Advancing Cancer Drug Development with Mechanistic Mathematical Modeling: Bridging the Gap between Theory and Practice

Abstract

Quantitative predictive modeling of cancer growth, progression, and individual response to therapy is a rapidly growing field. Researchers from mathematical modeling, systems biology, pharmaceutical industry, and regulatory bodies, are collaboratively working on predictive models that could be applied for drug development and, ultimately, the clinical management of cancer patients. A plethora of modeling paradigms and approaches have emerged, making it challenging to compile a comprehensive review across all subdisciplines. It is therefore critical to gauge fundamental design aspects against requirements, and weigh opportunities and limitations of the different model types. In this perspective, we discuss three fundamental types of cancer models: space-structured models, ecological models, and immune system focused models. For each type, it is our goal to illustrate which mechanisms contribute to variability and heterogeneity in cancer growth and response, so that the appropriate architecture and complexity of a new model becomes clearer. We present the main features addressed by each of the three exemplary modeling types through a subjective collection of literature and illustrative exercises to facilitate inspiration and exchange, with a focus on providing a didactic rather than exhaustive overview. We close by imagining a future multi-scale model design to impact critical decisions in oncology drug development.

1. Introduction

Cancer is a complex and multi-faceted disease that involves various biological processes occurring at different scales of space and time. There exists a handfull “hallmarks” of cancer, being top-level or meta-phenotypic properties that arise from the dysregulation and coordination of multiple aberrant processes across molecular, cellular, organ and system-level levels in space and time [1]. While in recent years, modern cancer medicine (e.g., precision oncology [2] or immuno-oncology [3], [4]) has technologically matured, has become more visible, and has produced tangible results, there are also sceptical voices raising the concern that currently pursued strategies are flawed (stating for example that “current strategies to control cancer are demonstrably not working” [5] and that “reductionist approaches to cancer chemotherapies including recent immunotherapy” are responsible for high mortality rates and failure rates in oncology drug development [6]). According to a recent “landscape” review [3] there seem to be two major limitations of the current development paradigm: first, targeted therapies tend to focus on specific genetic aberrations or mutations in smaller and smaller fractions of the population; second, immunotherapies rely on combination therapy approaches for broader effects, either with other immunotherapies or with more classical (and toxic) modalities such as chemotherapy or radiation therapy. These intrinsic limitations come with the ever-growing complexity clinical trial design that need to take into account testing multiple combinations of cancer and immune targets in different subpopulations of patients, and potentially many doses and drug regimens. Especially for combination therapy, it can be akin to finding a ”needle in the haystack”, as the number of possible permutations and combinations can be vast, making it difficult to systematically explore all potential options. These difficulties advocate the development and application of innovative approaches in clinical trial design, patient selection, and data analysis.

Mathematical modeling has been recognized as a powerful tool in oncology to improve the understanding of cancer dynamics, optimize treatment strategies, and guide clinical decision-making. The ultimate goal of cancer (drug) modeling is often defined as creating “predictive tools that are both patient-, and tumor-specific”. Further, “after refinement and parametrization with appropriate data”, in turn, these in silico tools should have the ”potential to be used in a clinical setting to aid in treatment protocols and decision-making” [7].

Over the last tens of years, a vast body of modeling works has emerged in the computational oncology literature and some attempts have been made to classify modeling types as statistical and inferred models, (bio)chemical and signaling networks, continuum and agent-based tumor, and tumor environment models [8]. Cancer models can range from simple data-driven models to complex knowledge-based (hypothesis-driven) models, and they can serve various purposes along the spectrum from fundamental mechanistic understanding to clinical decision-making [9]. They can contribute to the research and development as well as market access of new drugs. Examples of phenomena covered in the models are often the effect of cancer treatments, biomarkers of pharmacological activity, tumor evolution, as well as survival metrics (see a review by [10]), but also aspects of toxicity [11, 12].

Mathematical models in oncology can provide valuable digital evidence on the mechanism of action, pharmacological effect, and safety of new drugs (see e.g., [13]), which can play crucial roles in informing regulatory decision by major regulatory agencies such as the European Medicines Agency (EMA) ([14]) and the U.S. Food and Drug Administration (FDA) [15]. The model-informed drug development (MIDD) program by the FDA fosters the use of modeling and simulation as an essential ingredient of application dossiers, formally recognizing a dedicated model as an “important enabler of efficient and effective drug development”, as stated in the Prescription Drug User Fee Act (PDUFA) reauthorization performance goals and procedures for fiscal years 2018 through 2022 (PDUFA VI)[16]. The regulatory application of MIDD can be broadly classified into four categories: dose optimization, supportive evidence for efficacy, clinical trial design, and informing policy. The umbrella of MIDD is quite large and, while some model-based analyses have become routine and well accepted, others are still evolving and being evaluated[17]. Lately, FDA has initiated a program fostering to pivot from a “the more the better” paradigm towards dose optimization in early oncology drug development (Project Optimus). Here modeling will become instrumental [18].

The most widespread applications of quantitative oncology models use a rather pragmatic, data-driven paradigm: tumor growth dynamics (TGD) or tumor growth inhibition (TGI) models consist in stepwise regression or fitting of simplistic mechanistic models to population or individual longitudinal tumor or lesion size data (using e.g., exponential, logistic, Gompertz models, see [19]; for the latter, see also [20] or many others). The quantitative descriptive capacity of such models when combined with exposure dependent treatment effects (e.g., linear or exponential tumor cell killing depending on the drug’s pharmacokinetics parameters like the area under the curve), allow to support many clinical trial design and optimization tasks, to inform dosing strategies, and to – generally – support decision-making in both, early and late drug development.

TGI modeling for early clinical development leverages preclinical data to for example inform a first in human phase I trial design. The pre-clinical data available, consists mainly of in vitro (cell line, co-culture, primary human samples) experiments and also xenograft animal studies. Although these experimental models are only crude simplifications of human cancer, they are commonly used to assess drug exposure-response relationships and estimate potential antitumor activity and effective exposure in humans [21–23].

TGI models are also commonly used to support later regulatory decisions, such as end-of-phase II decisions, and inform the design of phase III studies. TGI models can be used to simulate survival outcomes based on tumor shrinkage data observed in early clinical studies, such as Phase Ib and II studies, and extrapolate these results to a phase III setting in patients with different baseline characteristics and treatment duration (see for example [24]) with the aim to optimize the design of such studies. Dose selection can be supported for example through dose-exposure response TGI modeling or (non)linear mixed effects modeling with stepwise covariate addition. Modeling survival and dropout in TGI models typically involves the use of hazard models, which take into account tumor size-related covariates and can capture the correlation between treatment response and dropout (patients that do not respond usually drop out quickly). As the tumor size data is the direct output of the deterministic, mechanistic model it can be employed as basis for these time-varying covariates. Hazard models[25], such as those used for modeling death and dropout in the context of immunotherapy with Durvalumab or other treatments, as well as parametric survival models based on performance status (e.g. measured by the Eastern Cooperative Oncology Group, ECOG, score [26]), baseline tumor size, and week 8 tumor size, are often phenomenological in nature and possess constrained function forms ([27]), and thus they are rather specific to the dataset applied to and less well suited feeding bottom-up evidence needed for translational predictions (e.g., from preclinical data of a related but new treatment modality).

Despite this undoubtedly important contribution to drug development, simple models cannot leverage the existent knowledge of different sources of heterogeneity in cancer. Cancer is a highly heterogeneous disease, and intra-tumoral heterogeneity, tumor structuration (spatial heterogeneity) and clonal or phenotypic diversity and evolution, and the tumor microenvironment are crucial features that contribute to the variability in prognosis and treatment response among patients. To capture the complexity of tumor behavior and treatment response, it is necessary to employ mechanistic models that incorporate these crucial features. Such models offer a more comprehensive and realistic representation of cancer dynamics, enabling the optimization of treatment strategies and the prediction of clinical outcomes. (cf Figure 1).

Fig. 1.

Different factors of heterogeneity within one cancer patient and even within one tumor. Figure created with Biorender.com

Complementary to a pragmatic and data driven top-down modeling approach, systems biology offers a valuable approach to cancer modeling that focuses on capturing the emergent behavior of cancer through a bottom-up understanding of the underlying biological processes. This approach seeks to integrate and model the complex interactions and dynamics of multiple components, such as genes, proteins, cells, and tissues, in order to understand the system-level behavior of cancer as a whole. Systems biology of cancer, however has several bottlenecks. First, cancer is a very complex disease with multiple interacting factors at different spatial and temporal scales, and therefore modellers need to “couple biological processes occurring at various spatial and temporal scales (from genes to tissues)”[28] in order to “capture all the dynamics of solid tumour growth and its progression”. Second, the tumor microenvironment and the intricate interplay between cancer cells and their surrounding microenvironment play a crucial role in cancer growth, progression, and response to treatment [29]. Understanding the mechanisms involved in the tumor microenvironment and how they impact cancer growth and microscopic events involved in metastasis, and the characterization of treatments provoking resistance by several mechanisms, is a challenging task that requires integration of experimental data, mathematical modeling, and computational simulation and trials [30].

A comprehensive cancer outcome prediction workflow has been proposed by [7], presenting a hybrid cellular automaton model that operates at multiple scales and use multiple cell-line–specific data to predict various pertinent tumor and treatment scenarios. Not always however, have modelers access to a complete data package to inform all parts of a multi-scale mechanistic framework, so that modelers have to revert to drastic approximations and simplifications. The intermediate granularity of a cancer model, which lies between the molecular and clinical scales, can also be influenced by data availability.

Different applications will require different levels of complexity in cancer modeling, depending on the specific research question, available data, and desired level of detail. For tumor growth dynamic models, a systematic way forward is to detail progressively physical processes underlying tumor growth and to overcome the limitations of the phenomenological models (i.e., linear or even saturated growth models). For example, incorporating cell-level behaviors such as proliferation, migration, and apoptosis, and their regulation by molecular signaling pathways, can provide a more mechanistic understanding of tumor growth dynamics. Including tissue mechanics, such as the effects of mechanical forces on tumor growth and invasion, can further enhance the realism of the model. However, it’s important to note that increasing the complexity of a model also presents challenges in terms of data availability, parameter estimation, and model validation. It may also be necessary to strike a balance between model complexity and model interpretability, depending on the specific research question and intended application of the model.

Complex multi-scale cancer models often require a large number of parameters to capture the intricate interactions and dynamics of biological processes occurring at different scales. Generally, the more complex a model is, the more parameters may need to be estimated and validated, the more experiments and clinical trials have to be conducted to determine the needed parameters of the model. On top of complexity of this task in complex multi-scale cancer models, patient-specific model parameterizations have to be used to account for realistic variabilities of the response to treatment and in order to identify responding patients in silico. Dedicated to this challenge, [31] report a clinically-relevant parameterization methodology for a patient-specific computational model of cervical cancer response to radiotherapy treatment with concomitant cisplatin. At this stage, however, the sensitivity to cisplatin as well as the extent of hypoxic regions responsible for resistance to radiotherapy are required to be supplied as input to the model rather than to be simulated as variables within model. In such cases, modelers may need to use empirical data or experimental measurements to estimate these parameters and provide them as inputs to the model. More generally it is important for mechanistic models to not only describe an average behavior but also inter-patient variability and how that variability evolves over time[32]. Capturing this variability in the modeling workflow is often termed a Virtual Population (Vpop) approach. Vpop based workflows typically comprise generation of (random) parameter co-distributions, and the subsequent calibration of either the the distribution meta-parameters or subsampling from a larger plausible Vpop with probability reweighing (see also [33]). Then, population predictions, subsampling, trial simulation, or individualized risk or therapeutic benefit [34] can be done. With respect to the latter, Susilo et al. have demonstrated the application of a bispecific T-cell engager QSP model and digital twins derived from Phase I oncology study to support dosing-regimen choices and biomarker selection[35]. Lately, the congruence of artificial intelligence and machine learning with QSP modeling has been discussed and for example workflows to boost Vpop calibration through model emulation techniques have emerged[36]. Nevertheless,there are still many challenges for virtual patient generation and the development of digital twins; especially in immunology[37]. Simply presenting ranges of parameter values does not fully address the important considerations of how to select which parameters to vary and how to refine these ranges. The methods for selecting parameters to vary (e.g., using Local Sensitivity Analysis (LSA), literature consensus, or experimental evidence) and refining these ranges are indeed critical features of a Vpop workflow. By incorporating these methodologies, a more robust and transparent approach to Vpop calibration can be provided.

In line with the more detailed understanding of cancer biology, complex mechanistic models also promise to contribute to the development of biomarkers for response, that would allow for better selection of the target population in clinical trials and the clinic. The impact of biomarker programs on clinical development cannot be overestimated; the probability of success (PoS) of a drug program evidently strongly depends on the availability of such markers. According to [38], ”using biomarkers for patient selection is effective in almost all diseases within oncology, raising the estimated overall PoS by an average of 13.3 percentage points”. Biomarkers identified by the single-gene level in one study may suffer from limited replicability in predicting patients’ outcomes under different settings. A recent suggestion to overcome this limitation is to employ (see [39]) pathway-based strategies to extract important features in genome-wide expression data. Here, more complex mechanistic models, such as systems pharmacology models or mechanistic systems biology models, are much more suited to support biomarker programs as they can integrate multiple data sources, including molecular, cellular, and clinical data, to capture the complex interactions and dynamics of the biological system. These models can provide a mechanistic underpinning to the relationships between measurable and non-measurable molecules, receptors, cells, and other components of the biological system, and can quantitatively explain the extent of treatment response.

By integrating complex mechanistic models with data from early stages of drug development, aligning them with the clinical data generation other potentially more classical MIDD techniques, a comprehensive and integrative understanding of the relationship between biomarkers, drug exposure, and clinical outcomes can be achieved. Despite the complementarity and potentially high return of investment, complex mechanistic modeling seems still underrepresented in oncology drug development. In this perspective report, we would therefore like to make the case for mechanistic modeling in clinical drug development – aiming to explain, rationalize, suggest and quantify factors determining response to therapies. Without delving into excessive mathematical and mechanistic details, it is our aim to shed light on the key working principles of multi-scale modeling culminating in the prediction of clinical outcomes at the single patient and whole population level. We will give a few examples of how complex models are currently used in cancer research and drug development and what potential applications such models could target in the future. Along this way, we suggest literature for further reading and detailed examples, in a subjective and non-exhaustive way.

2. Space structured models may inform therapy tailored to individual tumor morphology

2.1. Modeling transport of nutrients that limit tumor growth

As stated before, the most widespread application of tumor modeling are simplistic, data-driven models such as logistic models. These typically read $dN/dt=rN(1-N/K)$ where $N$ is the cancer cell population, $r$ is the growth rate scale and $K$ the carrying capacity. Already such a simple model has a mechanistic component: it takes growth saturation into account, but does not specify from which phenomenon this growth saturation is actually coming from.

One known limitation to solid tumor growth is the oxygen/nutrient deprivation [40, 41] of the cancer cells in the center of the growing tumor. One might therefore start modeling a more mechanistic tumor growth dynamics by including growth rate-limitation by the diffusion and consumption of nutrients, oxygen, and other relevant factors within the tumor microenvironment. Here, typically a partial differential equation (PDE) modelling approach is undertaken to describe the spatio-temporal distribution of these factors (see for example [42]). In the simplest case, we assume a radially symmetric tumor that stays non-vascularized so that its growth rate (as defined later) will be limited by spatial distribution of diffusible chemical species over time $c_i(r,t)$ short $c_i$ that serve as nutrients. In this simplified approach the concentrations of nutrientsuch as oxygen or glucose, $i=\text{glucose}$, ${\text{O}}_2$. will be described by a diffusion-reaction equation including a term of the consumption of oxygen and nutrients in the tumoral tissue.

$$\frac{\partial c_i}{\partial t}=\frac{D_i}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial c_i}{\partial r}\right)-{\text{Γ}}_i\left(c_i,r\right)$$ (1)

Here, and in the following $r$ is the spatial coordinate in spherical symmetry. The first term describes the diffusion with a diffusion constant for that species $D_i$ and the second term $-{\text{Γ}}_i\left(c_i,r\right)$ represents set of biochemical reactions (such as consumption, transformation, etc.) at a given point in space.

This PDE modeling approach can be transformed into a set of ordinary differential equations by finite differences discretization schemes (finite element methods or finite volume methods) that can be solved using numerical solvers and have been described extensively by [43].

In order to highlight the practicality of such an approach, we have devised the simplest possible discretized PDE nutrient diffusion approach encompassing oxygen supply from the outside of the tumor, oxygen diffusion through the bulk of the tumor and oxygen consumption due to metabolic activity, which itself is regulated by a transcriptional program (e.g., via HIF-$1\alpha$ expression).

$${\text{Γ}}_{O_2}\left(c_{O_2},r\right)=-V_{max}^{O_2}\cdot \frac{1}{1+{\left(\frac{c_{O_2}}{K_m^{HIF-1\alpha }}\right)}^{\omega }}$$ (2)

Note here, that $c_{O_2}=c_{O_2}(r,t)$. $-V_{max}^{O_2}$ is a (maximum) rate of consumption, $K_m^{HIF-1\alpha }$ is the setpoint of ${\text{O}}_2$ concentration where HIF-$1\alpha$ exerts half of its effect on switching from aerobic to anaerobic metabolism, and $\omega$ a shape parameter of that switching function. Note also that this term may have additional dependence on location $r$, which not detailed here.

With that diffusion-reaction model we have evaluated example oxygenation as a function of tumor size; the approach and results are given in Figure 2.

Fig. 2.

a) Simplified schema of the utilized diffusion-reaction partial differential equation approach (see equations 1 and 2) in spherical coordinates to calculate radius-dependent oxygenation profiles. b) example oxygenation profile of a $R=25mm$ tumor For variability, the metabolic regulation setpoint was varied setting $K_m^{HIF-1\alpha }=[0.15\text{\%},1\text{\%}]$ (light blue lines) and tissue density dependent oxygen diffusion coefficient was also varied by setting $D=[1\text{e}-9m^2/s,1\text{e}-7m^2/s]$ (dark blue lines). At the outside of the tumor $r<R$ Dirichlet boundary conditions apply assuming that the oxygen supply keeps tissue oxygenation at 5% (which is dependent on the tissue [147]). Inside the tumor at $r=0$ Neumann boundary conditions with no diffusion apply. $V_{max}^{O_2}$ has been set to 10e − 2 %/s and $\omega$ was 1. Initial conditions were 5% throughout, and solved to steady state oxygenation profiles. Hypoxia defined as $O_2$ concentration below 1% [147]. In c) and d) we compare the simulation of hypoxic volume of the tumors with experimental data reviewed by Hildingsson et al. [148] from which we digitized the 8 first studies reported in the supplementary information.

Generally, (as cancer cells need nutrients and oxygen) the tumor growth could be described with an integro-differential equation, e.g., for change in volume over time, that describes proliferation versus death per volume element whose properties are governed at a given time by the diffusion reaction problem. This growth equation in spherical coordinates could read:

$$\frac{d{V}^{Tumor}}{dt}={\int }_0^R{r}^{2}P\left(r,c_i\right)dr-{\int }_0^R{r}^{2}D\left(r,c_i\right)dr.$$ (3)

$\text{R}$ as upper integration limit determines the outer radius of the tumor. $P$ and $D$ are the growth and death rates (varying in space and time). Note, again that concentrations $c_i$ depend on space and time in general.

An equivalent formulation describes the growth of a cell population by per capita growth and death rates and then determine the radius and volume of the corresponding spherical tumor. Both formulations can be interconverted by introducing a single cell volume $V_c$, that is experimentally available for several histologies (and a descriptor quantifying the the stromal portion of the tumor) so that the number of cells per tumor volume (element) $N_c(t)=V^{Tumor}/\left(2V_c\right)$ (when half of the tumor volume is assumed to contain stromal material). The integral may be conveniently approximated by a sum over volume elements so that the whole growth model can be expressed as one ODE system (for a given discretization of the radial coordinate).

To illustrate the functioning of the reaction-diffusion model, we have evaluated tumor structuration by calculating $O_2$ profiles for a series of tumor sizes $R=[1mm,100mm],$ of which we show the one for a size of $R=25mm$ in Fig 2 b) We added potential between-patient variability (e.g., differences in HIF-$\alpha$ expression sensitivity and extent and tissue density, vascularization, respectively) under tumor growth. We thus can represent a vast variety of sources of spatial heterogeneity in potential patients.

On top of potentially explaining specific morphologic features for a given size and patient, such mechanistic model can also allow for more mechanistic growth modeling of a tumor. In fact an emprirically logistic (saturated) growth can be reproduced with the integro-differential one given in eq. 3 that for example has positive growth rate for normoxic regions, zero growth for hypoxic regions and net negative growth for necrotic regions. Compared with patient specific tumor size data(e.g. from pre-treatment CT series), the mechanistic and explicit diffusion reaction approach allows to include more markers for alignment with patient data. In particular the model allows to explore baseline or pre-treatment PET/CT imaging (indicating metabolic activity, structuration and size) and more generally radiomics as potential predictive markers for explaining differences in prognosis and response to treatment.

2.2. Modeling the limitation of transport of drugs into tumors

The spatial tumor framework introduced in subsection 2.1 can also be reformulated into other geometries to investigate phenomena within a tumor microenvironment, of which we here focus on intratumoral drug distribution post-vascularization. Assuming for the local environment of the tumor tissue connected to a capillary that 1), the capillary does not bend over the length scale of molecular diffusion; and 2), the tumor tissue is symmetric along the height and azimuth of the capillary; one can model movement of diffusible molecules from the capillary wall radially into the tumor in cylindrical coordinates, a framework called a Krogh cylinder [44–47]. Material passes through the capillary wall (at coordinate $R_{\mathit{cap}}$) and diffuses radially into the tumor to some maximum distance $R_{\mathit{kro}}$ (see e.g., Figure 1C in [47]). Similar to the example of subsection 2.1 the distal boundary of the tissue space is often enforced with a zero-flux (Neumann) boundary condition, while entry into the tissue space can be treated as either a concentration or transport flux. The concentration of the molecule of interest can therefore be described as

$$\frac{\partial M}{\partial t}=\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial M}{\partial r}\right)-V(T(t,r),M(t,r))$$ (4)

Where $M(t,r)$ is the concentration of the molecule of interest at position $r$ from the capillary wall at time $t$, $T(t,r)$ is the tumor cell density at the same position, and $V(T,M)$ represents the reaction terms driven by interaction between the molecule and the local cells (such as target binding/unbinding, pinocytosis, or others).

Tumor cell density, while a function of both space and time, need not be modeled as a PDE if tumor cell motility is not considered. Krogh cylinders have been applied not only to diffusion of nutrients analogous to the microcellular spheroid described previously [48], but also therapeutic agents [49, 50]. In typical applications $M(t,r)$ is obtained from a (physiologically based) pharmacokinetic model. This combination can effectively describe intra-tissue drug distribution as a function of in flow limitations and diffusion limitations, dosing regimen used, and the specific reactions (i.e. non-linear binding kinetics) of the drug. Applications of this geometry have generated novel and nonintuitive hypotheses about the ability of antibody-based directed toxin therapies such as antibody/drug conjugates (ADCs) and immunotoxins to penetrate a solid tumor mass (see for example [51, 52]). When a highly affine antibody is directed against a tumor-associated antigen with high expression levels, the antibody can paradoxically fail to diffuse far from the capillary wall in mechanistic models. This is due to high amounts of the antibody binding to its target immediately upon distribution through the capillary wall, termed the binding site barrier effect [52–56]. Simulations designed to better understand this effect have hypothesized that addition of unconjugated antibody along with the ADC leads to better overall penetration of toxin far from the capillary wall [55]. One may hope that such modeling will solve some of the questions around ADCs, for example why the payload normalized maximum tolerated doses of ADCs do not seem to differ with respect to the unconjugated small molecule drugs themselves [57] or what biodistribution (kinetics) – related molecular characteristics (affinities, linker stability, etc.) may explain the clinical trial (success vs. failure) landscape [58].

3. Ecological and evolutionary modeling to optimize treatment combination, regimen and dose

A widely accepted concept in cancer biology describes a set of fundamental traits or characteristics that are commonly observed in many cancer types [59] – the hallmarks of cancer. In parallel, the ecological perspective of cancer, as introduced by Peter Nowell [60] and often referred to as ”eco-oncology” or ”cancer ecology” [61]), has gained recognition in recent years as a valuable framework for understanding the complex dynamics of tumor growth and evolution and the ecological perspective of cancer. According to this perspective, a tumor is not just a mass of identical cancer cells, but rather a dynamic population of heterogeneous malignant cells that interact with each other and with their microenvironment in ways that influence their growth, survival, and evolution. Just like in ecological systems, interactions among different populations of cells within a tumor can involve competition for resources, cooperation among cells, predation by immune cells, and evolutionary diversification through genetic and epigenetic changes [62].

Recently, Somarelli proposed an intriguing perspective by considering the hallmarks of cancer as ecological fitness parameters [63] that influence the phenotype and trait distribution in the population of cancer cells (and which should be connected with the set of i.e. acquired somatic mutations). According to this viewpoint, these hallmarks of cancer, such as sustained proliferative signaling, evading growth suppressors, resisting cell death, can be considered as traits that confer fitness advantages to cancer cells within their microenvironment. Cancer cells with specific genetic or epigenetic changes that confer these fitness traits may have a higher likelihood of survival and proliferation, leading to a selective advantage and dominance within the tumor population (meaning that clones with increased fitness become more prevalent in the tumor mass over time). Therefore, understanding and modeling the dynamics of trait distribution and phenotype heterogeneity within the tumor population becomes important to inform tumor growth, progression, and response to therapy.

In many cancer types, driver mutations, which are somatic genetic alterations that contribute to tumor initiation and progression, tend to occur early in tumor development and are present in all tumor cells, referred to as truncal mutations [64]. These driver mutations can confer a constitutive fitness advantage to the tumor cells, allowing them to survive, proliferate, and propagate their genotype within the tumor population. Consequently, the evolution of the tumor shows a common pattern of early clonal genome doubling, followed by extensive subclonal diversification. This diversification leads to the emergence of distinct subpopulations of cancer cells with different genetic alterations and phenotypic traits, represented as branches in the phylogenetic tree of the tumor. Opposed to driver mutations (that confer a fitness advantage), passenger mutations are genetic alterations that do not confer any fitness advantage to tumor cells and do not directly contribute to tumor growth or progression.

During cancer treatment, an additional selection pressure is exerted onto the cancer cell population. Clones that harbor a trait (e.g., a driver mutation) that equips them with less sensitivity (or even resistance) to the treatment suddenly gain in fitness while they might have been out-competed in the treatment-free setting. Also, passenger mutations (that are numerous when for example genome instability drives the appearance of new mutations) have an indirect effect on resistance through increasing heterogeneity. The overall level of genetic diversity within the tumor population influences the potential phenotypic differences among tumor cells, thereby affecting the variability in the duration and extent of response to therapy.

Therefore, there is a need to better understand the roles played by both driver and passenger mutations in tumor evolution, fitness, and heterogeneity. It is important to understand their relationship in order to advance our understanding of cancer pathogenesis and treatment strategies. Incorporating population evolutionary mechanisms (especially the dynamics of genetic modification, selection, and competition among tumor cells) into cancer growth (inhibition) models is a very attractive goal: it could be a means to explore specific hypotheses of what drives the difference between response and refractiveness, absence or acquisition of resistance to cancer treatments from a bottom-up perspective and ultimately to establish a quantitative framework to design treatment regimens and guide development.

Connecting cellular genotype with phenotype (see [65]) remains a challenging task, particularly in the context of cancer genomes where the landscape of mutations can be complex and heterogeneous. Indeed, distinguishing between impactful and neutral missense mutations is an important aspect, as demonstrated in the functional interpretation of lung cancer genomes [66]. Discriminating impactful from neutral missense mutations requires a multi-faceted approach that combines computational predictions, experimental assays, comparative analysis, and integrative approaches, and remains a complex and ongoing challenge (see for example [67, 68]).

Integrating tumor evolutionary principles into treatment strategies can provide a more personalized and adaptive approach to cancer treatment [69]. Adaptive therapy, which is based on the principles of ecological competition among tumor clones, has been an area of active research for improving patient survival in cancer treatment [70]. Adaptive therapy aims to harness the evolutionary dynamics of tumors to improve treatment outcomes. It involves adjusting treatment strategies based on the changing dynamics of tumor populations during the course of treatment. The goal is to balance the control of tumor growth with the prevention of treatment resistance and to maintain the effectiveness of treatments over a longer period of time.

Mathematical models have been widely used to study and simulate the dynamics of adaptive therapy and clonal interactions in cancer. These models typically incorporate principles from population ecology, evolutionary biology, and cancer biology to simulate the dynamics of tumor growth and treatment response. These models can help researchers gain insights into how different factors, such as tumor heterogeneity, cellular turnover, mutation rates, clonal fitness, and treatment parameters, influence the outcomes of adaptive therapy. These models can also be applied to generate predictions of tumor growth trajectories under different treatment scenarios, explore the emergence of treatment resistance, and optimize treatment strategies to achieve the best possible outcomes for patients. In one example published by Chmielecki et al. a rather simple evolutionary mathematical model could suggest a therapeutic strategy of tyrosine kinase inhibitors (TKIs) alternative to the commonly used one that could prolong the clinical benefit in experimental models of epidermal growth factor receptor(EGFR)-mutant non-small cell lung cancer (NSCLC) by delaying the development of resistance [71].

The concept of ”more is better” has been traditionally applied in the use of cytotoxic chemotherapies at their maximum tolerated dose (MTD) to treat hematological malignancies and some solid tumors. However, this dogma is being challenged by evolutionary theory. Tumor evolutionary theory suggests that aggressive treatment strategies that aim to maximize drug exposure, such as high doses or continuous dosing, may initially suppress tumor growth but can also drive the emergence of drug-resistant clones due to selective pressure [72]. This is because high drug exposure can lead to increased mutation rates, genomic instability, and selection of pre-existing resistant clones, contributing to the development of resistance [73]. On the other hand, lower doses or intermittent dosing schedules may reduce selective pressure, allowing for the coexistence of drug-sensitive and drug-resistant clones, and potentially delaying or preventing the emergence of drug resistance.

Competition and interactions among different clones and subclones have a significant impact on the exposure-response relationships[69]. We have demonstrated this phenomenon using a simplified tumor model consisting of two subpopulations: one with genetic alterations that make it susceptible to a targeted therapy, and another without these alterations. The cells in the second population are inherently resistant to treatment and compete with the sensitive cells for limited resources. When the drug exposure is high, the sensitive cell populations are swiftly and extensively eliminated, reducing the competition faced by the resistant populations. This process, known as ”competitive release,” facilitates the growth of the resistant cells[74]. Depending on the nature and extent of interactions between the subpopulations, the exposure-response relationships can become shallower, flatter, or even inverted. To gain a better understanding of these exposure-response relationships for targeted therapies and to guide the selection of optimal doses and treatment regimens that offer the best long-term clinical benefits, evolutionary theory can be combined with traditional pharmacometric modeling approaches [75].

Integration of tumor evolution principles into exposure-response modeling can further enhance the understanding of the complex interactions between drug exposure, tumor dynamics, and treatment response, and facilitate the selection of optimal doses for molecularly targeted therapies and immunotherapies in oncology [76]. The impact of tumor evolution on exposure-response relationships has been explored in a theoretical framework, highlighting the complex dynamics of tumor evolution and the importance of considering tumor heterogeneity, clonal interactions, and the emergence of drug resistance in determining optimal dosing strategies.

Therefore, integrating tumor evolutionary principles into tumor models, such as considering tumor heterogeneity, clonal interactions, the emergence of drug resistance, and population extinction can inform appropriate dosing strategies that balance efficacy and toxicity and weigh short-term tumor response and long-term patient survival.

3.1. Evolutionary modeling to describe acquisition of treatment resistance

Resistance to treatments is the ultimate limitation to the success of current cancer therapies. Various mechanisms contribute to the development of resistance, including genetic, epigenetic, posttranslational, cellular, microenvironmental, and pharmacokinetic mechanisms. To gain a comprehensive understanding of this diverse range of phenomena, a wide variety of modeling approaches may offer valuable insights into the underlying mechanisms of resistance, facilitating the exploration of potential strategies to overcome treatment resistance and improve therapeutic efficacy [77].

Intra-tumor heterogeneity refers to the existence of distinct subpopulations of cancer cells within a tumor, as well as variations observed within different metastatic sites and between metastatic sites and primary foci [78]. These subpopulations of cancer cells can exhibit a range of phenotypic properties, including tumorigenicity, activation of signaling and metabolic pathways, evasion from antitumor immunity, induction of senescence, production of secreted factors, migration, metastasis, angiogenic capacity, genetic make-up, response to anticancer agents [78].

Traditionally, this phenotypic diversity was thought to arise from mixtures of cancer cell clones with distinct yet heritable phenotypes, and much effort has been devoted to understanding how mutational processes shape tumor evolution to inform strategies to limit tumor adaptation in the clinical settings [79]. Genetic instability, such as increased mutation rates or chromosomal instability, or both, can contribute to the increased diversity of cancer cell populations [80]. However, the current understanding of intra-tumor heterogeneity has evolved to encompass a more broader explanation that includes not only genetic and epigenetic aberrations but also regulatory states such as transcriptomic programs that determine functional or phenotypic properties with (or even without) spatial separation [81], including plasticity. Therefore, heterogeneity is not solely determined by cell intrinsic mechanisms but also by the dynamic tumor microenvironment, including factors such as angiogenesis, immune system, and fibroblasts [82]. Recent advancements in DNA barcoding techniques have enabled tracking of plasticity across hundreds of clones in a population of cancer cells [83]. This work revealed that combination chemotherapy selects for clones with increased phenotypic plasticity, which eventually is tied to worse prognosis, but the sensitivity of this effect to the dosage of chemotherapy remains to be determined.

Stochastic differential equations (SDE) [84] or game theory [85] are often used mathematical approaches to model phenotype plasticity and evolution over time in tumors, but they can be incompatible with typical ODE based simulation setups that are used to describe tumor growth at the organ or clinical level. SDEs are stochastic equations that account for randomness and variability in the system’s dynamics. SDEs can be used to describe how phenotypic plasticity and evolution occur in individual cells within a tumor, considering factors such as mutation rates, cellular interactions, and environmental fluctuations. Game theory, on the other hand, is a mathematical framework used to model strategic interactions between multiple entities, such as cells within a tumor, that make decisions based on their own payoffs or fitness. Game theory models can capture how different phenotypes or strategies may compete or cooperate with each other, and how these interactions can influence the evolution of phenotypic plasticity within a tumor. While SDEs and game theory can provide valuable insights into the mechanisms of phenotype plasticity and evolution in tumors, integrating these approaches into macroscopic models of tumor growth can be challenging, as they may require different mathematical formulations and computational approaches compared to traditional ODE-based models.

Champagnat et al. have made considerable efforts to connect single-agent stochastic dynamics with macroscopic population-scale models. By employing some fundamental approximations, they have derived the large population limit deterministic integro-differential or diffusion-reaction type of models [86]. In particular, phenotype-structured partial differential equations (PDEs) have been used to describe the density of cell populations with different phenotypes. This approach, as proposed by Clairambault et al. [87, 88], allows for a continuous approximation of gene frequencies and phenotypes, which are suitable for a diffusion-type of partial differential equations. This approximation is particularly relevant for large cell populations typically observed in advanced cancer stages where chemotherapy or immunotherapy may be the primary treatment option for patients. This approach further requires the assumption that genetic mutations result only in minor phenotypic changes.

In the following we would like to illustrate the case where a large heterogeneous cancer cell population can be described by the means of a diffusion reaction PDE. Let $n(x,t)$ be the population density at time $t$. Let us imagine a phenotype associated on-dimensional coordinate $x$ that can characterize cell viability or proliferation. For example, $x$ could be related to the total mutational burden of that cell, where very often the pathological state is characterized (on top of potential driver mutations) by a wide range of passenger-like genomic alterations, e.g. measurable as tumor mutational burden, TMB. One could imagine that resistance is associated with even more of these unidentified mutations, while too many will lead to complete dysfunction and non-viability of the cell. Here we could assume that $\text{x}$ is inversely correlated to such mutational burden: $x=1$ physiological mutational burden; $x=0$ very high mutational burden. We denote the average phenotypic state of the cell population as $\mu (t)=\frac{1}{\rho (t))}{\int }_0^{1}x\cdot n(x,t)dx$ where $\rho$ is the total number of cells at time $t$ that can be obtained by integrating over the phenotype space ${\int }_0^{1}n(x,t)dx$. The evolution of a cancer cell population under a treatment can be described as a time varying cell population density $n(x,t)$ in the frame of phenotype-structured equation with $c(t)$ being the concentration of the drug at time $t$:

$$\frac{\partial n}{\partial t}(x,t)=\beta \frac{{\partial }^{2}n}{\partial x^2}+S(x,c(t),n(x,t)).$$ (5)

$n(x,0)$ is continous and bound and thus integrable and zero flux boundary (Neumann) conditions apply here

The establishment of the cancer niche, which allows for the growth of cancer cells under the different conditions, implies evolutionary selection, by phenotype which itself is tied to the genotype through the emergence of selectively advantageous driver mutations, selectively neutral passenger mutations and deleterious mutations [89]. As the tumor grows, individual cells and their progeny compete for limited resources, such as space, nutrients, and oxygen, within the tumor microenvironment [89]. Cells with advantageous mutations that confer a fitness advantage are more likely to outcompete other cells and proliferate, leading to clonal expansion of these cells.

$S(x,c(t),n(x,t))$ represents the natural selection according to the evolutionary fitness of cancer cells in the phenotypic state $x$ at the time $t$ and with the concentration of cytotoxic drug $c(t)$. Mathematical models of cancer growth and evolution often incorporate both positive selection, which promotes the proliferation of cancer cells with advantageous phenotypic traits, and negative selection, which inhibits the growth of cancer cells with disadvantageous traits. For this example we have integrated positive and negative selection as a function of phenotype into a logistic growth equation where the total population size $\rho (t)$ will be limited to a carrying capacity $C$.

$$S(x,c(t),n(x,t))=r\cdot n(x,t)(P(x;c(t))-[\rho (t)/C+K(x,c(t))])$$ (6)

Here $P(x;c(t)$ quantifies the proliferating capacity for a given phenotype $x$ here assumed to be a Gaussian profile $u\cdot \mathcal{N}(\mu ,{\sigma }^2))$ with $\mu =0.7$ and $u=1$ without treatment $c(t)=0$ and $\mu =0.3$ and $u=0.85$ under treatment, ${\sigma }^2=0.1$ in both cases and $K(x;c(t)$ is the susceptibility of cells in phenotype $x$ to the killing by the drug, also a Gaussian profile $v\cdot \mathcal{N}(\mu ,{\sigma }^2))$, $v=1$ and $\mu =0.7$ under treatment $c(t)>0$ and $v=0$ otherwise.

The second order partial derivative $\beta \frac{{\partial }^{2}n}{\partial x^2}$ accounts for the diversification, e.g., a high rate of mutations due to genetic instability. Mutations are often assumed to be a random, non-directed, and independent process of our population in phenotype space [90] (in analogy to heat transfer or molecular diffusion [91]) and solutions can be found either analytically or numerically.

As shown in Figure 3, we have modeled tumor growth inhibition using a logistic differential equation and combined it with a phenotype structured approach with diversification, positive and negative selection; the main modeling objective is to explain between-patient variability of the time to tumor progression due to acquired resistance (Figure 3 a, tumor burden estimated from the overall population size $\rho (t)$ and cell size of 1000 μm³). For the logistic growth a multiplicative growth speed parameter $r$ is introduced (here $r=3\text{E}-6$), and a second order death term proportional to $\rho (t)/C$ is added to $K$ as additional death term representing overall growth limitations ($C$ set to 1E14 in this example, the phenomenological behavior can in fact be replaced by a space structured model of explicit nutrient limitation). Diversification uses a diffusion coefficient of $\beta =1\text{E}-9$. We have simplified the drug concentration to be either 0 or 1 depending on the time point $t$ lying within the treatment period, which in turn defines the baseline time point $t_{\mathit{\text{baseline}}}$. Treatment strength is represented by multiplicative scaling parameter in $K$ being $\gamma =5\text{E}-6$. $P(x,t)$ features a Gaussian profile centered around 0.7 and 0.3 (in case there is no treatment and under treatment, respectively) and with a standard deviation of 0.1. These profiles implement the hypothesis that there exists a phenotype that can proliferate despite treatment, but that will not be populated initially (Figure 3 b illustrates the selection term $S(x,\rho (t),c(t)))$ without scaling due to overall logistic growth limitation being – as $P-K$). However, it is important to note that this representation is highly simplistic, and real-world cancer dynamics are likely to be much more complex, involving multiple phenotypic traits, interactions, and feedback loops.

Fig. 3.

Simulation outputs of a discretized PDE model applying principles of phenotype structured modeling, eq. 5. a) Time course of total cancer cell population derived tumor burden (assumed average cell size being 1000μm³, blue line); starting from initial condition of 1E10 cells; treatment free context until baseline time (vertical black bar). Time under treatment $(c(t)=1)$ is indicated by an arrow. The time where tumor burden reaches baseline tumor burden is defined as time to first progression (TTP). b) Sum of proliferation term $K$ and killing term $P$ for the reaction part $S$ of the (integro-differential) reaction-diffusion equation system at two characteristic time points (before and after treatment has begun); shown is the positive proliferation fitness distribution minus negative (treatment induced) selection pressure distribution in phenotype space x. The time dependent whole population death term due to limited logistic growth is omitted for simplicity. c) resulting time dependent (colors from orange to blue) population density $n(x,t)$ across phenotype space d) subgroup analysis per quantile bin of baseline intratumor heterogeneity of a Virtual Population run simulating TTP where $r$ and $\beta$ were varied as patient descriptors according to lognormal distributions. For the definitions of subgroup the variance of baseline $n(x,t)$ was taken and the result binned in the percentiles indicated on the x-axis. For comparison with clinical response in intratumor heterogeneity (ITH) subgroups a “low ITH” and “high ITH” subgroup is defined as the baseline variance in $x$ being in the 25–50 percentile or 50–75 percentile, respectively (which itself depends on the phenotype diversification / diffusion coefficient). e) clinically observed and f) model-predicted ITH subgroup structured melanoma patient survival analysis (modeled result labeled with “Theor”). The web-based service build around the R script IPDfromKM [149] was used to digitize data from the study of Lin et al. [150] and replotted clinical and modeled data (and Greenwood 95% confidence intervals) using the Kaplan-Meier fitter of the lifelines python package [151].

Solving the resulting second-order differential equation, with the normalized total gradient of cancer cells, can be done using numerical methods such as finite difference schemes as the space structuration (see before). Finite difference methods discretize the space and approximate the derivatives in the differential equation using finite differences, which allows for numerical solution of the equation on a discrete grid. This approach can be computationally efficient and practical for solving phenotype-structured equations, especially when combined with appropriate boundary conditions that represent the biological context of the problem. In this example, we have used 20 phenotype space grid points. Example numerical simulations were conducted for 4 years and a treatment is applied between the 200^th day after $t0$ until the end of the simulation.

In Figure 3 c), the result of the phenotype structured population simulation under treatment pressure is visualized, indicating a shift from 0.7 to 0.3 in the phenotype space. This shift explains the attenuation of treatment effect seen in 3 a). In many more phenomenological tumor growth inhibition models, this attenuation is often considered by introducing a time-dependent treatment effect parameter, for example $\gamma (t)={\gamma }_0·ex{p}^{-\zeta t}$. In this case, the parameter $\zeta$ would represent the time to treatment effect attenuation.

To investigate potential inter-patient differences in that behavior, and to use evolutionary characteristics for prognostic predictions, we have set up a Virtual Population varying $r$ and $\beta$ by a multiplicative factor drawn from a lognormal distribution ($\mu =0$, $\sigma =1$). The time to progression (see Figure 3 a), was plotted for subgroups of baseline heterogeneity, represented by the variance of the discretized $n(x,t)$ at baseline time. Clearly, baseline heterogeneity, seen by the variance of baseline phenotype distribution and controlled by cancer niche selection profile versus diversification term (notably the magnitude of the diffusion coefficient) is negatively correlated with time to progression.

We further illustrate how such simulations could be compared with clinical data. Lin et al. investigated the impact of intratumor heterogeneity (ITH) on overall survival (OS) of melanoma patients under immunotherapy. We have digitized this data as Figure 3 e and compared it with the time to first progression multiplied it with a factor linking time between first progression and death that is normally distributed ($\mu =10$, $\sigma =3$). The resulting OS projections are given in 3 f. While much more work than this illustrative example needs to be done to investigate the hypothesis of ITH stratifying OS in melanoma, it seems plausible that the shown workflow and model type could be used for this very purpose

This approach not only provides a quantitative mechanistic framework, as opposed to a phenomenological model (e.g., to rationalize and align the model with available knowledge how heterogeneity affects efficacy), but also offers a more straightforward way to personalize model predictions with individual measurements of intratumor heterogeneity (e.g., imaging, liquid biopsy). By mechanistically translating baseline measurements into predicted time to progression, hypotheses derived from the literature can inform the design of prospective clinical studies. Moreover, the model itself may become a predictive biomarker for stratification and for individualized treatment schedules in the future, given appropriate regulatory qualifications.

3.2. Tumor extinction described by the low-density nonlinear Allee effect

The concept of the Allee effect, borrowed from ecological modeling, describes nonlinear effects observed at low population densities[92]. The basic assumption of the Allee effect is that the individual fitness of each element of the population initially increases with the population size. In the context of cancer growth, it means that the per capita growth rate increases with cell density for low values of cell densities. In particular, the per capita growth rate may be negative for very low values of tumor cell densities. The threshold at which the per capita growth rate becomes positive is called the Allee threshold. A tumor which cell density falls under this threshold (either after the initial stochastic events of spontaneous mutations, or under the pressure of a certain treatment) is thus destined to disappear.

The Allee effect in tumors is believed to originate from various mechanisms, which may differ depending on the type of cancer and the specific context. One such suggested mechanism is the diffusive autocrine signals released by cancer cells[93]. Another proposed mechanism is the individual cellular phenotype switching between cell migration and proliferation[94]. So far, the Allee effect has been relatively understudied in cancer modeling, and only a few experimental studies have been performed in vivo or in xenograft models to investigate the underlying mechanisms. The existing studies are all in support of the existence of an Allee effect, both in vitro [95, 96] and in in vivo xenograft animal models [97].

Let us consider a scenario where the tumor cell population is not a simplistic spherical solid tumor, as described in section 2 where tumor volume was proportional to the number of cancer cells and thus the cancer cell density remains constant. Instead, let’s assume that the tumor cell population is spatially distributed and composed of potentially different subtypes of cells, such as cancer cells (and different clones thereof), as well as stromal cells. This may be a justifiable approximation for several non-solid malignancies. When we choose a simple population growth like a logistic differential model, $\frac{dN}{dt}=rN(t)\left(1-\frac{N(t)}{K}\right)$, with $N(t)$ denoting the time-dependent population density, $K$ the carrying capacity for that density and with $r$ being the characteristic growth-rate determining parameter, the Allee effect can be introduced as a multiplicative term (see for example [98]). The logistic growth with Allee effect is then modified as follows:

$$\frac{dN}{dt}=rN(t)\left(1-\frac{N(t)}{K}\right)\cdot \left(\frac{N(t)}{A}-1\right).$$ (7)

Here, $A$ the so called Allee threshold which can render the per capita growth rate negative, even if the population size is below the carrying capacity (K)[92].

The presence of the Allee effect in tumor growth can have significant implications for various aspects of cancer dynamics and response to treatment. The behavior of tumors at low densities, where the per capita growth rate is negative, can determine the possibility of tumor eradication or complete clearance. The Allee effect allows the cancer-free equilibrium of the dynamical system to be always locally stable, which is not the case in classical models of tumor growth. This has significant implications for tumor eradication, resection, relapse, as well as clonal competition and selection. The potential outcomes of differential selection pressure exerted on competing clone lines are far more diverse by including an Allee effect. Each clone having a multi-stable behavior, meaning that the system allows to stably sustain its population or drives it to extinction depending only on the initial conditions. As a result, the interacting clones may be driven towards various combinations of population sizes, introducing greater complexity and diversity in tumor evolution.

The concept of ”two-strike therapy” is closely related to the Allee effect[70, 99]. The concept has been proposed and exploited in tumor models as a potential strategy to overcome the challenges posed by drug resistance and tumor heterogeneity. The first ”strike” typically targets the bulk of the tumor cells using conventional chemotherapy, radiation therapy, or targeted therapies. The second ”strike” involves the use of a different treatment modality or drug that targets specific mechanisms of drug resistance or vulnerabilities of residual tumor cells that survived the first treatment. The Allee effect could be exploited in the design of the second ”strike” treatments to target residual tumor cells that may have lower densities. Because of Allee effect, tumor cells with low densities may have reduced fitness, increased vulnerability to treatment, and decreased ability to recover from therapy-induced damage, making them more susceptible to the second strike of therapy, with a higher likelihood of eradicating the remaining tumor cells and achieving tumor eradication. This approach shares notable similarities with curative therapies observed in childhood acute lymphocytic leukemia, which have been empirically derived and proven effective.

4. Quantitative systems pharmacology models focusing on the immune system allow exploration of drug regimen and markers for drug response

Immunotherapies have emerged as a significant and rapidly growing pillar of cancer therapies in recent years [100]. Immunotherapy approaches leverage the body’s immune system to target and attack cancer cells, offering a promising avenue for cancer treatment with potentially fewer side effects compared to traditional treatments like chemotherapy and radiation therapy. Nevertheless, adverse events do occur [101] and in some cases can be severe or even fatal (for example stimulation of autoreactive T cells damaging host tissues with immune checkpoint inhibitors [102], cytokine release syndrome with bispecific T-cell engagers and chimeric antigen receptor T-cells [102]). Furthermore, some of these toxicities can have a late onset and be long-lasting [103], exhibit case-by-case variation, and potentially demonstrate non-linear dose-effect relationships [104]. The observation of a high number of severe adverse events during early stages of development, such as first in human (FIH) studies, motivate a more conservative approach that considers any potential effect when planning the FIH design (notably starting dose) [105]. In the development of immunotherapies, the minimum anticipated biological effect level (MABEL) has become a standard approach to complementing toxicity estimates and guide decision-making during this phase of development[106].

In fact, a limited predictive power of animal models for the effects on immunotherapies in humans has been formulated as the most important hurdle for the advancement in this class of treatments [107]. Therefore, in the context of immunotherapy, mechanistic modeling has the potential in providing both qualitative and quantitative insights into the intricate interactions between the immune system and cancer cells. These models can predict treatment responses, optimize dosing regimens, and inform (especially early phase) early-phase clinical trials where data from preclinical studies are limited [108, 109].

In zeroth order approximation, the role of the immune system in cancer can be summarized by two hallmarks A) immune stimulation and cytotoxic killing and B) immune suppression and effector function evasion.

CD8+ T cells constitute the main component responsible for killing cancer cells that present major histocompatibility complex (MHC) class I molecules on their cell surface; [110]. While also dendritic cells (DCs), natural killer (NK) cells, and CD4+ T cells play an important part in the activation and polarization of these cells, it is the mature, activated effector-type cytotoxic T lymphocyte (CTLs) that is able to release granules for the lysis of the cell membrane and that can activate FasL-mediated apoptosis; [110].

Many cancers evade the destruction through a variety of mechanisms, e.g., the recruitment of regulatory T cells, the attenuation of the initial antigen-dependent priming, or induced T-cell exhaustion [3]. These mechanisms avoid – under normal circumstances – that an over-aggressive immune reaction and auto-immunity is provoked. Programmed death-1 receptor (PD-1)-ligand (PD-L1) and CTL-associated antigen 4 (CTLA-4) are checkpoint receptors that – when targeted – can relieve exhaustion of CD8+ T cells and re-stimulate naive T-cell priming and activation. The re-invigoration of the T-cell response achieved by targeting these checkpoints through monoclonal antibodies has led to tremendous progress in treating cancer across indications with a number of treatments that have been approved in the last 10 years[111–113]. The direct implication of other immune cells (such as NK cells as representative innate cells) in tumor control has been discussed for a long time in the literature, but a clear picture is still somewhat missing; [114]. Despite those still open questions, new NK cell targeting immunotherapies are being developed with increasing effort; [115], [3].

Immunotherapies seem all to face a similar challenge, which is heterogeneity in patient’s response that can range from being absent or quickly vanishing (refractiveness and acquired resistance, respectively) to excessive (immune-related adverse events) and sometimes both at the same time.

The detailed understanding of immunological differences in patient’s tumors with respect to global features (“hot” versus “cold” tumors, composition of the tumor microenvironment) may enable to discriminate those situations where immunotherapies have a high chance of leading to response [116] (e.g. in monotherapy). A more detailed understanding of the intricate interplay between cancer and the immune system, often referred to as the cancer-immune function axis [3] may be needed to also render the “non-immunogenic” tumors like pancreatic adenocarcinoma [117] accessible by targeted combinations of immunotherapies. What markers can be used for such in-depth classification? The concept of the cancer “immunogram” [118] suggests the use of a comprehensive panel of markers (or omics data[119]) to assess the immune status of a patient’s tumor and guide immunotherapy decisions. Especially with respect to molecular and cellular markers characterizing details of patient specific immune tumor microenvironment, mechanistic mathematical models capturing the relationships between these different markers can be of enormous help.

On the other hand, when developing a mathematical model to describe the interaction between the immune system and cancer, it is important to strike a balance between granularity and practicability. The immune system is a highly complex and dynamic system, and incorporating all its intricacies into a mathematical model can be challenging. Therefore, it is necessary to choose a level of granularity that allows for practicality in terms of the number of variables and parameters, while still capturing the essential features of the immune system-cancer interaction.

A number of approaches have already been implemented to represent the immune system in its ability to control cancer on the one hand and the possibility that the tumor evades immune destruction on the other hand, as recently reviewed by [120] and [121]. Only those that are based on ordinary differential equations (ODE) are well suited for the integration with the overall ODE-type model even though agent based methods might be superior in their ability to describe stochasticity and rare events and are intrinsically focusing on single cells and tissue morphology at the same time.

The dynamics of CD8+ T-cell activation and effector function (see [110]), is often described by a dedicated submodel as part of the mechanistic immune system model. Additional cellular cytotoxicity can be added by extending the mathematical framework to natural killer (NK) cells. These cells of the innate immune system are thought to recognize altered cells independently of antigen presentation where the MHC) is downregulated in order to escape CD8+ killing. A model suggested by [122] simulates a minimal set of immune cell populations, notably NK cells and CD8+ T-cells in contact with altered self antigens – see [123] – to exert tumor control. Immunotherapy has been added in later work based on this model, see [124] and [125].

The fate of the tumor in individual patients (fundamental behavior of the model) and the quantitative growth rates are difficult to rationalize on just the immune system alone, and so it is not surprising that there are issues related to this reductionist approach e.g. by a strong dependence of this model’s results on the choice of initial immune cell composition described in the comment by [126]. A pragmatic approach to address this problem is to acknowledge that not enough knowledge exist to fully mechanistically model the tumor immune system interaction dependent of microenvironment of a given cancer type, stage of the disease, before and under different types of treatment and taking into account between-patient and between-lesion heterogeneity. A more phenomenological approach, expliciting tumor growth inhibition by a treatment and implicitly accounting for natural anti-tumor immunity in the initial conditions and parametrization, allow to model immune composition and markers evolving under the constraint of a known tumor growth profile that already implies a part of immune system activation, killing and evasion (as part of the time-dependent) growth rates. This approach is indicated where the immune system in the tumor microenvironment will affect a treatment response and effect size but where the ”natural“ immunity is not the focus of the model (which currently seems to be the majority of applications; note that this approach does not apply to for example personalized cancer vaccines ).

In order to illustrate the workflow and highlight opportunities from more complex immune system modeling that a parsimonious approach we embedded a slightly modified version of the model of Ref. [126], see Figure 4 a) into a simple tumor growth inhibition model of pembrolizumab. Chatterjee et al. investigated the tumor size observed in the phase I KEYNOTE-001 study where pembrolizumab demonstrated durable antitumor activity in patients with advanced non-small-cell lung cancer (NSCLC)[127]. In this analysis, a bi-exponential type tumor growth inhibition model was fitted to individual measurements. Tumor killing was characterized with respect to pembrolizumab exposure and covariates. Notably, the covariate analysis revealed that PDL1 expression was found to significantly impact the tumor killing term in the full covariate model. For our example mechanistic model, in the tumor growth inhibition structural model

$$\frac{\text{Tumorsize}}{dt}=(1-f)\cdot k_{growth}\cdot \text{Tumorsize}-f\cdot k_{death}\cdot \text{Tumorsize}$$ (8)

the term $k_{death}·\text{Tumorsize}$ has been replaced by its mechanistic counterpart.

Fig. 4.

a) Graph of the immune system - cancer interactions in the model in Ref. [122] modified to align with a simple model of bi-exponential tumor cell growth and killing (orange arrows). The “Tumor cells” degradation that is modulated by “CD8+ T”, “Anti-PD1” and ”PDL1” corresponds to equations 8 and 9. A dependence of the killing term by re-invigorated T-cells through tumor-expressed PDL1 was added to describe anti-PD1 mechanism of action. b) phenomenological (orange, one reference patient, no covariate effects) and more mechanistic tumor growth inhibition (range of PDL1 expression). c) comparison of simulated tumor size (varying $k_{death}$ in the data driven model to mimic the PDL1-expression covariate) and the explicit mechanistic model (including PDL-1 expression variability).

$$f\cdot \left({\gamma }_{PDL1}\cdot Tu\cdot \left(d\cdot {(L/Tu)}^l\right)/\left(s+{(L/Tu)}^l\right)\right))$$ (9)

Here, $Tu$ are the tumor cell numbers after conversion from the sum of longest diameter (SLD) coming from the empirical model and under the assumption of a single lesion and using an average cell size of 58μm³, $L$ are T-cells that can get re-invigorated by pembrolizumab (implying 100% PD1 expression). The equation for $L$ and values of the parameters $d$, $s$ and $l$ have been used as described in Refs. [122, 128]. We keep the notion of $f$ being the fraction of exposed cells. We neglected the explicit action of NK cells (assuming that the cancer has escaped natural immunity). Still, the indirect effect of NK-cells and cytokines of the mechanistic model on the action on reinvigorated T-cells has been kept. The initial conditions were 100 mm for SLD, $1E3$ for all immune cells, and 1 for cytokines (which however was reaching a steady state quickly, not shown). We incorporate the hypothesis that only those tumor cells can be killed through anti-PD1 antibody dependent T-cell re-invigoration which actually escape immunity through PDL1 presentation (and not those escaping via other mechanisms). Therefore our model features a multiplicative variable factor characterizing the fraction of PDL1-expressing tumor cells $\left({\gamma }_{PDL1}\right)$. In Figure 4 b) the tumor growth inhibition for a reference patient (roughly median $k_{growth}$ and $k_{death}$, $f=0.43$, no covariate impact) is depicted in orange. In blue, the range of profiles of the mechanistic version is given where PDL1 expression was varied from 0% to 100%, covering the results from thedata-driven model. The mechanistic version has, however a non-exponential killing profile that is due to killing term without adaptation to the different context of use, i.e. neglecting the emergence of any resistance to anti-PD1. To mimick the PDL1-expression covariate impact we have also modified $k_{death}$ in the more phenomenological model by a multiplicative factor between 0 and 100 %. Comparison between the two models (Figure 4 c) shows a marked correlation between both simulation methods, providing some validating evidence for our mechanistic assumption (PDL1 expression is a prerequisite for T-cell reinvigoration through PD1 inhibition). Despite being crude, this example highlights that a convergence of data-driven and mechanistic models can combine the robustness and credibility of inferential techniques while adding good explainability and explorative character to the table. In this regard, the hybrid approach could notably also be used to apply mechanistic knowledge about NK-cell, T-cell and cytokine relationship and stratify responses to pembrolizumab by recruitment efficiency and inflammatory context (”hot” vs. ”cold” tumors; [116]; notice that more sophisticated stratification might be possible depending on data availability and immune system model used, e.g. including also macrophage polarization) in other versions of the immune system model.

Since the advent of immunological models, progress has been made to render this technique practically applicable to specific questions in drug development. In fact, a whole class of models emerged that are termed immuno-oncology quantitative systems pharmacology models (IO-QSP). More generally, QSP models go beyond the complexity of traditional pharmacometric tools and provide a more mechanistic and holistic approach to understanding the pharmacology of drugs in the context of the disease and the human body and they are gaining increasing attention from academia, industry, and regulatory agencies. A recent survey by [129] highlights the growing acceptance and adoption of QSP models by pharmaceutical companies, especially larger-sized ones for a wide range of applications, including target identification and validation, drug efficacy and safety prediction, dose optimization, and clinical trial design. QSP-IO models typically incorporate multiple scales, from molecular and cellular levels to tissue and systemic levels, to capture the dynamic interactions between different components of the immune system and cancer cells. They often include detailed mechanisms of immune cell activation, proliferation, migration, and effector functions, as well as the effects of therapeutic interventions, such as immune checkpoint inhibitors, cancer vaccines, and adoptive cell therapies. In 2020, Chelliah et al. have performed a review of current QSP approaches, the legacy of mathematical models available to quantitative clinical pharmacologists describing interaction between tumor and immune system, and the recent development of QSP-IO platform models (141 models cited in the Supplementary Information of that paper) noting that virtual populations can be integrated for IO drug development approaches to increase the efficiency and effectiveness of the search for novel combination therapies[130].

While promising, QSP models often face several challenges in their application. These challenges stem from the complexity and heterogeneity of the immune system, the dynamic nature of tumor-immune interactions, the lack of standardized data and model validation criteria, and the need for interdisciplinary collaborations between mathematicians, immunologists, oncologists, and other stakeholders. Future perspectives in the field of mathematical modeling in immuno-oncology include the need for more mechanistic and patient-specific models, the integration of multi-omics data to improve model accuracy, the development of model-based biomarkers for patient stratification, and the use of machine learning and artificial intelligence techniques to enhance model predictive capabilities.

Addressing a more physiologically realistic modeling of tumor-immune interactions, agent based models (ABMs, see cross-disciplinary introduction in Ref. [131]) are an interesting alternative to (often ordinary) differential equation based QSP models. They can intrinsically resolve single agent (cell) interactions, display stochasticity, and have the built-in capacity for evolutionary modeling.

Pleyer et al. compares different ABM frameworks in view of spatial representation, microenvironment, intracellular and extracellular reactions, cellular properties, implementation, and practical use [132]. More specifically for cancer, the Popel group reviewed ABMs focusing on intra-tumor heterogeneity, and intratumoral interactions between cancer cells and stromal cells (including immune cells), the role of tumor-associated vasculature in the immune response, immune-related tumor mechanobiology, and immuno-oncology treatments [133] (see also the review by Metzcar [134]. As an application example, Cao et al. have used an ABM for T-cell tumor cell interaction to bridge the gap between population dynamics of immunological synapse formation induced by bispecific T cell engagers and clinical pharmacodynamics and emergence of treatment resistance [72].

Large ABMs, however can impose high computational costs and high-throughput application to create large virtual populations (i.e. for trial simulation) are currently less tractable. A potential avenue for a pragmatic use of such models would be a hierarchical one, using a few but detailed ABMs to inform the parametrization of the corresponding mean field (ODE) version of the model, which can be run more efficiently.

5. Conclusions and outlook for complex cancer models

There has been tremendous progress in our understanding of the hallmarks of cancer and factors leading to time dependent response, refraction and resistance to treatments, as well as their translation into clinical (trial) outcomes (see in particular [135]). As described in the illustrative cases before and the literature cited in these sections, it is nowadays possible to assemble complex mechanistic mathematical models that a) rationalize the causality and chain of events that could lead to tumor response and progression while b) aligning with quantitative data and more parsimonious MIDD techniques to improve model fidelity and predictability.

When putting together space-structuring, evolutionary dynamics, and immunological mechanisms, a rather “universal” multiscale cancer model could be imagined, for example as described in Figure 5. Such a model could implement

Fig. 5.

Schematic overview of potential submodels and phenomena (boxes) to be covered in a “universal” multiscale cancer model. Interfaces to treatment (i.e. (PB)PK) models and tumor size output being fed into time to event models are indicated on the left and the right, respectively. Potential measurements are indicated that could inform the model with individual patient data. For the description, see the main text.

• Treatment models describing the administration, biodistribution, metabolism and excretion of cancer drugs and drug candidates, informed with preclinical and clinical PK and drug metabolism/transporter studies.

• A multi-scale model of the primary tumor and potentially metastatic lesions, itself including

  • A submodel describing aberrant signaling pathways due to specific driver mutations and downstream effects such asconstitutive growth or tissue remodelling. This submodel could implement druggable driver mutations and their pharmacologic inhibition. The model would need inclusion of mutation specific data and could profit from individual patient data regarding genomic testing or sequencing of a patient’s tumor. Such submodel was not described in this paper but the literature holds plenty of such models (see e.g. [136].
  • A metabolic model describing different metabolic phenotypes specific to certain cancers and their microenvironments. In the literature, genome-wide metabolic models and flux balance analysis models (informed by RNA sequencing and proteomics) are suggested to take over such task [137] and those may predict mutation-specific overall metabolic phenotypes (e.g., lactate producing cancers, cancers showing the so called Warburg and inverse Warburg effect [138], as evidenced in functional PET imaging with labeled lactate of other metabolites).
  • The metabolic model then may be expanded into a space structured model (see section 2) to better describe size dependent nutrient and oxygenation limitation of growth, and the appearance of hypoxia and necrosis (and the phenomena triggered by these situations, e.g., tissue remodeling or resistance to radiotherapy). Here, again imaging techniques (CT or PET) help to identify patient’s tumor morphology in detail.
  • An immune system model to describe patient specific state of tumor control versus immune evasion of the cancer and immunotherapeutic mechanisms of action, see section 4. These mechanisms may be informed with immunohistochemistry, and cytometric data and in particular single cell sequencing data.
  • Orthogonal to any spatial coordinates, a continuous phenotype structured modeling approach can expand the description further to take into account heterogeneity of the cancer cell population and microenvironment and thereby allow for describing evolutionary processes during tumor progression and causal for the acquisition of treatment resistance, as described in subsection 3.1. While traditionally, heterogeneity had been measured by taking several biopsies, the advent of liquid biopsies [139] and radiomics analyses of imaging data [140] is promising to inform such phenotype structured model – quantitatively and individually.
  • Connected to both the signaling network and the phenotype structured model, a submodel of explicit handling of cell cycle progression versus apoptosis can output cancer specific growth rates of cells in different environments as well as their sensitivity to chemotherapy and external apoptosis signals (not discussed in this paper, see for example Ref. [141, 142].
  • All of the above cues directly or indirectly describe the integral lesion growth and shrinkage which conveniently outputs the longitudinal size of the primary tumor or metastatic lesion and (to be compared with CT or PET) other size-related tumor burden marker (circulating tumor DNA). When the model is supposed to describe the extinction of the tumor cell population, evolutionary approaches such as the Allee effect have to be accounted for here, see subsection 3.2.

• A parametric time-to-event model that translate time-varying tumor size into a hazard for death or dropout (in a clinical trial)

Obviously, this prototypical setup is still miles away from a universal cancer model describing all cancer hallmarks and hypothesized mechanisms of response, resistance and progression for different treatments and their combinations. Such a prototypical example would come with very high complexity and need for data for calibration of unknown parameters, as well as model identifiability issues and practical (computational cost) limitations.

Complex models face the dilemma that they can provide answers to many questions, including those that are impacting regulatory decision making. Regulators such as the FDA does recognize the potential of complex models (such as QSP models) as valuable tools for drug development and they to possess opportunities to exceed the descriptive limits of the primary dataset they have been calibrated against. Still, complex models have not yet been taken up as a standard practice in new drug applications to the regulators. One of the central open questions remains as to how to build credibility of such models in the absence of specific guidelines [143]. One of the most important points for future model-based tools will be the establishment of dedicated validation procedures, including the definition of a specific context of use (see for example one application of the ASME V&V 40 framework reported by [144] to validate and verify modeling and simulation in context of FDA approval for medical devices or application of this framework to physiology based pharmacokinetic modeling, PBPK, reported by [145]). Performing a qualification of the models for a given context of use through verification, technical and (prospective) clinical validation as well as comprehensive sensitivity and uncertainty quantification will help to establish credibility and create precedent use cases and ultimately allow for formal qualification of such models for use by regulatory agencies and public bodies.

The complexity of QSP models, particularly multiscale cancer ”disease/platform” models that integrate data from systems biology, requires interdisciplinary expertise and resources from academia, industry, and regulatory bodies. Academic researchers can contribute their knowledge of fundamental biological mechanisms and develop innovative modeling approaches, while industry partners can provide access to real-world data, resources for model development and validation, and regulatory expertise. Regulatory bodies play a critical role in evaluating the credibility and relevance of QSP models for regulatory decision-making and can provide guidance on the validation requirements and context of use for these models. The convergence of focus and collaboration between academia, industry and regulatory bodies (sometimes termed triple helix) can enable QSP models of increasing complexity to become feasible, credible and taken up, and thereby meld into the ecosystem so that these ultimately become mature enough to provide (supportive) evidence for patient care, drug development, and regulatory decision [146].