S Benzekry
email: sebastien.benzekry@inria.fr

Artificial intelligence and mechanistic modeling for clinical decision making in oncology

Keywords:

The amount of 'big' data generated in clinical oncology, whether from molecular, imaging, pharmacological or biological origin, brings novel challenges. To mine efficiently this source of information, mathematical models able to produce predictive algorithms and simulations are required, with applications for diagnosis, prognosis, drug development or prediction of the response to therapy. Such mathematical and computational constructs can be subdivided into two broad classes: biologically agnostic, statistical models using artificial intelligence techniques, and physiologically-based, mechanistic models. In this review, recent advances in the applications of such methods in clinical oncology are outlined. These include machine learning applied to big data (omics, imaging or electronic health records), pharmacometrics, quantitative systems pharmacology, tumor size kinetics, and metastasis modeling. Focus is set on studies with high potential of clinical translation, as well as applied to cancer immunotherapy. Perspectives are given in terms of combinations of the two approaches: 'mechanistic learning'.

Introduction

Cancer diseases are a major health concern in the modern society. Taken together, they were responsible for 9.6 million deaths in 2018, which represents nearly 1 in 6 deaths [START_REF] Bray | GLOBOCAN estimates of incidence and mortality worldwide for 36 cancers in 185 CA[END_REF] . In face of such a public health challenge and considering the "unreasonable effectiveness of mathematics in natural sciences" [START_REF] Wigner | Richard courant lecture in mathematical sciences delivered at New York University[END_REF] , one might wonder: can mathematical models be of help in oncology? Mathematical constructs have long been confined to the representation, understanding and prediction of physical systems. The first reported use of a mathematical model in the life sciences is an epidemiological model by D.

Bernoulli in 1760 who aimed to study the impact of "vaccination" (inoculation of a small amount of the disease) against smallpox on age distributions and life expectancy [START_REF] Bernoulli | Essai d'une nouvelle analyse de la mortalité causée par la petite vérole, et des avantages de l'inoculation pour la prévenir[END_REF] . His calculations predicted that "vaccination" would result in a net life expectancy gain of 3 years.

With the modern development of novel measurement methods (especially from molecular biology and imaging), accumulation of biological and clinical data is currently driving oncology towards a quantitative science, which raises the challenge of inferring general patterns and structures as well as extracting most information from these. Meanwhile, the number of mathematical models developed by theoreticians in the field of so-called "mathematical oncology" [START_REF] Gatenby | Mathematical oncology: cancer summed up[END_REF] has been exponentially growing in the last decades [START_REF] Barbolosi | Computational oncology-mathematical modelling of drug regimens for precision medicine[END_REF][START_REF] Altrock | The mathematics of cancer: integrating quantitative models[END_REF] . However, these formal constructs have often remained confined to qualitative conclusions and rarely been confronted to observations. Two possible aims can be conceived for mathematical modeling in oncology:

-better understand. By allowing the simulation (or analysis) of the implications of biological hypotheses, mathematical and computational models provide a way to test them against experimental data. This improves our understanding of the biology by discriminating theories that are able to describe the data and, perhaps more importantly, reject theories that are not.

-better predict. In this case, even if the model is biologically-agnostic (relying on no a priori biological knowledge), if its predictive power is properly validated, it can provide a powerful numerical tool for clinical applications. These can be divided into: (a) diagnosis, (b) prognosis/prediction of response, (c) drug development and (d) personalized medicine (see examples in Tables 1 and2).

In this review, we will only deal with models with the second aim. Such models can be broadly divided into two categories: artificial intelligence and mechanistic models (Figure 1). The first category implements biologically-agnostic statistical learning algorithms while the second deals with the modeling and simulation of actual physiological processes, in order to make predictions in a second phase.

Artificial intelligence for clinical oncology

Artificial intelligence (AI) is a sub-discipline of computer sciences that aims at reproducing, by means of algorithms (implemented in computers, i.e., machines), processes that usually require human intelligence (e.g. vision or language). A sub-branch of AI, termed machine learning (ML), refers to those algorithms that are largely built from data, in opposition to being explicitly programmed according to a set of predefined, static, rules. ML is an extension of traditional statistical techniques, with varying degree of human intervention along a continuous spectrum. ML algorithms relevant to clinical application can be divided into unsupervised and supervised learning. Unsupervised learning deals with the process of finding patterns from a given set of input data only (e.g., define a relevant classification of cancer from molecular expression data), while supervised learning tries to predict outcomes (e.g., benign versus malignant) from the input data (e.g., lesion images). It does so by learning the relationships between input and output from training sets, with the hope of being generalizable to other data unseen before. Classical approaches of ML comprise linear and logistic regression, decision trees, random forests, support vector machines and artificial neural networks, among others [START_REF] Kourou | Machine learning applications in cancer prognosis and prediction[END_REF] . In several instances relevant to clinical oncology, outputs to be predicted are time-to-event -thus censored -data, e.g. disease-free, progression-free or overall survival (OS). Consequently, these require adaptation of the ML algorithms for handling censoring. To deal with high-dimensional data, LASSO (i.e., L 1 penalization) or ridge (L 2 penalization) have been combined with classical Cox regression [START_REF] Simon | Regularization Paths for Cox's Proportional Hazards Model via Coordinate Descent[END_REF] . In the past few years, the field of machine learning has seen dramatic advances driven by impressive successes of deep learning (DL) [START_REF] Lecun | Deep learning[END_REF][START_REF] Topol | High-performance medicine: the convergence of human and artificial intelligence[END_REF] . DL algorithms are artificial neural networks with multiple hidden layers. In a given layer, the output of a neuron is made of the composition of a linear map of the outputs from the previous layer, and a nonlinear activation function (e.g., sigmoid). This nonlinear structure allows DL to capture complex mathematical relationships between the input data and the output. DL models can have millions of parameters and thus require massive amounts of data to be trained. The advent of 'big data' in contemporary years, combined with the increase and accessibility of computing power, has unleashed powerful applications of DL [START_REF] Topol | High-performance medicine: the convergence of human and artificial intelligence[END_REF] . In clinical oncology, such 'big data' come primarily from three sources: omics, images and electronic health records.

Omics data

Empowered by the public availability of large-scale, multi-omics cancer databases (e.g. The Cancer Genome Atlas (TCGA, https://www.cancer.gov/tcga) or Gene Expression Omnibus (GEO) [START_REF] Barrett | NCBI GEO: archive for functional genomics data sets-update[END_REF] ), ML analyses have the potential to drive omics-based precision medicine. The first ML studies implemented classification tasks. Given high-throughput molecular data (e.g., gene expression), ML methods have been applied to distinguish between myeloid and lymphoblastic acute leukemia [START_REF] Golub | Molecular classification of cancer: class discovery and class prediction by gene expression monitoring[END_REF] , determine cancer type 13 , establish a molecular classification of breast cancers [START_REF] Perou | Molecular portraits of human breast tumours[END_REF] or define gene expression signatures predictive of survival or metastatic relapse, for instance in breast [START_REF] Veer | Gene expression profiling predicts clinical outcome of breast cancer[END_REF] , lung [START_REF] Shedden | Gene expression-based survival prediction in lung adenocarcinoma: a multi-site, blinded validation study[END_REF] , colorectal [START_REF] Guinney | The consensus molecular subtypes of colorectal cancer[END_REF] or central nervous system [START_REF] Capper | DNA methylation-based classification of central nervous system tumours[END_REF] cancers. This was achieved using methods from unsupervised learning such as hierarchical clustering [START_REF] Eisen | Cluster analysis and display of genome-wide expression patterns[END_REF] , principal component analysis (PCA)-based clustering [START_REF] Alexe | Analysis of breast cancer progression using principal component analysis and clustering[END_REF] , k-means clustering or, more recently, t-distributed Stochastic Neighbor Embedding (t-SNE) [START_REF] Ijzendoorn | Machine learning analysis of gene expression data reveals novel diagnostic and prognostic biomarkers and identifies therapeutic targets for soft tissue sarcomas[END_REF] .

For tasks such as personal prediction of outcome (e.g., survival, or metastasisfree survival) from a patient's molecular profile, supervised learning has been employed, first using traditional ML techniques [START_REF] Kourou | Machine learning applications in cancer prognosis and prediction[END_REF] . For instance, Listgarten et al.

found that a support vector machines algorithm performed best at predicting breast cancer presence from single nucleotide polymorphisms data [START_REF] Listgarten | Predictive Models for Breast Cancer Susceptibility from Multiple Single Nucleotide Polymorphisms[END_REF] . Then, fueled by the success of deep learning in handling high dimensional data, recent studies have applied deep artificial neural networks with omics data as input [START_REF] Zhu | The Application of Deep Learning in Cancer Prognosis Prediction[END_REF] .

For instance, , or multiple data sets from the TCGA [START_REF] Yousefi | Predicting clinical outcomes from large scale cancer genomic profiles with deep survival models[END_REF] . In order to deal with the "curse of dimensionality" inherent to the very high number of features (consider that gene expression data can span approximately 20,000 genes), investigators have implemented various feature selection strategies to reduce dimensionality [START_REF] Huang | Survival Analysis Learning With Multi-Omics Neural Networks on Breast Cancer[END_REF] , including using neural networks themselves [START_REF] Ching | Cox-nnet: An artificial neural network method for prognosis prediction of high-throughput omics data[END_REF] . Of note, DL methods have been shown able to integrate multi-omics data spanning RNA and miRNA sequencing, methylation data and copy number alterations [START_REF] Chaudhary | Deep Learning-Based Multi-Omics Integration Robustly Predicts Survival in Liver Cancer[END_REF][START_REF] Huang | Survival Analysis Learning With Multi-Omics Neural Networks on Breast Cancer[END_REF] .

Imaging data

AI is arguably most proficient in leveraging high-dimensional data provided by images (high number of pixels/voxels). This is due to the successful application of a particular type of DL termed convolutional neural networks (CNN) to computer vision tasks. One emblematic success of AI in oncology in recent years has been to reach dermatologist-level performances for classification of skin cancer and diagnosis of melanoma from a photograph of the lesion [START_REF] Esteva | Dermatologist-level classification of skin cancer with deep neural networks[END_REF] .

However, the two main sources of images in clinical oncology are from digital pathology and radiological images.

Digital pathology

Applied to digital pathology slides, CNNs have been able to reach great success for the detection of the presence of lymph node metastases in breast cancer, with area under the roc curve (AUC) as high as 0.996 in an external validation set and a sensitivity of 91% at 1 false-positive per patient [START_REF] Liu | Artificial Intelligence-Based Breast Cancer Nodal Metastasis Detection: Insights Into the Black Box for Pathologists[END_REF] . Going further, such work led to the development of an augmented-reality microscope with real-time indication of susceptible areas [START_REF] Chen | An augmented reality microscope with real-time artificial intelligence integration for cancer diagnosis[END_REF] . Pushing the concept and using DL to "see" [START_REF] Kather | Deep learning can predict microsatellite instability directly from histology in gastrointestinal cancer[END_REF] , a finding similar to another study where a DL algorithm was used for classification of lung cancer histological subtypes [START_REF] Coudray | Classification and mutation prediction from non-small cell lung cancer histopathology images using deep learning[END_REF] .

Others also used DL for prediction of survival from histology in malignant mesothelioma and found a critical role of the stroma [START_REF] Courtiol | Deep learning-based classification of mesothelioma improves prediction of patient outcome[END_REF] , similar to Kather and colleagues [START_REF] Kather | Predicting survival from colorectal cancer histology slides using deep learning: A retrospective multicenter study[END_REF] .

AI for diagnosis and segmentation from radiological images

Another important source of high-dimensional data to feed AI algorithms is provided by radiological images, including radiography, computed tomography (CT), magnetic resonance imaging (MRI) and functional imaging (e.g. positron emission tomography (PET)). These can be used either for diagnosis (i.e., detecting the presence of a cancer), automatic segmentation (delineation of a tumor in an image) or prognosis/predictive (e.g., predicting response to therapy) purpose. In breast cancer, following the generalization of mammography screening, an important challenge is to be able to detect the presence of cancer lesions in the images. This task usually requires trained experts and can be time-consuming. To address this issue, recent advances have been made

whereby AI systems (deep CNNs) have been shown able of superior predictive power than human experts, in large (>20,000 subjects) databases [START_REF] Mckinney | International evaluation of an AI system for breast cancer screening[END_REF][START_REF] Wu | Deep Neural Networks Improve Radiologists' Performance in Breast Cancer Screening[END_REF] . Similar results have been obtained for reducing false positive and false negative rates in lung cancer screening, with an area under the ROC curve of 0.944 [START_REF] Ardila | End-to-end lung cancer screening with three-dimensional deep learning on low-dose chest computed tomography[END_REF] .

For segmentation tasks, multiple studies used a technique called transfer learning, that is, the adaptation to their purpose of a previously developed CNN for another (computer vision) task [START_REF] Shin | Deep Convolutional Neural Networks for Computer-Aided Detection: CNN Architectures, Dataset Characteristics and Transfer Learning[END_REF] . For instance, the U-Net model architecture, initially developed for segmentation of neuronal structures from microscopy images, has been successfully applied to the segmentation of pulmonary nodules from CT [START_REF] Setio | Validation, comparison, and combination of algorithms for automatic detection of pulmonary nodules in computed tomography images: The LUNA16 challenge[END_REF] , or the segmentation of liver metastases from colon cancer 40

.

Other methods -still relying on deep CNNs -have been used for the segmentation of brain tumors from MRI [START_REF] Pereira | Brain Tumor Segmentation Using Convolutional Neural Networks in MRI Images[END_REF] .

Radiomics

Fueled by the availability of open source libraries such as PyRadiomics [START_REF] Griethuysen | Computational Radiomics System to Decode the Radiographic Phenotype[END_REF] and free software [START_REF] Nioche | LIFEx: a freeware for radiomic feature calculation in multimodality imaging to accelerate advances in the characterization of tumor heterogeneity[END_REF] , the number of studies relying on radiological images as a source of data and applying computational medical imaging (radiomics) methods has been growing exponentially in recent years, since the introduction of the term 'radiomics' in 2012 [START_REF] Lambin | Radiomics: extracting more information from medical images using advanced feature analysis[END_REF] . In such works, quantitative features are computed from the images to serve as biomarkers. These can either be handcrafted -i.e. metrics related to the shape, intensity or heterogeneity of some region of interest -, or automatically extracted from the image, for instance from some layers of an artificial neural network [START_REF] Hosny | Artificial intelligence in radiology[END_REF] . Several radiomics studies have been conducted in lung cancer by the group of Aerts for either prediction of distant metastasis [START_REF] Coroller | CT-based radiomic signature predicts distant metastasis in lung adenocarcinoma[END_REF] , genetic mutations (EGFR and KRAS) [START_REF] Velazquez | Somatic Mutations Drive Distinct Imaging Phenotypes in Lung Cancer[END_REF] or gene expression profiles [START_REF] Grossmann | the biological basis of radiomic phenotypes in lung cancer[END_REF] . In sarcomas, Crombé et al. have established a deltaradiomics (i.e. difference of radiomic features between two time points) signature for prediction of response to neo-adjuvant chemotherapy [START_REF] Crombé | T2 -based MRI Delta-radiomics improve response prediction in soft-tissue sarcomas treated by neoadjuvant chemotherapy[END_REF] .

The first radiomic study for prediction of response to immunotherapy was provided by Sun et al. [START_REF] Sun | A radiomics approach to assess tumour-infiltrating CD8 cells and response to anti-PD-1 or anti-PD-L1 immunotherapy: an imaging biomarker, retrospective multicohort study[END_REF] . Using radiomic features from both the tumor center and peripheral area, in combination with clinical features, the authors developed a signature on an internal dataset and validated it in 3 external cohorts: 1) a TCGA dataset, which confirmed association of the radiomics signature and RNA-seq based quantification of CD8 T cells infiltration, 2) a histology dataset with direct assessment of immune infiltration and 3) a third dataset with phase I patients treated with immunotherapy (for predictive value). However, a closer examination of the results reveals that most of the predictive power was driven by non-radiomics variables (e.g. tumor location, or imaging machine setup) and not by the radiomics features themselves [START_REF] Sun | A radiomics approach to assess tumour-infiltrating CD8 cells and response to anti-PD-1 or anti-PD-L1 immunotherapy: an imaging biomarker, retrospective multicohort study[END_REF] . Similar observations have been made whereby radiomics signature has been found not to contain much additional value than the mere tumor volume [START_REF] Welch | Vulnerabilities of radiomic signature development: The need for safeguards[END_REF] . Thus the added value of radiomics should always be adjusted for possible confounding factors included in simpler variables.

A few radiomics studies have been published using DL and radiological images for prognosis or prediction of response to therapy. In these, DL models are applied directly on the images without pre-engineering of the image into welldefined features, as done in classical radiomics approaches. This allows an easier application of the algorithms, since it does not require fastidious segmentation of the area of interest. In addition, the AI analyzes the entire image which might contain informative patterns rather than the tumor itself (e.g.

surrounding tumor stroma). Applications include prediction of disease-free or overall survival in nasopharyngeal cancer [START_REF] Peng | Prognostic Value of Deep Learning PET/CT-based Radiomics: Potential Role for Future Individual Induction Chemotherapy in Advanced Nasopharyngeal Carcinoma[END_REF] from PET/CT images, in non-small cell lung carcinoma (NSCLC) from CT images [START_REF] Hosny | Deep learning for lung cancer prognostication: A retrospective multi-cohort radiomics study[END_REF] , and in glioblastoma from MRI

54

. The models are often used to derive a large number of DL-based features, subsequently plugged into a LASSO-Cox proportional hazard model.

Interestingly, in contrast to several reports of the superiority of DL in imaging in general [START_REF] Hosny | Artificial intelligence in radiology[END_REF] , in several comparative studies in oncology, investigators found no difference between DL and classical radiomics [START_REF] Wang | Comparison of machine learning methods for classifying mediastinal lymph node metastasis of non-small cell lung cancer from 18F-FDG PET/CT images[END_REF][START_REF] Cha | Bladder Cancer Treatment Response Assessment in CT using Radiomics with[END_REF] . For the highly agnostic DL models, using a single image -especially from non-functional, anatomical imaging such as CT-scans -can have limited prognosis value [START_REF] Xu | Deep Learning Predicts Lung Cancer Treatment Response from Serial Medical Imaging[END_REF] . In such cases, exploiting the information contained in longitudinal images can bring added value [START_REF] Xu | Deep Learning Predicts Lung Cancer Treatment Response from Serial Medical Imaging[END_REF] .

Electronic health records

Even though yielding promising results, the applications of AI mentioned above

have not yet reached the maturity to be transposed to the clinic, and only a handful of AI algorithms have been approved by the American food and drug administration [START_REF] Topol | High-performance medicine: the convergence of human and artificial intelligence[END_REF] . Meanwhile, the digitalization of health records also provides a source of massive, mineable data. Probably the most clinically advanced tool in the analysis of such data is IBM's cognitive decision-support system, Watson for health. Relying heavily on literature data (processed automatically using natural language processing) and expert knowledge (mostly from oncologists at the Memorial Sloan Kettering Cancer Center), IBM's tool aims at providing guidance in individualized treatment decisions. Although no prospective, randomized clinical study has been reported yet, IBM Watson has been deployed in multiple clinical institutions over the world. Testing concordance with actual clinical decisions, results have been contrasted, some reporting high concordance (93%) in breast cancer [START_REF] Somashekhar | Watson for Oncology and breast cancer treatment recommendations: agreement with an expert multidisciplinary tumor board[END_REF] , while others found a low concordance (48.9%) in colon cancer [START_REF] Lee | Concordance With Watson for Oncology, a Cognitive Computing Decision Support System for Colon Cancer Treatment in Korea[END_REF] . Overall, application of Watson has led to disappointing results so far, in terms of individual treatment planning with existing strategies as well as with potentially innovative treatments [START_REF]IBM pitched Watson as a revolution in cancer care. It's nowhere close[END_REF] .

Explanations have been proposed in that it mostly reflects the decisions of the experts who trained it [START_REF]IBM pitched Watson as a revolution in cancer care. It's nowhere close[END_REF] .

Mechanistic modeling

As seen above, AI has been very proficient at mining high-throughput data. In particular, the most impressive successes have been in imaging. This can be interpreted as a consequence of the actual structure of DL models, which mimicks actual neurons in the brain, directly involved in tasks such as vision.

However, despite this "physiologically-inspired" structure, AI algorithms suffer multiple limitations such as their lack of interpretability [START_REF] Hosny | Artificial intelligence in radiology[END_REF] or the requirement of very large annotated and properly curated data sets. Moreover, AI algorithms have so far been able to reach human performances but rarely outperformed them. For other tasks than vision such as clinical decision making and therapeutic management (e.g., predict the impact of a given change of dose or scheduling of a drug), AI has yet to demonstrate its applicability. Limitations in terms of interpretability of AI algorithms calls for incorporation of biological knowledge into the models, also called "visible" machine learning [START_REF] Yu | Visible Machine Learning for Biomedicine[END_REF] . In contrast to biologically-agnostic AI models, mechanistic approaches develop physiologically relevant models able not only to link input and output, but also to simulate the kinetics of oncogenic processes and treatment (Figure 1). They are thus linked to the type of relevant data amenable to quantification. The main categories of such data are pharmacological data (e.g., plasma drug concentrations) and tumor size (from radiological techniques). clearance or volume of distribution). Individual predictions from sparse data can further be obtained using empirical Bayesian estimation.

Population PK studies for ICI (monoclonal antibodies) have been performed

over the last 6 years and have found that two-compartmental models were often appropriate for description of the concentration time course of these drugs. They have revealed large inter-individual variability and low, possibly time-varying 64- 66 , clearance. Studies have been conducted for ipilimumab [START_REF] Feng | Model-based clinical pharmacology profiling of ipilimumab in patients with advanced melanoma[END_REF] , nivolumab (alone [START_REF] Bajaj | Model-Based Population Pharmacokinetic Analysis of Nivolumab in Patients With Solid Tumors[END_REF] or in combination with ipilimumab [START_REF] Sanghavi | Population Pharmacokinetics of Ipilimumab in Combination With Nivolumab in Patients With Advanced Solid Tumors[END_REF] ), pembrolizumab [START_REF] Ahamadi | Model-Based Characterization of the Pharmacokinetics of Pembrolizumab: A Humanized Anti-PD-1 Monoclonal Antibody in Advanced Solid Tumors[END_REF] , atezolizumab [START_REF] Stroh | Clinical Pharmacokinetics and Pharmacodynamics of Atezolizumab in Metastatic Urothelial Carcinoma[END_REF] and durvalumab [START_REF] Baverel | Population Pharmacokinetics of Durvalumab in Cancer Patients and Association With Longitudinal Biomarkers of Disease Status[END_REF] , in multiple tumor types. These studies allowed simulation of varying dosing regimens, and were generally supportive of similar PK exposures between weight-based and flat-dosing regimens. Nevertheless, due to the large inter-individual variability, the use of therapeutic drug monitoring to personally adapt the dose of monoclonal antibodies has been proposed, although not clinically applied so far [START_REF] Ratain | Time Is Money: Optimizing the Scheduling of Nivolumab[END_REF] .

Downstream the dose-concentration-exposure relationship, exposure-response (ER) studies aim at quantifying the dependency of efficacy (e.g., overall survival) on exposure metrics (e.g., AUC), see Figure 2. As with other monoclonal antibodies used in oncology, the ER relationship is complex, in particular due to 1) confounding baseline factors on clearance (better prognostic associated with lower clearance) and 2) for ICI at least, time-dependent clearance due to variable clearance with disease status 65,[START_REF] Li | Time dependent pharmacokinetics of pembrolizumab in patients with solid tumor and its correlation with best overall response[END_REF] . Adjustment to baseline prognostic in ER models is needed (to address 1) and use of early exposure metrics (first cycle) is for now the best approach to mitigate for 2) 65 .

ER modeling has been performed for ipilimumab [START_REF] Feng | Exposure-response relationships of the efficacy and safety of ipilimumab in patients with advanced melanoma[END_REF] , nivolumab [START_REF] Wang | Quantitative Characterization of the Exposure-Response Relationship for Cancer Immunotherapy: A Case Study of Nivolumab in Patients With Advanced Melanoma[END_REF][START_REF] Feng | Nivolumab Exposure-Response Analyses of Efficacy and Safety in Previously Treated Squamous or Nonsquamous Non-Small Cell Lung Cancer[END_REF] , pembrolizumab [START_REF] Turner | Pembrolizumab Exposure-Response Assessments Challenged by Association of Cancer Cachexia and Catabolic Clearance[END_REF] or atezolizumab [START_REF] Morrissey | Alternative dosing regimens for atezolizumab: an example of model-informed drug development in the postmarketing setting[END_REF] . For example, Feng et al. [START_REF] Feng | Nivolumab Exposure-Response Analyses of Efficacy and Safety in Previously Treated Squamous or Nonsquamous Non-Small Cell Lung Cancer[END_REF] .

Quantitative systems pharmacology in immunotherapy

Quantitative systems pharmacology (QSP) studies in oncology investigate the complex mechanisms of action of anti-cancer agents, in order to determine, for instance, safe and effective doses for first-in-human studies [START_REF] Zhao | Application of PK-PD Modeling and Simulation Approaches for Immuno-Oncology Drugs[END_REF] . Such QSP study of the PK/pharmacodynamics (PD) of pembrolizumab was performed by Lindauer et al. [START_REF] Lindauer | Translational pharmacokinetic/pharmacodynamic modeling of tumor growth inhibition supports dose-range selection of the anti-PD-1 antibody pembrolizumab[END_REF] . The authors combined compartmental PK for systemic distribution of the drug, physiologically-based (PB) PK for tumor-site distribution, a model of binding of pembrolizumab to its target PD1 and a tumor growth inhibition model for efficacy of anti-cancer agents in experimental systems (without the transit compartments that were found insignificant). The model was developed to predict the minimally effective dose maximizing the probability of at least 30% growth reduction. Validation was conducted in animals with several doses and measurements of receptor occupancy in the blood and at the tumor.

Translation to human was performed leveraging the PBPK components as well as replacing the compartmental PK module by a human one and using allometric scaling for some remaining PD parameters. The results suggested 2 mg/kg Q3W as the minimal dose, with no further improvements in the range 2-10 mg/kg. Interestingly, this flatness of the dose-response relationship was confirmed in the modeling analysis of the clinical studies [START_REF] Chatterjee | Population Pharmacokinetic/Pharmacodynamic Modeling of Tumor Size Dynamics in Pembrolizumab-Treated Advanced Melanoma[END_REF] . Optimal dosing of another anti-cancer agent using an immunocytokine was investigated by Ribba et al. 82 . Since their drug is able to bind to immune cells, the authors had to develop an advanced target-mediated drug disposition mathematical model, which they used to predict tumor uptake retrieved from imaging data. They found that a dose-dense schedule could improve the biodistribution of the drug and showed how their model could be used to individualize treatment plans.

Despite the important success of ICI treatments, long-term efficacy is only limited to a minority of patients [START_REF] Sharma | Adaptive, and Acquired Resistance to Cancer Immunotherapy[END_REF] . Therefore, combining ICI with other therapeutic modalities is an appealing strategy to harness anti-tumor immunity.

In www.smartcalculators.online/ied). For instance, the scheduling used in the PACIFIC study [START_REF] Antonia | Durvalumab after Chemoradiotherapy in Stage III Non-Small-Cell Lung Cancer[END_REF] (i.e., 5x2 Gy for 6 weeks) was predicted to have only modest IED efficacy (12%), whereas another scheduling with same biologically equivalent dose would increase IED efficacy (i.e., 74% with 3x8 Gy for 6 weeks).

This suggests that algorithmic strategies could lead to higher immunogenicity and achieve synergism between RT and ICI.

More generally, the question of the optimal combination between different anticancer agents with different modes of action is well-adapted for mathematical modeling [START_REF] Ciccolini | Mechanistic Learning for Combinatorial Strategies With Immuno-oncology Drugs: Can Model-Informed Designs Help Investigators?[END_REF] . For instance, several studies have investigated the combination of anti-angiogenics with chemotherapy [START_REF] Wilson | Modeling and predicting optimal treatment scheduling between the antiangiogenic drug sunitinib and irinotecan in preclinical settings[END_REF][START_REF] Imbs | Revisiting Bevacizumab + Cytotoxics Scheduling Using Mathematical Modeling: Proof of Concept Study in Experimental Non-Small Cell Lung Carcinoma[END_REF] . The biological rationale is that antiangiogenic agents such as bevacizumab induce a transient amelioration of the tortuous and poorly functional blood vessel network of a tumor, in a process called vascular normalization. In turn, this could improve the delivery of chemotherapy, which is always administered in combination with bevacizumab.

Design of early-phase clinical trials

Historical concepts

Historically, several concepts of clinical and pharmacological oncology have been developed in connection with mathematical models [START_REF] Benzekry | Metronomic reloaded: Theoretical models bringing chemotherapy into the era of precision medicine[END_REF] . For instance, the Maximum Tolerated Dose (MTD) paradigm emerged following the work of Skipper, Schabel and Wilcox in the 1960s [START_REF] Skipper | Experimental evaluation of potential anticancer agents XIII. On the criteria and kinetics associated with 'curability' of experimental leukemia[END_REF] . Working on leukemic cell lines, they observed exponential growth of the cell populations. In addition, they found that given concentrations of cytotoxic drugs (including 6-mercaptopurine, 5fluoruracil and vinblastine), instead of killing given amount of cells, were killing a constant fraction of cells, the so-called log-kill effect [START_REF] Skipper | Experimental evaluation of potential anticancer agents XIII. On the criteria and kinetics associated with 'curability' of experimental leukemia[END_REF] . This pharmacodynamic relationship between exposure and effect was further substantiated by Jusko [START_REF] Jusko | Pharmacodynamics of Chemotherapeutic Effects: Dose-Time-Response Relationships for Phase-Nonspecific Agents[END_REF] based on the premise of an irreversible reaction between the drug and the cells receptors, and possibly accounting for two cell subpopulations (sensitive and resistant) [START_REF] Jusko | A pharmacodynamic model for cell-cycle-specific chemotherapeutic agents[END_REF] . Following the further realization that the presence of as little as one single leukemic cell was sufficient to lead to the host death, they argued that the goal of the therapy should be to achieve complete cure of the disease, i.e. eradication of all malignant cells. In this context, they demonstrated that a largedose/short time (single administration) schedule was superior to a chronic (daily) low-dose schedule (with similar or larger total dose).

Later, Norton and Simon revised both the growth model and, consequently, the action of chemotherapy on tumor kinetics [START_REF] Norton | Tumor size, sensitivity to therapy, and design of treatment schedules[END_REF] . Arguing that for solid tumors (e.g., breast), there is a mechanical stress constraining growth, they favored a model of tumor growth exhibiting slowdown, for instance the Gompertz model [START_REF] Norton | A Gompertzian model of human breast cancer growth[END_REF] . Going further, since this consideration suggested a non-constant fraction of proliferative cells in the tumor, and considering that chemotherapy action is mostly based on anti-mitotic agents, they proposed and validated the Norton-Simon hypothesis, which consists in applying the killing term only to the actively proliferating fraction. Importantly, this theoretical concept suggested that densified regimen would achieve higher probability of cure, an idea that was further successfully tested clinically for adjuvant therapy in breast cancer [START_REF] Citron | Randomized trial of dose-dense versus conventionally scheduled and sequential versus concurrent combination chemotherapy as postoperative adjuvant treatment of node-positive primary breast cancer: first report of Intergroup Trial C9741/Cancer and Leukemia Group B Trial 9741[END_REF] .

Recent studies

Beyond the classical MTD paradigm, mathematical models have started to be applied to help the design of early-phase clinical trials. The first concern in such trials is safety, in particular controlling life-threatening hematological toxicities in the context of aggressive regimen such as the dose-dense schedules mentioned above. In this context, an opportunity for mechanistic modeling is given by the fact that dynamics of neutrophil counts and myelosuppression can be quantified 100 . To control severe neutropenia and make dose-dense regimen clinically safe, Meille et al. developed a combined PK-PD model incorporating both efficacy and toxicity 101 . This model was clinically applied in the MODEL1 trial (NCT02392845), to optimize the administration of a combinatorial regimen of docetaxel and epirubicin in breast cancer 102 . The algorithm was used to find the repartition of the two drugs during the densification process (from three to two weeks) that would ensure minimal tumor size while respecting toxicity constraints. Resulting from the optimization was a non-trivial schedule, which was also possible to be adapted individually, using measurements from the first cycle(s) and empirical Bayesian estimation. This study resulted in a successful densification of the protocol, with much lower hematological toxicity and higher efficacy than reported in other studies with standard dosing 102 . A similar strategy -although with a different underlying structural model -was employed to design the MetroVino trial (NCT02555007) for administration of metronomic vinorelbine in lung cancer [START_REF] Barbolosi | Computational oncology-mathematical modelling of drug regimens for precision medicine[END_REF]103 . However, it remains unclear how to articulate the relative benefits of dose intensification and metronomic administration.

The previous models did not account for relapse/resistance to therapy, a phenomenon widely observed in the treatment of advanced cancers. This is due to intra-(and inter-) tumor clonal heterogeneity 104 . Although this can be tackled using more complex models, e.g. with second-order kinetics 121 , biphasic zero/first-order kinetics models have been successfully adapted to the ICI context [START_REF] Chatterjee | Population Pharmacokinetic/Pharmacodynamic Modeling of Tumor Size Dynamics in Pembrolizumab-Treated Advanced Melanoma[END_REF]114,[122][123][124] .

TGI modeling for immunotherapy

To address the heterogeneity of response patterns to ICI, mixture modelswhereby some parameters are considered to depend on a latent variable defining multiple classes -have been considered, for instance for pembrolizumab [START_REF] Chatterjee | Population Pharmacokinetic/Pharmacodynamic Modeling of Tumor Size Dynamics in Pembrolizumab-Treated Advanced Melanoma[END_REF] and ipilimumab 114 , in advanced melanoma. This approach allows to account for multiple groups (e.g. fast, intermediate and no growth).

Going further, Chatterjee et al. also proposed a single model with no mixture component that was able to describe the entire population [START_REF] Chatterjee | Population Pharmacokinetic/Pharmacodynamic Modeling of Tumor Size Dynamics in Pembrolizumab-Treated Advanced Melanoma[END_REF] . The model is an extension of the Stein biexponential model 118 and writes:

#(%) = # ()*+,-.+ /(1 -2)3 4 56789: ⋅< + 23 >4 ?@A9: ⋅max(E,<>G+,)H) I,

where 2 is a parameter controlling the fraction affected by the shrinking effect and K3LMN is a time lag allowing to describe delayed response. developed the first model of this kind for cancer immunotherapy (atezolizumab in urothelial carcinoma) 124 . Using a similar model as equation ( 1) for tumor size kinetics, the authors identified two model-derived metrics as significantly associated with OS (time to growth and current SLD slope). Importantly, their model demonstrated good predictive ability of individual death times in an external validation data set, with time-dependent areas under the ROC curve all above 0.75, significantly higher than when the kinetic model metrics were not included in the model. Król et al. added an important supplemental layer to joint modeling of size kinetics and survival: the appearance of non-target lesion (recurrent event) 127 . This event is indeed a criterion for progression in the RECIST criteria and has an impact on survival, which is neglected in classical TGI-OS models. Using a link between the two types of time-to-event-data (=frailty), the authors demonstrated the utility of this approach to individually predict probability of death from on-treatment data.

Modeling metastasis

Inter-lesion heterogeneity

The vast majority of tumor size kinetics models describe the main quantitative output underlying RECIST: the sum of largest diameters of target lesions.

However, modeling this sum as a single mass prevents a truly mechanistic modeling of the disease growth, because the actual processes happen at the tumor level. As soon as the model is nonlinear, kinetics of the sum differ from the sum of the kinetics (e.g. the sum of two exponentials with different rates is not an exponential). Moreover, interactions between established tumors -a phenomenon termed 'concomitant resistance' -might blur the picture when modeling the total cancer burden 129 . Therefore, the question of modeling interlesion variability (ILV) within the same host is of important relevance and has started to attract the attention of mathematical modelers in recent years [130][131][132][133] .

In a model including a hierarchical layer for ILV for the kinetics of standard

Dynamics of metastastic birth

A cancer disease is not limited to growth processes. In the context of a population of multiple tumor lesions, the appearance of new tumors is a crucial step. While this process can be modeled using joint modeling as mentioned above 127 , mechanistic approaches have also been proposed [135][136][137] . These could address a critical clinical challenge for cancers diagnosed at early stages:

who should receive adjuvant therapy in addition to surgery, and how much? Growth suppression by the primary tumor -subsequently leading to postsurgery acceleration -was also mathematically inferred in another study by .

Metastatic appearance and growth has also been studied through the lens of clonal evolution 136 . In such models, each cell has, for a given time unit, a probability to mutate and acquire metastatic ability, and then a probability to leave the primary tumor and establish a distant metastatic colony. Haeno et al. 136 applied their model to comprehensive data of pancreatic cancer patients and found that metastases were likely to be present at diagnosis. The model predictions also suggested that chemotherapy alone was better than surgery alone to maximize survival.

Perspectives for combining AI and mathematical

1 modeling: mechanistic learning 2 While both dealing with relevant applications in clinical oncology, it is manifest 3 that the works described within the two classes of mathematical modeling used 4 above (AI and mechanistic modeling) have been performed by multiple 5 communities (AI/ML, pharmacometrics, statistics, mathematicians) that have 6 independently made progress but have remained partitioned. Bridging the gaps 7 between disciplines would surely lead to more powerful methods able to improve 8 the design of clinical trials and cancer patient care 142 . On one hand, the lack of 9 interpretability of AI algorithms calls to "open the black box" and incorporate 10 expert knowledge to design more biologically-informed models. On the other 11 hand, most of mechanistic modeling studies have thus far dealt with variables 12 in relatively low dimension but nevertheless face an increase in the amount of 13 quantitative data produced. Therefore, they could benefit from AI techniques to 14 integrate high-throughput data (e.g. omics or imaging data, beyond the mere 15 size or volume of the lesions). For instance, instead of relying almost exclusively 16 on linear covariate models (up to a transformation), highly nonlinear ML 17 algorithms could be used to relate covariates and parameters of the structural 18 model 143,144 . Another valuable approach is to derive quantitative metrics from 19 simulation outputs of mechanistic modeling and use them as ML inputs for 20 predictive purposes 145 . We propose to name such hybrid approaches 21 combining big data and machine learning with mechanistic modeling 22 "mechanistic learning" [START_REF] Ciccolini | Mechanistic Learning for Combinatorial Strategies With Immuno-oncology Drugs: Can Model-Informed Designs Help Investigators?[END_REF] survival analysis revealed similar predictive power among the methods. 30 However, the mechanistic approach presents the advantage to yield simulations 31 of the patient's metastatic state. 32 Furthermore, the dimension of longitudinal data is increasing. While imaging 33 data start to be analyzed longitudinally with techniques from AI (e.g., recurrent 34 neural networks) [START_REF] Xu | Deep Learning Predicts Lung Cancer Treatment Response from Serial Medical Imaging[END_REF] , the advent of measurements from circulating tumor DNA 35 provide an additional source of longitudinal high-dimensional data. So far, it 36 remains an open question to design mechanistic models able to integrate such 37 data. With the revolution of immunotherapy, increasing attention is given to 38 molecular and cellular immune players. Data gathered from immune-monitoring 39 (flow cytometry and seric markers) generate a large number ( > 500) of 40 quantitative variables measurable directly from blood samples, thus in a 41 minimally-invasive manner that can be repeated. For instance, the PIONeeR 42 project is a large-scale clinical study aiming at explaining primary or adaptive 43 resistance to PD-1 inhibitors 147 . Such immune-monitoring data will be collected 44 at multiple time points in patients during treatment. This data will further be 45 modeled using a mechanistic learning approach in the QUANTIC project 147 . 46 The objective is to build predictive tools able to leverage this high-dimensional 47 data for personal predictions of the kinetics of response to treatment. 48 For efficient translation to the clinic, current mathematical modeling faces 49 multiple challenges. First, it is highly dependent on data and requires large-scale 50 curated databases. Molecular data are usually made publicly available upon 51 publication of the results in online repositories such as the TCGA or GEO. 52

Imaging data are also increasingly shared in similar repositories (see the TCIA 53 148 for radiological images or The Digital Slide Archive 149 for pathology images). 54

However, individual-level data from clinical trials such as longitudinal lesion 55 sizes, PK or biomarkers have limited availability, which is limiting the power of 56 statistical and mechanistic modeling studies. Extending initiatives such as the 57 data sphere project 150 or Vivli (vivli.org), which proposes to share data from past 58 clinical trials, would surely be very beneficial. Second, data sharing brings two 59 issues: ethical and practical. The necessary ethical regulatory dispositions that 60 have been introduced in the US (HIPAA) and Europe (GDPR) preserve patient 61 privacy but also limit data sharing. Novel infrastructures enabling the 62 development and testing of the models while ensuring of the data have 63 to be created [START_REF] Hosny | Artificial intelligence in radiology[END_REF] . Finally, while the wide arsenal of available anti-cancer therapies 64 brings opportunities in terms of applications of mathematical modeling to 65 optimize the design of combinatorial clinical trials, a current challenge is to 66 develop mathematical models not only for clinical trials, but also for personalized 67 therapy (Figure 3). To achieve this, structured and unstructured real-world data 68 from routine management should be collected, organized, analyzed and made 69 available to researchers, not only by private companies (e.g. Flatiron Health), 70 but also public institutions 151 . Indeed, mathematical modeling is now well 71 established for applications drug development, but further advances are needed 72 to translate the use of the models at bedside. 73 74 75

Conclusion

Two hundred and sixty years after Bernoulli's model for vaccination, medicine, and particularly oncology, has never been closer to a quantitative science.

Mathematical modeling, either purely statistical (AI), or mechanistic, has demonstrated that it is able to provide clinically relevant tools, whether for the improvement of clinical trials or personalized medicine. Hopefully, today's mathematical models will have a better future than Bernoulli's and an increasing number of these will be implemented into clinically actionable softwares for better patient care at bedside.

Figure 2 recapitulates

 2 the classes of mechanistic models relevant to oncology. In recent years, cancer research and clinical oncology have focused on cancer immunotherapy, in particular with the advent of immune-checkpoint inhibitors (ICI) 62 . Consequently, mathematical modeling studies have addressed relevant issues relating to immune-oncology drug development or for clinical care. Modeling in (immune-)onco-pharmacology Pharmacometrics Historically, population pharmacokinetics (PK) modeling has been initiated by the pioneering work of L. Sheiner in the 1970's 63 . The aim of such quantitative approach is twofold. First, to translate the actual PK physiological processes into simplified, abstract mathematical constructs (the structural model, e.g. a compartmental model). Second, to capture inter-individual variability into statistical distributions of the parameters of the structural model, and to quantify the impact of covariates on such physiologically-inspired parameters (e.g.

86 .

 86 an elegant study, Palmer et al. demonstrated that in several cases of drug combinations (including ICI combinations), the population-level benefit over monotherapies observed in clinical trials could be explained by the concept of drug independence whereby there is no actual synergy or additivity at the individual-level but rather population-level superiority solely due to individual patients responding to the drugs they are most responsive to, with no added benefits from other drugs[START_REF] Palmer | Combination Cancer Therapy Can Confer Benefit via Patient-to-Patient Variability without Drug Additivity or Synergy[END_REF] 3. Meanwhile, mechanistic modeling could play an important role in determining the best modes of combination between ICI and established treatments (surgery, radiotherapy, chemotherapy, targeted therapy) 85,Modeling immunotherapy has recently been stretched to gain insights on optimal modalities for combinations with radiotherapy (RT)[START_REF] Serre | Mathematical Modeling of Cancer Immunotherapy and Its Synergy with Radiotherapy[END_REF][START_REF] Kosinsky | Radiation and PD-(L)1 treatment combinations: immune response and dose optimization via a predictive systems model[END_REF] . Based on the rationale that local radiotherapy increases systemic activation of the immune system through the release of neo-antigens -the 'abscopal effect' -these studies have translated the cancer-immune interplay into mathematical constructs to predict the impact of different therapeutic schemes. Based on a thorough understanding of immunological mechanisms, Kosinsky et al. used multiple experimental groups (of a murine colon carcinoma model), spanning different combination schedules, for development and validation of a detailed QSP model incorporating radiotherapy and anti-PD(L)1 therapies 88 . It included multiple processes such as immunogenic cell death and dendritic cell maturation, time course of PDL1 expression and Treg:CD8+ ratio. Despite its complexity, the model parameters were either found in the literature or identifiable from fitting the experimental data, and validation was carefully conducted in independent datasets. The model gave interesting insights about responders and non-responders, classified so from the only source of interanimal variability in the model, a parameter responsible for T cell infiltration into the tumor tissue. Simulation of various alternative combination regimens were conducted and suggested that optimal efficacy was obtained when anti-PD1 treatment was administered prior, or concomitant to the radiotherapy, using hypo-fractionated schemes. Such predictions could have value to guide clinical trials among a vast range of combination possibilities. Others proposed a simple model calculating an 'immunologically effective dose (IED)' proposed for radiotherapy fractionation schemes 89 . Relying only on two additional parameters to the classical ! and " of radiobiology -two physiological time constants that were derived from the literature -the model could reproduce preclinical data and gives a valuable tool for in silico exploration of

  Whereas clinical use is to give both drugs concomitantly, these observations suggest possible value to sequential administration, but an open question remains the optimal time interval between the two administrations. Imbs et al. used a first set of experiments to calibrate a model for this synergistic effect, and then demonstrated that the model predictions -an optimal time interval of 3 days between administration of bevacizumab and the chemotherapy doublet pemetrexed-cisplatin -were concordant with a second, validation experiment 92 .

  The link with exposure was provided by a dependence of the parameter O G+)<P on the individual area under the concentration curve. Contrasting implications about the dosing of ICI antibodies have been obtained from these dose/exposureresponse models. While Chatterjee et al. concluded to no difference between 1 mg/kg and 10 mg/kg Q3W 81 , Feng et al. found that the dose was associated with the mixture subpopulation, thus the response 114 .Analyzing trial data of atezolizumab compared to docetaxel in the treatment of NSCLC, Claret et al. demonstrated the applicability of TGI models to forecast the outcome of phase III trials 122 . Using only data from a phase II trial for calibration of their TGI-OS model (a sum of two exponentials, #(%) = 3 >G< + 3 Q< -1 similar to 118 , linked to a lognormal survival model), they identified the growth rate R as a highly significant covariate of OS and were able to predict the results (hazard ratios) of an independent phase III study, from the data of the size kinetics. They also demonstrated that such prediction could be achieved as soon as 40 weeks after study initiation, which corresponded to 30 weeks before what could be concluded from the observed data only. In addition, their analysis underscores the added value of TGI-OS models compared to RECIST alone, because the significant OS difference between the two arms was not detectable in ORR or PFS only.In the context of complex tumor-immune interactions, tumor size is not the only relevant longitudinal biomarker of response to therapy. With the development of immune-monitoring techniques, cellular and molecular players are amenable to quantification and it is thus appealing to design more advanced mechanistic models. In the first such study, Netterberg et al. selected interleukin (IL) 18 as having the most significant variation among 95 plasma biomarkers initially considered123 . They further built a mechanistic model integrating 1) PK of the drug (atezolizumab), 2) kinetics of IL18 (indirect response model) and 3) tumor size kinetics (similar to Claret et al.117 but with a zero order growth term). Their analysis unraveled the effects of exposure (area under the curve) on the magnitude of the response and those of relative change in IL18 on response duration, emphasizing the utility of immune-monitoring to predict response.However, a subsequent study demonstrated that IL18 had no significant impact on OS125 .Joint modelsTGI-OS models estimate the TGI and OS model parameters sequentially. This can be prone to bias because the survival time can be informative of the tumor kinetics (patients relapsing fast have a more aggressive disease) 126 . On the other hand, joint models propose to estimate simultaneously the parameters of the entire model, by defining and maximizing the likelihood of the full data set (longitudinal measurements + survival) in a single step. This joint modeling approach has recently flourished in oncology to analyze tumor size (or biomarker) kinetics and overall or disease-free survival124,127,128 . Tardivon et al.

  uptake values (SUV) from positron emission tomography (PET) imaging in sunitinib-treated gastrointestinal stromal tumors, Schindler et al. found a significant ILV for the drug effect, although smaller than inter-patient variability 134 . This finding was also observed in another study modeling ILV and comparing the OS predictive power of model metrics (time-to-growth) derived from either diameters or volumes (computed from semi-automated tumor segmentation) in 918 metastatic colorectal cancer patients 130 . Inter-lesion heterogeneity was further analyzed by Mistry et al., by means of mechanistic models of resistance (either de novo or acquired), leveraging multiple phase III trial data 131 .

  In a landmark paper, Iwata et al. proposed a structured partial differential equation for the size distribution of regional metastases from an hepatocellular carcinoma135 . Using a Gompertz growth law and a dissemination rate proportional to the tumor surface (i.e. S T/V if S is the tumor volume), they were able to accurately describe clinical data of visible metastases in one patient. Critically, their model is able to make inference about the number and size of invisible metastases, from the observations of the visible ones only. It has been further validated using mixed-effect techniques for inter-animal variability in nonsurgical 138 and surgical 139 animal models of metastasis. In the latter, the model was simplified: secondary metastases (metastases from metastases) were neglected and the dissemination rate was assumed to be proportional to the tumor volume. The model was also able to fit the probability of metastatic relapse as a function of primary tumor size in 2,648 breast cancer patients, with inter-patient variability captured in the statistical distribution of only one parameter 139 . Similar metastatic modeling was applied to clinical data of brain metastases in NSCLC137 , where it suggested periods of dormancy of the order of 5 months and demonstrated how prediction of metastasis could help to guide clinical decision for whole brain radiation therapy (Figure3). Mathematical modeling has also been used to shed light on a neglected, but nevertheless critical aspect of metastatic dynamics: post-surgery metastatic acceleration. Analyzing the patterns of relapse hazards following breast cancer surgery, Retsky et al. noticed a bimodal distribution that could be explained using a computational model incorporating surgery-driven acceleration 140 .

  (Figure 4).23 An example of such mechanistic learning study was recently published for the 24 prediction of metastatic relapse in early stage breast cancer146 . Instead of using 25 a biologically agnostic model for survival analysis (e.g. Cox or random survival 26 forest), Nicolò et al. derived a mechanistic model for the actual time of relapse. 27 the authors relied on ML for feature selection. Comparison with classical or ML 29

Figure 1 :

 1 Figure 1 : Artificial intelligence versus mechanistic modeling. Artificial intelligence algorithms act as poorly interpretable black boxes that process input 'big' data to generate predictive outputs, e.g for classification between high and low risk of survival or probability of response to therapy. In contrast, mechanistic models usually deal with lower-dimensional data (e.g., tumor size), but can provide simulations in addition to pure predictions. These include exploration of possible scenarii, for instance the impact of alternative scheduling regimen or the predicted amount of invisible metastases at time of diagnosis. TX = treatment.

Figure 2 :

 2 Figure 2 : Overview of mechanistic methods in oncology. Departing from the dose (known), such models accommodate for four main types of quantitative data: drug concentrations, efficacy (e.g., tumor size), safety (e.g. neutrophil counts) and survival data. Mathematical tools developed to mine these data have varying degree of complexity, from elementary statistical analysis tools (e.g. logistic or Cox proportional hazard regression) in dose/exposure-response modeling, to advanced dynamic models based on differential equations in quantitative systems pharmacology.

Figure 3 : 4 :

 34 Figure 3: Mechanistic modeling of brain metastasis in NSCLC 137 . (A) A mathematical model describing gompertzian primary tumor growth (two parameters), response to treatment through a tumor growth inhibition model (three parameters) and metastatic apparition and growth (two additional parameters) was able to describe longitudinal data on number and size of visible brain metastases from a patient with NSCLC. Time unit is months from

  Ching et al. proposed Cox-nnet, an artificial neural network for prediction of survival from transcriptomic data[START_REF] Ching | Cox-nnet: An artificial neural network method for prognosis prediction of high-throughput omics data[END_REF] . The model is composed of two layers: one hidden, fully connected layer and one Cox regression layer. The model was found to exhibit same or better predictive accuracy compared to Cox-LASSO, Cox-ridge, or random survival forests. Other examples include survival prognosis from multi-omics data in hepatocellular carcinoma[START_REF] Chaudhary | Deep Learning-Based Multi-Omics Integration Robustly Predicts Survival in Liver Cancer[END_REF] , breast cancer26

  .g., between 3 mg/kg and 10 mg/kg for ipilimumab 114 ). Therefore, strategies aiming at modeling the full time course of response are appealing, and might provide better predictors of OS. Such tumor kinetics models (also called tumor growth inhibition -TGI -models) have now been widely used since their initial introduction by Tham et al.115 , Wang et al. 116 , Claret et al. 117 and

	further tested in a phase I clinical trial where it unfortunately did not outperform the standard scheme, but nevertheless was well tolerated and prevented progression of central nervous system metastases 107 . Although basic PK modeling was considered, no extensive pharmacometrics (incorporating mixed-effects statistical learning) was included. This feature could have helped control variability in peak serum concentrations, which was proposed by the authors as different (eStein et al. 118 . They are often coupled to parametric or semi-parametric
	a likely explanation of the modest results obtained 107 . proportional hazard survival models for description and prediction of OS from
	Using similar eco-evolutionary principles for tumor heterogeneity, the concept model-derived covariates 116,117 .
	of 'adaptive therapy' was proposed by Gatenby 110 . Arguing that sensitive cells While TGI modeling is now more than a decade-old 119 , with models of
	compete with resistant cells within the tumor, it suggests to treat only upon increasing complexity to describe advanced growth processes related to, e.g.,
	progression, on a patient-by-patient basis. After confirmation of the superiority prolonged response from chemotherapy or radiotherapy in gliomas 120 , recent
	of this strategy over classical MTD or metronomic schedules in animal works have focused on applying the methodology to data from clinical trials
	experiments 111 , 'adaptive therapy' was successfully transposed at the bedside
	in metastatic castrate-resistant prostate cancer patients 112 .
	Tumor size kinetics
	Tumor growth inhibition modeling
	The generalization of computerized tomography for imaging tumor lesions
	allows to collect, non-invasively, quantitative measurements of lesions sizes at
	multiple time points. These radiologic measures are employed to evaluate
	response to therapy during follow-up by means of the Response Evaluation
	Criteria In Solid Tumors (RECIST) 113 , used nowadays to make go/no-go
	group for instance has transposed biological principles of cancer evolutionary decisions at the end of phase II trials. RECIST consists in following target
	dynamics into mathematical models relying on stochastic processes to lesions defined at baseline, and computing variations of the sum of their largest
	understand and predict the onset of resistance to anti-EGFR therapies in lung diameters to categorize response (progression, stable disease, complete or
	cancer 109 . Using models of polyclonal tumor growth in which each cell can partial response). By doing so, longitudinal and continuous information about
	randomly divide (with or without mutation), live or die, and preclinical data to the response to treatment is lost. Consequently, RECIST-derived metrics such
	calibrate their model, an alternative dosing strategy for erlotinib in the treatment as overall response rate (ORR) and progression-free survival (PFS) are only
	of EGFR-positive patients with NSCLC was proposed 106 . The strategy was modest surrogates of the gold standard for evaluating efficacy of a treatment,
	based on a combination of pulse dose twice weekly and daily low-dose. It was i.e. OS. Indeed, OS results can differ, even though PFS and ORR are not

. A tumor is composed of multiple clones with varying degrees of sensitivity to a given drug, either preexisting at the beginning of therapy or emerging during the course of treatment, which is only effective against the sensitive cells. It is thus a tempting avenue to hypothesize that different dosing regimens impact differently on the appearance of resistance. In particular, the classical MTD paradigm might not be best adapted since it might lead to eradication of all sensitive cells, thus leaving space and nutrients available for resistant clones. This eco-evolutionary view of tumor growth has led several investigators to design mathematical models aiming at controlling resistance phenomena, from early studies by Goldie and Coldman 105 to more recent works by Michor 106,107 or Gatenby 108 . The Michor involving ICI. Patterns of response to immunotherapy may often differ from those observed with more classical anti-cancer agents. Response to ICI may exhibit long term stabilization of disease or response, or conversely different patterns of progression (i.e. pseudo/hyper-progression)

Table 1 : Case studies of artificial intelligence in clinical oncology

 1

	Method	Aim	Ref

Table 2 : Case studies of mechanistic modeling in clinical oncology 1

 2 FIH: first in human. PK: pharmacokinetics. PD: pharmacodynamics. TGI: tumor growth inhibition. OS: overall survival 2

Moreover, in order to deal with the relatively large number of covariates[START_REF] Ijzendoorn | Machine learning analysis of gene expression data reveals novel diagnostic and prognostic biomarkers and identifies therapeutic targets for soft tissue sarcomas[END_REF],