How to provide robust decision guidance to reservoir managers challenging conflicting objectives?

Manager objectives: maintain the flow at monitoring stations ( ) between 2 extremes

How to provide robust decision guidance to reservoir managers challenging conflicting objectives?

Proactive (be ready for a future event) vs Reactive (Coping with an event)

The manager must balance between Proactive and Reactive Decisions… How to provide robust decision guidance to reservoir managers challenging many conflicting objectives?

Droughts Floods

From many objective decision… to a decision based on risk prioritization

How to get decisions that the manager can justify in the complex context of many conflicting objectives ?

The manager wants to minimize the risk to fail objectives… … but not all at the same time because some less important objectives can undermine more important ones

This means that the manager has to prioritize between acceptable and unacceptable risks for each objective depending on the situation A dashboard of the risks of future failure for each objective, taking into account the state of the system and the climate, would enable the manager to regain control of the decision-making process From single objective optimization… to many objective risk assessment

The problem is solved by dynamic programming:

• Modeling the minimum reservoir release (resp. filling) in reverse time for proactively handling flood (resp. drought) events

The model uses:

• A daily naturalized flows available at each monitoring station and at the reservoirs,

• a lag hydraulic routing model From single objective optimization… to many objective risk assessment

•
The risk assessment is based on evaluation of return period of extreme events by calculating optimal filling curves from daily naturalized flows How to convert optimal filling curves into risk assessment curves?

Credit: Julien Knez (https://kultt.fr)

Example of optimal minimum volume in Lake Yonne to achieve the objective of a minimum flow of 12,5 m 3 /s at Gurgy p. 8 From single objective optimization… to many objective risk assessment Sorted curves gives minimum (resp. maximum) stored volume associated with the probability of failing to achieve a future drought (resp. flood) event

How to convert optimal filling curves into risk assessment curves?

For each calendar day, we obtain the probability of future failure of the objective associated to the current stored volume

•

  3 rd : exceptionally flooded areas • 2 nd : frequently flooded areas • 1 st : bankfull discharge • Thresholds for low flows (based on water use restriction thresholds): • 1 st : vigilance • 2 nd : alert • 3 rd : reinforced alert • 4 th : crisis The case of the Seine lakes protecting for both floods and droughts p. 3 Drought and flood risk assessment of the Seine basin reservoir management under climate change Scientific Assembly of the IAHS, Montpellier, France / 1st June 2022 / Dorchies D. et al.

  Drought and flood risk assessment of the Seine basin reservoir management under climate change Scientific Assembly of the IAHS, Montpellier, France / 1st June 2022 / Dorchies D. et al.How to provide robust decision guidance to reservoir managers challenging conflicting objectives?The usual way (for scientists): the multi-objective optimization "Optimized decision" uptake of optimization tools by the water resource managers because:-limited understanding of the benefits and scope of the optimization -adopting excessively complex methodologies may generate practical problems that do not compensate for the benefits -lack of trust in the realism of models that leads to discarding optimization results (Pianosi et al. 2020) p. 5 Drought and flood risk assessment of the Seine basin reservoir management under climate change Scientific Assembly of the IAHS, Montpellier, France / 1st June 2022 / Dorchies D. et al.

  Drought and flood risk assessment of the Seine basin reservoir management under climate change Scientific Assembly of the IAHS, Montpellier, France / 1st June 2022 / Dorchies D. et al.

  constraint sets (physical constraints and local management rules such as minimum biological flow…) 315 optimizations performed (9 monitoring stations x 7 thresholds x 5 constraint sets) Example: Optimal release of the 4 reservoirs before the 1910 flood in Paris to maintain the flow under the 2 nd high flow threshold (1600 m 3 /s) How to compute optimal proactive decisions for a given objective? Drought and flood risk assessment of the Seine basin reservoir management under climate change Scientific Assembly of the IAHS, Montpellier, France / 1st June 2022 / Dorchies D. et al.

  Drought and flood risk assessment of the Seine basin reservoir management under climate change Scientific Assembly of the IAHS, Montpellier, France / 1st June 2022 / Dorchies D. et al.

Conclusion and perspectives

Current key points:

We are going to provide a decision risk assessment dashboard to the water resources managers:

• for multi-reservoir based on climatology statistics (historical or projections)

• this allows the manager to evaluate the risk of undermining proactive management when reactive decisions are taken

Future works:

• Derive real-time management rules from failure risk aversion defined by the water resources manager

• Test the robustness of the method in uncertain future

• Compare simulations of these management rules with current rules and other multi-objective optimizations


-adopting management practices aiming to improve animal health, -decreasing the inputs needed for production, -decreasing pollution by optimizing the metabolic functioning of farming systems, -enhancing diversity within livestock farming systems to strengthen their resilience, -preserving biological diversity in agroecosystems by adapting management practices

▪ "One Health/One Welfare" concepts -Animal health and welfare considered in a holistic approach integrating animals, humans, and their environment, and across scales (local, national, and worldwide)

▪

  Economic and political context ▪ Increasing demand for animal products (in the world) ▪ Intensification of production -Larger farms -> more animal / farmer and less time available / animal -Increasing productivity of animals and work ▪ Environmental impact -Energy consumption -Feed production -Excretion / emission of pollutants ▪ Increasing concerns about animal health and welfare ▪ New demands and views of livestock farming by consumers and society ▪ Social expectations of farmers (workload, attractiveness)

▪▪▪▪▪

  Revolution in digital new technologies -In society in general -In agriculture ▪ More and more advances electronic systems (wearable, autonomous, reactive) -IoT, connected objects -Robotic / cobotic -Virtual / augmented reality -Data and information flow / Big Data ▪ Methods for data analysis (high throughput , dynamic, AI, Importance of environmental impact of animal feed production  improve efficiency of feed utilisation to reduce inputs in livestock systems ▪ Identification (RFID), connected weigh scales, milking robots, automatic feed dispensers… as support for precision feeding = individual and daily adjustment of feed quantity and / or composition to individual nutritional requirements Decreasing the inputs needed for production ▪ Adaptation of feed composition depending on reproduction performance, feed intake, body composition… in gestating or lactating sows  reduction of 18.5 and 9% of nitrogen and phosphorus excretion and of 4% of feed cost per gestation Integrated management of farm animal health:  needs knowledge on -local environment -physiology and behaviour of animals ▪ Digital tools, based on a combination of various sensors, can be used to improve the acquisition, management, processing, and sharing of a complex information (different criteria, scales) in support to action, depending on algorithm quality -Assistance to improve prophylactic and curative treatments -Early detection of disease or welfare issues -Improvement of quality, quantity, targeting of treatments Panel of digital solutions to acquire measures to be aggregated -automates such as feeders, scales, milking robots… -wearable sensors (accelerometers, internal sensors…) -machine vision -sound analysis -To be adapted to purposes and characteristics of livestock system -for all systems: concerns on disease, heat stress… -pasture based systems: undernutrition, parasitism… -indoor production: postures, interactions between animals… -While avoiding overmedication and ensuring confidence in tools Well-being of farmers 10 One health / One Welfare includes wellbeing of farmer ▪ Automates, robots and sensors can allow farmers to save time by replacing physical tasks, simplifying monitoring while bringing flexibility and reducing stress through anticipation ▪ Changes in relationships with animals positively or negatively depending if saved time is reported to animal care ▪ Changes in the farmer's job -More attractive (new tools and skills, modern image) -Mental load regarding complexity, alerts, breakdowns -Loss of some skills and content if less close to animals…

  Mountain farming, pastoralism can benefit from such devices to help simplifying labor, monitoring and protect animals, managing resources (e.g. virtual fencing), traceability of products Potential for effective use to assist inputs and management at different scales (animals, waste production, market…) while being compatible with agroecology concepts) And thanks to Clément ALLAIN (Idele, FR) and Jean-Baptiste MENASSOL(Institut Agro, FR)

	Digital for a diversity of livestock systems farming Conclusion ▪ Conclusion ▪ More freedom for animals with outdoor access in livestock materials for plastic, electronic components) -negative environmental impacts (extraction, waste management…) -resource consumption (electricity…) and limitation of resources (raw Digital as a support but not a driver Digital within agroecological livestock farming systems: farming systems changing towards agroecological concepts  digital can help to continue to monitor animals and to understand effects of this evolution on welfare, health… (e.g., accelerometers, GPS, pedometers) for agroecological transition in livestock farming systems a balance to find between positive internalities and negative externalities ▪ Possible limitation of sustainability of digital solutions -lack of interoperability, risk of standardization (technics, animals…) Thank you!
	-costs/benefits for farmers
	▪  The integration of digital tools within agroecological livestock
	farming systems should be considered by taking into account Ingrand et al., 2018 this balance
	11 12 13

www.allflex.globalLelly

www.pellon.fi www.acofunki.fr www.octopusrobots.com www.tibot.fr


Antoine Zink

Dating group Elisa Porto

Christophe Falguères

Jean-Jacques Bahain

Anne Bouquillon

Yvan Coquinot

Benoit Mille

Lum

Benoît Mille

Christophe Falguère

With LumStimAge, the C2RMF will be able to offer a thermally stimulated (TL) and optically (OSL) luminescence imaging system focused on heritage materials (dating, characterization and provenance studies).

Imaging should be able to identify the most efficient zeroed grains while taking into account the effects produced locally by microdosimetry.

Recherche www.c2rmf.fr

LumStimAge project : TL/OsL imaging for heritage materials

OBJETS

LumStimAge project

Project description

The present project dedicated to TL / OSL imaging consists of the acquisition of the detection system (CCD camera and associated optics), and will include the physical optimization of the detection, calibration, signal-to-noise measurement system, but also the development or the adaptation of image processing algorithms to obtain a measurable intensity with a good spatial resolution.

The first applications will be devoted to composite materials, including raw architectural earth in connection with current research at C2RMF (stucco: ESPRIT project, raw bricks: MAFOUB project). Imaging should allow us to identify the grains that have undergone the most efficient zeroing during mortar fabrication, while taking into account the effects produced at the local level by microdosimetry.

Contribution to materials for heritage

The contribution of the imagery will lead to expand our knowledge of the mechanisms implemented in dating and of the properties of archaeomaterials (interaction between matter and radiation: effect of alpha and X radiation, characterization of archaeomaterials: luminescence, palaeoheating): > control of the homogeneity of the grains within a sample; > identification of alpha emitting hot spots -detection of artificial irradiations; > distribution of radioactivity -microdosimetry; > emission characterization (reproducibility, stability over time, sensitivity); > single grain dating; > research on hybrid materials, identification of phosphors and their dosimetric and luminescent characteristics; > evolution of the luminescence according to the degrees of heating; > study of the kinetics of luminescence.

TL / OSL imaging

Conclusion

The LumStimAge project proposes the acquisition of a TL / OSL imaging system to study stimulated luminescence phenomena (characterization, dating, provenance studies) in materials of heritage interest. In-depth research into image processing and algorithm development is needed.

  TL / OSL imaging, first developed internally by several laboratories (Clark-Balzan and Schwenninger, 2012), has recently been integrated as a module in commercial readers (Richter et al., 2013, Kook et al., 2015). The advantage of this method is that it can work directly on a raw sample, without grain separation, to evaluate the effects of micro-dosimetry by combining stimulated luminescence imaging with elementary imaging and autoradiography. On the other hand, there are optical aberration effects on produced images that distort measurements (Greilich et al., 2015). This requires "further exploration of image processing solutions based on more sophisticated algorithms" (Gribenski et al., 2015). This is what we plan to do by developing the potential of TL / OSL imaging to achieve sufficient sensitivity to quantify light intensity at each grain, even for young samples. Luminescence decay during optical stimulation (Top -spatial distribution ; bottom -intensity) • ©CAD A. Zink Image composed from the images obtained in IR-OSL (red channel) and in BL-OSL (blue channel). ; Fluvial sediment (Somme river, Epagne-Epagnette) irradiated at 300 Gy • ©CAD A. Zink Applications The dating of raw earth materials (mortar, brick) is a current issue both in radiocarbon and luminescence (see for example the works of GDRE Architectural terracotta and new dating methods associating the universities of Bordeaux, Milan, Durham and Catania ). As composite materials, they belong to the more general case of the dating of luminescent minerals in a complex matrix. Few people are interested in the dating of this type of material by luminescence, considering that the presence of different populations of grains resulting from partial restting or microdosimetric effects is problematic, although Urbanova (2015) shown the potential of dating mortars by single grain technique. Our TL / OSL imaging project is expected to go beyond the limits of the single grain technique (limited number of optical stimulations and need to work on isolated grains) by measuring grains in their environment. et al. 2015. Radiation Measurements. 81:171-177. Richter D et al.. 2013. Geochronometria. 40(4):220-228. Urbanová P. 2015. PhD Université Bordeaux III. 323p. Data treatment • ©CAD A.Zink

This work was supported by grants from Paris Ile-de-France Region -DIM "Matériaux anciens et patrimoniaux" Design N. Guyon -2018 DATATION


Annabelle Collin

Hadrien Bruhier

Jelena Kolosnjaj

Muriel Golzio

Marie-Pierre Rols

Clair Poignard

Spatial mechanistic modeling for prediction of 3D multicellular spheroids behavior upon exposure to high intensity pulsed electric fields

Keywords: PDE modeling, parameters estimation, electroporation, spheroids

de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Electroporation or electropermeabilization is a physical phenomenon that refers to the occurrence of permeant structures within the cell membranes upon cells exposure to well defined pulses of electric fields. Depending on the pulse parameters, such as the intensity of the electric field, their duration and the number (and the frequency) of the train of pulses, defects of different size may be induced in cell membrane, which make it permeable to non permeant molecules. This permeabilization can be either reversible (does not affect the cell viability) or irreversible (meaning that it can lead to cell death). In reversible electroporation, cell membranes reseal after several minutes and the cells continue their lifecycle. This process has led to a combined cancer therapy, referred to as electrochemotherapy, where electric pulses are used to permeabilize the cell and highly improve the local penetration of hydrophilic anticancer drugs [START_REF] Marty | Electrochemotherapyan easy, highly effective and safe treatment of cutaneous and subcutaneous metastases: Results of esope (european standard operating procedures of electrochemotherapy) study[END_REF]. Irreversible electroporation is the biophysical phenomenon that leads to cell death. This 1 process is currently under development in clinical oncology for nonsurgical and non thermal tumor ablation [START_REF] Miller | Cancer cells ablation with irreversible electroporation[END_REF]. It is worth noting that electric field induced membrane permeabilization is just one among many other phenomena [START_REF] Kolosnjaj-Tabi | Electric fieldresponsive nanoparticles and electric fields: physical, chemical, biological mechanisms and therapeutic prospects[END_REF], which can occur upon cells/tissue exposure to high electric field pulses.

While irreversible electroporation is efficient in most of the cases, with a complete ablation rate of about 75% of treated tumor nodules after one irreversible electroporation procedure [START_REF] Cannon | Safety and early efficacy of irreversible electroporation for hepatic tumors in proximity to vital structures[END_REF], it is well-known that for some patients the ablation is unsuccessful and event may lead to an accelerate growth of the tumor. In order to improve our understanding on the impact of the pulsed electric fields on the tumor response, we exploited multicellular tumor spheroids as an in vitro micro-tumor model. Multicellular tumor spheroids are three-dimensional structures composed of cancer cells that can accurately reproduce the behavior of small solid tumors in their preliminary avascular stage. Multicellular spheroids are becoming an essential tool in cancer research as they provide an intermediate complexity between 2D monolayer cell models and in vivo-grown solid tumors. Spheroids closely resemble solid micro-tumors in many aspects, such as the heterogeneous architecture, internal gradients of signaling factors, nutrients, and oxygenation. Moreover, the cells within the multicellular spheroids interact between each-other via cell-to cell interactions and their growth kinetics are similar to those observed in tumors in vivo. These similarities provide great potential for studying the biological properties of tumors, thus they are often used for drug screening as well as used to determine the therapeutic efficacy. Such models already demonstrated their relevance to address the electroporation phenomena [START_REF] Gibot | Antitumor drug delivery in multicellular spheroids by electropermeabilization[END_REF][START_REF] Frandsen | Calcium electroporation: evidence for differential effects in normal and malignant cell lines, evaluated in a 3d spheroid model[END_REF].

Standard clinical protocols of irreversible electroporation use about one hundred electric train of monopolar pulses of 100 µs duration of 1500 V/cm of nomimal amplitude at the frequency of 1 Hz. Our objective here was to quantify, thanks to mathematical modeling,the impact of different pulsed electric field of different nominal magnitudes on the growth of the spheroids. To mimick the pulses of the clinical protocols of IRE ablation, the train of pulses consist of 80 monopolar pulses of 100 µs duration at 1 Hz with an amplitude ranging from 500 to 2000 V•cm -1 . The aim was to characterize the multicellular spheroids regrowth after the pulse delivery with respect to the pulse magnitude. Throughout the paper the indicated field magnitudes are the nominal magnitudes, that is the voltage between the electrodes divided by their distance. It is worth noting it is not the electric field seen by the cells.

Thanks to mathematical models and calibration strategy, we fit the model with the experimental data to assess global behaviors among the spheroid population. Two approaches were considered. The first one consisted in studying only the resumption of the growth after the exposure to pulsed electric field with the Gompertz model, which is a classical ODE model for tumor growth [START_REF] Benzekry | Classical mathematical models for description and prediction of experimental tumor growth[END_REF][START_REF] Casey | The experimental alteration of malignancy with an homologous mammalian tumor material: I. results with intratesticular inoculation[END_REF][START_REF] Gompertz | XXIV. On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies[END_REF][START_REF] Laird | Dynamics of tumour growth[END_REF][START_REF] Laird | Dynamics of tumour growth: comparison of growth rates and extrapolation of growth curve to one cell[END_REF][START_REF] Norton | A gompertzian model of human breast cancer growth[END_REF]. Gompertz models are well-known to properly track the free growth of multicellular spheroids and tumors. The model consisted in a nonlinear ordinary differential equation written on the volume, where the multicellular spheroids are considered as a homogeneous entity. The second model is a more general model and is based on partial differential equations representing the evolution of populations of tumor cells controlled by a "carrying capacity". Three populations of tumor cells have been considered: the proliferative cells, the quiescent cells and the cells whose functioning is altered by the effect of the electric pulse. This more complex approach involves additional parameters but enables representing the heterogeneity of the response of the tumor cells within the multicellular spheroids.

The models were confronted to biological data corresponding to the volume evolution of multicellular spheroids submitted to electric pulses with different intensities. To fit the biological data with our models, we considered an estimation strategy of the parameters based on a population approach allowing to compensate for sparse sampling times and measurement uncertainties by constraining the variability of the parameters in the population.

The manuscript is divided as follows: in Section 2, the data containing various cohorts depending on the intensity of the electric field are presented. In Section 3, the mathematical models and the strategy to estimate the parameters using the biological data are introduced. Section 4 gives all the obtained results which are then discussed in Section 5.

Materials and method

Biological experiments

Biological experiments were conducted at the Institute of Pharmacology and structural biology (IPBS), Toulouse, France, and involved a human colorectal carcinoma cell line (HCT116) (ATCC ® CCL-247TM), expressing the green fluorescent protein (GFP) (HCT116-GFP). Cells were grown under standard conditions (5% CO 2 , 37 • C) in the Dulbecco's Modified Eagle Medium (DMEM, Gibco-Invitrogen, Carlsbad, CA, USA) containing 4.5 g/L glucose, L-Glutamine and pyruvate, 1% of penicillin/streptomycin, and 10% of fetal bovine serum. Multicellular cell spheroids were made by seeding 500 cells per well in Costar ® Corning ® Ultra-low attachment 96-well plates (Fisher Scientific, Illkirch, France). Plates were kept in 5% CO 2 humidified atmosphere at 37 • C. Multicellular spheroids acquired a cohesive structure and were submitted to pulsed electric field 3 days following the seeding. After the electric exposure (detailed below), the spheroids were repositioned in ultra-low attachment plates, in which they were kept for 10 days. Fresh medium was added to the spheroids at a 3 to 4 days interval. Multicellular spheroids growth was monitored by fluorescence and bright field videomicroscopy using the IncuCyte Live Cell Analysis System Microscope (Essen BioScience IncuCyte™, Herts, Welwyn Garden City, UK) at a magnification multiplied by × 10.

Electroporation was achieved following the delivery of 80 unipolar pulses at 0 V•cm -1 (control) or 500 V•cm -1 or 1000 V•cm -1 or 2000 V•cm -1 , the duration of a pulse was 100 µs, and the pulses were applied at a frequency of 1 Hz. The pulses were applied to multicellular spheroids in a low conduction pulsing medium (see [START_REF] Chopinet | First explanations for differences in electrotransfection efficiency in vitro and in vivo using spheroid model[END_REF][START_REF] Pelofy | Transfer of small interfering rna by electropermeabilization in tumor spheroids[END_REF]) with a parallel plate stainless steel electrode (the plate distance was 0.4 cm) connected to the Electrocell S 20 generator (Leroy Biotech, St Orens, France), following which the spheroids were rinsed in phosphate saline buffer and transferred to the cell growth medium. Once again let us emphasize that the values 500 V•cm -1 or 1000 V•cm -1 or 2000 V•cm -1 corresponds to the amplitudes of the nominal electric field. Presented results were obtained in two independent experiments. For each experiment, 6 to 8 spheroids were pulsed. Two control groups were used and are denoted either as free growth or 0 V•cm -1 . In free growth condition, the spheroids were kept in growth medium. For 0 V•cm -1 condition, the spheroids were transferred to the low conduction buffer for 10 minutes, following which the spheroids were rinsed in phosphate saline buffer and re-transferred to growth medium.

Images were analyzed with the ImageJ software (U.S. National Institute of Health, Bethesda, MD, USA). The software was used to determine the mean fluorescence intensity of spheroids, and to measure spheroids area in the equatorial plane. The representative micrographs of multicellular spheroids followed by fluorescence microscopy of control (0 V•cm -1 ), 500 V•cm -1 , 1000 V•cm -1 and 2000 V•cm -1 treated spheroids, immediately after the application of electric pulses are shown on Figure 1. The green color reflects the green fluorescing protein in living cells and the red color corresponds to propidium iodide, a red fluorescent probe that penetrates the cells when permeabilization occurs on cell membrane. Note that at 0 V•cm -1 no red fluorescence is visible, while at higher electric field the degree of red color increases, as increasingly more propidium iodide penetrates the cells illustrating the cell permeabilization. Figure 1-Top (resp. Bottom) shows the spheroids immediately (resp. 1 h, 4 h, 1 day and 10 days) after the shock. Note that 500 V•cm -1 pulses do not affect cell viability (as indicated by green fluorescence and the growth of the cellular mass), while at 1000 V•cm -1 we have a transient disappearance of green fluorescence, and at 2000 V•cm -1 we have a permanent disappearance of green fluorescence. Figure 2 shows the representative micrographs of treated spheroids (1000 V•cm -1 ), that illustrates the localization of spheroid regrowth following the pulse delivery. At 5 h following electroporation the green fluorescence generally begins to fade (complete extinction of fluorescence is only observed in one spheroid). Twenty four hours after treatment, the fluorescence is either extinct, or still observed inside the spheroid or in the periphery.

Data presentation

Figure 3 shows the evolution of the volume of the multicellular spheroids in control (Left panel) and electroporation (Right panel) configurations. In total 83 multicellular spheroids were tracked over 250 hours. Measurements, as plotted on Figure 3 were obtained from fluorescence micrographs as described in Section 2.1.

The cohort is divided as follows:

• 24 multicellular spheroids correspond to the control spheroids (free growth). We exclude the spheroids 3 and 24 as dynamics strongly differs from the others.

• 59 multicellular spheroids correspond to electroporated spheroids divided into 4 categories depending on the electric field strength.

Different groups are denoted EF0, EF500, EF1000 and EF2000, respectively, which corresponds to measures obtained in groups exposed to 0 Right-Electroporation experiment (59 cases divided into 4 categories: 13 for EF0, 16 for EF500, 14 for EF1000 and 16 for EF2000).

V•cm -1 , 500 V•cm -1 , 1000 V•cm -1 and 2000 V•cm -1 , respectively.

Elimination of the outliers

The cohort EF0 contains 13 multicellular spheroids. The cohort EF500 contains 16 multicellular spheroids among which 4 have been excluded: 2 because they did not see the electric field (spheroids 18 and 29) and 2 because their dynamics strongly differed from the others (spheroids 16 and 17). The cohort EF1000 contains 14 multicellular spheroids among which 3 have been excluded: 1 because there was no data after t > 90 h (spheroid 43), 1 because the time 90 h (which is a fundamental time for the estimation process) was missing (spheroid 37) and 1 because its dynamics strongly differed from the others (spheroid 38). The cohort EF2000 contains 16 multicellular spheroids among which 2 have been excluded: 1 because it did not see the electric field (spheroid 59) and 1 because its dynamics strongly differed from the others (spheroid 47).

Division of the cohort EF1000 into 2 subgroups

Data presented in Figure 4, indicates that according to multicellular spheroids response to the electric field, the cohort EF1000 could be split into 2 subgroups. As two kinds of behaviors appeared in cohort EF1000, we study them separately:

• Group A (6 spheroids): the multicellular spheroids were very damaged by the electric shock, the volume of green fluorescing cells reached almost 0 mm 3 after 48 hours,

• Group B (5 spheroids): the dynamics of the multicellular spheroids were very similar to the dynamics observed in cohort EF500.

Precisely, multicellular spheroids with a volume at 43h below (resp. above) 0.01 mm 3 could be grouped into Group A (resp. Group B), see Figure 4. 3 Methods

Modeling of volume evolution

Gompertz model for free growing volume Gompertz model is a well-known ordinary differential equation (ODE) for tumor growth, which fits well with the free growth of in vitro and in vivo tumors. The reader may refer to [START_REF] Benzekry | Classical mathematical models for description and prediction of experimental tumor growth[END_REF] for a comparison of different ODE models on in vivo data and [START_REF] Gompertz | XXIV. On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies[END_REF][START_REF] Casey | The experimental alteration of malignancy with an homologous mammalian tumor material: I. results with intratesticular inoculation[END_REF][START_REF] Laird | Dynamics of tumour growth[END_REF][START_REF] Laird | Dynamics of tumour growth: comparison of growth rates and extrapolation of growth curve to one cell[END_REF][START_REF] Norton | A gompertzian model of human breast cancer growth[END_REF][START_REF] Collin | Spatial mechanistic modeling for prediction of the growth of asymptomatic meningiomas[END_REF] for validations. The tumor volume arising from the Gompertz equation reads

V (t) = V 0 e a b (1-e -bt ) , (1)
where V 0 corresponds to the initial volume at time t = 0, b -1 corresponds to the characteristic time at which the tumor growth capacity decreases and a is a constant corresponding to the initial growth rate.

Population approach to fit the data To estimate the three parameters V 0 , a and b, a population approach has been chosen. The mixed-effect approach [START_REF] Ribba | A review of mixed-effects models of tumor growth and effects of anticancer drug treatment used in population analysis[END_REF] consists of pooling all the multicellular spheroids together and estimating a global distribution of the model parameters in the population. More precisely, the individual parameters a j and b j (where j denotes the individual) are assumed to be realizations of a random variable decomposed into two parts:

{V 0 , a, b} j = {V 0 , a, b} pop + { Ṽ0 , ã, b} j , ∀j ∈ [1, N s ],
where N s is the number of considered spheroids, V pop 0 , a pop and b pop correspond to the fixed effects and Ṽ0 j , ãj and bj correspond to the random effects and have been assumed with mean zero. The parameters that we want to estimate are not physiological and there are no priors for their means and standard deviations. Then, we assume that

a pop , b pop ∼ N (0, 0.1 2 ) and V pop 0 ∼ N (0, 0.05 2 ),
emphasizing that these assumptions do not seem too restrictive. The estimation strategy is not sensitive to these priors. We then assume that the measurement error for all volumes is proportional V j,M RI (t) = V j,estim (t)(1 + e), where V j,estim k corresponds to the estimated volume using ODE systems and e the measurement error. We assume that it follows a Gaussian law N (0, 0.1 2 ). The standard deviation is estimated.

This estimation strategy is implemented in the software Monolix [START_REF]Monolix version 2019R1[END_REF], which maximizes the likelihood using the stochastic approximation expectation maximization (SAEM) algorithm [START_REF] Kuhn | Maximum likelihood estimation in nonlinear mixed effects models[END_REF]. For all the results obtained with Monolix in this paper, the convergence of the algorithm has been reached.

The population estimation strategy for Gompertz model will be applied on the 22 multicellular spheroids of the control experiment and the 13 multicellular spheroids EF0 of the electroporation experiment in order to verify that the cohorts are similar. The objective is to validate the assumption that there is no impact of the electrodes on the growth.

A correlation between the parameters a and b in a population of tumors has been observed frequently in Gompertz model. In [START_REF] Vaghi | Population modeling of tumor growth curves and the reduced gompertz model improve prediction of the age of experimental tumors[END_REF], a reduced Gompertz model is introduced consisting in estimating V 0 and b as previously and k = a b as a constant parameter in the population i.e.

{V 0 , b} j = {V 0 , b} pop + { Ṽ0 , b} j , ∀j ∈ [1, N s ],

{k} j = {k} pop , ∀j ∈ [1, N s ].
The use of the reduced Gompertz model improves the stability of the estimation process and will be used in what follows. There were fewer measurements in the electroporation experiment than for the control experiment. It is therefore necessary to check whether the SAEM algorithm works correctly with a low number of spheroids and a low number of measurements. To do this, we create a cohort with a certain number of individuals denoted by y, drawn at random from the control group. Then, we randomly select x consecutive times to estimate the parameters. This operation is performed 50 times in order to minimize errors caused by chance. The results are given in Appendix.

Gompertz model for multicellular spheroids after electroporation pulses In this paragraph, we want to study the growth of multicellular spheroids after the electric shock in order to answer the following questions.

• After the electric shock, is there a resumption of growth?

• Is it still a Gompertz growth?

• What are the impacts on the parameters of the model? First of all, for most of the spheroids which have been electroporated (excepted of spheroids for cohort EF2000), a resumption of the growth occurs, see Figure 3. Letting t init , the time of resumption and t last the time of the last measurement, we rewrite the reduced Gompertz model in an inverted form

V (t) = V last e ke -bt last (1-e b(t last -t) )
.

For the 3 cohorts EF500, EF1000-A and EF1000-B, we will estimate the parameters V last , b and k for each fixed t last using the population approach presented previously (in which V 0 plays the role of V last ). The parameter t last will be estimated by considering the minimal value given reasonable errors. This approach based only on the volume evolution will show some limitations motivating us to consider 3D evolution of the multicellular spheroids.

Modeling the 3D evolution of spheroid

In this section, we complexify the growth model to introduce a heterogeneity in the cell population. The model has been designed to describe the 3D evolution of the spheroids with two constraints: (1) to be compatible with the Gompertz model at the volume scale, (2) to depend on few parameters which are identifiable with the available data.

General equations We will consider 3 compartments of cells: the proliferative ones whose the density denoted by P , the quiescent ones whose the density denoted by Q and the cells with a functioning altered by the impact of the electrical shock whose the density denoted by F . By denoting the tumor spheroid domain at time t by Ω(t), the evolution of the density P is supposed to satisfy the following equation

∂ t P + ∇ • ( vP ) = f (t, P, Q), Ω(t), (2)
and the density of Q follows

∂ t Q + ∇ • ( vQ) = g(t, P, Q), Ω(t), (3)
where v denotes the velocity field that describes the motion of tumor cells. The evolution of the density F is supposed to satisfy the following equation

∂ t F + ∇ • ( vF ) = 0, Ω(t). (4)
Remark 1. In the case of free growth (when there is no electroporation), the density F equals to 0 for all t and for all x.

Using the saturation hypothesis P +Q+F = 1 in Ω(t), we have ∇•v = f (t, P, Q)+g(t, P, Q). This implies that the volume defined by

V (t) = Ω(t) dX verifies V (t) = d dt Ω(t) dX = ∂Ω(t) (v • n)dX = Ω(t) ∇ • vdX = Ω(t)
[f (t, P, Q) + g(t, P, Q)]dX.

Our objective is to have the volume following a Gompertz law when there is no electroporation (then P + Q = 1, Ω(t)). One can prove -following the proof given in [START_REF] Collin | Spatial mechanistic modeling for prediction of the growth of asymptomatic meningiomas[END_REF] -that a way to do so consists in imposing

f (t, P, Q) + g(t, P, Q) = τ G (t)(P (t, x) + Q(t, x)), ∀t, (5)
where τ G (t) = ae -bt corresponds to the growth rate.

Modeling of the tumor heterogeneity

The function g has to be defined in order to obtain realistic behavior. Assuming that proliferative cells become quiescent cells, we assume that g(t, P, Q) = τ P toQ P . The function τ P toQ depending on (t, x) has to be chosen to describe the appearance of a high quiescent proportion of cells in the center of the spheroid.

Electroporation modeling When electroporation pulses occur, three phenomenas are considered:

(1) a part of cells is destroyed,

(2) the functioning of a part of the cells is modified i.e. a part of proliferative and quiescent cells changes their functioning,

(3) the initial growth rate is modified. Denoting by t as the time just after the electrical shock, these three phenomenas can be mathematically modeled by:

(1) the death of proliferative and quiescent cells leads to P (t as , x) = 0, for x ∈ ω P ⊂ Ω(t < t as ) and Q(t as , x) = 0, for x ∈ ω Q ⊂ Ω(t < t as ) then Ω(t as ) decreases,

(2) the functioning modification of proliferative and quiescent cells leads to F (t as , x) = λ(P (t as , x) + Q(t as , x)), for x ∈ Ω(t as ),

(3) the value of the parameter is decomposed into two parts:

a = a bs , ∀t < t as , ma bs , ∀t ≥ t as , (6)
where m is a constant (always estimated superior to 1) which can be interpreted as a boost of the growth after the electrical shock.

Radial equation for free growth Inspiring from the work of [START_REF] Michel | Mathematical modeling of the proliferation gradient in multicellular tumor spheroids[END_REF], the spherical symmetry of spheroid can be used to rewrite the model in radial and relative coordinates. Interestingly, this implies that the velocity field v is entirely determined by its divergence, therefore no assumption on the rheology of the tumor spheroid is needed. Since the volume V = 4 3 πR 3 follows a Gompertz law when free growth is considered as ( 5) is verified then simple calculations shows that the radius of the spheroid R follows also a Gompertz law: R(t

) = R 0 e a 3b (1-e -bt ) with R 0 = 3 4π V 0 1 3 since it satisfies R (t) = τ G (t) 3 R(t). (7)
For all x ∈ Ω(t), we denote the normalized radial coordinate (r = 0 at the center of the spheroid, r = 1 at the surface)

r(t, x) = x R(t)
.

Under the hypothesis of invariance by rotation, we can then define, for all (t, x): ũ(t, r(t, x)) = u(t, x), for unknowns P, Q, τ P toQ and v.

We denote by ṽ the radial component of the velocity ˜ v and the invariance by rotation implies that ˜ v = ṽ e r . As the explicit formula of R is known, it is sufficient to compute the unknown P and then Q can be deduced by Q = 0 outside the tumor and Q = (1 -P ) inside the tumor. One can show, see Appendix for the proof-that P follows for each r ∈ [0, R(t)], the following ordinary differential equation P = τ G (1 -P ) -τP toQ P .

To close the system, the evolution of the rate τP toQ has to be described. Seen as representing the lack of oxygen, one can consider the following logistic function τP toQ (t, r) = τ begin P toQ -

τ begin P toQ -τ end P toQ 1 + e (R(t)(1-r)-d) s
.

The parameter d corresponds to the distance from the tumor front for which the oxygen is easily accessible. The parameter s corresponds to the slope of the function. The parameters τ begin P toQ (resp. τ end P toQ ) corresponds to the value of τP toQ far from (resp. close to) the tumor boundary.

Remark 2. When the hypothesis of spherical symmetry is not valid anymore, the equation on the velocity ∇ • v = f + g is not sufficient enough to close the system. For example, in [START_REF] Collin | Spatial mechanistic modeling for prediction of the growth of asymptomatic meningiomas[END_REF], a Darcy law v = -∇π where π is the pressure exerted by tumor cells to healthy cells is considered. Furthermore, the evolution of the rate τ P toQ is not valid anymore as it directly depends on R and r. One could consider the distance to the tumor front instead of the radius. The study of the stability of the results for almost radial situations is referred to future work.

Radial equation for electroporation process When considering electric pulses, the volume V (resp. the radius R) does not follow Gompertz laws as we have:

V (t) = τ G (t) Ω(t) (P (t, x) + Q(t, x))dX.
We denote by I the following function of t and r

I(t, r) = r 0 ( P (t, r) + Q(t, r))r 2 dr
Using radial coordinate and the fact that V = 4 3π R 3 , one can show that

R (t) = R(t)τ G (t)I(t, 1).
Following the proof given in Appendix , one can show that

∂ t P (t, r(t, x)) + τ G (r -2 I(t, r) -rI(t, 1))∂ r P (t, r(t, x)) = τ G (1 -F )(1 -P ) -τP toQ P . (8)
and

∂ t F (t, r(t, x)) + τ G (r -2 I(t, r) -rI(t, 1))∂ r F (t, r(t, x)) = -τ G (1 -F ) F , (9)
then Q can be determined using the fact that Q = 1 -( P + F ). Six parameters have to be estimated: V 0 , k, b, p, m and λ using volume evolution. As previously, the electroporation data are separated in four groups: EF500, EF1000-A, EF1000-B and EF2000. To properly estimate the values of the parameters, we do not consider the 4 times juste after pulses: t = 18, 24, 30, 36 h. Indeed, due to to the osmotic shock in cells constituting the spheroids after the electrical shock, very large values of volumes have been measured. As our model does not integrate this phenomenon, only the values after the rebuilding of the multicellular spheroids (t > 36 h) will be used to estimate the parameters. The software Monolix [START_REF]Monolix version 2019R1[END_REF] cannot be used here to estimate the parameters as our model is a PDE system. We then will use a Matlab library called nlmefitsa in which the SAEM algorithm is implemented.

Considered errors

To validate all our estimation strategy, we will consider the mean squared error (MSE) defined by

MSE = 1 N s Ns s=1 1 N s t N s t i=1 |V i ds -V (t i s )| 2 , (10)
and the l 1 -norm defined by

err 1 = 1 N s Ns s=1 1 N s t N s t i=1 |V i ds -V (t i s )|, (11)
where N s t corresponds to the number of observations for each spheroid s ∈ [1, N s ], V i ds corresponds to the data at time t i s for each spheroid s and each observation i ∈ [1, N s t ] and V (t i s ) to the predicted volume value at time t i s .

4 Results

Volume fits

Free growth Table 1 gives the mean and the standard deviation of the estimated values of V 0 , a and b for the cohort of control experiment (second column) and for the cohort EF0 of electroporation experiment (third column). As expected, the values are very close meaning that the electrodes do not perturb the growth. Figure 5-Top shows the corresponded fits. The mean and the standard deviation of the estimated values of V 0 , a and b coupling the 2 cohorts are given in the last column of Table 1. Figure 5-Bottom-Left shows the strong correlation between parameters a and b (p = 6.02 × 10 -9 ) validating the use of the reduced Gompertz model hence the fits using the coupling of the 2 cohorts are given in Figure 5-Bottom-Right and the values of the estimated parameters in Table 2. Electroporation Figure 6 shows MSE associated to each considered initial time for EF500, EF1000-B and EF1000-A. Using this data, checking the stability of the estimated parameters and the quality of fits, we fix the value at t init = 90 h (resp. 90 h and 43 h) for cohort EF500 (resp. EF1000-B and EF1000-A). The corresponding fits are given in Figure 7 and the corresponding estimated parameters in Table 3. Figure 8-Left shows the Gompertz evolution for the three cohorts using the mean of estimated parameters between 0 h and 250 h. This figure can be directly compared to the data given in Figure 3-Right.

3D spheroid fits

For the sake of simplicity, the tilde notations will be removed in this section.

Free growth The parameters a = kb and b have been fixed at the means of the estimated parameters obtained using the 2 cohorts of control, see Table 2. To reproduce heterogeneity similar to the data available in [START_REF] Michel | Mathematical modeling of the proliferation gradient in multicellular tumor spheroids[END_REF] (Figure 6, CAPAN-2 control), we fix the values d = 0.3, τ begin P toQ = 0.04, τ end P toQ = 0 and s = 0.05. We consider an initial constant value of P of 0.8. None of these 3 parameters (d, τ begin P toQ , τ end P toQ and s) or of the initial condition P (0, •) can be estimated using only the volume of the spheroid as the equation on R (or on V ) does not depend on them. Figure 9 shows the evolution of r → P (•, r) at different time snapshots. Figure 10 (first column) shows the spheroid evolution with distribution of proliferative cells. These simulations are obtained by using the mean values of estimated values . Electroporation process Table 4 gives the estimation of the 6 parameters for the 4 As expected, values of parameters V 0 , k and b are close to the values estimated with control data. The estimated parameters of cohort EF1000-B are very similar to the parameters of cohort EF500. The values of λ increase with the intensity of the pulse. Concerning the value of p, it is interesting to note that there is no (resp. small) decrease of the volume for EF500 (resp. EF1000-B V•cm -1 ) while for the group 1000-A V•cm -1 (resp. 2000 V•cm -1 ), a decreasing of 80% (resp. almost 100%) is observed. Concerning m, the value increases with the value of the electrical field. Concerning the MSE and the l 1 -norm, the values are close to the values obtained with the free growth cohort, see Table 2 for example. Fits are given in Figure 11. Figure 10 shows the spheroid evolutions with distribution of proliferative cells for the 4 cohorts (column 2 to column 5). Figure 8-Right shows the Gompertz evolution for the four cohorts using the mean of estimated parameters between 0 h and 250 h. This figure can be directly compared to the data given in Figure 3-Right.

Discussion and Conclusion

We first focused on the control cohort and we validated the reduced Gompertz model a more stable version of the very classical Gompertz model (1) using biological data corresponding to multicellular spheroids in free growth. We also showed that the presence of the electrodes did not impact the growth by comparing the control cohort with the data in the electroporation cohort, where electric pulses were not applied. It is possible to obtain reasonable fits using the reduced Gompertz model, see Figure 7. Even if this volume approach shows encouraging preliminary results, this approach did not allow to properly quantify the impact of the electroporation and did not integrate the heterogeneity of the tumor, i.e. the fact that the tumor was composed of proliferative and quiescent cells. These two limitations motivate the introduction of a PDE system enabled to represent the tumor evolution upon submission to electrical shocks. Three densities of tumor cells were considered: the proliferative cells, the quiescent cells and the cells whose functioning was altered by the effect of electric field. This model was confronted to biological data corresponding to multicellular spheroids submitted to electric pulses with different intensities. The quality of the fits on the spheroid volumes see Figure 11 and the errors see Table 4 validated our model. These results allowed determining the percentage of destroyed cells and the percentage of cells whose functioning was altered by the effect of electroporation for each value of the electrical field. One could see that for an electrical field close to 1000 V•cm -1 (partially irreversible electroporation and partially reversible electroporation), the increasing of the electrical field seemed to boost the resumption of spheroids growth, see Figure 8-Right.

Our results provide a good insight into the impact of electroporation on tumor growth. In clinics, when irreversible electroporation is used to ablate tumors, the treatment may induce cell death by irreversible electroporation. This effect is observed in a specific zone, namely, the zone where the electric field strength is the highest at the optimal place in respect to the electrodes. The efficacy of completely irreversible electroporation was well observed in spheroids group exposed to 2000 V•cm -1 .

Nevertheless, adjacent to the epicenter of the highest electric field, where the tumor is ablated, we have zones that underwent milder electric field strengths exposure, and we have the occurrence of partially irreversible electroporation. Within this shadow zone, not all cells are killed, and the ones that survive will resume their growth. This effect was clearly present in the multicellular spheroids exposed 1000 V•cm -1 . This field amplitude generated thus in two radically different behaviors: either death or growth boost. We hypothesize that this variable response rate is due to the non-homogeneity of the electric field. If the spheroid (or tumor) was not efficiently attained, we have a relapse. If the positioning of the spheroid (or tumor) is in the center of the electrodes, the treatment is efficient. This is a recurrent problem also in clinics, where clinicians observe a different response rate either efficient treatment of a patient with a decreased tumor growth or tumor relapse [START_REF] Scheffer | Irreversible electroporation for nonthermal tumor ablation in the clinical setting: a systematic review of safety and efficacy[END_REF]. The advantage of our model, is that soon after the electroporation, we can detect the trend of tumor growth. In practice, this model can thus allow to assess which tumors were inefficiently treated, and that the patient is on his or her way to relapse. This would indicate to the treating doctor that the treatment was not efficient and that an additional cycle of electroporation is necessary to re-ablate the tumor. Concerning the growth boost, it is particularly intriguing and certainly requires further inquiries, from the biological, physical and mathematical perspective.

Figure 1 :

 1 Figure1: Bright field micrographs of multicellular spheroids followed by fluorescence microscopy of control (0 V•cm -1 ), 500 V•cm -1 , 1000 V•cm -1 and 2000 V•cm -1 -treated spheroids, after the application of electric pulses (N = 80, t = 100 µs, ν = 1 Hz). Green color: green fluorescing protein in living cells and red color: propidium iodide. Top: immediately after pulses application. Bottom: 1 h, 4 h, 1 day and 10 days after pulses application.

Figure 2 :

 2 Figure 2: Localization of spheroid regrowth following electroporation at 1000 V•cm -1 . Top: micrographs of representative spheroids 5 h post treatment. The green fluorescence generally begins to fade (complete extinction of fluorescence is only observed in one spheroid-top right panel). Middle: same spheroids as in top panels at 24 h after treatment. The fluorescence is either still observed inside the spheroid or in the periphery (first and second middle micrographs, respectively, or is completely extinct). Bottom: micrographs of same spheroids as on top panels, at 7 days after treatment, all exhibiting green fluorescence that indicate spheroids viability and growth. The regrown zones can be either attached to the treated spheroids or scattered in space.

Figure 3 :

 3 Figure 3: Volume evolutions of multicellular spheroids. Left-Control experiment (24 cases). Right-Electroporation experiment (59 cases divided into 4 categories: 13 for EF0, 16 for EF500, 14 for EF1000 and 16 for EF2000).

Figure 4 :

 4 Figure 4: Volumes after exposure to electric pulses (t = 43 h) for cohort EF1000 (logarithmic scale).

Figure 5 :

 5 Figure 5: Volume evolution in case of free growth. Left-Top: fits using classical Gompertz model for the cohort of control experiment. Right-Top: fits using classical Gompertz model for the cohort EF0 of electroporation experiment. Left-Bottom: parameter a in function of parameter using classical Gompertz model for the fusion of the 2 free growth cohorts. Right-Bottom: fits using reduced Gompertz model for the fusion of the 2 free growth cohorts.

Figure 6 :

 6 Figure 6: Mean squared error (MSE) depending on t init using reduced Gompertz model. Left: EF500. Middle: EF1000-B. Right: EF1000-A.

Figure 7 :

 7 Figure7: Volume evolution after electrical shock. Left: fits using reduced Gompertz model for EF500 (t init = 90 h). Middle: fits using reduced Gompertz model for EF1000-B (t init = 90 h). Right: fits using reduced Gompertz model for EF1000-A (t init = 43 h).

Figure 8 :

 8 Figure 8: Simulations using the mean of estimated parameters for the 3 cohorts. Top: volume models (Gompertz after electroporation process). Bottom: 3D models.

Figure 9 :

 9 Figure 9: Free growth -time snapshots of r → P (•, r).

Figure 10 :

 10 Figure 10: Spheroid evolutions with distribution of proliferative cells. Black (resp white) = 100 (resp. 0) % of proliferative cells. Column 1: free growth, column 2: EF500, column 3: EF1000B and column 4: EF1000A and column 5: EF2000.

Figure 11 :

 11 Figure 11: Fits of electroporation system (8)-(9) using volume data for the 4 cohorts: EF500 (top-left), EF1000B (top-right), EF1000A (bottom-left) and EF2000 (bottom-right).

Table 1 :

 1 Estimated parameters of Gompertz model in case of free growth.

	Parameters	Control Experiment	Electroporation experiment EF0	Cohorts coupling
	V 0	0.052 (mean) -0.0024 (std)	0.055 (mean) -0.00069 (std) 0.050 (mean) -0.0024 (std)
	a	0.024 (mean) -0.0023 (std)	0.019 (mean) -0.0017 (std)	0.023 (mean) -0.0027 (std)
	b	0.011 (mean) -0.00093 (std) 0.0091 (mean) -0.00055 (std) 0.011 (mean) -0.0012 (std)
	MSE	0.00016	0.00011	0.00014
	err 1	0.0089	0.0079	0.0086

Table 2 :

 2 Estimated parameters of reduced Gompertz model in case of free growth.

	Parameters Free growth (control case and EF0)
	V 0	0.051 (mean) -0.0029 (std)
	k	2.18 (mean) -0.0 (std)
	b	0.010 (mean) -0.0017 (std)
	MSE	0.00023
	err 1	0.010

Table 3 :

 3 Estimated parameters of reversed reduced Gompertz model after electrical shock. Left: EF500. Middle: EF1000B. Right: EF1000A.

	Parameters	EF500	EF1000-B	EF1000-A
	t init	90h	90h	43h
	V last	0.24 (mean) -0.039 (std)	0.24 (mean) -0.017 (std)	0.19 (mean) -0.051 (std)
	k	3.17

Table 4 :

 4 Estimated parameters of electroporation system (8)-(9) using volume data for for the 4 cohorts: EF500, EF1000B, EF1000A and EF2000.

	Parameters	EF500	EF1000-B
	V 0	0.063 (mean) -2.14 ×10 -6 (std)	0.062 (mean) -0.00011 (std)
	b	0.0071 (mean) -0.0014 (std)	0.0086 (mean) -0.00055 (std)
	k	2.12 (mean) -0 (std)	1.83 (mean) -0 (std)
	λ	0.63 (mean) -0.015 (std)	0.66 (mean) -0.061 (std)
	p	0.99 (mean) -9.89 ×10 -5 (std)	0.86 (mean) -0.15 (std)
	m	1.36 (mean) -0.039 (std)	1.98 (mean) -0.32 (std)
	MSE	0.00013	6.17 ×10 -5
	err 1	0.0089	0.0060
	Parameters	EF1000-A	EF2000
	V 0	0.055 (mean) -1.10 ×10 -5 (std)	0.057 (mean) -0.0014 (std)
	b	0.011 (mean) -6.04 ×10 -5 (std)	0.0078 (mean) -0.0019 (std)
	k	2.30 (mean) -0 (std)	2.46 (mean) -0 (std)
	λ	0.90 (mean) -0.0029 (std)	0.99 (mean) -5.36 ×10 -6 (std)
	p	0.20 (mean) -0.059 (std)	0.0012 (mean) -0.00019 (std)
	m	4.89 (mean) -0.67 (std)	5.81 (mean) -6.43 (std)
	MSE	4.38 ×10 -5	4.83 ×10 -7
	err 1	0.0044	0.00024

Acknowledgements

This work has been partly performed in the scope of the projects NUMEP (PC201615) and MECI n • 21CM119-00 granted by Plan Cancer Inserm.

Conflict of interest

The Authors declare that there is no conflict of interest.

Appendix Validation of the SAEM algorithm using few measurements

To check whether the SAEM algorithm works correctly with a low number of multicellular spheroids and a low number of measurements, Figure 12 shows the error V 0 -V obj 0 , k -k obj , bb obj 2 using various cohorts with y individuals, drawn at random from the control group and x consecutive times. The values {V 0 , k, b} obj correspond to the values estimated with the maximum number of measurements and multicellular spheroids (top, right of Figure 12).

Radial equation of multicellular spheroids evolution in case of free growth

When considering free growth, we have F (t, x) = 0, ∀t, ∀x leading to study the following system

coupled with the dynamics of τ P toQ .

The first equation of System 12 can be rewritten as

Simplified equation for the density of proliferating cells -We will search the simplified equation followed Concerning the gradient term ∇P , we have ∇P (t, x) = ∇ P (t, r(t, x)) = ∇r(t, x)∂ r P (t, r(t, x)).

As ∇r(t, x) = 1 R(t)

x

x , we obtain

and using the fact that the velocity is radial and Equation 13, we obtain

Using Eq. ( 7), this equation can be rewritten

Equation for the radial velocity -The equation ∇

1 r 2 ∂ r (r 2 ṽ(t, r)) = τ G ( P + Q).

By integration of the previous equation between 0 and r (the invariance by rotation implies that ṽ(t, 0) = 0), we obtain ṽ(t, r) R(t) = 1 r 2 r 0 r2 τ G ( P + Q)dr.

and this gives as P + Q = 1 inside the tumor ṽ(t, r) R(t) = r 3 τ G (t).

This implies that P follows for each r ∈ [0, R(t)], the following EDO P = τ G (1 -P ) -τP toQ P .