between antigen density and cytotoxicity across multiple donors and CAR designs, indicating that modeling NK cell activation through CAR antigen expression alone is not sufficient to account for the observed cytotoxicity (Fig. 1D). We observed similar nonmonotonicity of CAR-NK cell cytotoxicity with both adhesion (ICAM-1) and inhibitory (HLA-A/B/C) ligand expression on target cells (SI Appendix, Fig. S1). Thus, we reasoned that a model which considers multiple ligand and receptor interactions would be necessary to predict CAR-NK cell cytotoxicity.

Fig. 1.

CD33 CAR-NK cell cytotoxicity displays nonmonotonicity with CD33 expression across cell lines. (A) Representative schematic of CAR-T activation as a function of CAR antigen expression. Data compiled from RA Hernandez-Lopez et al. (17). (B) Representative schematic of CAR-NK cell activation as a function of CAR antigen expression within the same cell line, Raji. Data compiled from Rahnama et al. (21). (C) Linear relationship between CD33 expression and CAR-T cell cytotoxicity across OCI-AML, MOML-14, and U937 cell lines. Data compiled from Freeman et al. (22). (D) Nonlinear relationship between CD33 expression and CD33 CAR-NK cell cytotoxicity against K562, Kasumi-1, and HL-60 cell lines. CD33 expression was quantified by flow cytometry. N = 3 donors. Gen2 and Gen4v2 represent two CARs that differ in signaling domains. In panels (A–C), CAR1, CAR2, CAR3 labels represent cells expressing different CAR constructs. CAR constructs are unique panel to panel.

Model Predicts Killing of Tumor Cell Lines by CAR-NK Cells in a Donor Specific Manner.

We developed a multiscale mechanistic in silico model which integrates signals arising from interaction of CARs, inhibitory, activating, and adhesion NK cell receptors (NKRs) with their cognate ligands expressed on target cells and predicts the cytotoxicity of CD33CAR-NK cells against target cells. In this model, we include key processes involving interaction of NK cells with target cells (e.g., tumor or healthy cells) through processes that occur in three different scales: molecular, subcellular, and cell population (Fig. 2A).

Fig. 2.

Multiscale in silico model design and prediction of CAR-NK cell tumor lysis. (A) Representation of CAR-NK activation and cytotoxicity from molecular to subcellular to cell population scales: t1 receptor–ligand interaction, t2 membrane proximal signaling events, and t3 tumor cell lysis. (B) Representative schematics of modeled biochemical reactions at the molecular and subcellular scale and tumor cell lysis kinetics at the cell population scale. Stars represent phosphorylation and activation states. See “Solutions of the ODE Model” in the Materials and Methods section for further details. (C) (Top) Expressions of CD33, ICAM-1, and HLA-A/B/C on Kasumi-1 and HL-60. (Bottom) CD33CAR, LFA-1, and iKIR expressions on CD33CAR-NK and WT NK cells. PE/APC denote fluorophores used in flow cytometry staining. (D) CAR designs evaluated in (C–E). (E) (Top and Middle) Training data and model fit with CD33CAR-NK cell and WT NK cell cytotoxicity against Kasumi-1 and WT NK cell cytotoxicity against HL-60. (Bottom) Evaluation of model’s prediction of CD33CAR-NK cell cytotoxicity against HL-60 using biological data. (F) (Top) Expressions of Her2, ICAM-1, and HLA-A/B/C on DIPG36 and SKOV3. (Bottom) Her2CAR, LFA-1, and iKIR expressions on Her2CAR-NK and WT NK cells. (G) CAR design evaluated in (F–H). (H) (Top and Middle) Training data and model fit with Her2CAR-NK cell and WT NK cell cytotoxicity against DIPG36 and WT NK cell cytotoxicity against SKOV3. (Bottom) Evaluation of model’s prediction of Her2CAR-NK cell cytotoxicity against SKOV3 using biological data. In (E and H), R² coefficient of determination was calculated for goodness of fit with the data. 4 h in-vitro cytotoxicity measured by calcein release. Error bars in cytotoxicity data represent SD for n = 3 technical replicates.

Molecular scale.

In the model, CD33CAR, adhesion receptor LFA-1, and inhibitory KIRs (iKIRs) bind their cognate ligands CD33, ICAM-1, and HLA-A/B/C, respectively. These interactions are described by second-order binding-unbinding reactions with their respective binding and unbinding rates. The effects of other activating NKRs are included implicitly (Fig. 2B and Materials and Methods). The CD33CAR constructs (Gen2 and Gen4v2) we considered here contain CD3ζ and costimulatory domains such as CD28 and 2B4 (Fig. 2D). We account for the cell–cell variations of receptors and their cognate ligands in the model by considering the distribution of their expressions instead of their average value (SI Appendix, Table S1 and Fig. S2). To reduce computational complexity while maintaining the characteristics of their distributions, we binned receptor and ligand expression values into five discrete bins (Materials and Methods).

Subcellular scale.

The formation of the ligand–receptor complexes initiates different downstream reactions within the NK cell (23). Adaptor proteins such as CD3ζ in the intracellular domain of the CARs are phosphorylated by Src family kinases leading to recruitment of Syk family kinases (24). The phosphorylated Syk family kinases induce further downstream signaling reactions leading to phosphorylation of the guanine exchange factor Vav1. Phosphorylated Vav1 leads to the release of lytic granules onto the target cells (25, 26). Upon binding with ligand ICAM-1, the adhesion receptor LFA-1, promotes phosphorylation of Vav1 (27). Thus, both CAR and the adhesion receptor contribute toward Vav1 phosphorylation. The inhibitory KIRs are associated with immunoreceptor tyrosine-based inhibition motifs (ITIMs) in the transmembrane domain which are phosphorylated by Src family kinases upon binding with class I MHC molecules. Phosphorylated ITIMs recruit the phosphatase SHP-1 which deactivates Vav1 via dephosphorylation (28). Other cytosolic serine, threonine, and tyrosine phosphatases act to regulate NK cell homeostasis and activity (29–31). We used a series of first-order reactions to represent the above chemical modifications as different “states” of the receptor ligand complex (Fig. 2B and Materials and Methods). The CAR, adhesion, and the inhibitory KIR signaling lead to formation of the end complexes, C_N1, C_N3, and C_N4, respectively. We did not model the activating NKR signaling explicitly as there are multiple activating NKRs in the donor NK cells which could be stimulated by uncharacterized ligands on target cells. Therefore, we introduced the end complex corresponding to all other activating receptor signaling as the parameter C_N2. In the model, the end complexes C_N1, C_N2, and C_N3 phosphorylate Vav1 as enzymes. The end complexes C_N4, corresponding to inhibitory KIR signaling, and $E_2^{\prime }$, corresponding to unspecified phosphatases not bound to inhibitory KIRs, dephosphorylate phosphorylated Vav1 enzymatically. The details of the model are provided in the Materials and Methods.

Cell population scale.

We consider a population of target and NK cells interacting where each interaction between a NK cell and a target cell leads to production of phosphorylated Vav1 and subsequent lysis of the target cell with a rate proportional to the abundance of phosphorylated Vav1 generated during the interaction. We also assume each NK cell can interact with any target cell with equal probability. The tumor target cells can proliferate, however we assume the NK cells do not proliferate or die within the time scale of interest (~4 to ~48 h). The population kinetics of target and NK cells are given by a set of coupled ordinary differential equations (ODEs) (Fig. 2B and Materials and Methods). The in silico model contains 10 parameters ${\alpha }_1,C_{N_2},{\alpha }_3,{\alpha }_4,E_2^{\prime },V_c,K_1,K_2,{\lambda }_c,r_c$ that are estimated during model training using single cell abundances of the receptors and ligands and percentage of lysed target cells in in-vitro cytotoxicity assays. The list of the parameters along with their interpretations are shown in Table 1.

Table 1.

List of processes and parameter values used in Fig. 4

	Parameter	Description	Units	Range used	Kasumi-1 donor H estimate (CI)
1	${\alpha }_1$	forward probability of active complex of CAR33 and CD33	Unitless	0 to 1	0.6 (0.518-0.596)
2	$C_{N_2}$	concentration of activating receptor–ligand active complex	Molecules/cell	300 to 7,000	1637.0 (693.8-1028.2)
3	${\alpha }_3$	forward probability of active complex of LFA-1 and ICAM-1	Unitless	0 to 1	0.6 (0.61-0.70)
4	${\alpha }_4$	forward probability of active complex of iKIR and HLA-ABC	Unitless	0 to 1	0.6 (0.777-0.913)
5	$E_2^{\prime }$	Concentration of phosphatase	Molecules/cell	50 to 2,000	1190.0 (1300-1494.63)
5	$V_c$	Relative rate of pVav1 formation	Unitless	${10}^{-3} \mathrm{to} {10}^{-1}$	0.075 (0.094-0.118)
6	$K_1$	Scaled Michaelis constant for transition of Vav1 to pVav1	Unitless	$0.01 \mathrm{to} 1$	$0.117$ (0.162-0.210)
7	$K_2$	Scaled Michaelis constant for transition of pVav1 to Vav1	Unitless	$0.01 \mathrm{to} 1$	$0.01$ (0.01,0.0101)
8	${\lambda }_c$	Lysis constant	${\mathrm{h}}^{-1}$	${10}^{-4} \mathrm{to} {10}^{-7}$	5.911e-5 (5.97e-5,7.40e-5)
9	$r_c$	Proliferation rate of tumor cells	${\mathrm{h}}^{-1}$	$0 \mathrm{to} {10}^{-3}$	$0$ *

^*denotes fixed values.

We trained the above multiscale in silico model and then evaluated the model’s prediction of short term (4 h) cytotoxicity of CD33CAR-NK cells to novel cell lines. We performed quantitative flow cytometry to approximate the molecular abundances of CD33, ICAM-1, and HLA-A/B/C on acute myeloid leukemia (AML) cell lines Kasumi-1 and HL-60 as well as CD33CAR, LFA-1, and iKIR on NK cells (Fig. 2C and SI Appendix, Table S1). We predicted the cytotoxicity of two CD33CARs, Gen2 and Gen4v2, which differ in their signaling domains and expression (6). We split cytotoxicity data into train and test datasets and trained the model by first fitting the lysis of WT NK and CAR-NK cells against Kasumi-1 to estimate nine parameters ${\alpha }_1,C_{N_2},{\alpha }_3,{\alpha }_4,E_2^{\prime },V_c,K_1,K_2$, and ${\lambda }_c$. As the doubling time for AML cell lines is between 1 and 2 d, $r_c$, the parameter reflecting the proliferation rate of target cells, was set to 0 for models trained on short-term (4 h) cytotoxicity assays (32). Using the common parameter values identified, we then re-estimated $C_{N_2}$, and ${\lambda }_c$, for HL-60 to account for the differences in the activating ligand repertoire and lysis sensitivity of HL-60. Parameter estimation of ${\alpha }_1,C_{N_2},{\alpha }_3,{\alpha }_4,E_2^{\prime },V_c,K_1,K_2$, and ${\lambda }_c$ was performed through minimizing the least square residual cost function (Materials and Methods). The estimated values of the parameters are listed in SI Appendix, Table S2. We then set up an in silico cytotoxicity assay with a mixture of 10,000 target cells, varying the number of effector cells to recapitulate the E:T ratios used experimentally. The NK cells and target cells interacted in the model using the previously identified best fit parameters (SI Appendix, Table S2). We then predicted the percentage of HL-60 tumor cells lysed by Gen2 and Gen4v2 CD33CAR-NK cell after 4 h of coculture. We calculated goodness of fit of our model prediction using the $R^2$ coefficient of determination and observed that the predictions agreed with the data for both Gen2 ($R^2$: 0.92, 0.63) and Gen4v2 ($R^2$: 0.84, 0.40) CD33CAR designs in two NK cell donors (Fig. 2E). These results show that our multiscale in silico model accurately predicts short term (4 h) CD33CAR-NK cell lysis of novel tumor cells.

To demonstrate that this model is generalizable to predict CAR-NK cell cytotoxicity against diverse cancer types and tumor antigens, we also applied the model to predict the cytotoxicity of CAR-NK cells targeting a solid tumor antigen, Her2 (Fig. 2G). We performed quantitative flow cytometry to assess expression of Her2, ICAM-1, and HLA-A/B/C on tumor cells and Her2CAR, LFA-1, and iKIR on NK cells (Fig. 2F and SI Appendix, Table S1). We trained the model on in-vitro cytotoxicity data against DIPG36, a diffuse midline glioma cell line, and predicted the lysis of CAR-NK cells against ovarian carcinoma cell line SKOV3. We observed high agreement ($R^2$: 0.92, 0.92) between the model prediction and in-vitro cytotoxicity data (Fig. 2H and SI Appendix, Table S3).

Model Predicts Long Term Cytotoxicity of CD33CAR-NK Cells.

We next evaluated the ability of our model to predict the long-term (~48 to 72 h) cytotoxicity of CAR-NK cells. Long-term cytotoxicity assays spanning multiple days better capture CAR-NK cell serial killing (33). NK cell cytotoxicity mechanisms captured in long-term cytotoxicity assays include perforin/granzyme mediated lysis, secretion of soluble factors such as IFN-γ, and death receptor interactions such as Fas-FasL (34). Although perforin-granzyme mediated cytotoxicity is the dominant signal encoded by the CAR and is the major mechanism of NK cell cytotoxicity, the secondary effects of cytokine secretion and antigen-independent death receptor signaling also play an important role in tumor control (35, 36). It was unknown how well the model would capture these additional functions of NK cells in a long-term cytotoxicity assay. To adapt the model to predict long-term CAR-NK cytotoxicity, we included a nonzero value of parameter $r_c$ to represent the proliferation rate of tumor cells. To assess the long-term killing ability of CAR-NK cells, we performed cytotoxicity assays with CD33CAR-NK cells and MV4-11 tumor cells and measured target cell death after 48 and 72 h of coculture using propidium iodide flow cytometry. We evaluated the cytotoxicity of two CD33CAR constructs, V5 and V6, which differ in their orientation of the light and heavy chains of the scFv, which could impact ligand affinity, but are otherwise identical in all other domains (Fig. 3B). We split the cytotoxicity data into test and train datasets and trained the model using CD33CAR-NK cell and WT NK cell cytotoxicity of MV4-11 target cells at 48 h. Molecular abundances of the modeled receptor and ligands were approximated using quantitative flow cytometry (Fig. 3A and SI Appendix, Table S1). After fitting the model to the data and parameter estimation (SI Appendix, Table S4), we set up an in silico cytotoxicity assay and predicted the percent lysis of MV4-11 cells by CD33CAR-NK cells and WT NK cells at 72 h. We observed good agreement between the in silico model prediction and the data (V5 $R^2$: 0.83, 0.88, 0.95, and V6 $R^2$: 0.97, 0.99, 0.98, WT $R^2$: 0.64, 0.52, 0.88) from in-vitro experiments of three independent donors, showing that our multiscale model robustly captures NK serial killing and multiple mechanisms of tumor cytotoxicity (Fig. 3C).

Fig. 3.

Model predicts long-term in vitro and in-vivo CD33CAR-NK cell cytotoxicity. (A) Distributions of CD33, ICAM-1, and HLA-A/B/C expression on MV4-11 tumor cells and representative distributions of CD33CAR expression by flow cytometry. (B) CAR designs evaluated in (A–C). CAR designs differ in scFv heavy and light chain orientation. (C) (Top) Training data and model fit with CD33CAR-NK and WT NK cell cytotoxicity against MV4-11 at 48 h. (Bottom) Evaluation of model’s prediction of CD33CAR-NK cell and WT NK cell cytotoxicity against MV4-11 at 72 h using biological data. 48 and 72 h in vitro cytotoxicity was measured by propidium iodide flow cytometry. Error bars in cytotoxicity data represent SD for n = 3 technical replicates. (D) In-vivo tumor burden of MV4-11 bearing NSG mice injected with 1 × 10⁶ MV4-11 cells on day 0 and 1 × 10⁷ WT NK or CD33CAR-NK cells on day 3, 7, 10, 14. Tumor burden measured by bioluminescence imaging. (E) (Left) Model fit and estimation of tumor cell number in tumor only mice and mice treated with WT NK cells. (Right) Evaluation of model’s in vivo prediction of CD33CAR-NK cell tumor control in MV4-11 bearing mice using biological data. Error bars represent SD of mouse bioluminescence. In (C and E), R² coefficient of determination calculated for goodness of fit with the data.

Model Predicts CD33CAR-NK Cell Control of Tumor In Vivo.

We next assessed whether in-vitro predictions accurately related CD33CAR-NK cell cytotoxicity to in vivo data. For the in vivo studies, we generated CD38^KO CD33CAR-NK cells due to the metabolic benefits of CD38 deletion (37). We previously observed that CD38^KO did not affect short-term or long-term in vitro cytotoxicity suggesting that our in vitro cytotoxicity modeling would not be affected by CD38^KO (37). We performed an in vivo study testing the tumor control of CD38^KO CD33CAR-NK cells in a human xenograft mouse model using MV4-11 leukemia cells. NK cells were administered in four doses and tumor burden was measured by bioluminescence imaging every 2 to 3 d (Fig. 3D). To determine the model’s ability to predict in vivo tumor cytotoxicity, we predicted tumor growth kinetics for the first 22 d. We first estimated the in vivo tumor growth rate $r_c$ from bioluminescence in the tumor only group (Fig. 3E). To account for NK cell decay and repeated dosing, we incorporated a decay parameter $r_{\mathit{nk}}$ and performed stepwise integration to account for the time delay between NK cell dosing and bioluminescence imaging (Materials and Methods). We then estimated $C_{N_2},{\alpha }_3,{\alpha }_4,E_2^{\prime },V_c,K_1,K_2$, and ${\lambda }_c$ by fitting the bioluminescence of mice treated with WT NK cells on day 4, 8, 11, 15, 18, and 22 to model outputs on those respective days (SI Appendix, Table S5). We then applied the ${\alpha }_1$ parameter estimated from in-vitro cytotoxicity assays corresponding to the signaling of the same CAR design and predicted the tumor growth of MV4-11 tumor bearing mice treated with CD38^KO CD33CAR-NK cells. We observed an excellent agreement ($R^2$: 0.84) between model predictions and in vivo bioluminescence measurements showing that this in-vitro model framework can be extended to predict in vivo tumor lysis (Fig. 3E).

CD33CAR Expression Results in Targeting of Healthy Monocytes.

On-target off-tumor toxicities arise when CAR-expressing immune cells target nonmalignant tissues that express the targeted antigen (38). In many cancers, including AML, there is a lack of ideal antigens that are exclusively expressed on cancer cells and absent from healthy cells. For example, CAR therapies designed to target myeloid antigens such as CD33, CD123, and CD38 inadvertently result in cytotoxicity against healthy myeloid cells which express low to moderate amounts of the cognate antigen (39–42). Mistargeting of CD33, specifically, results in lysis of healthy cells such as neutrophils and monocytes which are essential for controlling infection (43, 44). These on-target off-tumor toxicities have posed significant clinical risk in CAR-T therapies (45, 46). The toxicities of CAR-NK cell therapies, including those associated with on-target off-tumor activity, have been less severe than those observed in CAR-T cell therapies, possibly due to the stimulation of inhibitory KIRs by MHC molecules (47, 48). However, these studies have been done in systems where the targeted antigen exhibits significantly lower expression on healthy tissues than the targeted tumor cell (48, 49). Similar on- versus off-tumor studies for targeted antigens which have equivalent expression on healthy and tumor cells have not been performed for CAR-NK cells. To evaluate the on-tumor vs. off-tumor activity of CD33CAR-NK cells, we isolated primary monocytes from healthy donor PBMCs as a representative cell type of the myeloid lineage. We quantified ligand expression on monocytes using quantitative flow cytometry and found comparable levels of CD33 expression on monocytes and AML cell lines (Fig. 4A and SI Appendix, Table S1). One day after isolation, we performed calcein cytotoxicity assays using CD38^KO CD33CAR-NK cells or WT NK cells without prior HLA matching. We observed that WT NK cells had low (4 to 25% lysis) cytotoxic activity against monocytes at all effector-to-target ratios. CD38^KO CD33CAR-NK cells showed improved cytotoxicity against Kasumi-1 tumor cells but also markedly increased cytotoxicity against healthy monocytes despite high levels of MHC class I expression (Fig. 4B).

Fig. 4.

CD33CAR expression results in targeting of healthy monocytes. (A) Expression of CD33 on monocytes, lymphocytes, and AML tumor cell lines by quantitative flow cytometry. (B) 4 h calcein cytotoxicity assay of CD33CAR-NK and WT NK cells against Kasumi-1 and primary monocytes. (C) (Left) Training data and model fit of CD33CAR-NK cell and WT NK cell cytotoxicity against Kasumi-1 and WT NK cell cytotoxicity against monocytes. (Right) Evaluation of model’s prediction of CD33CAR-NK cell cytotoxicity against monocytes using biological data. R² coefficient of determination calculated for goodness of fit with the data. In (B and C), error bars in cytotoxicity data represent SD for n = 3 technical replicates.

Given these results, we first asked whether the model could predict the level of monocyte lysis observed in our experimental data. We employed the aforementioned workflow and trained the model using CD38^KO CD33CAR-NK cell and WT NK cell cytotoxicity data against Kasumi-1 targets along with WT NK cell cytotoxicity against monocytes from an unmatched donor. We fit the model to the data by estimating the parameters described (Table 1) and predicted the cytotoxicity of CD38^KO CD33CAR-NK cells to monocytes. We observed an excellent agreement between our model prediction and the experimental data (R²: 0.98), indicating that the modeled parameters and processes could be extended to describe NK cell cytolytic activity to nonmalignant primary cells as well as tumor cells (Fig. 4C).

Predicting Optimal CAR Expression and Signaling to Minimize On-Target Off-Tumor Cytotoxicity.

After validating the model’s ability to describe nontumor cytotoxicity, we aimed to manipulate parameters within the model framework to predict the optimal CAR expression and signaling kinetics to maximize cancer cell killing while sparing lysis of healthy tissues. In our experimental cytotoxicity data, we observed a tradeoff between cancer cell lysis and monocyte lysis (Fig. 5A) which suggested the system can be suitable for multiobjective parameter optimization. In multiobjective optimization, there are multiple desired outcomes which must be simultaneously considered, for example, maximizing lysis of cancer cells while minimizing lysis of healthy cells. Multiobjective optimization will produce a set of solutions which lie on a front known as the Pareto optimal front on which all points are equivalently optimal. Our data suggested that WT NK cells best minimized on-target off-tumor cytotoxicity while maintaining tumor-specific lysis, but it was unclear whether WT NK cells represented the Pareto optimal front and which signaling parameters produced this front (Fig. 5A). We employed an in silico two-objective Pareto optimization scheme to explore the signaling parameters of CAR-NK and WT NK cells and calculate the Pareto optimal front, beyond which either lysis of cancer cells cannot be maximized and lysis of healthy cells cannot be minimized without sacrificing the other. In our two-objective optimization problem, we simultaneously minimized lysis of monocytes while maximizing lysis of Kasumi-1 tumor cells using in silico cytotoxicity assays (50, 51). Effector NK cells were introduced at t = 0 and allowed to interact with target cells for 4 h while varying parameters to evaluate the Pareto fronts.

Fig. 5.

Pareto optimization reveals optimal design parameters of WT and CD33CAR-NK cells against leukemic and healthy cells. (A) On-tumor vs. off-tumor cytotoxicity of CD33CAR-NK and WT NK cells. The line denotes % lysis of monocytes = % lysis of Kasumi-1. 0%, 50%, 75%, and 100% CAR represents WT NK cells, 1:1 CD33CAR-NK:WT NK mixture, 3:1 CD33CAR-NK:WT NK mixture, and CD33CAR-NK cells respectively. (B) (Top) Model fit of CD33CAR-NK and WT NK cytotoxicity against Kasumi-1 and WT NK cytotoxicity against monocytes. (Bottom) Prediction of 3:1 CD33CAR-NK cell:WT NK cell cytotoxicity and 1:1 CD33CAR-NK cell:WT NK cell against Kasumi-1 and monocytes. Model mixtures created using estimated Ω values. R² coefficient of determination calculated for goodness of fit with the data. Error bars in cytotoxicity data represent SD for n = 3 technical replicates. (C) Pareto fronts for CD33CAR-NK cell % lysis of monocytes and % lysis of Kasumi-1. The Pareto fronts are calculated for different E:T ratios, varying Ω as the controlled parameter. All other parameters are fixed to estimated values. (D) Pareto fronts for CD33CAR-NK cell % lysis of monocytes and % lysis of Kasumi-1. The Pareto fronts are calculated for different E:T ratios, varying ${\alpha }_1$ as the controlled parameter. All other parameters are fixed to estimated values. (E) Pareto fronts for CD33CAR-NK cell % lysis of monocytes and % lysis of Kasumi-1. The Pareto fronts are calculated for a fixed E:T ratio (10:1), varying ${\alpha }_1$, ${\alpha }_4$, and Ω as the controlled parameters. ${\alpha }_1$, ${\alpha }_4$, and Ω represent the signaling propensity of CD33CAR, the signaling propensity of iKIRs, and the CAR transduction efficiency respectively.

First, we aimed to analyze the effect of CAR expression in minimizing off-tumor toxicity while maintaining on-tumor killing. The CD33CAR expression on CAR-NK cells showed a bimodal distribution which can be modeled as a mixture of two log-normal distributions with modes centered around low (0.42 MFI $\equiv$ 467 molecules) and high (16.4 MFI $\equiv$ 17,188 molecules) MFI values (SI Appendix, Fig. S3 A–C). We created a parameter Ω which determines the percentage of CD33CAR positive NK cells and is assumed to be biologically equivalent to the transduction efficiency of CAR generation via CRISPR-AAV site-directed insertion. Given the presence of background cellular autofluorescence and nonspecific antibody binding, we reasoned that the log-normal distribution centered around the lower MFI in the CAR-NK cells is equivalent to nontransduced WT NK cells. Similarly, we reasoned that the lognormal distribution centered around the higher MFI represents the CAR expression in transduced NK cells. We modeled the cell surface expression of CD33CAR on CAR-NK cells and WT NK cells as a mixture of two lognormal distributions and estimated the distribution parameters including Ω using maximum likelihood estimation (SI Appendix, Fig. S3 A–C, See Materials and Methods).

Next, we evaluated the ability of our in silico model to describe and predict target cell lysis generated by the mixture of WT and CAR-NK cells. We considered the following ratios of WT and CAR-NK mixtures (CD33CAR-NK alone, 3:1 CD33CAR-NK cell:WT NK cell-, 1:1 CD33CAR-NK cell:WT NK cell, and WT NK cell alone) and measured cytotoxic responses of those mixtures against healthy and tumor target cells. We determined the value of Ω in the experimental mixtures through maximum likelihood estimation (SI Appendix, Fig. S3 A–C). We performed in silico cytotoxicity assays using 10,000 target cells while varying $\mathrm{\Omega }$. Our model predictions varying Ω showed strong agreement with the biological data using both Kasumi-1 (3:1 CD33CAR-NK:WT NK, $\mathrm{\Omega }=0.54, R^2=0.72$; 1:1 CD33CAR-NK:WT NK, $\mathrm{\Omega }=0.39, R^2:0.96$) and monocytes (CD33CAR-NK, $\mathrm{\Omega }=0.80, R^2=0.99$; 3:1 CD33CAR-NK:WT NK, $\mathrm{\Omega }=0.54, R^2:0.95$; 1:1 CD33CAR-NK:WT NK, $\mathrm{\Omega }=0.39,$ $R^2=0.97$) as the target cell for all NK cell conditions (Fig. 5B). Similarly, the model was also able to predict cytotoxicity generated by a mixture of cells when the mixture of effector cells was generated by sampling rather than $\mathrm{\Omega }$ estimation (SI Appendix, Fig. S3D). Therefore, the in silico model can successfully describe target cell lysis by CD33CAR-NK cells with different CAR expression efficiency.

We then computationally investigated the effect of manipulating transduction efficiency, or Ω, on the trade-off between healthy cell and tumor cell lysis through performing in silico cytotoxicity assays while varying Ω. Concordant with our experimental data, we found that increasing Ω increased lysis of Kasumi-1 cells but also disproportionally increased lysis of healthy monocytes (Fig. 5C).

Next, we performed Pareto optimization while varying signaling parameters ${\alpha }_1$ and ${\alpha }_4$ to investigate the signaling mechanisms which produce the Pareto optimal front. ${\alpha }_1$ represents the signaling propensity of CD33CAR and we found that increasing ${\alpha }_1$ increased lysis of Kasumi-1 cells but disproportionally affected lysis of healthy monocytes (Fig. 5D). ${\alpha }_4$ represents the signaling propensity of inhibitory receptors. Simultaneously varying ${\alpha }_1, {\alpha }_4, \mathrm{and} \mathrm{\Omega }$ revealed that NK cells which are representative of WT NK cells (i.e., cells with either low CAR transduction or low CAR signaling) best optimized the objective, confirming that WT NK cells represent the Pareto optimal front (Fig. 5E).

Discussion

We developed a framework integrating quantitative flow cytometry measurements of receptor and ligand expressions and in vitro cytotoxicity assays with multiscale mechanistic computational modeling that predicts CAR-NK cell cytotoxicity across different tumor cell lines. We utilized a coarse-grained modeling approach which implicitly includes subcellular signaling processes into the model parameters, allowing us to focus our quantitative analysis on select receptor and ligand abundances and signaling parameters which are important for CAR-NK cell cytotoxicity.

The in silico model integrates opposing signaling initiated by CAR, activating NKRs, adhesion receptors, and inhibitory NKRs and correctly captures the nonmonotonicity of CAR-NK cell cytotoxicity against tumor cells expressing various molecules of both hematologic and solid tumor CAR antigens (e.g., CD33, Her2). The model also successfully describes cytotoxicity of CAR-NK cells containing different CAR signaling domains such as CD28 in Gen2 CAR and 2B4 in Gen4v2 CAR, as well as short- (4 h), long-term (48 to 72 h), and in vivo cytotoxicity of the CAR-NK cells. In addition, the model also correctly predicts cytotoxic response of WT NK cells and the mixture of WT and CAR-NK cells. The ability of our model to describe lysis data and predict cytotoxicity outside of the training data with high accuracy shows the validity of our integrated framework. The framework can be straightforwardly extended to describe cytotoxicity of other CAR constructs and target cells. Together, these studies show that CAR-NK cell cytotoxicity can be accurately modeled with a limited number of experimentally measurable variables and a system of ordinary differential equations.

The multiscale model implicitly describes subcellular signaling processes including immunological synapse formation (52) or inside-out LFA1-ICAM signaling (53) and cytotoxic responses using coarse-grained processes. Values of the associated model parameters can potentially capture the changes in the CAR transmembrane domain or any involvement of FasL–Fas interactions in late-time cytotoxicity. The model parameters account for unknown contributors to cytotoxicity such as intrinsic tumor resistance mechanisms against NK cell cytotoxicity (54, 55). Poor model predictions were largely due to saturations in percentage lysis in in-vitro cytotoxicity assays where the experimentally measured cytotoxicity plateaued at high E:T ratios (Fig. 2E). In the model, we assumed all NK cells can interact with all tumor cells with equal probability, thus as the in silico E:T ratios increase, the model will predict higher lysis values until maximal lysis (100% target cell lysis) is achieved. Given this assumption, the model would not be able to predict nonmaximal plateaus of cytotoxicity. Incorporating the spatial distribution of NK cells and tumor cells in future models may improve model predictions. Moreover, investigating poor model predictions may uncover nondominant mechanisms or discover unknown mechanisms of CAR-NK cell cytotoxicity relevant to therapeutic efficacy.

We observed that per cell molecules of CAR and LFA-1 vary over 100-fold and 10-fold in CAR-NK cells, respectively, and the per cell molecules of tumor ligands (CD33, Her2, ICAM-1, HLA-A/B/C) vary between 0.5 and 50-fold across tumor and nontumor targets. The cell–cell variability in receptor and ligand expression can also give rise to cell–cell variations of cytotoxic responses of CAR and WT NK cells. Our model captures these cell–cell variations of cytotoxicity arising due to variations of per cell molecules of CAR, activating and inhibitory NKRs, and adhesion receptors in NK cells and abundances of cognate ligands on target cells. The excellent agreement of the model predictions for the percentage lysis with the cytotoxicity assays suggests the relevance of such cell–cell heterogeneity in determining bulk cytotoxicity response. A model devoid of cell–cell variations in receptors and ligands expression and instead trained only by the mean expression values shows lower predictive capacity (SI Appendix, Fig. S2). This finding supports the relevance of cell–cell heterogeneity in affecting the bulk cytotoxic response.

We applied our model to explore the on-tumor and off-tumor cytotoxic activity of CAR-NK cells. We performed multiobjective optimization to optimize the on-target cytotoxicity of CAR-NK cells. Our analysis manipulating parameters that reflect the efficiency of CAR transduction, CAR signaling strength, and inhibitory receptor signaling strength all converge and agree with the biologic data that WT NK cells maximize on-tumor toxicity while minimizing off-tumor toxicity. This analytical approach is valuable for NK cell-based cancer immunotherapies where the cell and the CAR both contribute distinct and substantial antitumor cytotoxicity.

Limitations of This Study.

Our model assumes limited genotypic and phenotypic polymorphisms in the NK cell receptor repertoire and does not specify contributions of cytotoxicity from all known NK cell receptor–ligand interactions with tumor cells. This assumption is most directly reflected in the $C_{N_2}$ parameter, which must be re-estimated for each additional target cell. We show that the granularity and number of the ligand–receptor pairs represented in the model is sufficient for a minimal and generalizable framework for predicting CAR-NK cytotoxicity. Although other identified receptor–ligand interactions play a role in NK cell cytotoxicity, our understanding of the hierarchy of signal integration as determinants of NK cell cytotoxicity is incomplete. Further work characterizing the NK-intrinsic and tumor-intrinsic factors that mediate NK cell cytotoxicity is necessary to develop more powerful predictive models.

Our model only takes in selected signaling kinetics and population dynamics parameters to predict cytotoxicity such as binding affinity and the distribution of NK cells and tumor cells in a mixed coculture. Experimentally, we did not directly measure NK cell activation and instead utilized cytotoxicity as a downstream surrogate for activation. As cytotoxicity is a combined function dependent on both effector activation and target response, there may be NK cell-intrinsic signaling overlooked by our approach and instead compensated by other parameters. To adapt this model to intricately study NK cell signaling, incorporating data which directly measure NK activation (i.e., Ca2+ flux, signaling proteins) would be necessary.

To reduce computational complexity while capturing characteristics of NK cell and tumor cell receptor–ligand expression, we elected to bin single-cell protein expression distributions into histograms of five bins. In this process, granular information about protein expression and protein coexpression at the single cell level may be lost. Future models incorporating this information will allow more detailed mechanistic exploration of CAR-NK cell cytotoxicity.

The model system used for Pareto optimization is not representative of all situations in which off-tumor toxicities would arise. In many cases of off-tumor toxicities, the targeted antigen is lowly or moderately expressed on healthy tissues and highly expressed on the targeted tumor, contrasting with the system of Kasumi-1 and primary monocytes we utilize in which the tumor cells have lower expression of CD33 (38). Pareto analysis using a tumor cell line (i.e., MV4-11) which abundantly expresses the targeted antigen and is resistant to innate WT NK cell killing may reveal a Pareto optimal front that favors CAR-NK cells. Furthermore, the clinical problem of simultaneously targeting healthy tissues in CAR therapies is also pertinent in solid tumors. In the future, applying this model to evaluate the on-target off-tumor cytotoxicity of solid tumor-targeting CAR-NK cells is both exciting and clinically relevant.

Tumor microenvironment modulation of immune cell activity hinders the success of cellular immunotherapies and is an area of active investigation (56, 57). While not assessed in this study, this model can be adapted to predict the impact of tumor microenvironment factors (immunosuppressive cytokines, hypoxia, tumor immune checkpoints) on NK cell cytotoxicity through the addition of parameters and associated kinetic processes. For example, predicting NK cell cytotoxicity responses to TGFβ using the model can be achieved by adding an intratumoral TGFβ concentration parameter and a TGFβ receptor parameter and training on the appropriate in-vitro or in-vivo data. Other biological variables such as immune cell migration and antigen escape are also important for clinical responses to adoptive T and NK cell therapies (57–59). Capturing this biology in silico will require spatially resolved and stochastic models.

Materials and Methods

Quantitative Flow Cytometry and Receptor and Ligand Quantification.

Kasumi-1, HL-60, SKOV3, DIPG36, WT NK, and CAR-NK cells were stained with Tonbo Ghost Dye (Tonbo Biosciences, San Diego, CA) in PBS and Fc blocked (Miltenyi 130-059-901) prior to staining with either anti-CD33 (BD 561816), anti-Her2 (Biolegend 324406), anti-ICAM-1 (BD 560971), anti-HLA-A/B/C (BD 567582), anti-CD11a (BD 550851), anti-KIR2DL2/DL3 (Miltenyi), anti-KIR2DL1 (RnD FAB1844P), anti-KIR3DL1 (RnD FAB12251P), anti-cmyc (abcam ab72468), or anti-Whitlow linker (Cell Signaling 62405S) antibodies in flow buffer (PBS +2% FBS). Cell surface protein expression was quantified using the PE phycoerythrin fluorescence quantitation kit according to the manufacturer’s instructions (BD 340495). To import flow data into the model and reduce model complexity, histograms of receptor and ligand expressions were binned into five bins of equal size. For CAR distributions, one bin was dedicated to nontransduced cells identified by the MFI threshold set by flow cytometry. For all other distributions, bins were uniformly distributed. For cells and donors that could not be directly stained quantitatively, flow cytometry MFI histograms were scaled by the average expression of Donor H to obtain estimates for CD33CAR expression. For these cells and donors, KIR and LFA-1 expressions were taken from Donor H. These are denoted by * in SI Appendix, Table S1.

CAR-NK Generation and Expansion.

CD33 CAR-NK and Her2 CAR-NK cells were generated as described previously (6). Briefly, NK cells were isolated from PBMC buffy coats from healthy human donors using RosetteSep Human NK cell Enrichment Cocktail (STEMCELL Technologies, Vancouver, BC, Canada). All human samples were deidentified prior to use. Isolated NK cells were expanded with irradiated mbIL-21 and 4-1BBL expressing K562 feeder cells at a 2:1 ratio for 1 wk prior to CAR generation as previously described (60). On day 7, NK cells were electroporated with Cas9-RNPs using HiFi SpCas9 Nuclease V3 (Integrated DNA Technologies, Coralville, IA) and gRNAs targeting AAVS1 or CD38. 30 min after electroporation, NK cells were transduced with AAV6 virus (Andelyn Biosciences) encoding the CD33 CAR or Her2 CAR construct. Two days after electroporation, cells stained for CAR expression and were expanded twice with irradiated mbIL-21 and 4-1BBL expressing K562 feeder cells at a 1:1 ratio.

Monocyte Isolation.

Primary monocytes were isolated from healthy donor buffy coats using RosetteSep Human Monocyte Enrichment Cocktail (STEMCELL Technologies, Vancouver, BC, Canada). All human samples were deidentified prior to use. Isolated monocytes were cultured for 24 h in complete RPMI (RPMI + 10% FBS) media prior to use in calcein cytotoxicity assays.

Flow Cytometry Based Cytotoxicity Assays.

Target cells were washed in complete RPMI (RPMI + 10% FBS) and resuspended to a final concentration of 0.1e6 cells/mL. NK cells were resuspended in complete RPMI at a concentration of 1e6 cells/mL and serially diluted in a twofold dilution in a 96-well plate. Cells were plated in triplicate. 1e4 labeled target cells were added to each well of NK cells to achieve the final effector to target ratios 10:1, 5:1, 0.625:1, 0.3125:1. Target cells and NK cells were incubated for the allotted time at 37 °C. After incubation, propidium iodide was added to each well and flow cytometric analysis was performed to identify dead target cells.

Calcein Cytotoxicity Assays.

Target cells were loaded with calcein AM (2 μg/mL) (Fisher Scientific, Hampton, NH). Target cells were incubated with serially diluted effector cells to achieve the final effector to target ratios 10:1, 5:1, 0.625:1, 0.3125:1. Maximum release was achieved with media containing 2% Triton-X 100. Spontaneous release was achieved by incubating target cells with 100 μL of complete RPMI media. Cells were incubated at 37 °C for 4 h and supernatant was collected for reading on a fluorescence plate reader. Cytotoxicity is plotted as percent specific lysis [(experimental release−spontaneous release)/(maximum release−spontaneous release)]×100 (60).

In Vivo Mouse Modeling.

10 to 12-wk-old NSG mice (Jackson Laboratory, Bar Harbor, ME) were injected with 10e6 firefly luciferase-bearing MV4-11 AML tumor cells by tail-vein injection on day 0. On days 3, 7, 10, and 14 10e7 WT or CD33CAR-NK cells were administered by tail-vein injection along with 3 μg IL-2. Untreated mice were injected with 3 μg IL-2. Tumor growth was measured by bioluminescent imaging following intraperitoneal injection of luciferin twice a week.

Solutions of the ODE Model.

In this model, we have assumed a simplistic scenario where the NK cells interact with the tumor cells across three different timescales (from seconds to hours) as shown in Fig. 2 A and B. In the first stage (at Molecular scale), we have considered the interaction between four types of NK receptors (NKRs): the CARs (CD33CAR or Her2CAR), activating receptors, adhesive receptors (LFA-1) and inhibitory receptors (KIR) with their cognate ligands expressed on tumor cells, CD33 or Her2, stress ligands, ICAM-1, and HLA-A/B/C, respectively. These receptor–ligand interactions lead to formation of the activated complexes. In the Subcellular scale, the activated end complexes formed by binding of CAR, activating NKRs, and LFA-1 with their cognate ligands act as kinases that phosphorylate Vav1 to phosphorylated Vav1 (pVav1). Conversely, activated end complexes generated by the KIR-HLA-A/B/C interaction behave as phosphatases that dephosphorylate pVav1 to Vav1. Our model assumes that the increased subcellular concentration of pVav1 leads to the lysis of target cells by activating the NK cell at the Cell Population scale.

Because the lysis of target cells occurs at a much slower timescale (~hours) than the rapid receptor signaling reactions (such as receptor–ligand interactions, interconversion of Vav1 and pVav1) which happen on order of seconds, we have correlated the steady-state conditions of the signaling species (here, the steady state concentration of pVav1) with the lysis of the tumor cells.

Molecular scale (t1).

The cytotoxic process of NK cells, starting with binding of a receptor to its cognate ligand and culminating in the killing of a tumor cell, involves numerous biochemical reactions within the signaling pathway. In our model, we have employed a kinetic proofreading scheme, wherein all the complexes or modified forms of signaling proteins (e.g., pVav1) generated in the signaling pathway become free and revert to their initial forms when the receptor unbinds from the cognate ligand (61).

The following (Reaction scheme R1) is the signaling scheme of a specific type of receptor (R_i) expressed on a NK cell interacting with its cognate ligand (L_i) present on a tumor cell:

where index “i” represents the types of the receptor R_i (CD33CAR, activating NKRs, LFA-1, inhibitory KIRs) that binds with its cognate ligand $L_i$ (CD33, stress ligands, ICAM, and MHC1) to form the initial complex $C_{0_i}$ with rate $k_{o{n}_i}$. The initial complex $C_{0_i}$ can either unbind with rate $k_{o{\mathit{ff}}_i}$ or it can undergo $N$ sequential modifications to form activated complex $C_{N_i}$. Additionally, we assume that, at any state, the phosphorylated complex $C_{j_i}$ can disintegrate with rate $k_{o{\mathit{ff}}_i}$ which resets the complex to the molecular form of the receptor and ligand following the kinetic proof-reading scheme.

The time-dependent concentrations of the complexes $C_{0_i}, {C_1}_i, \dots {C_N}_i$ within the signaling pathway are described by the following ordinary differential equations (ODE):

$$\frac{d{C}_{0_i}}{\mathit{dt}}=k_{o{n}_i}[R_i^{\mathit{free}}][L_i^{\mathit{free}}]-\left(k_{of{f}_i}+k_{p_i}\right)C_{0_i},$$ [1a]

$$\frac{d{C}_{j_i}}{\mathit{dt}}=k_{p_i}{C}_{j-1_i}-\left(k_{of{f}_i}+k_{p_i}\right)C_{j_i} jϵ(1,N-1),$$ [1b]

$$\frac{d{C}_{N_i}}{\mathit{dt}}=k_{p_i}{C}_{N-1_i}-\left(k_{of{f}_i}\right)C_{N_i}.$$ [1c]

Eqs. 1a–c lead to the following solutions at the steady-state limit

$$C_{0_i}=\left(\frac{k_{o{n}_i}}{{k_{\mathit{off}}}_i+{k_p}_i}\right)[R_i^{\mathit{free}}][L_i^{\mathit{free}}],$$ [2a]

$$C_{j_i}={\stackrel{\sim }{\alpha }}_i{C}_{j-1_i}={\stackrel{\sim }{\alpha }}_i^j{C}_{0_i},$$ [2b]

Using Eq. 2b, we get

$$C_{N_i}=\frac{k_{p_i}}{k_{of{f}_i}}{C}_{N-1_i}=\left(\frac{k_{p_i}}{{k_{\mathit{off}}}_i}\right){\stackrel{\sim }{\alpha }}_i^{N-1}{C}_{0_i}=\frac{{\stackrel{\sim }{\alpha }}_i^N}{1-{\stackrel{\sim }{\alpha }}_i}{C}_{0_i},$$ [2c]

where ${\stackrel{\sim }{\alpha }}_i=\frac{k_{p_i}}{{k_{\mathit{off}}}_i+{k_p}_i}$ is the fraction of the concentration of the intermediate complex going in the forward direction before it disintegrates. The concentrations of free receptors and ligands in the steady state are ${[R}_i^{\mathit{free}}]=$${[R}_{0_i}]-\left(C_{0_i}+C_{1_i}+\cdots +C_{N_i}\right)={[R}_{0_i}]-\frac{{C_0}_i}{1-{\stackrel{\sim }{\alpha }}_i}$ and ${[L}_i^{\mathit{free}}]={[L}_{0_i}]-\frac{{C_0}_i}{1-{\stackrel{\sim }{\alpha }}_i}$. Substituting these into Eq. 2a leads to the following solution of ${C_0}_i$ in the steady state

$$\begin{array}{c}{C}_{0_i}=\frac{(1-{\stackrel{\sim }{\alpha }}_i)\left(\left[R_{0_i}\right]+\left[L_{0_i}\right]+K_{D_i}\right)}{2}\left(1-\sqrt{1-\frac{4\left[R_{0_i}\right]\left[L_{0_i}\right]}{{\left(\left[R_{0_i}\right]+\left[L_{0_i}\right]+K_{D_i}\right)}^2}}\right)\\ =\left(1-{\stackrel{\sim }{\alpha }}_i\right){{\stackrel{\sim }{C}}_0}_i,\end{array}$$ [3]

where the dissociation constant $K_{D_i}=\frac{{k_{\mathit{off}}}_i}{{k_{\mathit{on}}}_i}$. Thus, in the steady state we get the solution of the final product using Eqs. 2c and 3,

$$C_{N_i}={\stackrel{\sim }{\alpha }}_i^N{{\stackrel{\sim }{C}}_0}_i=\tilde{\alpha }{{\stackrel{\sim }{C}}_0}_i.$$ [4]

where ${\alpha }_i={\stackrel{\sim }{\alpha }}_i^N$

Subcellular scale (t2).

We have assumed that the reversible interconversion between Vav1 $\left(W\right)$ and pVav1 $\left(W^{\ast }\right)$ occurs by the kinase and phosphatase $E_1$ and $E_2$, respectively (Reaction schemes R2 and R3).

$$W+E_1\underset{d_1}{\stackrel{a_1}{\rightleftharpoons }}W{E}_1\stackrel{k_1}{\to }{W}^{\ast }+E_1,$$ [R2]

$$W^{*}+E_2\underset{d_2}{\stackrel{a_2}{\rightleftharpoons }}{W}^{\ast }{E}_2\stackrel{k_2}{\to }W+E_2.$$ [R3]

Enzyme $E_1$ is the sum of the final activated complexes generated from CAR, activating, and adhesive receptor–ligand interactions (i.e., $E_1=C_{N_1}+C_{N_2}+C_{N_3}$) and the enzyme $E_2$ is the sum of the final activated complex of inhibitory receptor–ligand interaction and unspecified phosphatases $E_2^{\prime }$ (i.e., $E_2=C_{N_4}+E_2^{\prime }$), where $C_{N_i}$ is given by Eq. 4.

The time-dependent coupled ODEs representing the kinetics of concentrations are:

$$\frac{d[W]}{\mathit{dt}}=-a_1\left[W\right]\left[E_1\right]+d_1\left[W{E}_1\right]+k_2[W^{\ast }{E}_2],$$ [5a]

$$\frac{d[W{E}_1]}{\mathit{dt}}=a_1\left[W\right]\left[E_1\right]-(d_1+k_1)[W{E}_1],$$ [5b]

$$\frac{d[W^{\ast }{E}_2]}{\mathit{dt}}=a_2\left[W^{\ast }\right]\left[E_2\right]-(d_2+k_2)[W^{\ast }{E}_2],$$ [5c]

$$\frac{d[W^{\ast }]}{\mathit{dt}}=-a_2\left[W^{\ast }\right]\left[E_2\right]+d_2\left[W^{\ast }{E}_2\right]+k_1[W{E}_1],$$ [5d]

where [$W], [W^{\ast }], [E_1], [E_2], [W{E}_1]$ and $[W^{\ast }{E}_2]$ are the concentrations of Vav1, pVav1, kinase, phosphatase, intermediate complexes (in the phosphorylation and dephosphorylation reactions), respectively. At any time, the total concentration of Vav1 (unphosphorylated or phosphorylated or complex forms) is conserved $W_T=\left[W\right]+\left[W^{\ast }\right]+\left[W{E}_1\right]+[W^{\ast }{E}_2]$. Solving for the steady state of Eqs. 5 A–D, we get the fraction of pVav1 compared to total Vav1 as following (62)

$$\begin{array}{c}{W}^{\ast }=\frac{\left[W^{\ast }\right]}{W_T}\hfill \\ =\frac{\left(V_r-1\right)-K_2\left(K_r+V_r\right)+\sqrt{{\left(V_r-1-K_2\left(K_r+V_r\right)\right)}^2+4K_2{V}_r(V_r-1)}}{2(V_r-1)},\hfill \end{array}$$ [6]

where $K_r=\frac{K_1}{K_2}$ is constructed with the scaled Michaelis constant $K_i=\frac{d_i+k_i}{a_i{W}_T}$ and $V_r=\frac{V_1}{V_2}=\frac{k_1{E}_1}{k_2{E}_2}=V_c\left(\frac{E_1}{E_2}\right)$.

Cell population scale (t3).

The formation of pVav1 leads to the activation of the NK cell, which, when interacting with target cells, results in their lysis over time. Our model assumes the proliferation of the target cells with time. At any time t, the rate of change in the number of target cells $T_L$ expressing the ligand abundances (molecules/cell) $L\equiv (L_1,L_2,L_3,L_4)$ is

$$\frac{d{T}_L}{\mathit{dt}}=-{\displaystyle \sum _R}{\lambda }_{\mathit{RL}}{E}_R{T}_L+r_c{T}_L,$$ [7]

where E_R is the number of effector cells (NK cells) expressing receptor abundances $R\equiv \left\{R_1,R_2,R_3,R_4\right\}$ (with indices 1-4 corresponding to the types of receptors or ligands used in the model). The first term on the right-hand side is the loss term due to the interactions between target cells ($T_L$) and NK cells ($E_R$) which leads to the lysis of the target cell with the proportionality constant ${\lambda }_{\mathit{RL}}$ determined by the distributions of the receptors and ligands. Our model assumes the constant ${\lambda }_{\mathit{RL}}$ is proportional to the subcellular concentration of pVav1 in the steady state, represented as ${\lambda }_{\mathit{RL}}={\lambda }_c{W}^{\ast }$ with a lysis constant λ&, where $W^{\ast }$ is given by Eq. 6${\lambda }_c$. The summation is considered over different NK cells expressing various receptor types and quantities. The limits of the sum are decided by the number of the receptors or ligands based on their distributions on NK cells or the target cell. The second term is the gain in the target cell population $T_L$ due to its proliferation where $r_c$ is proliferation rate assumed to be constant.

Given the initial conditions $\left\{T_L\left(0\right),E_R\left(0\right)\right\},$ we solved Eq. 7 for given parameters $\theta \equiv \theta \left({\alpha }_1,C_{N_2},{\alpha }_3,{{\alpha }_4,E_2^{\prime },V}_c,K_1,K_2,{\lambda }_c,r_c\right)$. We used the molecular distributions of three receptors (CAR, LFA-1, and iKIR) for CAR-NK cells and the corresponding ligands (CAR antigen, ICAM, MHC-1) for target cells as input. We numerically solved Eq. 7 using the function solve_ivp with “RK45” method from the Python scipy.integrate library and applied the following formula Eq. 8 to calculate the percentage of target cell lysis at time t.

$$\%lysis\left(t\right)=\left(1-\frac{{\sum }_L{T}_L\left(t\right)}{{\sum }_L{T}_L\left(0\right)}\right)\ast 100.$$ [8]

Parameter estimation.

We trained our model to estimate the parameters $\theta \left({\alpha }_1,C_{N_2},{\alpha }_3,{{\alpha }_4,E_2^{\prime },V}_c,K_1,K_2,{\lambda }_c,r_c\right).$ We minimized the cost function for both CAR-NK and WT NK simultaneously to find the optimal parameter values.

$$\begin{array}{c}cost={\left(\%lysisCARN{K}_{\mathit{expt}}-\%lysisCARN{K}_{\mathit{model}}\right)}^2\hfill \\ +{\left(\%lysisWTN{K}_{\mathit{expt}}-\%lysisWTN{K}_{\mathit{model}}\right)}^2.\hfill \end{array}$$

Minimization of cost function was performed using least_squares function of scipy (a python module) that uses “Trust Region Reflective algorithm.” The ranges of parameters for the search space are presented in Table 1.

R² coefficient of determination was calculated using the r2_score function in scikit learn (63).

Modeling In Vivo Tumor Lysis.

We have adapted the model to reflect in vivo experimental conditions including tumor cell growth and repeated dosing of NK cells. NK cells have transient persistence after being infused in vivo, with the majority of cells persisting between 7 and 10 d (64, 65). In addition to modeling the tumor population as described in Eq. 7, we incorporated the dynamics of NK cell loss into our model, as in vivo tumor lysis was observed over longer periods (a few weeks). We applied an NK cell loss parameter, $r_{\mathit{nk}}$, that represents NK cell death, exhaustion, or exit from the tumor microenvironment. At any time t, the rate of change in the number of NK cells E_R expressing receptor $R\equiv (R_1,R_2,R_3,R_4)$ is given by

$$\frac{d{E}_R}{\mathit{dt}}=-r_{\mathit{nk}}{E}_R.$$ [9]

We solved Eqs. 7 and 9 simultaneously to compute the number of tumor cells (T_L) on the observations days 4, 8, 11, 15, 18, and 22. All tumor cells were injected into the mice on day 0 and NK cells were injected intermittently on days 3, 7, 10, and 14. In our model, we assumed day 3 as the initial time point where 10⁷ NK cells and ~1.6 × 10⁶ tumor cells are present. The number of tumor cells on day 3 was computed by using Eq. 7 between day 0 and 3 in the absence of NK cells. Since the days of tumor observation and NK cell injection are different in the experiment, we used a step wise integration approach which accounts for the decay of existing NK cells and the injection of NK cells during the intervals between the observation days (Table 2). For instance, we used the NK and tumor cell numbers from day 4 as the initial condition and solved the ODEs (Eqs. 7 and 9) in the duration of 3 d to obtain the tumor and NK cell numbers at day 7. The numbers of NK and tumor cells were computed on day 4 by solving the same ODEs in the interval of 1 d using the tumor and NK cell numbers at day 3 which includes the injection of NK cells at day 3 as the initial condition. We followed the same method to compute the tumor cell number (T_L) at days 11, 15, 18, 22 respectively.

Table 2.

Schedule of in-vivo mouse model experiments and in-silico modeling

	Day 0	Day 3	Day 4	Day 7	Day 8	Day 10	Day 11	Day 14	Day 15	Day 18	Day 22
Tumor injection	x
NK or control injection		x		x		x		x
Bioluminescent imaging			x		x		x		x	x	x
Integration interval (# of days)			1	3	1	2	1	3	1	3	4
Model output for fitting and prediction			x		x		x		x	x	x

Maximum likelihood estimation.

The CD33CAR expressions on CAR-NK cells showed a bimodal distribution (SI Appendix, Fig. S3A). We modeled this using the superposition of two log-normal distributions $p_1({\mu }_1,{\sigma }_1)$ and $p_2({\mu }_2,{\sigma }_2)$, where mean $({\mu }_1,{\mu }_2)$ and sigma $({\sigma }_1,{\sigma }_2)$ are corresponding to respective log-normal distributions. We used a linear combination of these log-normal distributions: ${(1-\mathrm{\Omega })\times p}_1({\mu }_1,{\sigma }_1){+\mathrm{\Omega }\times p}_2({\mu }_2,{\sigma }_2)$ with weights $(1-\mathrm{\Omega })$ and $\mathrm{\Omega }$ for WT and CD33CAR-NK, respectively. We estimated five parameters $({\mu }_1,{\sigma }_1,{\mu }_2,{\sigma }_2,\mathrm{\Omega })$ that maximize the likelihood. We used a function least squares from scipy (a python module) that uses “Trust Region Reflective algorithm”.

Pareto optimization.

For targeted cancer immunotherapies like CAR-NK cell therapy there is a tradeoff between maximizing lysis of tumor cells while minimizing lysis of healthy cells. For example, CAR-NK cell can recognize and lyse healthy cells which express the targeted cognate antigen (i.e., CD33). However, healthy cells will express high amounts of major histocompatibility complex proteins that inhibit NK cell lysis. We set up a multiobjective optimization task to explore the signaling and receptor expression parameters in our model and calculate a Pareto front. This Pareto front represents the set of optimal parameters values where no objective can be further maximized without sacrificing the other. In our Pareto optimization scheme, we set two conflicting objectives: 1) minimization of healthy lysis and 2) maximization of tumor lysis.

We utilized the Non-dominated Sorting Genetic Algorithm II (NSGA-II) (51), a population-based method, for Pareto optimization to obtain the nondominated solutions. This approach allows us to efficiently explore and identify optimal trade-offs among the above two objectives. We called functions NSGAII and Problem from platypus python library. We explored the optimal front within the possible range of controlled parameters (${\alpha }_1,{\alpha }_4,\mathrm{\Omega }$).

Supplementary Material

Appendix 01 (PDF)

Data, Materials, and Software Availability

Cytotoxicity data, flow cytometry data, python code data have been deposited in github (https://github.com/drsamalik/CAR_NK_Modeling) (66).