Mathematical modeling of cancer immunotherapy for personalized clinical translation

Abstract

Encouraging advances are being made in cancer immunotherapy modeling, especially in the key areas of developing personalized treatment strategies based on individual patient parameters, predicting treatment outcomes and optimizing immunotherapy synergy when used in combination with other treatment approaches. Here we present a focused review of the most recent mathematical modeling work on cancer immunotherapy with a focus on clinical translatability. It can be seen that this field is transitioning from pure basic science to applications that can make impactful differences in patients’ lives. We discuss how researchers are integrating experimental and clinical data to fully inform models so that they can be applied for clinical predictions, and present the challenges that remain to be overcome if widespread clinical adaptation is to be realized. Lastly, we discuss the most promising future applications and areas that are expected to be the focus of extensive upcoming modeling studies.

Both adaptive and innate immune responses play key roles in the elimination of cancer precursor cells, but if these cells are able to evade these mechanisms, it may lead to tumor formation^1,2. Cancer immunotherapy seeks to resensitize or reactivate the patient’s own immune system to fight their disease³. Many methods for initiating immune sensitization towards the tumor have been developed⁴, including vaccination, exogenous cytokine therapy, adoptive cell transfer (ACT) and immune checkpoint inhibitor (ICI) therapy. For example, cytokine therapies deliver small-molecule immune modulators, including interleukin-2 (IL-2) to activate natural killer cells or to increase tumor-infiltrating lymphocyte cytotoxicity⁵, or interferon-α to increase antigen presentation and dendritic cell activation⁶, thereby increasing the immune response. Many other cytokines are now being investigated for immunotherapy applications⁷. In ACT immunotherapy, either immune cells that are already sensitive to the patient’s tumors, or non-sensitive immune cells that are modified in the laboratory to become tumor sensitive, are expanded ex vivo and then reintroduced into the patient for tumor eradication⁸. ICI therapy is based on immune checkpoints, which are receptor–ligand pairs that allow for host immune recognition and immune suppression and are exploited by tumor cells to avoid immune detection. ICIs are antibodies that block these immune evasion pathways, and have already been approved for clinical inhibition of the immune checkpoint molecules programmed cell death protein 1 (PD-1), its ligand PD-L1, and cytotoxic T-lymphocyte associated protein 4 (CTLA4)⁹.

Increasing evidence is emerging that immunotherapy may be used synergistically with traditional cancer therapeutics, such as chemotherapy¹⁰ or radiation therapy¹¹. This expansive treatment landscape encompasses more than 600 individual cancer drugs, including over 30 Food and Drug Administration-approved immunotherapy agents¹², and many more may be reasonably expected to emerge as the field progresses. This represents an immense number of possible treatment combinations. Thus, identifying the optimal combination treatment strategy using clinical trials alone is not practical or sustainable, and neither is relying on each physician to consider all possible combinations when determining the clinical treatment for each individual patient. To overcome this problem, mathematical modeling has emerged as a powerful tool for describing and quantifying key biological and physical tumor–immune interaction mechanisms and their effects on treatment outcome^13–16.

Advances in modeling techniques and computational power, biological understanding and biomarker libraries have enabled mathematical models to grow in complexity. While valuable in revealing better mechanistic understanding by covering a larger parameter space, these models may have surpassed the threshold for practical clinical applicability, thereby concurrently hindering the likelihood of clinical adoption. This may occur due to inclusion of parameters that are impractical or impossible to measure clinically, requiring procedures that patients are unlikely to endure (such as excessive biopsies, which increase patient withdrawal from clinical trials¹⁷); disconnects between mathematical and clinical vocabularies and collaboration; and lack of feasible clinical integration, among others^18,19. However, methods for integrating mathematical modeling into clinical workflows to aid in decision-making are being developed²⁰, and practical, feasible methods that may stand up to scrutiny when tested in real-world clinical settings are emerging. In this Review, we highlight the current advances and how mathematical and computational biologists are partnering with clinicians and scientists to overcome the challenges of transitioning theory into real-world applications.

The state of the art in cancer modeling

The field of mathematical cancer modeling endeavors to mathematically describe and functionally link the key physical and biological factors involved in cancer growth, invasion, metastasis, response to treatment and therapeutic resistance. In the case of immunotherapy for cancer, the key players to be described include the tumor cells, key immune components (immune cells such as T cells, B cells, natural killer cells or macrophages; key molecular factors such as cytokines; and so forth), immunotherapy drugs to be studied (and their delivery), and the interactions between these players; examples of some of the most commonly included factors are depicted in Fig. 1. In the same way that these biological factors span multiple scales, from system (or organism)-wide drug delivery, to single immune or tumor cells, and down to the scale of molecular interactions, mathematical models may also be designed to include one or more relevant scales of interest. This is often accomplished using different approaches that are best suited to the scale(s) to be studied, which may then be mathematically hybridized into multiscale models as needed. We offer a brief summary of different modeling approaches below and in Table 1; interested readers are referred to more extensive discussions elsewhere^21–28.

Fig. 1 |. Selected key tumor–immune interaction processes for consideration in model design.

Many of the interactions that are thought to be influential in the tumor–immune system are depicted; these are also shown for ICI immunotherapy. Although a substantial simplification of the true complexity of the tumor–immune system (for example, there are many other types of immune cell), a subset of the processes shown are commonly selected for model inclusion, depending on the biological or clinical problem to be studied. We have depicted the ICIs anti-CTLA-4 and anti-PD-1 as an example of immunotherapy factors that may be included. NK, natural killer cell; EMT, epithelial–mesenchymal transition; APC, antigen presenting cell; M1, classically activated, anti-tumor, pro-inflammatory macrophage; M2, alternatively activated, tumor-associated macrophages; TCR, T-cell receptor; MHC, major histocompatibility complex; B7, B7 receptor (an integral membrane protein); CD28, Cluster of Differentiation 28 protein. See refs. 120–122 for further discussions of these and other processes.

Table 1.

Common modeling approaches and applications

Model type	Common uses	Advantages	Disadvantages
Continuum PDE	Modeling spatiotemporal behaviors and feedback between populations	• Computationally low cost • Relatively easier to develop • Model large-scale processes that are impractical for discrete representations • Allows data-driven empirical description or mechanistic formalism of the system	• Difficult to characterize heterogeneity • Deterministic solutions • Phenomenological model parameters (if used) may not have direct biological translation • Parameters may be difficult to measure experimentally
Continuum ODE, compartmental modeling	Modeling time-dependent population shifts and feedback between populations	• Computationally low cost • Relatively easier to develop • Model large-scale processes that are impractical for discrete representations • Allows data-driven empirical description or mechanistic formalism of the system • High potential for prospective clinical use or supporting regulatory approval applications	• Does not describe spatial effects • Difficult to characterize heterogeneity • Deterministic solutions • Phenomenological model parameters (if used) may not have direct biological translation • Parameters may be difficult to measure experimentally
Agent-based modeling	Describe spatiotemporal dynamics of heterogeneous entities, such as molecules, cells, tissue and organs	• Explicit description of heterogeneous properties and relevant underlying mechanisms (biological and physical) • Allows stochasticity and are non-deterministic	• Difficult to develop and implement • Computationally intensive • May be impractical for describing large-scale systems • Multiscale modeling may be difficult
Hybrid modeling	Describe spatiotemporal dynamics of both continuous and heterogeneous entities and feedback between populations and across scales	• Explicit description of heterogeneous properties and relevant underlying mechanisms (biological and physical) • Allows stochasticity • Explores complexity and diversity of cancer • Multiscale in nature	• Difficult to design • Computationally intensive • Extensive numerical techniques required • Model parameterization may be difficult

Each model type has advantages and disadvantages, and is thus better suited for some studies and applications than others. The decision of which modeling approach to use must be carefully considered by modelers to optimize efficacy, accuracy and relevance of model results.

Briefly, the type of model chosen for a particular application is usually based on the physical and time scales to be studied, computational costs and complexity of model implementation. Discrete models generally operate at the resolution of individual cells, and they can easily incorporate biological or physical mechanisms at the cell or molecular level that have been revealed experimentally, such as those defining (immune and cancer) cell–cell interactions. However, modeling each cell in fine detail is computationally intensive, which limits the model to a relatively small number of cells. In this case, continuum models become the preferred approach for modeling larger-scale systems, because they reduce computational cost and complexity by representing the spatiotemporal averages of underlying (smaller scale) factors and processes. However, it is difficult to use continuum models to explore tumor heterogeneity when the cell properties vary over small spatiotemporal scales. When inclusion of both large- and small-scale factors is desired, modelers often implement a hybrid approach, wherein discrete representations of key factors are included as needed and continuum approximations are used when possible, which are hybridized together into a single, cohesive, complete model, enabling studies that are inaccessible when using either approach alone.

At the tissue or organism scale, ordinary or partial differential equations (ODEs or PDEs) are often used to describe the time-dependent transport, distribution, and concentration profiles of model parameters²⁹. At the (human) system scale, pharmacokinetic (PK) models based on ODEs describe the key factors and processes that govern drug concentration throughout the body after delivery^30,31. Once a drug is delivered to the vascular boundary of a target tissue, the time-dependent drug penetration into and distribution within the tissue may be obtained, for instance, via Fick’s law-based continuum descriptions^32,33, which are often solved via numerical methods, either taken as an average across the tissue (ODEs) or described explicitly across the organ geometry (PDEs). Upon delivery to the site of action, drug effects are often described via pharmacodynamic (PD) equations of dose–response relationships^34,35. Continuum modeling may be also generalized to exclude spatial descriptions in compartmental models, thereby instead only focusing on the interplay between states (for example, T-cell stimulation versus suppression³⁶). These are useful to understand overarching trends, such as total tumor burden over time or the dynamics of immune transition between states (for example, naive, activated, exhausted), while offering the advantage of low computational costs³⁷.

An organ or a piece of tissue may be further discretized; as an example, individual entities such as cells may be explicitly represented as discrete objects (called agents) in an agent-based modeling (ABM) approach^24,38,39. These can be used to study the effects of heterogeneity, for instance, heterogeneous vasculature structures, phenotypic cellular distributions, genetic mutations and drug sensitivities. While continuum models are often deterministic, agent-based models impose a set of biological and biophysical rules on each agent and often allow for stochasticity among decision probabilities (a cell may move along a stochastically determined path). As a result, the behavior observed at the tissue scale emerges non-deterministically from these individual decisions, allowing for detailed bottom-up studies of how lower-level processes affect the higher-level system trajectory. As an example, NetLogo (an ABM platform) has been successfully adapted to describe the interactions between the immune system and tumor cells under both immunosurveillance⁴⁰ and escape^41,42 conditions.

As aforementioned, continuum and discrete modeling approaches can be hybridized into single models, allowing for the study of heterogeneous effects (for instance, individual tumor cell drug sensitivity) under conditions better suited for continuum descriptions (for instance, drug diffusion within this tumor). As another example, Gong et al. developed a three-dimensional, on-lattice model of early-stage generic (meaning not type-specific) cancer development (a discrete ABM) with continuum descriptions of molecular scales, enabling direct representation of IL-2-mediated T-cell recruitment and delivery of anti-PD-1 or anti-PD-L1 therapy⁴³, yielding valuable insights into the roles of inflammation and tumor mutational burden on the emergence of the immune-suppressive anti-PD-L1 phenotype. Open-source platforms for multiscale ABM have become available in recent years; as a notable example, PhysiCell, which enables rapid development of multiscale, lattice-free ABMs, and has been used to study mechanisms of escape of tumors from the adaptive immune response^44,45.

Towards optimizing clinical outcomes

Figure 2 shows an overview of key mathematical modeling areas contributing towards immunotherapy clinical applications, with examples of recently developed mathematical models that we discuss in this Review. Regardless of the area studied, many applications involve the following key development processes: identification of key biophysical processes and factors for model inclusion, model development and refinement, verification of model predictions, and post-simulation analysis to determine predictive accuracy and establish usage guidelines. Some applications have also attempted to identify additional biological surrogates for model parameters to ensure clinical usability, and some offered predictive platforms in which predicted clinical outcomes can be weighed against potential side effects or treatment risks, enabling the selection of the treatment option that results in the greatest therapeutic ratio, meaning maximal benefit with minimal toxicity. Importantly, the biological and clinical insights gained from these mathematical models and computer simulations (as detailed below) together have enabled experimental and clinical scientists to have a better quantitative understanding of different aspects of cancer immunology and immunotherapy, and offer a pathway towards clinical integration of personalized modeling predictions.

Fig. 2 |. Overview of cancer immunotherapy modeling applications in the four key research areas examined in this Review.

Only representative modeling examples with their respective approaches are shown for each topic. These modeling efforts are paving the way for development of new, accurate clinical decision tools for predicting and optimizing cancer immunotherapy outcomes for patients. (refs. 43, 46, 50, 56, 59, 63, 67, 69, 72, 73, 75, 76, 79, 82, 84, 90, 123) We apologize to colleagues whose work we could not cite in this Review due to space constraints and editorial considerations (mainly focusing on recently published work).

Modeling the tumor–immune interaction

Fundamentally, any model for studying antitumor immunotherapy must first generate correct descriptions of the tumor–immune interaction without drug intervention^46,47. Modelers must carefully consider which factors to include in the model, and identify only those absolutely necessary to properly characterize and capture the overall immune response (this is referred to as feature selection⁴⁸). This reduces the complexity of the system so that it may be described mathematically, increases the likelihood of obtaining unique solutions and reduces the risk of overfitting.

The earliest models of tumor–immune interactions were simple and often only theoretical (that is, not validated with experimental data) predator–prey-type models⁴⁹, wherein competition between the predator (immune cells) and the prey (tumor cells) determines system stability and the transition between immune elimination, evasion or escape. These models may be modified to include the effects of immunotherapy, such as increased T-cell recruitment and tumor invasion, drug-mediated immune activation, recognition of tumor cells and cytotoxic activity (among other mechanisms), thus enabling mathematical studies of immunotherapy intervention. Many modeling teams are now transitioning their work away from descriptions of tumor–immune systems for basic science purposes, and instead are moving towards immunotherapy models for clinical applications⁵⁰. Notable advances are emerging towards modeling-derived design of personalized treatment strategies based on individual patient measurements and parameters, including prediction of patient outcomes, optimized drug selection and increasing immunotherapy synergy when used in combination with other treatment strategies^51–55.

Testing cancer treatments in silico

Mathematical models offer powerful tools to mechanistically link perpatient-measured values with outcomes of interest, and to optimize personalized therapeutic strategies for maximum efficacy. This was demonstrated in a study by Lindauer and colleagues⁵⁶, who implemented a compartmental PK/PD model combined with a tissue compartment model⁵⁷ and tumor growth model⁵⁸ to predict the optimal dosing range of prembrolizumab (an anti-PD-1 ICI) in humans based on data collected in mouse models (in this study, a continuum approach was implemented so that key factors at the whole-body scale could be described; this is often the case for PK/PD models). From extensive in silico dosing perturbation studies, they observed that the maximal patient response was achieved at dosages of 2.0 mg per kg (mg of drug per kg of body mass) when delivered every 3 weeks, with only minor additional therapeutic effects observed with increased dosing frequency (to every 2 weeks) or at higher dosages.

Other groups are going a step further towards predicting not only effective clinical dosage but also toxicity of new drugs before any testing in humans has occurred. Chen, Haddish-Berhane and colleagues performed such a PK/PD modeling study using both in vitro and in vivo models to measure the pharmacological activity of the bispecific immunomodulatory P-cadherin LP-DART⁵⁹, a drug that binds to both tumor and T cells. They were able to either directly or allometrically translate experimental data to in silico studies of human dosage to determine safe but effective dosage ranges before the start of clinical trials. This represents a notable step towards mathematical modeling-supported clinical adaptation of newly developed immunotherapies by considering not only dose but also non-target drug toxicity—an important factor commonly omitted in modeling studies—because such multi-functional drugs may not easily upscale from animal to humans via standard approaches such as highest non-severely toxic dose^60,61. Such modeling platforms not only provide valuable foundations to study the mechanisms underlying the delicate balance between treatment success and failure but also provide quantitative tools to transition the treatment optimization process from approaches based on trial and error to engineered design.

Maintaining the balance between anti-tumorigenic and pro-tumorigenic immune cells is also critical for effective immunotherapy, which may fail if recruited or delivered immune cells then transition to a tumor-promoting state⁶². Accordingly, combination therapies are often employed to address both recruitment and activation. Combination therapy using immunotherapy, chemotherapy and migration inhibitors in colorectal cancer was recently investigated in a modeling study by Kather and associates⁶³ with a three-dimensional agent-based model containing discrete tumor cells, lymphocytes, macrophages and stroma, and continuum representations of drug and oxygen concentration profiles. In this work, an ABM was used so that discrete interactions and distributions among different cell types could be explicitly studied, which were compared with and validated against immunohistochemistry (IHC)-based observations in colorectal liver metastases collected from patients. Their model analysis revealed that there is probably not a ‘one size fits all’ combination treatment, and that immunotherapy when used as a monotherapy performs best in high-antigenicity tumors, while low-antigenicity tumors will respond better when chemotherapy is co-delivered. This result is supported by emerging evidence from clinical trials testing ICIs + chemotherapy⁶⁴; however, direct comparison between model prediction of treatment efficacy and measured outcomes remains outstanding, and this result remains purely in silico at this time.

Correlating biomarkers with patient outcomes

Predicting outcomes for patients with cancer before or shortly after the start of treatment is much needed in clinical practice. The tragic reality is that many patients with cancer will not survive long enough to endure several rounds of unsuccessful treatments before finding one that works, not to mention the immense physical, functional, financial and emotional burden this process imposes on the patient. Although treatment optimization is the stated goal of many mathematical models for oncology applications, the ability to inform models exclusively from already-available clinical measures and make reliable predictions of patient outcomes remains a considerable challenge, as models may fail when even a single critical parameter is not measurable via existing techniques. This relegates many technically advanced models to purely hypothesis-generating roles. Through collaborative efforts between clinical and modeling teams, this gap is now being bridged, and we have seen predictive models that can be informed using standard clinical measures being tested and validated towards clinical adaptation.

As previously mentioned, Gong et al. implemented a three-dimensional hybrid agent-based model of tumor growth and response to anti-PD-1 or anti-PD-L1 therapy⁴³. This multiscale model allowed for studying the spatiotemporal dynamics of T cells and cancer cells (with per-cell ABM descriptions), molecular heterogeneities (using continuum descriptions) and how heterogeneous tumors respond to anti-PD-1/anti-PD-L1 therapy. By simulating various treatment strategies across a range of tumor heterogeneities (for instance, cell phenotype distributions, mutational burdens, antigen strengths and tumor architectures), the authors were able to generate treatment outcome spectrums that can be used to predict and design individualized therapeutic strategies (however, experimental or clinical validation of these strategies remains outstanding). This work also raises an important point: that models containing parameters that vary between different cancer types often need to be adjusted to a specific cancer type, and that these parameters must be measurable on a per-patient basis for personalized applications. This was demonstrated in an in silico study by Ruiz-Martinez and colleagues⁶⁵, who applied a quantitative systems pharmacology (in this case, ODE-based) tumor model with ABM descriptions of tumor and immune cells to simulate the effects of anti-PD-1 immunotherapy on triple-negative breast cancer. In this work, distributions of cell types of interest (tumor cells, cytotoxic T lymphocytes, regulatory T cells and so on) were generated and then validated against IHC from triple-negative tumor slides under different tumor growth conditions with and without immunotherapy. In a separate study using this same modeling platform, Wang et al.⁶⁶ demonstrated in a simulated patient population how such data collected from IHC and other tumor-specific measures could be applied to the model for investigating the impact of tumor-associated macrophages on the cancer–immune cell interactions. It is important to note that their modularized modeling platform can be expanded for other cancer systems and other forms of treatment modalities, and could be applied in clinical trial simulation.

Ribba and colleagues⁶⁷ conducted a first-in-human trial to quantitatively measure in vivo tumor delivery of ⁸⁹Zr-radiolabeled CEA-IL2v (an immunocytokine) via both blood plasma sampling (natural killer cells, B cells, CD4⁺ and CD8⁺ T cells in n = 74 patients) and by positron emission tomography(PET) imaging (n = 14 patients) in patients on clinical trial NCT02004106. By integrating a PK model with a previously published diffusion-based ODE model of solid tumor drug distribution and uptake⁶⁸, the authors generated a complete tumor delivery model (a whole-body scale model), which they informed using blood plasma measurements. Their analysis revealed that frequent drug doses are often necessary to account for increasing IL-2R⁺ populations and their associated drug consumption, and that increased dosage frequency can be used to stimulate higher T-cell activation and expansion in blood. Patient-measured blood markers have also been demonstrated to provide reliable ICI response predictions in an ODE-based, ecologically inspired tumor–immune interaction model developed by Griffiths et al., who used per-patient measures of phenotype-specific circulating immune cells to predict tumor response to anti-PD-1 therapy⁶⁹. Other groups have also shown a reasonable level of accuracy in predicting disease response and time to progression using imaging data from melanoma patients treated with pembrolizumab^70,71.

Notable strides in tumor response prediction using clinically available imaging measures have also been achieved. Butner, Cristini, Wang and colleagues^72,73 recently published a mechanistic mathematical model of solid tumor ICI therapy built on several key biological and physical factors and processes involved in ICI-mediated immune–tumor recruitment and response. It is important to note that the final model, which quantifies the total tumor burden over time after immunotherapy intervention via a simple ODE (derived from a system of PDEs), can be fully informed solely using non-invasive computerized tomography (CT) or magnetic resonance imaging (MRI) imaging^72,73. Using time-course tumor burden data obtained via CT, the authors calibrated the model to data from n = 28 patients from a basket study (representing multiple different cancer types and ICI drugs), and determined that model parameters were able to sort patients not only based on long-term treatment response but also based on the clinical gold standard of patient survival (to 50% population survival time), which was obtainable by the time of first patient follow-up after treatment initiation. These results were confirmed using data from an additional n = 273 patients^72,74. More importantly, the authors have also, at least in principle, demonstrated methods to inform key model parameters directly from IHC measures of PD-L1 or CD8⁺ T cells obtained at the start of treatment⁷⁵. This provides multiple methods for informing model parameters, thereby increasing the likelihood of model usability in real clinical settings where incomplete or missing data is an ongoing challenge (for instance, imaging may not be available in all hospitals).

Notably, these results provide predictive readouts as simple, scalar values that are easily and quickly interpretable, which is important in fast-paced clinical environments. A similar result was achieved by Mueller-Schoell and colleagues, who have discovered a clinical composite score from an ODE-based quantitative systems pharmacology model of CD19-specific chimeric antigen receptor (CAR)-T-cell immunotherapy for non-Hodgkin lymphoma⁷⁶. Their model, which takes as inputs CAR-T phenotypic measurements, patient-derived plasma cytokine counts and metabolic tumor volume measurements, was demonstrated to reliably predict tumor–immune infiltration and CD19⁺ metabolic volume changes based on T-cell expansion in n = 19 patients. Further analysis revealed a scalar composite score, defined as the ratio of maximum CAR-T counts divided by baseline tumor volume, which was shown to be highly predictive of patient survival in this preliminary cohort.

By demonstrating validation against clinical data and by establishing simpler numerical readouts of model outputs (when possible), these examples represent a notable step towards establishing clinical trust in modeling-based prediction of immunotherapy outcome. That is, at least in part, they present results and validation in formats clinicians are more used to (or expect), thereby taking a proactive role in bridging the gap between these distinct fields. Therefore, presenting model results in clinical language is highly important for increasing clinical trust in mathematical modeling, fostering new modeler–clinician collaborations, and ultimately clinical translation of modeling methods.

Hybrid treatment strategies

Engineered or computer-aided design of the optimal combination strategy requires identification of not only the best drug or drugs but also the timing, order and dosages in which each should be delivered—a problem that is particularly well suited for mathematical modeling optimization^55,77,78. Mpekris and colleagues developed an ODE and PDE-based model to study the effect of doxil (nanoparticle formulated doxorubicin, which can prime the tumor for enhanced immunotherapy delivery by lowering stiffness and repairing vasculature) delivered alone or in combination with anti-PD-L1 that was validated with experimental data collected from murine T41 and MCA205 models, which were monitored via ultrasound for tumor elastic modulus assessment⁷⁹. Their model results revealed that doxil pretreatment led to improved performance of anti-PD-L1, with greatest performance improvement observed with high-frequency, low-dose delivery. Interestingly, the value of high-frequency, low dosage treatment scheduling agrees with other modeling work, which has demonstrated that this strategy also leads to therapeutically favorable tumor-microenvironment remodeling^80,81, and has shown that therapeutically driven vasculature remodeling before immunotherapy improves therapeutic efficacy.

The ever-expanding array of existing treatment options requires treatment optimization to consider two or more possible therapies for use in combination. Towards this end, Coletti et al. used a PDE-based two-compartment (prostate and lymphoid) model to examine in silico multimodal combinations of seven different treatments in castration-resistant prostate cancer⁸². The therapeutic effects of androgen deprivation, anti-IL-1, anti-regulatory T cells, anti-myeloid derived suppressor cells, immunotherapy vaccine, natural killer cell injection and ICI therapy, used alone or in combination, and their interactions, were scored by the Bliss combination index⁸³. Their model analysis enabled development of a theoretical treatment decision tree, highlighting how complex treatment combinations can be mathematically optimized, which could be possible for individual patients in the future.

ICI therapy can also be personalized through secondary treatment with non-traditional molecular-based drug interventions. Benchaib et al.⁸⁴ examined tumor–immune interactions under anti-PD-L1 treatment with a focus on epithelial growth factor (EGF)-mediated tumor cell proliferation (high EGF) or dormancy (low EGF) conditions within the lymph node compartment in an in silico study. Using both a population and a multiscale model (wherein a lattice-free ABM was hybridized with PDE-based descriptions of molecular concentration profiles), they identified key levels of anti-PD-1/anti-PD-L1 and EGF necessary to achieve tumor elimination. Interestingly, they observed that even though EGF promotes tumor proliferation, at high levels it is also able to supplement lower levels of anti-PD-1/anti-PD-L1 therapy, possibly by overcoming cell dormancy-induced drug resistance. This suggests the counterintuitive treatment strategy that, in some circumstances, exogenous EGF delivery may actually improve ICI therapy in secondary tumors within the lymph node. In another study, Wei and colleagues examined the mechanisms of cytotoxic CD4⁺ T cells and tumor-suppressing IL-4 cytokine therapy used alone or in combination at various tumor stages, dosing regimens and levels of immune system health⁸⁵. This model revealed that regions of tumor response instability may exist within some treatment dosage schedules or ranges, and the difference between success or failure of treatment may hinge sharply on very small changes in the treatment strategy. Other groups have examined in silico the influence of promoting tumor vasculature to facilitate tumor infiltration by subsequently delivered T-cell-based immunotherapies on maximizing therapeutic response^86,87, highlighting the importance of tailoring dosage scheduling to the specific mechanisms and responses of individual cancer types.

Another combination therapy approach to improve immunotherapy outcome is ancillary radiation therapy (RT)¹¹. RT reduces tumor mass via DNA and subcellular damage, and upregulates the release of tumor-specific antigens that sensitize the immune response to the tumor⁸⁸. RT can also be immunosuppressive, which can upregulate PD-L1 expression⁸⁹. The complex interplay of ICI therapy when delivered alongside RT-mediated pro- and antitumor effects presents substantial challenges when it comes to personalizing treatment strategies: a problem that has been extensively studied by Serre and colleagues using mathematical modeling⁵¹. This work includes separate biological mechanisms for anti-PD-1/anti-PD-L1 and anti-CTLA4 therapy, and for RT therapy, and accounts for both primary and secondary immune responses in a system of five linked discrete time equations. Through model analysis, they were able to reproduce in silico a set of experimental and clinical results, including tumor mass after treatment, probability of metastasis rejection and patient survival. This work also highlights the importance of treatment scheduling when using ICI + RT combination for maximizing synergy and treatment efficacy. The impact of treatment scheduling was later explored in greater detail by Kosinsky et al.⁹⁰, who used a quantitative systems pharmacology ODE model, this time calibrated to experimental data, to further explain the importance of scheduling and dosing for ICI + RT as a function of the level of disease progression and yielding mechanistic insights as to why suboptimal treatment design may fail.

Perspectives and future directions

Moving forwards, we expect that such engineered, computation-based immunotherapy treatment strategies will become a critical part of the next-generation therapies by facilitating clinical translation of new drugs⁵⁹ and optimizing personalized treatment strategies for maximized therapeutic success^91,92. We believe that such applications will include identifying new biomarkers, utilizing systems approaches⁹³ to expedite drug development and deployment pipelines, and enabling clinical design of personalized treatment strategies.

Discovering new biomarkers

Physiologically relevant mechanistic mathematical models are able to not only link inputs and outputs but also describe, simulate and predict the kinetics of oncological processes and treatment from first principles of biophysics. Mechanistic modeling has shown impact on the discovery of novel, accurate imaging-based biophysical markers of patient outcome of immunotherapy treatment^72,75, as this approach facilitates rapid testing of therapeutic outcomes across a wide parameter space of possible host and tumor conditions. Upon application of parameter sensitivity analysis, the key players and bottlenecks in disease response or escape can be identified^94–96. Because the relationship between model parameters and the biological processes or mechanisms they represent is known, such analysis provides valuable guidance on optimal biomarker selection, either by direct measurement of the represented biology, or by more readily measurable related parameters (for instance, blood neutrophil and lymphocyte counts as surrogates for tumor–immune infiltration^97,98).

Alternatively, mechanistic models can provide a platform to design quantitative metrics (we call them mathematical biomarkers) containing information from multiple key underlying factors into a small number of easy-to-interpret scalars. As shown before^72,73,75,76, through biologically valid assumptions, a set of reliable, broadly applicable biomarkers of immunotherapy response were derived from converting a full model (composed of multiple equations) into a simplified version. These mathematical biomarkers were found to be predictive of patient response and patient survival. More importantly, quantification of these mathematical biomarkers relies only on standard-of-care imaging or biofluid measurements. Hence, these mathematical biomarkers can be potentially used in the clinic to provide timely identification of the most effective immunotherapy treatment protocol for an individual patient, which would allow physicians to select or fine-tune the treatment strategy to maximize efficacy. The use of easy-to-interpret scalar measures remains the foundation for many health measures, from basic health (for instance, body temperature, body mass index) to complex assessment of disease response (for instance, iRECIST⁹⁹ for cancer response assessment), and it is likely that similar standard criteria will more readily achieve clinical adaptation in the future.

Integrating with artificial intelligence

The technology enabling measurement of increasing numbers of biomarkers has led to the push for large-scale data collection, and multimodal and multi-omics data¹⁰⁰ (from high-throughput sequencing to automated screening) in both pre-clinical and clinical settings is becoming more readily available. As a result, the need for readily interpretable measures or markers, which can simply communicate overarching conclusions from integrated analysis of large datasets, becomes paramount to future clinical adaptions. Precision oncology is already moving to incorporate the latest ‘omics’, including genomics, proteomics, metabolomics and epigenomics, as well as therapeutic response monitoring, imaging technologies and analysis of the resulting combined high-dimensional datasets^101,102. Analysis of these large datasets will require new, complementary and synergistic approaches based on the quantitative sciences to identify the most important diagnostic information and new treatment options.

Artificial intelligence (AI) has substantial potential to enable precision oncology¹⁰³. AI methods are being applied to multi-omics or imaging datasets across many diagnostic, prognostic and predictive domains in oncology¹⁰⁴. In immunotherapy treatment, self-learning methods have been used by Cuplov and colleagues to further understand the complex immune–tumor interplay in combination (immunotherapy + chemotherapy) treatment¹⁰⁵. The ability of AI has also been reported to meet or exceed the performance of human experts in analyzing medical images^106–108. Despite offering powerful predictive methods for clinically relevant applications, the ‘black box’ nature of standard, well-established machine learning models¹⁰⁹ remains at odds with basic science research, which must capture and understand the underlying mechanistic causal links to identify specific biological targets for further study and to design drugs to control these molecular pathways.

Generally speaking, incorporation of biological knowledge into a modeling structure is not particularly a strength of existing data-driven AI models, essentially because the focus of which is on learning with data (using algorithms) to deduce the input–output mappings^110,111. The lack of mechanistic knowledge in such models can also limit predictive accuracy, as this makes intelligent refinement of model design and choice of data inputs difficult¹¹². Mechanistic modeling of the cancer–immune response can help overcome these shortcomings by incorporating the biological and physical laws and functional relationships known a priori (for instance, regarding the complex mechanisms of the cancer system and the action of drug agents), as well as novel biological hypotheses, into the modeling structure. Conversely, AI approaches could be used to identify the most important factors to include in mechanistic models. We believe it is likely that AI and mechanistic modeling will be combined into ‘mechanistic AI’ models that take advantage of the strengths of both approaches to overcome the weaknesses inherent to each method, yielding powerful new strategies to optimize clinical predictions. This work is already underway and showing promising results, especially in the area of physics-inspired neural networks^113,114. This new class of methods incorporates mechanistic modeling by minimizing residuals to ensure a mechanistic equation is being solved.

Towards real-world application

Mathematical modeling continues to advance, and we have witnessed that many immunotherapy models are moving from a tool used exclusively for hypothesis generation in basic sciences towards clinical applications. This transition necessitates changes in how models are designed, so that from conception they are engineered to be usable in real-world settings. In the clinical world, there are many more constraints on what can be done with patients in terms of measurements, and there are only limited numbers of measurable parameters available for modeling use (compared with basic science work and wet-lab experimental work). Thus, models must be able to be fully characterized by existing measurement techniques¹⁹, and should provide outputs that motivate widespread adoption to be accepted by frontline treating physicians in clinical oncology. Computational scientists should have a greater understanding of this fact when they work with physicians and oncologists to have a chance to translate their modeling work to the clinic. In Fig. 3, we present a conceptual strategy for developing and simplifying mathematical models to accommodate different clinical situations, which essentially suggests the need for models built on fewer—but clinically measurable—parameters. This strategy takes advantage of current advances in both AI (efficient in handling the massive amounts of data) and biology/immunology (for example, new discoveries regarding the complex mechanisms of the cancer system and the action of anticancer immunotherapy agents) in the process of model simplification.

Fig. 3 |. Conceptual design of a strategy for simplifying mathematical models for clinical application by accounting for current advances in both AI and cancer biology.

The strategy has two overarching phases: modeling and application. In the modeling phase, a full model is developed; at this stage, the model can be complex, attempting to capture many processes in cancer immunotherapy intervention. Please note that we present only a general form of a nonspecific multivariate model function for illustration purposes here, and it should not be taken to be explicitly representative of any specific system. The development of the full model is based on the clinical question at hand and what data (multimodal and multiscale) are available. Both AI and prior knowledge or hypotheses regarding the cancer biology (or immunology) can be used to help simplify the model for clinical use. At the same time, valid biological, physiological and/or physical assumptions^72,73,75 can be made to remove unimportant processes or unmeasurable parameters from the full model. Used together, these two methods may prove to be more efficient. The simplified model must be validated with retrospective or prospective data, in an iterative process between model simplification and validation (with data) to minimize model complexity while maintaining the predictive power. In the application phase, clinical oncologists use the simplified model to make predictions about a certain treatment plan (dose, dosing schedules, drug combinations and so on) for a given patient. There will be a minimum data availability requirement for using the model, for example, standard-of-care MRI has to be available. This is important, because not all data will be available in all patients in different clinical situations, for example, one patient may only have MRI data (situation 1 with x), and another patient may have both MRI and pro-inflammatory markers data (situation 2 with x and y). In any situation, the model can be run to render predictions within a reasonable time window (for example, on the scale of minutes), which is critical in current clinical practice. This strategy can also be applied to investigation of other biological or treatment systems.

Using this ‘less is better’ strategy (primarily, the simpler the model is, the easier the model or computational tool can be accepted in the clinic) reduces over-parameterization and overfitting while promoting structural and practical identifiability¹¹⁵. Powerful methods to ensure model identifiability are being studied¹¹⁵ and are proving useful for model design, for instance, as demonstrated by Okuneye and colleagues in a recently developed, experimentally validated ODE model of combination anti-PD-1/anti-PD-L1 with anti-FGFR3 (fibroblast growth factor receptor 3) immunotherapy¹¹⁶. Arguably, this may represent a transition towards models that are less mechanistic and are more phenomenological; however, it should be recognized that at some level all mechanistic models contain some portion of phenomenological descriptions due to modeling limitations, unmeasurable (or estimated) parameters, and our incomplete understanding of the mechanisms (and their interplay) in biological systems. Ultimately, the balance that is necessary to achieve successful clinical translation with sufficient modeling detail for accurate mathematical predictions must be discovered and perfected by modelers.

Moreover, experimental validation of mathematical models is essential to verify their ability to provide accurate predictions¹¹⁷, and decisions should not be based on predictions from poorly validated models. It is generally more difficult to perform experiments to validate models as parameter numbers increase. This underlines the importance of carefully designing models with parameters that can be assigned values that later can be obtained from experiments. The modeling field is now transitioning to prospective validation, where verified models are used to customize treatment strategies (different dosage, dosing schedules) that are then validated in vivo^118,119. Indeed, translation of mathematical modeling to clinical applications will probably follow a similar path to other tools and drugs, from conception to in vitro and in vivo testing, and to full clinical translation only after sufficient experimental validation is complete. Another key step towards clinical translation is creating open-source repositories (of data, model and protocols) where mathematical models can be tested by other groups independently.

Going forwards, mathematical modeling may inspire new measurements, and similarly, new measurements in the clinic may arise from the development of new technologies in the clinic, which will probably enable inclusion of additional parameters at that time. If the modeling field is to achieve its stated goal of clinical adoption and improvement of personalized treatment strategies, then modelers must recognize that they are in fact designing for acustomer: the physician, the patient or the wet-lab drug developer. We hope that computational scientists will continue to strive for more collaborative efforts with physicians and oncologists to bridge this gap and increase the likelihood of translating their modeling work to the clinic.