# A Spatial Quantitative Systems Pharmacology Platform spQSP-IO for Simulations of Tumor–Immune Interactions and Effects of Checkpoint Inhibitor Immunotherapy

^{1}

^{2}

^{3}

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## Abstract

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## Simple Summary

## Abstract

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Agent-Based Model Rules

#### 2.1.1. Environment

^{3}volume comprises 50 × 50 × 50 voxels. Each voxel can host one cancer cell. The number of T cells allowed to coexist in the same voxel depends on whether or not a cancer cell occupies the voxel. We allow up to eight T cells to reside in the same voxel without a cancer cell, and one if there is a cancer cell. These numbers are determined based on diameters of cancer cells (~20 micron) and lymphocytes (~10 microns) and can be adjusted in the simulations; they can be cancer type dependent. Cells are allowed to move into voxels in their von Neumann neighborhood (defined as the six voxels directly adjacent to current voxel in three dimensions). When a cell scans its surroundings for possible interactions, the Moore neighborhood (defined as the cubic neighborhood of 26 voxels surrounding current voxel) is used, similar to our previous model [43]. The cell agents are subject to no-flux boundary conditions at the volume boundaries.

#### 2.1.2. Cancer Cells

#### 2.1.3. CD8+ T Cells

#### 2.1.4. Regulatory T Cells

#### 2.2. QSP-ABM Coupling

#### 2.2.1. Converting QSP Model to a spQSP-IO Module

#### 2.2.2. Deriving ABM Parameters from QSP Parameters

#### Cancer Cell Population Dynamics

_{s}. A fraction, k, of the divisions is asymmetric, generating one progenitor cell and one stem-like cell; the other 1 − k of all divisions is symmetric and generates two daughter stem-like cells. Progenitor cells divide at a rate r

_{p}, and they have a limited number of divisions (d

_{max}). When the maximum number of divisions is reached, the daughter cells become senescent. Senescent cells do not proliferate and have a fractional death rate of µ. The proliferation, differentiation and death of different cancer cell subtypes are stochastically determined by probabilities in the ABM simulation; the aforementioned rate parameter r

_{s}is calculated from these probabilities so that the overall cancer cell population dynamics is consistent with the deterministic QSP module.

_{c}, P

_{i}and S

_{n}denote cancer stem-like cell, progenitor cell after i divisions, and senescent cells, respectively.

_{1}and pS

_{c}approaches 0 as tumor grows. Similar relationships can be derived between other cancer cell types. The system asymptotically approaches a state where all species grow with the same rate r = (1 − k)r

_{s}, such that the ratios of the corresponding species approach constant at large t:

_{s}and k, we set k value in the parameter file and calculate CSC growth rate r

_{s}= r/(1 − k). r

_{p}and µ can be set independently.

Species | S_{c} | P_{i} | S_{n} |

Fraction | $\frac{1}{q}$ | $\frac{p{l}_{1}^{i-1}}{q}$ | $\frac{p{l}_{1}^{dmax-1}{l}_{2}}{q}$ |

#### Modifier Function of PD-1–PD-LY Interaction

_{2}, then the system can be written as:

_{2}in immune synapse can be calculated during simulation when each Teff interacts with another cell. It can be shown that there exist one and only one real root in the interval 0 < x < 1 (See the Supplementary Text S1).

#### Cytotoxic T Cell (Teff) Killing of Cancer Cells

#### Effector T Cell Exhaustion

- From PD-L1 interaction (Reaction 61):

- 2.
- Inhibition by Treg (Reaction 80):

#### Recruitment to Cancer Cells from Blood

_{r}has to be chosen carefully so that maximal $nTeff\_rec$ and $nTreg\_rec$ is close to but smaller than 1 to reduce the number of entry points examined in each simulation time step and save computing time.

#### 2.2.3. Integrating QSP and ABM Modules

^{c}) are simulated using the ODE system from the QSP module. For the species from set A of the tumor compartment, a fraction w

_{QSP}is tracked by the ODE in the QSP module, and the remaining 1 − w

_{QSP}is handled by the ABM module. Additionally, mechanisms resulting in the exchange of material between set A and set A

^{c}species are reflected in the ABM module. Teff and Treg recruitment from the central compartment to the tumor is scaled by w

_{QSP}, while recruitment of T cells to the ABM tumor compartment is deducted from the central compartment. In this study, we simulate the application of an anti-PD-1 agent, following the published QSP study applied to NSCLC [20]; the agent used there was nivolumab. Here, we use the nivolumab concentration from the tumor compartment of the QSP module in the ABM in the calculation of PD-1–PD-L1 bond numbers. When cancer cells die in the ABM module, their death number is recorded and proportionally added to the tumor-specific antigen species in the QSP module to determine the peptide-MHC number on antigen-presenting cells (APCs), which in turn drives T cell priming in the lymph node compartment.

_{QSP}of the entire tumor. However, due to the limited volume size, the raw number of cells in each volume needs to be scaled. The scaling factor for volume i, s

_{i}, can be calculated using the following equation:

_{i}is the fraction of the type of region represented by region type i in the entire tumor; C

_{QSP}and C

_{i}are cancer cell number in the tumor compartment of QSP module and cancer cell number in ABM module i, respectively. k

_{i}depends on the shape and size of the tumor. For tumors with a smoother surface, k

_{2}(invasive front) is relatively small compared with tumors with irregular shapes and fingering structures. When the tumor grows larger, k

_{2}can decrease as the invasive front comprises a smaller fraction of the tumor volume. When tumor resection is performed, the majority of the core is removed from the tumor, and k

_{2}becomes much larger (depending on how much of the tissue is surgically removed). When calculating the recruitment of T cells and generation of tumor antigens to update the QSP module, the numbers recorded in volume i are multiplied by s

_{i}to reflect changes induced by 1 − w

_{QSP}of the entire tumor.

#### 2.3. Diffusion-Reaction in the Spatial Compartments

#### 2.4. Simulating Patient Responses to Immune Checkpoint Treatment

#### 2.5. Parameter Sensitivity Analysis

#### 2.6. Biomarker Analysis Using Variable Selection

#### 2.6.1. Tumor Eiameter

_{i}is the number of cells of type i, and V

_{i}is cell size, and f

_{vol_tum}is the interstitial fraction of tumor (void fraction).

#### 2.6.2. Responsiveness

_{t}is tumor diameter at time t and d

_{0}is the pretreatment tumor diameter.

#### 2.6.3. Time to Progression

_{1,2, …, n}and the endpoint of interest Y. Let set A be the set of selected biomarkers, and start with a null set with the intercept being the only term in the regression model. The procedure can be represented in a pseudo code (available in the Supplementary Materials, Text S3).

## 3. Results

#### 3.1. Studying Tumor Spatial Characteristics Using spQSP-IO

#### 3.2. Simulating Patient Response to Nivolumab Treatment with spQSP-IO

#### 3.3. Parameter Sensitivity Analysis in spQSP-IO

#### 3.4. Assessment of Impact of Tumor Vascular Density and Its Heterogeneity Using spQSP-IO

#### 3.5. Identifying Biomarker for Different Simulated Clinical Endpoints Using spQSP-IO

## 4. Discussion

^{2}area is comparable to whole-slide image data, but still only large enough to host a section of a small-sized tumor, and the small depth used in the simulation might skew the result toward a 2D simulation. In the case of the multi-volume simulations, in order to keep the invasive front volume at the invasive front of the tumor, the window is shifted towards the normal tissue when cancer cell number increases so that the volume is not filled with the growing tumor, and towards the tumor side when cancer cell number drops to avoid eliminating cancer cells at the invasive front entirely. In the latter case, new cells are introduced to populate the opening space created after shifting, and the arrangement of these cells can bring bias to the simulation. To strike a balance between better capturing the spatial heterogeneity and maintaining adequate computational performance, improvement in the coupling method of the ABM and QSP modules need to be made in future developments of such hybrid models.

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Transport and interaction of cells and molecules in different compartments. On the left: QSP model: central compartment, tumor-draining lymph node compartment, peripheral compartment, and tumor compartment are included in the model. Mature antigen-presenting cells process tumor antigen collected in the tumor compartment and transport them through lymphatic vessels to the lymph node compartment to prime naïve cytotoxic T lymphocytes (CTL) and T regulatory cell (Treg). After clone expansion, these cells are trafficked through blood circulation (central compartment) and extravasate into the tumor microenvironment. On the right: spatio-temporal ABM module partially replaces the tumor compartment from the QSP model, capturing dynamics of different subtypes of cancer cells and T cells and their interactions with higher granularity and a spatial resolution. Spatio-temporal distribution of soluble cytokines IL-2 and IFNγ are described by PDEs. Details of these interactions are described in the method section.

**Figure 2.**Coupling of QSP and ABM modules during simulation. (

**A**) A group of species from the tumor compartment (Set A) including cancer cells, effector T cells (active and exhausted) and regulatory T cells (Tregs) are partially represented with the ABM tumor module. The fraction of tumor tracked in the QSP module is w

_{QSP}, and that weight is applied to the initial condition of these species and their transmigration into the tumor compartment. The other 1 − w

_{QSP}of the tumor compartment is represented by the ABM module (A1 and A2). The rest of the species (set AC) are fully accounted for by the QSP module, the partially weighted QSP is referred to here as QSP_w. (

**B**) Cell counts in the ABM modules are scaled up to account for 1 − w

_{QSP}of the tumor when their contribution to the QSP is calculated during the simulation, depending on the number of ROIs chosen and the fraction of each tumor region in the whole-tumor volume. k

_{i}could vary in time when the tumor changes in size, shape or undergoes resection during treatment. (

**C**) Flow of control during simulations. At the beginning of each time step, some ABM variables are updated from the QSP module, including central compartment concentration of Teff, Treg and anti-PD-1 agent. Then, the ABM module is simulated over time interval dt, followed by the simulation of QSP_w. At the end of the time step, relevant QSP variables in set AC are updated according to ABM simulation results.

**Figure 3.**Tumor morphology affected by cancer stem-like cell mechanisms. Part of the tumor in the images is removed to make the interior of the tumor visible. Brown: progenitor cells; red: cancer stem cells (CSC); grey: senescent cells. (

**A**) Low asymmetric division probability, low migration rate; (

**B**) low asymmetric division probability, high migration rate; (

**C**) high asymmetric division probability, low migration rate; (

**D**) high asymmetric division probability, high migration rate. In the 4th column (Stem ID, in which a unique ID is assigned to each stem-like cancer cell and its descendants), each color represents progenitor cells originating from the same CSC. A time-lapse video showing the spatio-temporal tumor development with pseudo colors corresponding to cancer cells’ stem ID is included in the Supplementary Materials (Supplement_CSC.mp4). Each box represents 3 × 3 × 3 mm volume.

**Figure 4.**Visualization of spatio-temporal tumor dynamics. (

**A**) Simulated tumor in 3 × 3 × 3 mm volume. Cancer cells: brown: progenitor cells; red: cancer stem cells (CSC); grey: senescent cells; T cells: blue: Tregs; green, red, and yellow: CD8+ T cell, as newly recruited, enhanced, and exhausted states. (

**B**) Simulated tumor in 10 × 10 × 0.2 mm volume, with or without anti-PD-1 treatment. (

**C**–

**L**) Snapshots of tumor at day 125 after treatment began. (

**C**) 3D visualization. (

**D**) Close-Up 3D visualization of cells. (

**E**) Virtual multiplex immunofluorescence (mIF) (red: Treg; blue: cancer cells; green: Teff). (

**F**) Close-Up virtual mIF. (

**G**) Simulated IL-2 concentration distribution, ng/mL. (

**H**) Virtual immunohistochemistry (IHC), CD8+. (

**I**) Virtual IHC, FoxP3+. (

**J**) Virtual IHC, PD-L1+. (

**K**) Close-Up of J. (

**L**) Simulated IFNγ concentration distribution, ng/mL.

**Figure 5.**Response to treatment in a virtual cohort and parameter sensitivity. (

**A**) Tumor diameter change, relative to diameter prior to first drug administration. Colors correspond to tumor growth rate, with red representing high growth rate and purple representing low growth rate. (

**B**) Best response measured as minimum tumor diameter changes from 8 weeks after the first drug administration. Colors represent tumor mutational burden, where red indicates the patient has a mutational burden higher than median value among the cohort and blue indicates a mutational burden lower than median. (

**C**) PRCC of simulation output against different parameters. *, ** and *** indicate p-values smaller than 10

^{−3}, 10

^{−6}and 10

^{−9}.

**Figure 6.**Spatial distribution of different cell types and concentration of IL-2 and IFNγ in tumor core and invasive Figure 4. Cytokine concentration (ng/mL) reflects the concentration in the x–z plane cross section at y = 0.5 mm location. Cell distributions are pretreatment, early treatment and day 200, while cytokine concentrations shown are pretreatment. (

**A**) Baseline vascular densities. (

**B**) Increased vascular density in the tumor core. (

**C**) Increased vascular density in tumor at the invasive front. (

**D**) Increased vascular density in normal tissue outside of the invasive front.

**Figure 7.**Virtual IHC of pretreatment CD8+, FoxP3+ and PD-L1+ distribution from core and invasive front ROIs of simulated tumor. Light blue: nuclei, indicating location of nucleated cells; brown: indicated labels of CD8, FoxP3 or PD-L1, with the darkness representing their intensities. Time points chosen are pretreatment, early treatment and day 200. (

**A**) Baseline vascular densities. (

**B**) Increased vascular density in the tumor core. (

**C**) Increased vascular density in tumor at the invasive front. (

**D**) Increased vascular density in normal tissue outside of the invasive front.

**Figure 8.**Tumor dynamics with different vascular densities in various sub-regions of tumor. Red: baseline vascular densities. Green: increased vascular density in the tumor core. Blue: increased vascular density in tumor at the invasive front. Yellow: increased vascular density in normal tissue outside of the invasive front.

**Figure 9.**Tumor diameter in patient subgroups based on indicated biomarkers. On the right-hand side, waterfall plots are shown of the lower and upper half of the virtual population with regard to each biomarker.

**Figure 10.**ORR in patient subgroups based on indicated biomarkers. On the right-hand side, the proportion of outcome (progressive disease, stable disease and partial/complete response) is plotted against the percentile of each biomarker.

**Figure 11.**Hazard-ratio of progression in patient subgroups based on indicated biomarkers. On the right-hand side, the proportion of patients whose tumor have progressed is plotted against time after treatment started. Blue and orange corresponds to the upper and lower half of the virtual population with regard to each biomarker.

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**MDPI and ACS Style**

Gong, C.; Ruiz-Martinez, A.; Kimko, H.; Popel, A.S.
A Spatial Quantitative Systems Pharmacology Platform spQSP-IO for Simulations of Tumor–Immune Interactions and Effects of Checkpoint Inhibitor Immunotherapy. *Cancers* **2021**, *13*, 3751.
https://doi.org/10.3390/cancers13153751

**AMA Style**

Gong C, Ruiz-Martinez A, Kimko H, Popel AS.
A Spatial Quantitative Systems Pharmacology Platform spQSP-IO for Simulations of Tumor–Immune Interactions and Effects of Checkpoint Inhibitor Immunotherapy. *Cancers*. 2021; 13(15):3751.
https://doi.org/10.3390/cancers13153751

**Chicago/Turabian Style**

Gong, Chang, Alvaro Ruiz-Martinez, Holly Kimko, and Aleksander S. Popel.
2021. "A Spatial Quantitative Systems Pharmacology Platform spQSP-IO for Simulations of Tumor–Immune Interactions and Effects of Checkpoint Inhibitor Immunotherapy" *Cancers* 13, no. 15: 3751.
https://doi.org/10.3390/cancers13153751