Data-driven modelling of biological multi-scale processes

Biological processes involve a variety of spatial and temporal scales. A holistic understanding of many biological processes therefore requires multi-scale models which capture the relevant properties on all these scales. In this manuscript we review mathematical modelling approaches used to describe the individual spatial scales and how they are integrated into holistic models. We discuss the relation between spatial and temporal scales and the implication of that on multi-scale modelling. Based upon this overview over state-of-the-art modelling approaches, we formulate key challenges in mathematical and computational modelling of biological multi-scale and multi-physics processes. In particular, we considered the availability of analysis tools for multi-scale models and model-based multi-scale data integration. We provide a compact review of methods for model-based data integration and model-based hypothesis testing. Furthermore, novel approaches and recent trends are discussed, including computation time reduction using reduced order and surrogate models, which contribute to the solution of inference problems. We conclude the manuscript by providing a few ideas for the development of tailored multi-scale inference methods.Comment: This manuscript will appear in the Journal of Coupled Systems and Multiscale Dynamics (American Scientific Publishers

A proof-of-concept pipeline to guide evaluation of tumor tissue perfusion by dynamic contrast-agent imaging: Direct simulation and inverse tracer-kinetic procedures

Dynamic contrast-enhanced (DCE) perfusion imaging has shown great potential to non-invasively assess cancer development and its treatment by their characteristic tissue signatures. Different tracer kinetics models are being applied to estimate tissue and tumor perfusion parameters from DCE perfusion imaging. The goal of this work is to provide an in silico model-based pipeline to evaluate how these DCE imaging parameters may relate to the true tissue parameters. As histology data provides detailed microstructural but not functional parameters, this work can also help to better interpret such data. To this aim in silico vasculatures are constructed and the spread of contrast agent in the tissue is simulated. As a proof of principle we show the evaluation procedure of two tracer kinetic models from in silico contrast-agent perfusion data after a bolus injection. Representative microvascular arterial and venous trees are constructed in silico. Blood flow is computed in the different vessels. Contrast-agent input in the feeding artery, intra-vascular transport, intra-extravascular exchange and diffusion within the interstitial space are modeled. From this spatiotemporal model, intensity maps are computed leading to in silico dynamic perfusion images. Various tumor vascularizations (architecture and function) are studied and show spatiotemporal contrast imaging dynamics characteristic of in vivo tumor morphotypes. The Brix II also called 2CXM, and extended Tofts tracer-kinetics models common in DCE imaging are then applied to recover perfusion parameters that are compared with the ground truth parameters of the in silico spatiotemporal models. The results show that tumor features can be well identified for a certain permeability range. The simulation results in this work indicate that taking into account space explicitly to estimate perfusion parameters may lead to significant improvements in the perfusion interpretation of the current tracer-kinetics models

Parameterization des modeles tumoral bases sur des maillages des donnees experimentaux.

In order to establish a predictive model for in-vivo tumor growth and therapy, a multiscale model has to be set-up and calibrated individually in a stepwise process to a targeted cell type and di erent environments (in-vitro and in-vivo). As a proof of principle we will present the process chain of model construction and parametrization from di erent data sources for the avascular growth of the EMT6/Ro and the SK-MES-1 cell line. In a rst step, a multiscale and individual-based model has been built up and validated with EMT6/Ro mouse mammary carcinoma multi-cellular cell spheroid data from literature. For this cell line it predicted the growth kinetics to be controlled by a combination of spatial restrains and nutrient limitation. ATP was found to be the critical resource which the cells try to keep constant over a wide range of oxygen and glucose medium concentrations by switching between aerobic and anaerobic metabolism. Only if both, oxygen and glucose are very limiting saturation was observed which the model could explain by cell-cell-adhesion-driven migration. In a second step, the model was adapted to the SK-MES-1 cell line. The growth kinetics was calibrated quantitatively in comparison with growth curves and qualitatively by image analysis of spheroid cryosections stained for apoptosis and proliferation. Beside ATP, lactate was identi ed to control the size of the necrotic core. For the transition to the in-vivo situation, we propose a model extension introducing a blood vessel network and angiogenesis. In order to parametrize the functional vessel properties and to validate angiogenesis rules, we study the parameter inference from contrast enhanced perfusion images. As a benchmark, we rst solve the direct problem of contrast agent perfusion along a network of either permeable or non-permeable vessels. Then by voxel-wisely solving the inverse problem and direct comparison between recovered and original parameter maps we study its predictive e ciency for di erent cases.Dans le but d'établir un modèle prédictif pour la croissance tumorale in-vivo et la thérapie, le modèle multi-échelle doit être élaboré et calibré par étape et de façon individuelle pour chaque type de cellule ciblé et pour di érents environnements (in-vitro et in-vivo). Nous présenterons, en tant que preuve de concept et à partir de di érentes sources de données, les étapes de la construction et de la paramétrisation du modèle de la croissance avasculaire des lignées de cellules EMT6/Ro et SK-MES-1. Dans une première étape, un modèle multi-échelle à base d'agents a été construit et validé avec des données provenant de la littérature sur les sphéroïdes multicellulaires de carcinomes mammaires de souris EMT6/Ro. Pour cette lignée de cellules, il a pu prédire que la cinétique de croissance est contrôlée par une combinaison de contraintes spatiales et de limitation des nutriments. Il a été trouvé que l'ATP est la ressource critique que les cellules essayent de garder constante en permutant d'un métabolisme aérobique à anaérobique et ce pour de larges plages de concentrations d'oxygène et de glucose. La saturation de la croissance a été observé uniquement dans le cas de faibles concentrations d'oxygène et de glucose ce que le modèle a pu expliqué par une migration guidée par l'adhésion de cellule à cellule. Dans une seconde étape, le modèle a été adapté à la lignée cellulaire SK-MES-1. Nous avons calibré la cinétique de croissance qualitativement en analysant des images de cryosections de sphéroïdes marquées pour l'apoptose et la prolifération et quantitativement en la comparant des courbes de croissance. Au delà de l'ATP, le lactate a été identi é comme contrôlant la taille du noyau nécrotique. Pour rendre compte de la situation in-vivo, nous proposons une extension du modèle qui prend en compte un réseau de vaisseaux sanguins et le phénomène de l'angiogenèse associé. A n de paramétrer les propriétés des vaisseaux fonctionnels et dans le but de valider les lois de l'angiogenèse, nous menons à partir d'images de perfusion d'agents de contraste une étude de sensibilité aux paramètres. Dans un premier temps, nous résolvons le problème direct de la perfusion des agents de contraste dans un réseau de vaisseaux perméables ou non. Ensuite, nous résolvons le problème inverse rigoureusement et, grâce à des comparaisons directes entre les paramètres originaux et ceux récupérés, nous étudions la capacité de prédiction du modèle dans di érents cas

Parameterization of lattice-based tumor models from data

Dans le but d établir un modèle prédictif pour la croissance tumorale in-vivo et la thérapie, le modèle multi-échelle doit être élaboré et calibré par étape et de façon individuelle pour chaque type de cellule ciblé et pour différents environnements (in-vitro et in-vivo).Nous présenterons, en tant que preuve de concept et à partir de différentes sources de données, les étapes de la construction et de la paramétrisation du modèle de la croissance avasculaire des lignées de cellules EMT6/Ro et SK-MES-1.Dans une première étape, un modèle multi-échelle à base d agents a été construit et validé avec des données provenant de la littérature sur les sphéroïdes multicellulaires de carcinomes mammaires de souris EMT6/Ro. Pour cette lignée de cellules, il a pu prédire que la cinétique de croissance est contrôlée par une combinaison de contraintes spatiales et de limitation des nutriments. Il a été trouvé que l ATP est la ressource critique que les cellules essayent de garder constante en permutant d un métabolisme aérobique à anaérobique et ce pour de larges plages de concentrations d oxygène et de glucose. La saturation de la croissance a été observé uniquement dans le cas de faibles concentrations d oxygène et de glucose ce que le modèle a pu expliqué par une migration guidée par l adhésion de cellule à cellule.Dans une seconde étape, le modèle a été adapté à la lignée cellulaire SK-MES-1. Nous avons calibré la cinétique de croissance qualitativement en analysant des images de cryosections de sphéroïdes marquées pour l apoptose et la prolifération et quantitativement en la comparant des courbes de croissance. Au delà de l ATP, le lactate a été identifié comme contrôlant la taille du noyau nécrotique.Pour rendre compte de la situation in-vivo, nous proposons une extension du modèle qui prend en compte un réseau de vaisseaux sanguins et le phénomène de l angiogenèse associé. Afin de paramétrer les propriétés des vaisseaux fonctionnels et dans le but de valider les lois de l angiogenèse, nous menons à partir d images de perfusion d agents de contraste une étude de sensibilité aux paramètres.Dans un premier temps, nous résolvons le problème direct de la perfusion des agents de contraste dans un réseau de vaisseaux perméables ou non. Ensuite, nous résolvons le problème inverse rigoureusement et, grâce à des comparaisons directes entre les paramètres originaux et ceux récupérés, nous étudions la capacité de prédiction du modèle dans différents casPARIS-BIUSJ-Mathématiques rech (751052111) / SudocSudocFranceF

Estimating Dose Painting Effects in Radiotherapy: AMathematical Model

International audienceTumor heterogeneity is widely considered to be a determinant factor in tumor progression and in particular in its recurrence after therapy. Unfortunately, current medical techniques are unable to deduce clinically relevant information about tumor heterogeneity by means of non-invasive methods. As a consequence, when radiotherapy is used as a treatment of choice, radiation dosimetries are prescribed under the assumption that the malignancy targeted is of a homogeneous nature. In this work we discuss the effects of different radiation dose distributions on heterogeneous tumors by means of an individual cell-based model. To that end, a case is considered where two tumor cell phenotypes are present, which we assume to strongly differ in their respective cell cycle duration and radiosensitivity properties. We show herein that, as a result of such differences, the spatial distribution of the corresponding phenotypes, whence the resulting tumor heterogeneity can be predicted as growth proceeds. In particular, we show that if we start from a situation where a majority of ordinary cancer cells (CCs) and a minority of cancer stem cells (CSCs) are randomly distributed, and we assume that the length of CSC cycle is significantly longer than that of CCs, then CSCs become concentrated at an inner region as tumor grows. As a consequence we obtain that if CSCs are assumed to be more resistant to radiation than CCs, heterogeneous dosimetries can be selected to enhance tumor control by boosting radiation in the region occupied by the more radioresistant tumor cell phenotype. It is also shown that, when compared with homogeneous dose distributions as thosebeing currently delivered in clinical practice, such heterogeneous radiation dosimetries fare always better than theirhomogeneous counterparts. Finally, limitations to our assumptions and their resulting clinical implications will be discussed

Modeling Steps from a Begnin Tumor to an Invasive Cancer: Examples of Instrinsically Multiscale Problems

The step from benign tumors to invasive cancer is characterized by neovascularization, detachment of cells from the main tumor and eventually invasion of cells into the surrounding tissue and blood vessels leading to distant metastases. We will for each of these steps show how experimental observations can be explained by the interplay of processes on the molecular and the cellular scale within a framework using individual-based models. The representation of the cell in the models permits us to represent physical, particularly biomechanical, constraints.We first study how a neoformation of blood vessels can affect the development of tumor size and shape if the nutrients transported in the vessels control the growth rate of the individual cells. Cell detachment is often triggered by a misfunctioning of the beta-catenin-degrading apparatus in the cytosol.We demonstrate how an elevated beta-catenin concentration in one cell can trigger a cascade of other cells stepwise detaching as well, which then can migrate freely into the surrounding tissue. Before the cells can form distant metastasis, they need to invade blood vessels.We show how the competition between N-CAM and VE-cadherin bonds can facilitate invasion of a cancer cell into a blood vessel if the involved pathways have defects

Inferring Growth Control Mechanisms in Growing Multi-cellular Spheroids of NSCLC Cells from Spatial-Temporal Image Data

International audienceWe develop a quantitative single cell-based mathematical model for multi-cellular tumor spheroids (MCTS) of SK-MES-1 cells, a non-small cell lung cancer (NSCLC) cell line, growing under various nutrient conditions: we confront the simulations performed with this model with data on the growth kinetics and spatial labeling patterns for cell proliferation, extracellular matrix (ECM), cell distribution and cell death. We start with a simple model capturing part of the experimental observations. We then show, by performing a sensitivity analysis at each development stage of the model that its complexity needs to be stepwise increased to account for further experimental growth conditions. We thus ultimately arrive at a model that mimics the MCTS growth under multiple conditions to a great extent. Interestingly, the final model, is a minimal model capable of explaining all data simultaneously in the sense, that the number of mechanisms it contains is sufficient to explain the data and missing out any of its mechanisms did not permit fit between all data and the model within physiological parameter ranges. Nevertheless, compared to earlier models it is quite complex i.e., it includes a wide range of mechanisms discussed in biological literature. In this model, the cells lacking oxygen switch from aerobe to anaerobe glycolysis and produce lactate. Too high concentrations of lactate or too low concentrations of ATP promote cell death. Only if the extracellular matrix density overcomes a certain threshold, cells are able to enter the cell cycle. Dying cells produce a diffusive growth inhibitor. Missing out the spatial information would not permit to infer the mechanisms at work. Our findings suggest that this iterative data integration together with intermediate model sensitivity analysis at each model development stage, provide a promising strategy to infer predictive yet minimal (in the above sense) quantitative models of tumor growth, as prospectively of other tissue organization processes. Importantly, calibrating the model with two nutriment-rich growth conditions, the outcome for two nutriment-poor growth conditions could be predicted. As the final model is however quite complex, incorporating many mechanisms, space, time, and stochastic processes, parameter identification is a challenge. This calls for more efficient strategies of imaging and image analysis, as well as of parameter identification in stochastic agent-based simulations

Radial organization of spheroids.

<p>Combination of images of a cross-section of a tumor simulated with the model presented in this article depicting the spatial organization of cellular phenotypes (proliferating, dying) and the molecular agents considered by the proposed model as main resources (oxygen, glucose), growth/viability promotors (GP/VP) or growth/viability inhibitors (GI/VI). The arrows point into the direction from high to low concentrations. The image shall be compared with the corresponding scheme in ref. [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004412#pcbi.1004412.ref008" target="_blank">8</a>], which shows a combination of images of spheroid median sections studied with different technologies: autoradiography, TUNEL assay, bioluminescence imaging, and probing with oxygen micro-electrodes.</p

Model comparison.

<p>The table shows the log likelihoods (A, B) and the Akaike information criterion (D) comparing three models to experimental data. The growth curves of all nutrient conditions and the radial profiles of proliferation, death and ECM at a specific time were compared individually (A) as well as a whole (B, D). We also calculated the values according to the Bayesian information criterion (BIC), which led to the same ordering (not shown). Notice that the penalty terms in the AIC (2<i>k</i>) and BIC (<i>k</i> log(<i>n</i>), <i>n</i>: number of data points) play no role in the evaluation as the difference in the penalty terms are much (4–5 orders of magnitude) smaller than ln <i>L</i>. (Δ AIC = AIC(Model row-index)—AIC(Model column-index).)</p

Cell density and cell size estimation.

<p>Left: The Delaunay triangulation of all segmented nuclei serves to estimate the cell sizes via its dual, the Voronoi diagram. Right: Average cell diameter as a function of distance to the spheroid border. The black curve is the average profile of six images (condition III, T3) with bin size 1<i>μm</i> and the red curve is the gliding average with window size 10<i>μm</i>.</p