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Mechanistic characterization of oscillatory patterns in unperturbed tumor growth dynamics: The interplay between cancer cells and components of tumor microenvironment [1]

['Aymara Sancho-Araiz', 'Pharmacometrics', 'Systems Pharmacology Group', 'Department Of Pharmaceutical Technology', 'Chemistry', 'School Of Pharmacy', 'Nutrition', 'University Of Navarra', 'Pamplona', 'Idisna']

Date: 2023-12

Abstract Mathematical modeling of unperturbed and perturbed tumor growth dynamics (TGD) in preclinical experiments provides an opportunity to establish translational frameworks. The most commonly used unperturbed tumor growth models (i.e. linear, exponential, Gompertz and Simeoni) describe a monotonic increase and although they capture the mean trend of the data reasonably well, systematic model misspecifications can be identified. This represents an opportunity to investigate possible underlying mechanisms controlling tumor growth dynamics through a mathematical framework. The overall goal of this work is to develop a data-driven semi-mechanistic model describing non-monotonic tumor growth in untreated mice. For this purpose, longitudinal tumor volume profiles from different tumor types and cell lines were pooled together and analyzed using the population approach. After characterizing the oscillatory patterns (oscillator half-periods between 8–11 days) and confirming that they were systematically observed across the different preclinical experiments available (p<10−9), a tumor growth model was built including the interplay between resources (i.e. oxygen or nutrients), angiogenesis and cancer cells. The new structure, in addition to improving the model diagnostic compared to the previously used tumor growth models (i.e. AIC reduction of 71.48 and absence of autocorrelation in the residuals (p>0.05)), allows the evaluation of the different oncologic treatments in a mechanistic way. Drug effects can potentially, be included in relevant processes taking place during tumor growth. In brief, the new model, in addition to describing non-monotonic tumor growth and the interaction between biological factors of the tumor microenvironment, can be used to explore different drug scenarios in monotherapy or combination during preclinical drug development.

Author summary Mathematical models for tumor growth kinetics have been widely used for several decades, including among others the exponential, the Gompertz and the Simeoni model. However, as the knowledge of the multiple processes taking place during tumor microenvironment increases, models including plausible mechanisms are becoming increasingly important. In this work, we highlight the oscillatory dynamics observed in the tumor growth over time curves and we propose a novel semi-mechanistic model capable of describing the non-monotonic growth including the interaction between cancer cells, angiogenesis, and resources such as nutrients or oxygen, in the tumor microenvironment. Our model, with respect to previous literature models, improves diagnostic plots such as weighted residuals versus time plots and residuals lag plots, and individual predictions. Additionally, the framework allows the incorporation of anticancer treatments considering their mechanisms of action. Therefore, the model constitutes a valuable tool in the development of therapeutic strategies, supporting the rational design and selection of different treatment scenarios in preclinical drug development.

Citation: Sancho-Araiz A, Parra-Guillen ZP, Bragard J, Ardanza S, Mangas-Sanjuan V, Trocóniz IF (2023) Mechanistic characterization of oscillatory patterns in unperturbed tumor growth dynamics: The interplay between cancer cells and components of tumor microenvironment. PLoS Comput Biol 19(10): e1011507. https://doi.org/10.1371/journal.pcbi.1011507 Editor: James Gallo, University at Buffalo - The State University of New York, UNITED STATES Received: May 23, 2023; Accepted: September 11, 2023; Published: October 4, 2023 Copyright: © 2023 Sancho-Araiz et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Data Availability: All relevant data are within the paper and its supporting Information files. Funding: JB acknowledges partial financial support from research project with Ref. PID2020-116927RB-C22 from the State Research Agency, Ministry of Economy and Competitivity (Spain).The other authors did not receive any specific funding for this work. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Competing interests: The authors have declared that no competing interests exist.

Introduction Tumor growth dynamic (TGD) modeling represents a key element of model-based drug discovery and development in oncology [1]. Specifically, among other applications, it has been used to select promising drug candidates, assist the selection of the first-in-human dose, generate predictive quantitative translational frameworks, leverage clinical data, and identify relevant biomarkers [2–5]. Over the years, different models have been developed to describe tumor progression and tumor shrinkage effects of different anticancer therapeutic strategies [6–9]. From a high-level perspective, these tumor growth inhibition (TGI) models incorporate two main components: (i) disease (unperturbed tumor) progression and (ii) treatment effects. The latter comprises all aspects driving the link between drug exposure and drug effects. On the other hand, disease progression models describe the natural growth of cancer cells in the absence of treatment [10]. Currently, different model structures have been proposed to characterize unperturbed tumor dynamics relying on reasonable assumptions and providing adequate descriptive and predictive power [11–13] (see methods section for a detailed description of the most used models in data-driven analysis). The main characteristic of these model structures is their monotonic nature (i.e. predictions are entirely non-increasing, or entirely non-decreasing). This is due to the fact that most currently accepted models are variants of the exponential model with scarce mechanistic support. However, data from the literature indicate that, even in the absence of treatment, tumor growth rate may increase or decrease at different stages generating oscillatory patterns [13–15], which cannot be satisfactorily described by the aforementioned models (Fig 1). PPT PowerPoint slide

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TIFF original image Download: Fig 1. Tumor volume over time of a single mouse from a control group which was inoculated with lung H1975 tumor cells. The raw data (observations represented with points and joined with dashed lines) shows an apparent oscillatory profile in contrast with the predictions obtained from the commonly used tumor growth dynamics models (solid colored lines). https://doi.org/10.1371/journal.pcbi.1011507.g001 Tumor growth is a complex and dynamic process involving cell-cell and cell-extracellular matrix interactions that allow cancer cells proliferation, drug resistance and metastasis. During the development of solid tumors, a large amount of nutrients is consumed due to the rapid proliferation of tumor cells [16]. Moreover, high oxygen consumption, lack of nutrients and accumulation of metabolic substances in cells can create an oxygen-deficient microenvironment that is not suitable for tumor cell growth [17]. Tumor hypoxia-induced responses include, among others, enhanced angiogenesis and vasculogenesis [18]. Moreover, hypoxia also contributes to a reduced anti-tumor immune response through different pathways [19]. These processes can result in patterns of growth showing deceleration followed by acceleration (oscillatory tumor growth patterns). Mathematical model structures can be constructed to reveal key mechanisms generating subtle states of imbalance that could explain the oscillatory patterns in tumor growth. However, to the best of our knowledge, limited mathematical models are available describing these types of dynamics and (i) including the interplay of the multiple elements forming the tumor microenvironment (TME) in a mechanistic fashion, (ii) using a few sets of ordinary differential equations, and (iii) estimating precise model parameters in absence of anticancer treatment. From a theoretical perspective [20–22], very recent reports suggest that the prey-predator concept, largely used in ecology, could resemble the interaction between cancer cells and immune cells [20–22]. And, although the dynamics of immune cells and target cells present some discrepancies with the classical ecological framework, it inspires a new perspective on the possible mechanisms involved in tumor progression. Particularly, the work carried out by Kareva et al. [20] proposes a prey-predator-based model to describe the interplay between a heterogeneous population of tumor cells (prey) and cytotoxic immune cells (predator), and compares different types of interactions (i.e. mutualism, competition, prey-predator), with the multiple mechanisms undertaken in the TME. However, results are based on multiple simulations using literature parameters and do not include experimental observations supporting the theoretical framework. The development of model structures capable of characterizing non-monotonic behaviors of tumor growth is highly relevant, particularly at the preclinical stage, since due to its translational capacity, a more accurate estimation of the parameters that govern cancer cells proliferation could necessarily affect the design of dosage schemes or treatment combinations that guarantee optimal efficacy results. The objective of the current investigation is to develop a data-driven model characterized by a structure describing oscillatory tumor growth profiles and untangling the underlying mechanisms in untreated mice. To fulfil that objective, xenograft-derived unperturbed tumor growth longitudinal data obtained from a large set of tumor cell lines and several cancer types were analyzed using the non-linear mixed effect approach. As a corollary, the capability of the model to incorporate the effect of different treatments is also explored.

Discussion The main contribution of this work is the development of a simple but mechanistic based tumor growth model that retrieves precise model parameter estimates using only tumor longitudinal observations gathered from standard experimental settings in unperturbed mice. Furthermore, although the Gompertz expression has demonstrated a high predictive power [11,13] and adequately fits both preclinical [12,38] and clinical data [39], the proposed model showed increased adequacy in all the numerical and graphical diagnostics performed. In this respect, the lag plots, scarcely used as a goodness of fit plot to compare among competitor models, emerge as a powerful diagnostics tool to support the selected model. The development of this novel tumor growth model was motivated by the oscillatory patterns observed in unperturbed (i.e., in the absence of anti-cancer treatment) tumor growth profiles obtained in a variety of xenograft studies. The new model proposal describing non-monotonic growth assumes: (i) angiogenesis induced by cancer cells, (ii) the increase of tumor growing resources (i.e. nutrients and oxygen) as a result of angiogenesis, (iii) tumor growth dependent on the available resources, and (iv) consumption of resources by cancer cells. Remarkably, the possibility that those patterns were the consequence of random variations was discarded based on the results of a statistical analysis performed comparing the distribution of random HP oscillation and the observed HP (KS test with a p-value lower than 10−9). In addition, the statistical analysis also revealed that the characteristics of the oscillations were evenly distributed across the different cell lines. No pattern suggesting that a particular tumor type was more prone to show non-constant growth rates was observed. It is also important to note that, although a formal analysis was not performed, intuitively, we can conclude that in those cases where there is a sparse sampling, the detection of the oscillatory behavior might be jeopardized. Therefore, highlighting the relevance of the study design and the sampling schedule to identify the optimal times for tumor volume measurements that will inform as maximum as possible the different model parameters within a given model structure [23,40]. Even though different authors have pointed out that even in the absence of treatment, tumor growth rate can vary over time as a consequence of the TME [20,41,42], limited mathematical models have been developed characterizing this complex dynamic. There are a few models describing oscillatory tumor growth profiles in untreated groups through complex mathematical frameworks or from a more theoretical approach [41–44]. However, interestingly, the vast majority of models that are being currently used to describe the aforementioned profiles using the non-linear mixed effects approach treat the deviations around the monotonic growth as noise (residual error). The oscillatory patterns are then described as a result of treatment administration, and once the treatment is rescinded, tumor growth returns to follow a monotonic increase. Thus, even though our work is not the first attempt of studying this complex pattern, to the best of our knowledge, it is the first data-driven modelling exercise that describes non-monotonic tumor growth profiles in the absence of treatment through a simple system of ordinary differential equations, easily implemented in any modelling platform. Nevertheless, note that one of the limitations of this structure is that only tumor volume measures were available and therefore, no experimental data were available to confirm the mechanisms behind the oscillatory patterns. On the one hand, systemic biomarkers reflecting the multiple mechanisms taking place in the tumor microenvironment are hardly available as longitudinal observations. And on the other hand, complex designed experiments including the administration in monotherapy and in combination with multiple anti-cancer drug acting on different mechanisms will be needed in order for the tumor volume data to reflect tumor growth mechanisms. In our model, the initial tumor volume (TV 0 ) and the tumor growth rate constant (λ), estimated to 69.06 mm3 and 0.046 day-1, respectively, were within the range obtained with reference models. This highlights the fact that although the classical TG models are not able to capture the oscillatory behavior, they are generally capable of describing the mean trend of the data. In addition, the previous publication [23] in which the same experimental data was analyzed using the Simeoni model suggested that certain model parameters were different across cell lines. However, in the current work, although we investigated the different HP for each tumor type, we did not consider these differences during model development. The present analysis should be extended in order to further study the non-monotonic pattern in each tumor type or cell line. With the proposed model, which includes key biological mechanisms involved in tumor progression, emerges the possibility of locating drug effects more mechanistically on different parameter targets. Figs 8 and 9 of the main text suggest that both treatments which decrease k res and k ang or increase k consumption and k death , present slower tumor growth profiles or even show tumor shrinkage. Therefore, the importance of this approach lies in its potential translational impact on drug combinations from different angles. Even in more mechanistic models, the drug effect is only included in tumor death [7]. Furthermore, the aspect of optimizing the dosing schedules in combination therapies remains elusive. Increasing the mechanistic understanding of the system gives the opportunity to explore whether different dosing schemes can increase the benefit and lower toxicity. One example of hematological toxicity in chemotherapy is the model built for diflomotecan [45] which includes the coexistence of different states of cancer cells (proliferative and quiescent). In summary, the work presents a new semi-mechanistic model able to describe the non-monotonic growth and the interaction between the tumor, angiogenesis and resources in the TME. This framework constitutes a valuable tool to explore different mechanisms of action in a more mechanistic fashion and thus supporting the rational design and selection of drug candidates in different scenarios.

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[1] Url: https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1011507

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