Delphine Le Guennec
email: delphine.le_guennec@uca.fr

Victor Hatte

Marie-Chantal Farges

Stéphanie Rougé

Marie Goepp

Florence Caldefie-Chezet

Marie-Paule Vasson

Adrien Rossary

Modulation of inter-organ signalling in obese mice by spontaneous physical activity during mammary cancer development

Accumulative evidence links breast cancer development to excess weight and obesity. During obesity, dysregulations of adipose tissue induce an increase in pro-inflammatory adipokine secretions, such as leptin and oestrogen secretions. furthermore, a raise in oxidative stress, along with a decrease in antioxidant capacity, induces and maintains chronic inflammation, which creates a permissive environment for cancer development. physical activity is recommended as a non-pharmacological therapy in both obese and cancer situations. physical activity is associated with a moderation of acute inflammation, higher antioxidant defences and adipokine regulation, linked to a decrease of tumour-cell proliferation. However, the biological mechanisms underlying the relationship between oxidative stress, low-grade inflammation, carcinogenesis, obesity and physical activity are poorly understood. Our study is based on old, ovariectomised mice (C57BL/6J mice, 33 weeks old), fed with a high fat diet which increases adipose tissue favouring overweight and obesity, and housed in either an enriched environment, promoting physical activity and social interactions, or a standard environment constituting close to sedentary conditions. our model of mammary carcinogenesis allowed for the exploration of tissue secretions and signalling pathway activation as well as the oxidative status in tumours to clarify the mechanisms involved in a multiple factorial analysis of the data set. the multiple factorial analysis demonstrated that the most important variables linked to moderate, spontaneous physical activity were the increase in growth factor (epithelial growth factor (EGF), hepatocyte growth factor (HGF)) and the activation of the signalling pathways (STAT3, c-jun n-terminal kinases (JNK), EKR1/2, nuclear factor-kappa B (NF-κB)) in the gastrocnemius (G). In inguinal adipose tissue, the NF-κB inflammation pathway was activated, increasing the IL-6 content. The adiponectin plasma (P) level increased and presented an inverse correlation with tumour oxidative status. Altogether, these results demonstrated that spontaneous physical activity in obesity conditions could slow down tumour growth through crosstalk between muscle, adipose tissue and tumour. A spontaneous moderate physical activity was able to modify the inter-organ exchange in a paracrine manner. The different tissues changed their signalling pathways and adipokine/cytokine secretions, such as adiponectin and leptin, resulting in a decrease in anti-oxidative response and inflammation in the tumour environment. This model showed that moderate, spontaneous physical activity suppresses tumour growth via a dialogue between the organs close to the tumour.

Breast cancer is the most common female cancer in the world (25% of female cancers with 1.7 million new diagnoses in 2012) and the second leading cause of death by cancer for women 1 . Some of the strongest associated risk factors for breast cancer development are becoming overweight after menopause and obesity at any age. These factors are also determinants of poor prognosis and high recurrence. In the context of the global epidemic of obesity with a growing prevalence (15,9% in Europe in 2014) 2 , the expansion of adipose tissue as a result of obesity

induces an increase in lipolysis, chronic low-grade inflammation and dysregulation of energy homeostasis [START_REF] Zahid | Inflammation, dysregulated metabolism and aromatase in obesity and breast cancer[END_REF] . However, lipids play a recognized role in tumour growth and development, notably by modulating aromatase activity in the adipose tissue and selective oestrogen receptors in tumours [START_REF] Zahid | Inflammation, dysregulated metabolism and aromatase in obesity and breast cancer[END_REF] . Moreover, an altered adipokine profile is also observed in excess of weight and adiposity linked to obesity. The secretion of leptin, a pro-inflammatory adipokine, increases differentiation, proliferation and the survival of cancer cells via the Janus Kinase (JAK)/ Signal Transducers and Activators of Transcription 3 (STAT3), Mitogen-activated protein kinases (MAPK)/extracellular signal regulated kinases (ERK) and phosphoinositide 3-kinase (PI3K)/protein kinase b (AKT) signalling pathways [START_REF] Argolo | The Impact of Obesity on Breast Cancer[END_REF] . On the contrary, adiponectin, a hormone inversely correlated with obesity, is protective against breast cancer through its anti-inflammatory, insulin sensibilization and pro-apoptotic effects [START_REF] Choi | Adipocyte biology in breast cancer: From silent bystander to active facilitator[END_REF] . These two hormones, secreted by the adipose tissue (TA), regulate the whole-body metabolism [START_REF] Stadler | Cholesterol metabolism and breast cancer[END_REF] . Physical activity (PA) is recognised as a protective factor against the development of breast cancer, especially in the case of obesity [START_REF] Figueira | Efficacy of Exercise on Breast Cancer Outcomes: A Systematic Review and Meta-analysis of Preclinical Data[END_REF] . However, the mechanisms of interaction between physical activity and tumour development are poorly understood.

Cell metabolic dysregulations and reprogramming are hallmarks of cancer cells [START_REF] Hanahan | The Hallmarks of Cancer[END_REF][START_REF] Hanahan | Hallmarks of Cancer: The Next Generation[END_REF] . The uncontrolled cell proliferation of neoplastic cells involves metabolic adjustments such as the Warburg effect. In addition, hypoxia is induced by cell neoplasia faster than neovascularization. This situation modifies the metabolism, creating a symbiosis between hypoxic cells that need lactate and Warburg cells that secrete lactate. This physiological mechanism is derived from muscles. During exercise, lactate accumulates in the cytosol of muscle cells. This lactate can be oxidized in the mitochondria or transported out of the muscle fibres in order to be oxidized in other tissues, such as the myocardium, the muscles oxidative fibres or the liver. The inter-organ exchange of metabolites allows the optimal function of tissues in all conditions [START_REF] Kennedy | Tumor metabolism of lactate: the influence and therapeutic potential for MCT and CD147 regulation[END_REF][START_REF] Feron | Pyruvate into lactate and back: from the Warburg effect to symbiotic energy fuel exchange in cancer cells[END_REF][START_REF] Semenza | Tumor metabolism: cancer cells give and take lactate[END_REF] . Tumour cells and muscle cells have other similarities, linked to hypoxia. During exercise and cancer development, a high production of ROS is observed. Indeed, in cancer cells, an increase in oxidative stress and a diminution of antioxidant capacity induces a chronic inflammatory environment favouring tumour growth [START_REF] Reuter | Oxidative stress, inflammation, and cancer: how are they linked?[END_REF] . Contrary to cancer cells, in muscle cells, a moderate and transient oxidative stress is observed during exercise, promoting an increase in antioxidant capacity and an acute inflammatory response [START_REF] Catalán | Adipose tissue immunity and cancer[END_REF] .

We hypothesise that physical activity, by increasing the needs of muscle cells and by its acute anti-inflammatory response, can induce a competition between metabolites and inflammation status through the inter-organ exchanges with the tumour cells. This hypothesis was sustained by the improvement in the lipid and carbohydrate profiles observed in patients with physical activity therapy [START_REF] Mika | Effect of Exercise on Fatty Acid Metabolism and Adipokine Secretion in Adipose Tissue[END_REF] . Our purpose was to explore the impact of physical activity on the tissue environment in a model of breast carcinogenesis. We based our study on old and ovariectomised mice (C57BL/6 J mice, 33 weeks old) fed with a high fat diet and housed in either an enriched environment, promoting physical activity and social interactions, or a standard environment constituting close to sedentary condition. The metabolic exchange was analysed between most involved organs around the tumour environment, including healthy mammary gland, inguinal adipose tissue (IAT), gastrocnemius muscle and tumours. Analysed metabolites were adipokines (adiponectin, leptin, resistine), estradiol, interleukin 6 (IL-6), signalling pathways and tumour anti-oxidative response markers. To have a new global approach, a Multi Factorial Analysis (MFA) was conducted on the data set obtained, along with the correlations.

Materials and methods

Study design: diet and animals. Female C57BL/6 mice (33 weeks; 29,6 ± 2,2 g) were purchased from the Charles River Laboratory (Lyon, France), housed at 22 °C ± 2 °C in standard laboratory conditions (12-h light and 12-h dark cycle on a reverse light cycle) with ad libitum access to diet and water. After 2 weeks of acclimatization, the mice were ovariectomised and randomized into two groups (n = 10), in a standard environment (SE) or in an enriched environment (EE), both groups had a high fat diet until sacrifice. The high-fat diet (4.3 kcal/g) was prepared by SAFE (SAFE, Augy, France) according to the American Institute of Nutrition 93 (AIN-93G) recommendations for laboratory rodent purified diets [START_REF] Reeves | AIN-93 purified diets for laboratory rodents: final report of the American Institute of Nutrition ad hoc writing committee on the reformulation of the AIN-76A rodent diet[END_REF] . Diet composition is detailed in Supplemental Table 1. Enrichment of the environment was obtained by housing the mice in a larger cage (60 * 38 * 20 cm) and providing additional accessories (wheels, nests, tunnels…).

Body weight and spontaneous food intake were measured twice a week throughout the experimental period. Body composition (n = 5) in the non-anaesthetized mice was individually measured twice during the experiment (Weeks 5 and 8) by quantitative magnetic resonance imaging (MRI) using an EchoMRI 3-in-1 composition analyser (Echo Medical Systems, Houston, TX). Whole-body energy metabolism and physical activity of a group, which comprises five mice, were measured by indirect calorimetry twice during the experiment, before and after tumour implantation, using a two-cage TSE System PhenoMaster/LabMaster (TSE System, Bad Homburg, Germany) [START_REF] Even | Indirect calorimetry in laboratory mice and rats: principles, practical considerations, interpretation and perspectives[END_REF] . The mice were placed in calorimetry cages (n = 5 per cage) for 4 days with free access to their diet and water (22 ± 2 °C, 12 h daylight cycle).

The mouse sacrifice day is determined by either a tumour volume (2 cm 3 ) limit or a 38-day maximum after implantation. For the control mice, sacrifice occurred randomly during the same time period. The Mice were sacrificed by injection with ketamine/xylazine (i.p., 100/10 mg/kg of body weight) (Sigma Aldrich) and by cardiac puncture. After blood centrifugation (13,800 G for 10 minutes at 4 °C in heparinized tubes), plasma was collected, aliquoted and stored at -80 °C until analysis Several organs were harvested: leg muscle, inguinal adipose tissue, mammary gland (MG) and tumour. The organs were weighed before being frozen at -80 °C until analysis.

Mammary adenocarcinoma cell line and fat pad implantation. C57BL/6 syngeneic EO771 mammary tumour cells (Center for Stem Cell Research, Houston, TX) is a medullary breast adenocarcinoma cell line isolated from spontaneous tumours in C57BL/6 mice [START_REF] Sugiura | Studies in a tumor spectrum. I. Comparison of the action of methylbis(2-chloroethyl)amine and 3-bis(2chloroethyl)aminomethyl-4-methoxymethyl -5-hydroxy-6-methylpyridine on the growth of a variety of mouse and rat tumors[END_REF] .

The cells were cultured in complete RPMI 1640 medium (Biowest, Nouaille, France) supplemented with 10% fetal calf serum (Biowest), 100 μg/ml of streptomycin (Sigma-Aldrich), 100 U/mL of penicillin (Sigma-Aldrich), 2 mM of glutamine (Sigma-Aldrich) at 37 °C in a humid atmosphere at 5% CO 2 .

Prior to implantation, the cells were detached with trypsin, filtered to prevent cell clumping, added to 100 μl of Matrigel reduced in growth factor (BD Matrigel ™ Matrix, BD Biosciences, Bedford, MA) with a density of 5 × 10 5 cells per 100 μl and kept on ice until administration to mice.

After 5 weeks, mammary neoplastic cells were orthotopically implanted into the fourth right mammary gland using the fat pad technique, with 100 μl of cell suspension (6 mice per group) or with vehicle alone (4 mice per group) (see section below).

Three times a week, the size of the tumours was determined by measuring the perpendicular diameter with a digital calliper. The tumour volume was calculated using the formula V = 4/3π × (width/2) 2 × (length/2), where width is the smaller of the two measurements.

Tissue preparation and plasma quantification. Tissue preparation. Tumours, inguinal adipose tissue and the right gastrocnemius were cut with a scalpel before being milled with ultra-turrax in an ice bucket. The tissues were sonicated before being frozen at -80 °C for a minimum of 10 minutes. After thawing, the samples were homogenized with ultra-turrax and centrifuged at 500 G for 5 minutes. The supernatant was filtered (40 μm filter) before being aliquoted and then frozen before analysis at -80 °C Intra-tissue protein quantities and plasma proteins were assayed with a BC kit assays (Interchim, Montluçon, France) based on the Biuret method in multiskan (ThermoFisher Scientific, Villebon sur Yvette, France) at 550 nm.

Quantification of biomarkers.

Total cholesterol, triglycerides and glucose were quantified with commercial kits from ABX Pentra Horiba (Montpellier, France), according to the manufacturer's instructions.

Plasma levels of 17 β-oestradiol was assayed with the immunoassay EIA kits according to the manufacturer's instructions (Cayman Chemical, Ann Arbor, MI) with a microplate spectrophotometer reader (Multiskan FC, Thermo Scientific, Waltham, MA).

Using Multiplex Biomarker Immunoassays (cat. MADCYMAG-72K-05) according to the manufacturer's instructions, metabolic hormones (adiponectin, leptin, resistin, IL-6) were determined in plasma and tissue after appropriate dilution. The mean fluorescence intensity (MFI) was detected by the Multiplex plate reader for all measurements (Luminex System, Bio-Rad Laboratories, Germany) using a Luminex system, Bio-Rad Laboratories software version 4.2.

Using Multiplex Biomarker Immunoassays (cat. kit 48-680MAG and 48-681MA) according to the manufacturer's instructions, both total and phosphorylated forms of signalling pathways (cAMP response element-binding protein (CREB), JNK, NFκB, p38, ERK1/2, AKT, p70S6K, STAT3 and STAT5) were determined in the tumours after appropriate dilution. The mean fluorescence intensity (MFI) was detected by the Multiplex plate reader for all measurements (Luminex System, Bio-Rad Laboratories, Germany) using a Luminex system, Bio-Rad Laboratories software version 4.2.

Quantification of oxidative status markers in tumours. Determination of total glutathione. Total glutathione (GSH) content was determined by the method of Cereser et al. [START_REF] Cereser | Quantitation of reduced and total glutathione at the femtomole level by high-performance liquid chromatography with fluorescence detection: application to red blood cells and cultured fibroblasts[END_REF] . Briefly, dithiothreitol reduced tumour homogenate for 10 minutes at room temperature, and glutathione ethyl ester was added as an internal standard. After protein precipitation, the supernatant was derivatized by adding ortho-phthalaldehyde (OPA) (Sigma-Aldrich, Saint-Quentin-Fallavier, France). The HPLC separation of GSH-OPA adducts were performed by a UP3 HDO silica-based, reversed-phase C18 column (150 × 3.60 mm, particle size 3 µm) from Phenomenex (Interchim, Montluçon, France) maintained at 37 °C, followed by fluorimetric detection at 420 nm after excitation at 340 nm (Summit HPLC system, Dionex SA, Courtaboeuf, France). Derivatives were eluted using a 10-50% acetonitrile gradient in a 25 mM phosphate buffer at pH 6 for 5 minutes. The flow rate was 0.25 ml/min for an elution run of 20 minutes. Chromatograms were integrated using Chromeleon software from Dionex (Version 6.80, Dionex SA, Courtaboeuf, France). GSH content, calculated using a standard curve plotted under the same conditions, was expressed in µmol/g of protein.

Determination of protein thiols. Protein thiols were assayed using the method described by Himmelfarb et al. [START_REF] Himmelfarb | Plasma protein thiol oxidation and carbonyl formation in chronic renal failure[END_REF] . Free thiol groups oxidized by dithiobis-2-nitrobenzoic acid (Sigma-Aldrich, Saint-Quentin-Fallavier, France) were measured at 405 nm on a microplate spectrophotometer reader, expressed as a ratio to protein content in µmol/g. Determination of lipid hydroperoxides. The amounts of tumour lipid hydroperoxides were determined by the method described by Arab and Steghens [START_REF] Arab | Plasma lipid hydroperoxides measurement by an automated xylenol orange method[END_REF] . Tissue lysates were treated with buffer reagent (40 mM H 2 SO 4 , 20 mM formic acid, 150 µM iron D -gluconate and 120 µM xylenol orange in glycerol) (Sigma-Aldrich, Saint-Quentin-Fallavier, France). A standard curve was obtained using a tert-Butyl hydroperoxide solution.

Measurements were made at 570 nm on a microplate spectrophotometer reader. The concentration of lipid hydroperoxides, normalized to the protein content, was in µmol/g.

Determination of isoprostanes.

The level of tumour isoprostanes was measured with an Elisa kit (No. 516351, Cayman chemical, USA) according to the manufacturer's instructions and read at 405 nm on a microplate spectrophotometer reader.

Glutathione reductase activity. Glutathione reductase (GR) activity was determined as previously described [START_REF] Arab | Docosahexaenoic acid enhances the antioxidant response of human fibroblasts by upregulating gamma-glutamyl-cysteinyl ligase and glutathione reductase[END_REF] . The tissue lysate was incubated with buffer reagent (100 mM Tris-HCl, 1 mM EDTA, 0.16 mM NADPH and 4.6 mM oxidized glutathione (GSSG), pH 7.4) (Sigma-Aldrich, Saint-Quentin-Fallavier, France). Kinetic NADPH oxidation was followed at 340 nm and 37 °C for 3 minutes in a microplate spectrophotometer reader. GR activity, normalized to the protein content, was in IU/g. Glutathione peroxidase activity. Glutathione peroxidase (GPx) activity resulted in the oxidation of GSH in the presence of tert-Butyl hydroperoxide. Secondarily, GR recycled GSSG in the presence of NADPH [START_REF] Cheng | Overexpression of cellular glutathione peroxidase does not affect expression of plasma glutathione peroxidase or phospholipid hydroperoxide glutathione peroxidase in mice offered diets adequate or deficient in selenium[END_REF] . The tissue lysate was incubated with reagents (100 mM Tris-HCl, 1 mM EDTA, 22 mM tert-Butyl hydroperoxide, 5 mM GSH, 0.1 IU/ml GR, 2 mM NADPH, pH 7.4) (Sigma-Aldrich, Saint-Quentin-Fallavier, France). Kinetic NADPH oxidation due to GSH recycling was followed at 340 nm and 37 °C in a microplate spectrophotometer reader. GPx activity, normalized to the protein content, was in IU/g. Glutathione S-transferase activity. Glutathione S-transferase (GST) activity was quantified as previously described [START_REF] Arab | Docosahexaenoic acid enhances the antioxidant response of human fibroblasts by upregulating gamma-glutamyl-cysteinyl ligase and glutathione reductase[END_REF] using the conjugation reaction of GSH with artificial substrate 1-chloro-2,4-dinitrobenzene. The tissue lysate was incubated with reagents (50 mM HEPES, 5 mM GSH, 1 mM 1-chloro-2,4-dinitrobenzene, pH 7.4) (Sigma-Aldrich, Saint-Quentin-Fallavier, France). The 1-chloro-2,4-dinitrobenzene-glutathionylation kinetics were followed at 340 nm and 37 °C in a microplate spectrophotometer reader. GST activity, normalized to the protein content, was in IU/g. Cyclooxygenase activities. The activity of cyclooxygenase (COX) 2 and 1 were measured with a kit (Cayman Chemical Company, Ann Arbor, USA, No 760151) according to the manufacturer's instructions. Kinetic measurements were performed at 590 nm and 37 °C in a microplate spectrophotometer reader. COX-2 and 1 activities, normalized to the protein content, were in IU/g. Determination of thioredoxin reductase activity. The activity of the enzyme thioredoxin reductase was analysed in the tumours with a kit CAK1042 (Bioscience Cohesion, London, UK) according to the manufacturer's instructions. The reading was performed by Multiskan at 405 nm for 5 minutes. Thioredoxin reductase activity, normalised to the protein content, was in UI/g.

Heme oxygenase 1 activity. The catalytic activity of heme oxygenase 1 was measured by the mixture of the cofactor NADPH, H + (2 mg/mL) (Sigma Aldrich) with ferric heme in a Tris buffer solution (pH 7.4, Tris-Hcl at 100 mM and MgCl 2 at 2 mM) (Tris: Biosolve, Dieuze, France; MgCl 2 : Prolabo, Nantes, France). The mixture was read by spectrophotometry at 405 nm.

Data analyses.

The first part of the statistical analyses were calculated using R, version 3.2.2. 5,93% of the data were missing. This problem was handled by imputation of the missing values using Expectation-Maximization algorithm [START_REF] Ott | Counting methods (EM algorithm) in human pedigree analysis: linkage and segregation analysis[END_REF] implemented in the MissMDA R package. Data were centered and scaled.

A Multi-Factor Analysis [START_REF] Escofier | Analyses factorielles simples et multiples[END_REF][START_REF] Pagès | Multiple Factor Analysis by Example Using R. Chapman and Hall/CRC[END_REF] was performed using the R package FactomineR as described in [START_REF] Lê | An R Package for Multivariate Analysis[END_REF][START_REF] De Tayrac | Simultaneous analysis of distinct Omics data sets with integration of biological knowledge: Multiple Factor Analysis approach[END_REF] . Eight groups were identified: muscle masses, adipose tissue masses, tumour growth, tumour oxidative status, tumour biology, inguinal adipose tissue biology, right gastrocnemius biology and plasma biology (composition of groups of variables in supplemental Table 2). Correlations were calculated using the corrplot R package.

In order to obtain the most relevant metabolites from the first statistical analysis, a second statistical analysis was performed. The second phase of statistical analyses was performed with GraphPad Prism 5 software (GraphPad Software, Inc., San Diego, CA). Pairwise comparisons were calculated with a Mann Whitney t test. Comparisons of three or more groups were made with a 1-way ANOVA when only one parameter was changed or with a 2-way ANOVA when 2 factors were modified. These comparisons were followed by a Bonferroni post test. Statistical means are p ≤ 0.05 (*), p < 0.001 (**) and p < 0.0001 (***). All data were presented as mean ± SEM.

ethics approval. This study was approved by the Animal Experimental Committee (Comité Régional d'Ethique sur l'Expérimentation Animale, No. 01095.02, Clermont-Ferrand, France) and carried out in accordance with the ethical guidelines.

Results

Body composition, physical activity and mammary tumour growth. All of the mice used in the study were fed with a high-fat diet. Only the housing environment was different with a standard environment (SE) and an enriched environment (EE), the latter favouring physical activity (PA) and social interactions.

Respiratory quotient was similar between the groups. Food intake and energy expenditure were also the same regardless of the environment (data not shown). Despite PA, no changes were observed in the body weights or body composition of the mice. In both groups, weight increased throughout the experiment, mostly due to the high-fat diet (data not shown), without differences in lean mass, adipose mass (data not shown) or distribution.

As expected, PA was different between the two groups. The level of physical activity, represented by the total energy expenditure divided by the minimum energy expenditure, was higher in the EE than in the SE (Fig. 1A). Physical activity recording over a 24 h period revealed an increase in mean distance covered per hour and per mouse in the EE group (Fig. 1B), resulting in a higher total distance covered per mouse in the EE than in the SE (Fig. 1C).

After EO771 cancer cell implantation, tumour growth was significantly lower for the mice housed in the EE vs. the SE (Fig. 2A). Mammary tumour growth took longer to reach 2 cm 3 for the group housed in the EE than in the SE (Fig. 2B), resulting in a better survival rate for the mice in the EE group than in the SE, with respect to the experimental end-points (Fig. 2C).

Multiple factorial analyses.

No differences were perceived in terms of anatomy or weight between the groups housed in standard or enriched environments. However, even in the case of obesity, the enriched environment slowed down tumour growth. This is why we have focused on the biological effects, especially the inter-organ exchange. A particular interest was placed on tissue implied in physical activity, adiposity and mammary carcinoma (i.e. gastrocnemius, inguinal adipose tissue, plasma and tumour). In view of the relatively large amount of variables, a MFA was performed in order to have a comprehensive and global approach to the effects of moderate physical activity on mammary cancer development. Data were sorted into height groups of quantitative variables and one qualitative variable corresponding to the environment to improve readability without losing information. The MFA defined 11 principal components, called dimension (Dim), to explain the data set variability. The qualitative variable "environment" showed a high contributive value in each of the first two Dims, which explained 34,4% of the total variance. This association of Dim1 and Dim2 discriminated, in the individual plot, the mice into two distinct groups according to their housing environment (Fig. 3A). In the first two Dims, among the variable groups, the main contributors were the biological parameters for plasma, gastrocnemius and inguinal adipose tissue (Fig. 3B), for which enriched environment (EE) presented a significant impact. Moreover, Dim1 was mainly represented by the muscular mass and the tumour growth (Fig. 3B and Table 1) whereas Dim2 was more affected by the adipose tissue masses and the environment (Fig. 3B and Table 2). Tables 1 and2 resume the significant contributive variables inside each variable group for the two Dims in terms of correlation factor and p-value. To investigate additional differences between the mouse groups and the Dim signification, the MFA parameters representing each variable in our multi-dimensional space (cos2) were also plotted against the two Dims (Fig. 3C). For more clarity, only the variables with cos2 > 0.5 were represented on Fig. 3C.

The qualitative variable "environment enriched" was associated negatively with Dim1 (r² = -1.2721, p = 0.0447) and positively with Dim2 (r² = 1.4431, p = 0.0049). Concerning Dim1, among the 25 variables associated, the most important variables and the main positive contributors were variables of the tumour oxidative status with 8 variables (total glutathione (r² = 0.8352, p = 7.00E-04), total COX activity (r² = 0.7584, p = 0.0043), heme oxygenase activity (r² = 0.6612, p = 0.0192) and thiols (r² = 0.5927, p = 0.0423). The gastrocnemius was also highly positive contributor in Dim1 with 6 variables mainly represented by the gastrocnemius biology (leptin (r² = 0.6761, p = 0.0158), PECAM soluble (r² = 0.5787, p = 0.0487)) and the muscle masses. The last positive contributors were the inguinal adipose tissue biology, notably, the activated signalling pathway of IAT (AKT (r² = 0.6905, p = 0.0129), p38 (r² = 0.6555, p = 0.0206), CREB (r² = 0.6342, p = 0.0268)). Five of the 25 variables were negatively associated in Dim1 (days to reach limit point, gastrocnemius EGF, IAT NFκB and TNF-α). The plasma amount of adiponectin was the last negatively associated variable in Dim1 (r² = -0.7437, p = 0.0056) and the only plasma variable involved in the definition of these Dims. Concerning Dim2, only 11 variables were implicated. As for Dim1, adipose tissue masses (total adipose tissues (r² = 0.6781, p = 0.0154), IAT (r² = 0.7506, p = 0.0049), visceral adipose tissue (r² = 0.6653, p = 0.0182)) and the gastrocnemius signalling pathway (NFκB (r² = 0.7554, p = 0.0045) and STAT3 (r² = 0.6983, p = 0.0115)) were major positive contributors. Tumour oxidative status was represented by the isoprostanes levels (r² = 0.6168, p = 0.0326). Three variables were negatively associated with this Dim and reflected tissue remodelling (IAT MMP2 (r² = -0.6283, p = 0.0287), tumour MMP2 and MMP3 (r² = 60.6606, p = 0.0194; r² = -0.6790, p = 0.0152 respectively)).

To conclude, MFA demonstrated that the most important variables for the impact of spontaneous physical activity during breast cancer development in situations of obesity were the adipose tissue mass linked to the circulating adiponectin level, the tumour oxidative status and the tissue signalling pathways. These variables could explained the differences observed between the mouse groups and highlight the metabolic pathways implied. correlation matrix. For more clarity, the correlation matrix was performed on the significant MFA variables and only the significant correlations (p-value ≤ 0.05) were represented (Fig. 4). The correlation matrix plot was insightful in highlighting the relations between the array of variables considered in our model.

Among the plasma parameters only adiponectin emerged from the MFA. Plasma adiponectin had a significant inverse correlation with all of the adipose tissue masses (i.e. total, visceral and inguinal). The amount of plasma adiponectin correlated inversely with the activation of proliferation signalling pathways such as tumour ERK1/2 and IAT p38. Circulating adiponectin presented an inverse correlation with gastrocnemius IL-6 and several tumour antioxidant factors (total glutathione, glutathione S-transferase and total COXs activities).

All the adipose tissue masses had a positive correlation with each other. The different masses of adipose tissue were positively correlated with different inflammation and oxidative markers: the activation of NFκB in the gastrocnemius; the amount of thiol, reduced glutathione, glutathione reductase activity and COX-1 in the tumour.

The mass of the right gastrocnemius was positively correlated: firstly, with the masses of adipose tissue (total, inguinal and visceral) and with the masses of the leg muscles (right and left); secondly, with the production of IL-6 in the right gastrocnemius; thirdly, with different tumour oxidative protection markers (protein thiols, total and reduced glutathione amount, glutathione S-transferase and glutathione reductase activity, heme oxygenase and glutathione peroxidase activity) as well as two oxidative stress markers which were total COXs and COX-1. Finally, the mass of the right gastrocnemius was inversely correlated with the amount of plasma adiponectin.

MFA analyses highlighted the importance of the secreted parameters (myo-adipo-cytokines and growth factors) from the right gastrocnemius in conjunction with the environment. Firstly, the amount of IL-6 in the gastrocnemius was positively correlated with leg masses but also fat tissue masses (total, inguinal and visceral). An inverse correlation was found between the amount of muscle IL-6, the amount of circulating adiponectin and the amount of tumour MMP3. Gastrocnemius IL-6 was also positively associated with different markers of tumour oxidative response (total and reduced glutathione, enzymatic activity of glutathione reductase, COXs total and COX-1). Secondly, the amount of leptin in the gastrocnemius correlated positively with the amount of soluble PECAM and MMP2 in the same tissue as well as with different actors of the tumour oxidative status (total glutathione, glutathione-S-transferase, glutathione reductase, thioredoxin and COX-2). An inverse correlation was found between the amount of leptin and the amount of EGF and of HGF in the gastrocnemius. Thirdly, the growth factors EGF and HGF in the gastrocnemius have the same correlation profile: positive between them and with the activation of the NFκB inflammation pathway in inguinal adipose tissue, and also with the amount of tumour EGF. Inverse correlations were found with the amount of MMP2 in gastrocnemius and the activation of the CREB pathway in inguinal adipose tissue. Lastly, gastrocnemius sPECAM appeared to have mainly local correlations. Positive correlations were observed with leptin and MMP2. On the other hand, negative correlations were observed with growth factors (EGF and HGF) and the activation of signalling pathways (NFκB and STAT3). Among the tumour oxidative status, all the effectors of the antioxidant response correlated positively together, excepted the amount of tumour isoprostanes. The amount of tumour isoprostanes had a singular behaviour and uncorrelated with tumour oxidative status (except for reduced glutathione). Tumour isoprostanes were positively associated with EGF in tumours and with HGF and NFκB activation in the gastrocnemius.

Discussion

The aim of this study was to investigate the impact of spontaneous, moderate physical activity on the development of mammary cancer in old, ovariectomised and obese mice. Mice, fed with the same high fat diet, were then divided into two groups, one housed in a standard environment and the other in an enriched environment.

As expected, the mice housed in an enriched environment had a greater physical activity without change in energy expenditure, body composition or circulating metabolites such as glucose, triglycerides or cholesterol. This lack of change can be explained by the short time frame of the experience, there being no more than 12 weeks until the sacrifice. It is currently recognised in the literature, which mainly focuses on high intensive training, that sequential intensive training over a period of more than 12 weeks has a positive impact in diet-induced obesity models [START_REF] Kazeminasab | A comparative study on the effects of high-fat diet and endurance training on the PGC-1α-FNDC5/irisin pathway in obese and nonobese male C57BL/6 mice[END_REF] . In such conditions, physical activity has demonstrated the ability to correct biological and body parameters in obese mice [START_REF] Kazeminasab | A comparative study on the effects of high-fat diet and endurance training on the PGC-1α-FNDC5/irisin pathway in obese and nonobese male C57BL/6 mice[END_REF] . In our conditions, despite an adiposity increase of around 20% of body composition, the biological disorders related to metabolic syndrome only just have time to appear. Further, the physical activity in our study is not imposed but spontaneous. In terms of intensity, our physical activity intervention improved regular movement throughout the day, which only corresponds to moderate intensity. In this regard, we did not expect an impact of the spontaneous moderate physical activity on these parameters in our model as it is less stressful and intensive. Yet, our physical activity intervention did, in fact, enable a change in tumour growth. It is for this reason that we decided to focus on the inter-organ dialogue around the tumour environment.

To highlight the effects of physical exercise, we choose to focus on the right gastrocnemius. This muscle is the largest of the paw and has glycolytic and hybrid mixt myosin [START_REF] Delghingaro-Augusto | Skeletal muscle fiber types in C57BL6J mice[END_REF] . Moreover, the gastrocnemius manifests an important response to physical activity notably myokine secretions and biological effects. Finally, the right gastrocnemius is close to the tumour and could impact the tumour environment. In parallel, the impact of adipose tissues on carcinogenesis and on the whole-body homeostasis was evaluated principally by the biological parameters of the inguinal adipose tissue in our study. This tissue is the closest fat deposition to the tumour and appears to be a part of the tumour environment crosstalk [START_REF] Guaita-Esteruelas | The peritumoural adipose tissue microenvironment and cancer. The roles of fatty acid binding protein 4 and fatty acid binding protein 5[END_REF][START_REF] Wang | Friend or foe: Multiple roles of adipose tissue in cancer formation and progression[END_REF] . Lastly, the left fourth mammary gland of the mouse was considered in order to measure the impact of our intervention on healthy tissue. The inter-organ crosstalk between adipose tissue, muscle, mammary tumour and plasma was investigated in terms of inflammation, oxidative stress, tissue secretions and energy metabolites. Among the 133 variables studied in our model, a few of them are statistically related to the moderate physical activity ensured by the enriched environment. It is interesting to note that most of the variables associated with the enriched environment are related to cytokine and hormonal crosstalk. Surprisingly, the main cytokines, IL-6 and TNF-α, usually assayed in cancer models, presented no changes in our conditions [START_REF] Reuter | Oxidative stress, inflammation, and cancer: how are they linked?[END_REF] . Conversely, the two major adipokines, adiponectin and leptin, as well as some growth factors (EGF, VEGF-α and HGF) seemed to play a major role in the tissue crosstalk around the tumour. Other factors that plays a role in tissue remodelling and angiogenesis (MMPs), and usually observed in cancer models [START_REF] Hanahan | Hallmarks of Cancer: The Next Generation[END_REF] , also presented changes in our study.

It appears that moderate physical activity could change inflammation and hormonal parameters both at the systemic and tissue levels [START_REF] Figueira | Efficacy of Exercise on Breast Cancer Outcomes: A Systematic Review and Meta-analysis of Preclinical Data[END_REF][START_REF] Belloum | Cancer-induced cardiac cachexia: Pathogenesis and impact of physical activity (Review)[END_REF] . Physical activity is recognized as being able to modify adipose tissue secretions notably in a situation of obesity in animal models and clinical trials [START_REF] Mika | Effect of Exercise on Fatty Acid Metabolism and Adipokine Secretion in Adipose Tissue[END_REF][START_REF] Belloum | Cancer-induced cardiac cachexia: Pathogenesis and impact of physical activity (Review)[END_REF][START_REF] Trujillo | Adipose Tissue-Derived Factors: Impact on Health and Disease[END_REF] . Adipose tissue have a major role in inflammation both at systemic and tissue levels in various chronic diseases (i.e. cardiovascular, metabolic, cancer) that is why the amount of fat tissue is an important factor in chronic inflammation, particularly when due to obesity [START_REF] Zahid | Inflammation, dysregulated metabolism and aromatase in obesity and breast cancer[END_REF][START_REF] Trujillo | Adipose Tissue-Derived Factors: Impact on Health and Disease[END_REF][START_REF] Nicholson | The role of adipokines in skeletal muscle inflammation and insulin sensitivity[END_REF] . Our results showed that despite regular physical activity, the amount of adipose tissue plays a very strong role in tumour development, particularly through adipokine production (adiponectin, leptin). Interestingly, in our model, a localized inflammation is presented in the inguinal adipose tissue. The amount of TNF-α is linked to an increase in the activation of the NFκB signalling pathway in this tissue. Conversely, a decrease in activation is observed in the IAT signalling pathways AKT, CREB, p38. In the literature, obesity promotes CREB activation resulting in a decrease in adiponectin secretion [START_REF] Qi | Adipocyte CREB Promotes Insulin Resistance in Obesity[END_REF] . This previous observation is in agreement with the increase in the amount of adiponectin in the inguinal adipose tissue and in the plasma observed in our study in response to the decrease of CREB activation. In our model, these results suggest that a part of the anti-inflammatory and the anti-carcinogenesis effects of physical activity is supported by the anti-inflammatory effects of adiponectin. This observation is reinforced by the inverse correlations between circulating adiponectin and the various tumour antioxidant markers. What is more, another adipokine is modified by physical activity leptin in our model. The gastrocnemius and tumour leptin levels decrease without changes in IAT or plasma levels. According to the literature, physical activity induces a more important leptin captation by muscular cells [START_REF] Zheng | Leptin-induced basal Akt phosphorylation and its implication in exercise-mediated improvement of insulin sensitivity[END_REF] . Regular exercise decreases plasma levels of leptin and improves insulin sensitivity, mainly due to the leptin receptors in muscle cells [START_REF] Perez-Suarez | Severe energy deficit upregulates leptin receptors, leptin signaling, and PTP1B in human skeletal muscle[END_REF] . In this context of muscle exercise, leptin signalling will increase the signal to insulin and trigger a better uptake of the glucose required for physical activity [START_REF] Zheng | Leptin-induced basal Akt phosphorylation and its implication in exercise-mediated improvement of insulin sensitivity[END_REF] . In parallel, leptin, an important pro-carcinogenesis adipokine, is strongly captured by tumour cells because it promotes cell proliferation by activating main signalling pathways such as JNK, AKT, p38 [START_REF] Choi | Adipocyte biology in breast cancer: From silent bystander to active facilitator[END_REF] . We could hypothesise that a competition between muscles and tumour appears. Leptin is not available for tumour growth while physical activity consumes it. This is highlighted by the gastrocnemius leptin decrease and the tumour oxidative and inflammatory status changes. Physical activity also induces changes in muscle. Notably, the STAT3 cell-signalling pathway, particularly responsive to leptin, is activated to a greater extent. This result confirms the strong and increased consumption of leptin in the gastrocnemius due to physical activity. In parallel, the inflammation-signalling pathway, NFκB, is activated more in the gastrocnemius of the mice housed in the enriched environment. This result is in agreement with the literature, which describes an acute and local inflammatory response attributed to physical activity. As found in the literature [START_REF] Accattato | Effects of acute physical exercise on oxidative stress and inflammatory status in young, sedentary obese subjects[END_REF] , and in our model, there is also an increase in the amount of growth factors like HGF and EGF in the gastrocnemius. Whereas, and conversely to the literature, tissue remodelling is reduced with a decrease in soluble PECAM and in MMP2 muscle contents. Nevertheless, the right gastrocnemius mass is positively correlated with 9 markers of oxidative stress and anti-inflammatory response in the tumours. All together, these results support the hypothesis that physical activity reduces tumour growth through a modification of the tumour environment in terms of adipokine and MMPs as well as inflammatory and antioxidant status. Finally, changes in the tumour environment resulte in crosstalk between IAT and muscles. An enriched environment is linked to a decrease in the tumour proliferative pathway ERK1/2 associated with the decrease in MMP2 and 3 as well as the tumour anti-oxidative response and COX total activity. As described in the literature, inflammation and oxidative stress are two major hallmarks of cancer and are essential with the modulation of MMP activities to sustain tumour growth and expansion [START_REF] Hanahan | Hallmarks of Cancer: The Next Generation[END_REF] . Among the antioxidant response biomarkers, thioredoxin, GST, heme oxygenase and COX-2 activities decrease contrary to the tendency for the damage markers related to a high lipid per oxidation (isoprostanes) to increase in the mice housed in an EE. The positive correlations between all markers of antioxidant response, such as glutathione-s-transferase and COX-2, indicate that physical activity has a global effect on the tumour antioxidant response. Furthermore, physical activity is also linked to a decrease in intra-tumour NFκB, which is the major signalling pathway responding and controlling inflammation, as well as oxidative stress. At last, the isoprostane increase reflects cell stress [START_REF] Czerska | Isoprostanes -A novel major group of oxidative stress markers[END_REF] , confirming that physical activity plays a major role in tumour environment oxidative status.

conclusion

This original study allows us to highlight the changes induced by excess weight and an active daily life in mammary carcinogenesis. Our study demonstrates that, similar to intensive physical activity training, moderate spontaneous physical activity is able to counteract tumour growth. The increase of physical activity due to an enriched environment led to a decrease in inflammation and in the antioxidant response in the tumour environment. This effect does not seem to act directly on the tumour but by virtue of the surrounding tissues, through cytokine, growth factor and inflammation changes. However, large metabolites (cholesterol, triglycerides, glucose) are not altered by spontaneous physical activity, demonstrating that the hormonal signals are preponderant in tumour growth and in the activation of intracellular signalling pathway. Despite the changes observed in each tissue, moderate physical activity had no impact on plasma concentrations of inflammatory biomarkers except for adiponectin, highlighting a paracrine crosstalk between tissues. Our model confirms that one of the most important modulators in carcinogenesis is tumour oxidative stress, as previously cited in the literature [START_REF] Acharya | Redox regulation in cancer: a double-edged sword with therapeutic potential[END_REF][START_REF] Nourazarian | Roles of oxidative stress in the development and progression of breast cancer[END_REF] , and, further, that it could be modulated by spontaneous moderate physical activity in a paracrine manner [START_REF] Guennec | The interrelationship between physical activity and metabolic regulation of breast cancer progression in obesity via cytokine control[END_REF] .

Figure 1 .

 1 Figure 1. Physical activity in SE and EE. (A) The level of physical activity. (A) corresponds to total energy expenditure divided by the minimum energy expenditure, for a mouse. Results are mean ± SEM (n = 5/group). Data were analysed by Mann Whitney t-test. Representative data are shown: *p < 0.05, SE vs. EE. (B) Physical activity recording over a 24 h period. Results, expressed in mean distance covered per hour and per mouse, are mean ± SEM (n = 5/group). Data were analysed by ANOVA repeated measures. Representative data are shown: *p < 0.05, SE vs. EE. (C) Total distance per mouse per day. Results (meters) are mean ± SEM (n = 5/group). Data were analysed by Mann Whitney t-test. Representative data are shown: *p < 0.05, SE vs. EE.

Figure 2 .

 2 Figure 2. Tumour development in SE and EE. (A): Tumour volume at day 20 in mm 3 depending on the environment. Results are mean ± SEM (n = 10/group). Data were analysed by Mann Whitney t-test. Representative data are shown: *p <0.05, SE vs. EE. (B) Time course of tumour volume to reach 2000 mm 3 . Results are mean ± SEM (n = 10/group). Data were analysed by ANOVA repeated measures. Representative data are shown: *p < 0.05, **p < 0.01, SE vs. EE. (C) Time course survival in terms of end-point to sacrifice. Results are mean ± SEM (n = 10/group). Data were analysed by Mantel-Cox . Representative data are shown: *p < 0.05, SE vs. EE.

Figure 3 .

 3 Figure 3. Multiple factorial Analyses -Contribution of the variables in the two dimensions. (A) Individual plot according to the first 2 dimensions. Group in green is the SE and group in red is the EE. (B) Variable groups plot according to the first 2 dimensions. (C) Most important variables in dimensions 1 and 2. Only the variables having a cos2 greater than 0.5 were shown.

Figure 4 .

 4 Figure 4. Correlation matrix. Correlations between significant variables contributing to Dim1 and 2. Only significant correlations with a p-value of less than 0.05 were shown.

Table 1 .

 1 Contributions of variables to dimension 1. Results are mean ± SEM (n = 6/group). Data were analysed by multiple factorial analysis.

	Groups	Variables		Correlations	P-value
		Environment		0.3448	0.0447
	Environment	ES		1.2721	0.0447
		EE		-1.2721	0.0447
	Tumour growth	Days to reach limit point	-0.6698	0.0172
		Glutathione total		0.8352	7.00E-04
		Glutathione reductase	0.7932	0.0021
		COX total		0.7584	0.0043
	Tumour oxidative status	COX-1 Glutathione-S-transferase	0.7401 0.7074	0.0059 0.0101
		Heme oxygenase		0.6612	0.0192
		Glutathione reduced	0.6119	0.0345
		Thiols		0.5927	0.0423
	Tumour biology	ERK1/2		0.6171	0.0325
	Adipose tissue masses	visceral mass total mass		0.5937 0.5811	0.0418 0.0475
		AKT		0.6905	0.0129
		CREB		0.6342	0.0268
	Inguinal adipose tissue biology	p38 MAPK		0.6555	0.0206
		TNF-α		-0.5852	0.0456
		NFkB		-0.6071	0.0363
	Plasma biology	Adiponectin		-0.7437	0.0056
		MMP2		0.8228	0.001
	Gastrocnemius biology	Leptin PECAM soluble		0.6761 0.5787	0.0158 0.0487
		EGF		-0.7671	0.0036
		Right G mass		0.7594	0.0042
	Muscles masses	Left leg mass		0.6704	0.017
		Right leg mass		0.5998	0.0393
	Groups	Variables	Correlations P-value
		Environment	0.5628	0.0049
	Environment	EE	1.4431	0.0049
		ES	-1.4431	0.0049
	Tumour oxidative status	Isoprostanes	0.6168	0.0326
		EGF	0.6665	0.0179
	Tumour biology	MMP2	-0.6606	0.0194
		MMP3	-0.679	0.0152
		Inguinal mass	0.7506	0.0049
	Adipose tissue masses	total mass	0.6781	0.0154
		Visceral mass	0.6653	0.0182
	Inguinal adipose tissue biology	MMP2	-0.6283	0.0287
		NFκB	0.7554	0.0045
	Gastrocnemius biology	STAT3	0.6983	0.0115
		HGF	0.6455	0.0234

Table 2 .

 2 Contributions of variables to dimension 2. Results are mean ± SEM (n = 6/group). Data were analysed by multiple factorial analysis.

Scientific RepoRtS |(2020) 10:8794 | https://doi.org/10.1038/s41598-020-65131-9

Acknowledgements

We thank Christophe Montaurier for the indirect calorimetric measurements. We thank Marion Brandolini-Bunlon and Melanie Petera for their advice on the multiple factorial analysis. We thank François Senejoux, Catherine Felgines, Caroline Decombat, Laetitia Delort, Marion Vermerie, Marjolaine Vareille-Delarbre, Sophie Garcin, Nicolas Goncalves-Mendes and Jérémie Talvas for their help with the animal sacrifices. We thank La Ligue Contre le Cancer and their committee for the PhD funding of DLG. This work was supported by a PhD fellowship (Delphine Le Guennec) from le Comité national de La Ligue contre le cancer and by the local councils (Allier, Cantal, Haute-Vienne and Haute-Loire) and l'Institut national du cancer (INCA: projet MammAdipo; PLBIO 13-106).

Data availability

The datasets used and/or analysed during the current study are available from the corresponding author upon reasonable request.

Author contributions

competing interests

The authors declare no competing interests.

Additional information

Supplementary information is available for this paper at https://doi.org/10.1038/s41598-020-65131-9.

Correspondence and requests for materials should be addressed to D.L.

Reprints and permissions information is available at www.nature.com/reprints.

Publisher's note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.


Ankush Aggarwal

Damiano Lombardi

Sanjay Pant
email: sanjay.pant@swansea.ac.uk

An information-theoretic framework for optimal design: analysis of protocols for estimating soft tissue parameters in biaxial experiments

Keywords: optimal design, soft tissue mechanics, mutual information, biaxial experiment, inverse 14 problems, information theory. 15

A new framework for optimal design based on the information-theoretic measures of 1 mutual information, conditional mutual information, and their combination is proposed. The 2 framework is tested on the analysis of protocols-combination of angles along which strain 3 measurements can be acquired-in a biaxial experiment of soft tissues for the estimation of 4 hyperelastic constitutive model parameters. The proposed framework sees information gain about 5 the parameters from the experiment as the key criterion to be maximised which can be directly 6 used for optimal design. Information gain is computed through k-nearest neighbour algorithms 7 applied to the joint samples of the parameters and measurements produced by the forward and 8 observation models. For biaxial experiments, the results show that low angles have relatively low 9 information content compared to high angles. They also show that fewer number of angles with 10 suitably chosen combinations can result in higher information gains when compared to a larger 11 number of angles which are poorly combined. Finally, it is shown that the proposed framework is 12 consistent with classical approaches, particularly the D-optimal design.

Introduction

Soft tissues exhibit complex biomechanical behaviour, including nonlinearity, anisotropy 17 and heterogeneity [START_REF] Holzapfel | Nonlinear solid mechanics[END_REF]. Moreover, the tissues also demonstrate inelastic properties, such 18 as rate-dependence, hysteresis and permanent set. The important link between the 19 biomechanics and their physiological function have motivated a large number of ex-vivo 20 studies aimed at characterizing their biomechanical properties. Given the complex 21 interplay between the different aspects of their biomechanical properties, the experi-22 mental design of ex-vivo soft tissues is extremely challenging and has been a subject of 23 investigation with a variety of experiments been proposed [START_REF] Zhang | A generalized method for the analysis of planar biaxial mechanical data using tethered testing configurations[END_REF][START_REF] Labrosse | Planar biaxial testing of heart valve cusp replacement biomaterials: Experiments, theory and material constants[END_REF][START_REF] Humphrey | On constitutive relations and finite deformations of passive cardiac tissue: I. A pseudostrain-energy function[END_REF][START_REF] Laurence | An investigation of regional variations in the biaxial mechanical properties and stress relaxation behaviors of porcine atrioventricular heart valve leaflets[END_REF][START_REF] Jett | Integration of polarized spatial frequency domain imaging (pSFDI) with a biaxial mechanical testing system for quantification of load-dependent collagen architecture in soft collagenous tissues[END_REF].

24

Since a variety of soft tissues are thin, e.g., blood vessels, heart valves and skin, biax-25 ial testing is a widely used experimental technique that allows independently stretching 26 the tissue in two orthogonal directions and measuring the corresponding forces [START_REF] Billiar | Biaxial mechanical properties of the native and glutaraldehyde-treated aortic valve cusp: part II-a structural constitutive model[END_REF][START_REF] Ross | Biaxial Mechanical Characterizations of Atrioventricular Heart Valves[END_REF]. A variety of hyperelastic models have been developed to describe the anisotropic 33 and nonlinear elastic properties of specific soft tissues [START_REF] Humphrey | On constitutive relations and finite deformations of passive cardiac tissue: I. A pseudostrain-energy function[END_REF][START_REF] Maurel | Constitutive Modeling[END_REF][START_REF] Holzapfel | A new constitutive framework for arterial wall mechanics and a comparative study of material models[END_REF][START_REF] May-Newman | A constitutive law for mitral valve tissue[END_REF]. The biaxial experimental 34 data is commonly fit to these models in order to determine the model parameters. As the 35 unknown parameters depend on specific model, the choice of experimental setup-the 36 problem of optimal design-might depend on the choice of model. However, in practice, 37 a predetermined set of experimental protocols are used.

38

In the present work, an optimal design problem is defined as finding the most 39 suitable protocol in view of estimating the parameters of the material model. A compre-40 hensive overview of the optimal design problem can be found in [START_REF] Pukelsheim | Optimal design of experiments[END_REF][START_REF] Banks | Comparison of optimal design methods in inverse problems[END_REF], and several 41 criteria for optimal design have been proposed in the literature, often based on the 42 minimisation of the variance of the parameters and sensitivities [START_REF] Banks | Generalized sensitivities and optimal experimental design[END_REF][START_REF] Banks | Optimal design techniques for distributed parameter systems[END_REF]. In the present 43 work, we investigate a criterion based on information theoretic quantities, in the spirit of 44 what has been proposed in the work [START_REF] Sebastiani | Maximum entropy sampling and optimal Bayesian experimental design[END_REF] (from a Bayesian point of view) and [START_REF] Capellari | Parameter identifiability through information theory[END_REF]. The 45 goal is to characterise the amount of information the measurements convey about the 46 parameters we would like to estimate and then maximise that information. Estimating 47 information theoretic quantities is in general a challenging problem, and this is especially 48 the case in high-dimensional settings. In the present work, a model reduction method is 49 coupled with a non-parametric sample based mutual information estimation in order to 50 provide a pertinent estimation of the information theoretic quantities involved in the 51 optimal design problem and then applied to the biaxial testing of soft tissues.

52

The structure of the work is as follows: in Section 2 the model and information the-

58

The second part of the Section is devoted to the description of the information-59 theoretic framework for solving the optimal design problem. In Section 2.2.1 we intro-60 duce the problem, in Sections 2.2.2 and 2.2.3 the information-theoretic quantities and 61 their numerical estimation are detailed. We then present the reduce order modeling 62 method used and how to validate the results obtained by the proposed approach. The 63 section ends with an overview of the method.

64

The results and the discussion are presented in Section 3, followed by the conclusion 65 and the perspectives. We begin by defining the notation: a material point at its reference position X ∈ R 3 72 moves to x ∈ R 3 after deformation. The elastic behavior of soft tissues is described using 73 hyperelastic strain energy density Ψ which depends on the deformation gradient tensor 74 F = ∇ X x. The ratio of volume after deformation to that before deformation is given by 75 J = det(F). Soft tissues are commonly regarded as incompressible due to high water 76 content, i.e. J is constrained to be unity.

77

We consider the hyperelastic model proposed by Gasser et al. [START_REF] Gasser | Hyperelastic modelling of arterial layers with distributed collagen fibre orientations[END_REF] which defines the 78 strain energy density

79 Ψ = k 1 2k 2 e k 2 (κI 1 +(1-3κ)I 4 -1) -1 + µ(I 1 -3), (1)
where I 1 = tr(F F) is the first invariant of the right Cauchy-Green strain tensor C = F F 80 and I 4 = M • CM is the fourth invariant representing stretch along fiber direction M.

81

The resulting Cauchy stress is given by

82 σ = 2F • ∂Ψ ∂C • F -pI, (2)
where p acts as the Lagrange multiplier to enforce incompressibility and I is the identity 83 matrix.

84

For this model, the set of unknown parameters can be written as {k 1 , k 2 , κ, µ} 85 assuming that the fiber direction M is known a priori (based on another experiment, 86 e.g., light scattering [START_REF] Jett | Integration of polarized spatial frequency domain imaging (pSFDI) with a biaxial mechanical testing system for quantification of load-dependent collagen architecture in soft collagenous tissues[END_REF]). In this paper, in order to simplify the problem, we assume that 87 κ = 0.1 and µ = 0.1 kPa are known and fixed. Thus, the aim of an ex-vivo biomechanical 88 experiment is to determine parameters k 1 ∈ [5, 100] kPa and k 2 ∈ [5, 80] robustly and 89 with high confidence [START_REF] Aggarwal | An improved parameter estimation and comparison for soft tissue constitutive models containing an exponential function[END_REF][START_REF] Aggarwal | Effect of Residual and Transformation Choice on Computational Aspects of Biomechanical Parameter Estimation of Soft Tissues[END_REF]. A commonly used experiment called the biaxial testing is

F = diag λ 1 , λ 2 , 1 λ 1 λ 2 , (3)
where λ 1 is the stretch along first in-plane direction and λ 2 is the stretch along 98 second in-plane direction. The fiber direction M is generally aligned with the first 99 coordinate axis, which results in only normal stress components. As no force is applied 100 along the thickness of the tissue, σ 33 = 0 is used to determine the Lagrange multiplier p.

101 Thus, we get

102 σ 11 = 2 ∂Ψ ∂I 1 λ 2 1 - 1 λ 2 1 λ 2 2 + 2 ∂Ψ ∂I 4 λ 2 1 ( 4
)
σ 22 = 2 ∂Ψ ∂I 1 λ 2 2 - 1 λ 2 1 λ 2 2 . ( 5
)
The applied stresses σ 11 , σ 22 are controlled using load cells. The resulting strains defined 103 as e 1 := λ 1 -1 and e 2 := λ 2 -1 are measured from the marker positions 1 . It is important 104 to note that a homogeneous stress and strain state is assumed in the middle of the sample 105 (Fig. 1a). Therefore, an implicit assumption is that the material properties and sample A single-angle biaxial protocol is defined as a straight line in the σ 11 -σ 22 space (Fig. 116 1b). That is, the ratio between the two stresses is kept constant while the applied forces 117 are increased until a maximum value σ max = 200 kPa. Thus, for a chosen angle φ, we 118 apply:

119

σ 11 =    σ if φ ≤ π 4 tan(φ)σ else (6) σ 22 =    cot(φ)σ if φ ≤ π 4 σ else , (7)
where σ ∈ [0, σ max ]. For σ, 100 linearly spaced observation points between zero and 120 the maximum stress (σ max = 200 kPa) are used. The resulting strains are calculated 121 by iteratively solving equation (5) for λ 1,2 and thereby obtaining e 1,2 . In practice, a 122 combination of angles can be successively tested. We refer to this combination as the 123 experimental protocol that needs to be optimally designed. The problem of optimal design typically refers to the choice of a design of ex-126 periments such that the design is optimal with respect to a pre-determined statistical 127 criterion. We propose that the information-theoretic measures naturally define such 128 statistical criteria. The central idea is that information gain [START_REF] Pant | An information-theoretic approach to assess practical identifiability of parametric dynamical systems[END_REF][START_REF] Pant | Information sensitivity functions to assess parameter information gain and identifiability of dynamical systems[END_REF] from an experiment 129 or protocol-as quantified by the information-theoretic quantities of mutual information 130 and conditional mutual information-can be directly used as a reasonable statistical 131 criterion for optimal design. These quantities are described next after presenting the 132 framework for optimal design. Consider the following general model:

y = M(θ), (8)
where M denotes for the forward model that takes θ ∈ R m and outputs y ∈ R n . Note 135 that θ may contain initial and boundary conditions of the model and that y may subsume the output at many time-points in the case of a dynamical system. Subsequently, consider 137 that the measurement model is as follows

138 z = H p (y, θ) + ε, (9)
where H p represents the observation operator, z ∈ R d represents the measurement vector, S(H p ) to be maximised, the optimal design is given by

143 Ĥp = arg max H p S(H p ). (10)
In the case of the biaxial experiments, the model M represents the model for the

Φ = {φ 0 , φ 1 , • • • , φ α }. (11)
The possible set of protocols is then given by any combination of elements in Φ with the 151 restriction that the number of elements in a protocol be limited to C. Thus, if Φ ⊂ Φ is a 152 subset of angles representing a protocol, our set of protocols is given by 153

H p = Φ ⊂ Φ | 1 ≤ |Φ| ≤ C , ( 12
)
where | • | represents the number of elements in the set. In other words, we choose at 154 least 1 and up to C elements from Φ, with the total elements in H p being and Z, respectively, the mutual information (MI) between the parameters Θ and the 160 measurements Z is defined as [START_REF] Pant | An information-theoretic approach to assess practical identifiability of parametric dynamical systems[END_REF] 161

155 |H p | = α 1 + α 2 • • • + α C . ( 13
I(Θ; Z) = X Θ ×X Z p Θ,Z (θ, z) p Θ,Z (θ, z) p Θ (θ)p Z (z) dθdz, (14)
where p X (x) represents the probability density of a random variable X with a realisation 162 X = x and support X X . The mutual information

where τ > 0 is a regularisation parameter.

197 Note that the above list is not exhaustive, and based on the interpretations of MI 198 and CMI, other criteria may be constructed based on the desired sense of optimality. 205 Denoting these N s samples as

θ (i) , i = {1, 2, • • • N s }, each θ (i)
can be propagated through 206 the forward and observation models of equations ( 8) and ( 9) to produce corresponding 207 samples of Z, denoted as z (i) . The samples of θ (i) and z (i) can subsequently be used on 208 non-parametric estimators of MI and CMI. Such non-parametric estimators can broadly 209 be classified into two categories: kernel density estimators (KDE) [START_REF] Moon | Estimation of mutual information using kernel density estimators[END_REF] and k-nearest 210 neighbour (kNN) estimators [START_REF] Kraskov | Estimating mutual information[END_REF][START_REF] Lombardi | Nonparametric k-nearest-neighbor entropy estimator[END_REF]. For an overview of such methods, we refer to [START_REF] Beirlant | Nonparametric entropy estimation: An overview[END_REF].

211

While the estimator proposed by Kraskov et. al. [START_REF] Kraskov | Estimating mutual information[END_REF] is widely used and performs very 212 well across a range of scenarios, one of its drawbacks is that it suffers from higher errors 213 when extreme correlations are present between the variables and/or when the the data 214 effectively lives in a lower-dimensional manifold. Since we are working with models that 215 specify explicit relationships between the variables through the forward and observation 216 model, this is likely to be true for the data-set of (θ (i) , z (i) ). Thus, in this study, we employ the local non-uniformity correction (LNC) proposed in [START_REF] Gao | Efficient estimation of mutual information for strongly dependent variables[END_REF], which includes a 218 correction term to the original estimator by Kraskov et. al. [START_REF] Kraskov | Estimating mutual information[END_REF]. This term accounts for 219 strong dependencies between the variables through local principle component analysis 220 [START_REF] Gao | Efficient estimation of mutual information for strongly dependent variables[END_REF]. The method of [START_REF] Gao | Efficient estimation of mutual information for strongly dependent variables[END_REF] is used for estimation of all MIs, and CMIs are estimated from 221 the difference of two MIs, see equation [START_REF] Banks | Optimal design techniques for distributed parameter systems[END_REF]. structure of a set of data (a comprehensive review of the topic can be found in [START_REF] Benner | A survey of projection-based model reduction methods for parametric dynamical systems[END_REF][START_REF] Benner | Model reduction and approximation: theory and algorithms[END_REF][START_REF] Quarteroni | Reduced order methods for modeling and computational reduction[END_REF][START_REF] Ma | Manifold learning theory and applications[END_REF]).

230

A large spectrum of methods have been proposed in the literature. In the present con-231 tribution, we adopt a local reduced-basis method (similar, in the spirit, to the methods

232

proposed in [START_REF] Amsallem | PEBL-ROM: Projection-error based local reduced-order models[END_REF][START_REF] Maday | Locally adaptive greedy approximations for anisotropic parameter reduced basis spaces[END_REF]). Let the strains computed by the model be e

235 e 1,2 ≈ n ∑ i=1 η i r i (σ, φ)s i (k 1 , k 2 , φ), (17)
which is well defined by virtue of the Eckart-Young theorem. First, let us observe that 236 a given protocol consists in a set of known angles Φ. An efficient way to construct the 237 local reduced basis is hence to introduce a Proper Orthogonal Decomposition (POD) for 238 each of the angles φ j ∈ Φ. This corresponds to looking for an approximation of the form:

239 e (j)
1,2 (σ;

φ j ; k 1 , k 2 ) ≈ n ∑ i=1 η (j) i r (j) i (σ)s (j) i (k 1 , k 2 ), (18)
where r

(j) i , r (j)
k Ω σ = δ ik and s

(j) i , s (j)
k Ω k = δ ik ( •, • Ω σ ,Ω k being the standard L 2 scalar 240 product). The error in the approximation is related to the number n of modes retained:

241 e (j) 1,2 - n ∑ i=1 η (j) i r (j) i (σ)s (j) i (k 1 , k 2 ) 2 L 2 (Ω σ ×Ω k ) = ∞ ∑ i=n+1 η (j) i 2 (19)
In the present work, a number n = 4 modes proved to be sufficient in order to get errors smaller than 10 -3 in L 2 norm on the solution reconstruction. This means that the set of elements e 1,2 (σ; φ j ; k 1 , k 2 ) are close to the linear subspace spanned by the first n = 4 modes r (j) i . Henceforth, instead of considering the discretised e 1,2 we will consider their coordinates in the subspace, given by:

z (j) 1,2 | i = e 1,2 , r (j) i Ω σ = η (j) i s (j) i (k 1 , k 2 ). (20)

Validation of results against existing methods

242

Several methods and criteria to define and reach an optimal design of experiments 243 were proposed [START_REF] Pukelsheim | Optimal design of experiments[END_REF]. Among them, D-optimality criterion attempts at maximising

H = E Θ [∂ 2 θ f | θ * ], (21)
where θ * is the value of the parameter minimising the loss function.

267

For the estimation of MI and CMI, a total of N = 10, 000 values of (K 1 , K 2 ) are 268 uniformly distributed in the parametric space. For each such sample (k

(i) 1 , k (i)
2 ), the 269 numerical model of the biaxial experiment is run to produce e

(i)
1 and e

(i)
2 , which are 270 then projected on to the reduced basis, giving z (i) . The N triplets of (k

(i) 1 , k (i) 2 , z (i) ) are 271
subsequently used for estimation of MI and CMI through the LNC estimator (see section 272 2.2.3). In equation ( 16) we use τ = 1. Generally, all the three information criteria increase with increasing number of 284 angles in the protocol. Intuitively, this is expected as higher number of angles imply 285 more measurement data and hence a higher potential for improved estimation 286 of the parameters. This observation is true for the maximum information gain, 287 minimum information gain, and the mean information gain.

3.

From Figures 2 and3 it is observed that the angular combinations that maximise 293 information gain for K 1 are not identical-and vary significantly when more than 2

294

angles are simultaneously used-to those that maximise information gain for K 2 .

295

This further motivates the use of a criterion that balances information gains in both 296 the parameters while minimising their interdependence. Figure 4 shows that the best combinations that maximises a balanced criterion such 298 as I(K 1 ; Z) + I(K 2 ; Z) -I(K 1 ; K 2 |Z) are a trade-off between the combinations of angles that maximises I(K 1 ; Z) and I(K 2 ; Z) individually. For example, when 5 300 angles are considered, the angles that maximise I(K 1 ; Z) are φ a = [66, 72, 78, 84, 90],

301 those that maximise I(K 2 ; Z) are φ b = [START_REF] Quarteroni | Reduced order methods for modeling and computational reduction[END_REF]36,42,48,54], while the combination that 302 maximises I(K 1 ; Z) + I(K 2 ; Z) -I(K 1 ; K 2 |Z) is [START_REF] Quarteroni | Reduced order methods for modeling and computational reduction[END_REF]36,48,78,90], which has two

5.

Finally, it is observed that the worst combinations are all low angles: [0, [START_REF] Jett | Integration of polarized spatial frequency domain imaging (pSFDI) with a biaxial mechanical testing system for quantification of load-dependent collagen architecture in soft collagenous tissues[END_REF][START_REF] Pukelsheim | Optimal design of experiments[END_REF][START_REF] Gasser | Hyperelastic modelling of arterial layers with distributed collagen fibre orientations[END_REF][START_REF] Kraskov | Estimating mutual information[END_REF].

307

This can be related to the fact that at low angles, the applied stress is largely aligned 308 along the stiff fibers of the tissue, thus resulting in lower strain values. Thus, the 309 lower angles provide a small range of the observations, while the larger angles 310 provide a larger range (Fig. 9a), thereby containing more information about the 311 parameters. From this point onward, we present results only for the balanced information 313 criterion S = I(K 1 ; Z) + I(K 2 ; Z) -I(K 1 ; K 2 |Z). Figure 5 shows the variation in S 314 across all the combinations (x-axis, and in log-scale to capture the spread) grouped 315 by the number of angles in a protocol and sorted in increasing order of S within each 316 such group. Within each group, when observing the minimum and maximum values 317 of S, it shows that a better choice of angles can lead to more than 100% increase in 318 the information gain compared to a poor choice. Furthermore, it shows that good 319 combinations of a lower number of angles can lead to higher information gain compared 320 to higher number of angles with bad combinations. For example, the maximum S 321 when only one angle is used is higher than many combinations with 2-4 angles. This 322 emphasises the utility of optimal design and the proposed framework. and is also related to the fiber dispersion in our constitutive model (equation (1), [START_REF] Gasser | Hyperelastic modelling of arterial layers with distributed collagen fibre orientations[END_REF]). It 339 is remarkable and encouraging that the information-theoretic framework captures the 340 physics of the problem without explicitly considering it in the framework. While for 341 simpler low-dimensional models the association between physics and optimal design 342 may be relatively easy to see, inferring such behaviour is, in general, not trivial for more 343 complex and higher-dimensional models.

S = I(K 1 ; Z) + I(K 2 ; Z) -I(K 1 ; K 2 |Z)

344

Similarly to Figure 8, the results of the information-theoretic optimal design and 345 further be analysed for higher number of angles. To further illustrate the validity of the information-theoretic approach, comparison 355 with a classical method, see section 2.2.5, is presented. For one and two angles in a 356 protocol, Figure 11 shows a comparison between S and the log of the determinant of

27

  Applying different stretches in two directions allows characterization of the in-plane 28 anisotropic behavior of a given tissue, while a range of stretches provides us its nonlinear 29 elastic response. However, even with this relatively simple set of options, the choices of 30 what stretches to apply are unclear. Moreover, it is not obvious what these choices will 31 depend on.

53

  oretical aspects of the problem are introduced. In particular, in Section 2.1 we detail the 54 mathematical model of the biaxial experiments for soft tissues: after having introduced 55 the notation and the non-linear elasticity model, in Section 2.1.1 we particularise it to 56 the biaxial testing experimental setup. In Section 2.1.2 we introduce the experimental 57 protocol definition.

66 2 .70 2 . 1 .

 221 Methods67The methodological aspects are divided into two broad categories: the mathemat-68 ical model of the biaxial experiments; and the information-theoretic optimal design 69 framework. Mathematical model of the biaxial experiments 71

124 2 . 2 .

 22 Information-theoretic framework for optimal design 125

133 2 . 2 . 1 .

 221 Optimal design problem134

139 and ε

  represents the vector of measurement error/noise. Note that the the observation 140 operator H p depends on the design of experiments, which specifies which quantities are 141 measured. Given a set of possible H p = {H 1 , H 2 , • • • , H h }, and a statistical criterion142

144

  force controlled experiment (sections 2.1.1 and 2.1.2) and H p essentially denotes the 145 experimental protocol, see section 2.1.2, representing the combination of angles-each 146 representing a straight line in the σ 11 -σ 22 plane-along which the strain measurements 147 of e 1 and e 2 are acquired. With the possible variation of each angle between 0 and π/2, 148 the set Φ of possible angles φ are constructed through a uniform discretisation of the 149 space between 0 and π/2 into α levels, thus 150

182 2 .

 2 The joint mutual information may be maximised, giving S = I(Θ; Z). In the sense 183 of classical optimal design this can be interpreted as the D-optimal design. This is 184 because D-optimal designs minimise the determinant of the Fisher Information 185 Matrix inverse, and S = I(Θ; Z) measures the information gain in the joint Θ 186 space. 187 3.The sum of individual parameter mutual information may be be maximised, giving 188 S = ∑ m i=1 I(Θ i ; Z). In the sense of classical optimal design this can be interpreted 189 as the A-optimal design. This is because the the A-optimal design minimises the 190 trace of the Fisher Information Matrix inverse, and S = ∑ m i=1 I(Θ i ; Z) measures 191 the sum of the information gains for all the parameters.

  may seek to maximise individual parameter information gain 193 while minimising pairwise CMI, thus seeking both small posterior variances and 194 minimising pairwise correlations between the parameters. In this case, the statistical i ; Θ j |Z),

199 2 . 2 . 3 .

 223 Estimating mutual information200In general, the forward model in equation (8) is non-linear, and hence even if the 201 observation operator is linear (implying linear combinations of the state are measured),202the analytical computation of mutual information is intractable. Thus, the information-203 theoretic quantities of MI and CMI must be estimated. A common method is to generate 204 samples of Θ through specification of an appropriate prior probability density p Θ (θ).

222 2 . 2 . 4 .

 224 Dimensionality reduction for the biaxial experiment 223 One of the main difficulties in estimating information theoretic quantities is re-224 lated to the data dimension. The non-parametric estimation is particularly challenging 225 whenever the data are close to manifolds embedded in high-dimensional spaces. This is 226 indeed the case when a physical model relates parameters and observable quantities. 227 One of the possible ways to overcome this difficulty or, at least, to mitigate it, is (dimen-228 sion or) model reduction, which aims at discovering the underlying low-dimensional 229

244

  the determinant of the information matrix. In the present case, this is equivalent to 245 minimize the determinant of the inverse of the average Hessian of the loss function we 246 would introduce in a classical parameter estimation method. In a noisy setting, and, 247 in particular, when the noise is Gaussian, this cost function is equivalent to minus the 248 logarithm of the likelihood function. Let the misfit function be f (θ) and E Θ denote the 249 expectation operator. The average of the Hessian reads:

251 2 . 2 . 6 .

 226 Overview of approach for the biaxial experiments 252 In the context of the biaxial experiments, the parameters are k 1 and k 2 , represented 253 as random variables K 1 and K 2 , respectively. The variability in these parameters is 254 considered to be uniform (thus imposing a uniform prior distribution) in the following 255 intervals: k 1 ∈ [5, 100] kPa and k 2 ∈ [5, 80]. For a single value of angle φ, the measure-256 ments are the strain values e 1 and e 2 and measured at 100 points along the line defined 257 by the angle φ. Here, we consider α = 16 discrete values of possible measurement angles 258 φ uniformly distributed between, and including, 0 • and 90 • . For each angle φ, separate 259 reduced bases of 4 modes for e 1 and e 2 are constructed through POD over 400 values 260 of (K 1 , K 2 ) sampled uniformly in the aforementioned parametric space. Thus, for any 261 angle φ, the dimensionality reduction approach projects e 1 and e 2 measured at 100 points 262 along the line defined by φ to a basis of 4+4 modes. For a given protocol consisting of 263 multiple angles, the measurement vector z (with corresponding random variable Z), is 264 the collection of all the reduced basis representation of e 1 and e 2 along the angles in the 265 protocol. Lastly, the maximum number of angles in a protocol is restricted to C = 5, 266 giving a total of 6,884 unique combinations of the α = 16 angles.

273 3 . Results and discussion 274 For all the 6 ,

 32746 884 combinations of angles, three statistical criteria are evaluated: (i) 275 I(A; Z); (ii) I(B; Z); and (iii) I(K 1 ; Z) + I(K 2 ; Z) -I(K 1 ; K 2 |Z). While the first two 276 criteria aim to maximise information gain about A and B individually, the third criterion 277 aims to maximise information gain about A and B simultaneously while minimising 278 the information dependence between them. Figures 2-4 show the variation in these 279 three criterion when grouped by the number of angles in a protocol. In these Figures the 280 values of information criterion when using two approaches to uniformly discretise the 281 angular space within protocols is also presented. Observations from these plots are as

288 2 .

 2 Across all the three criteria, it is observed that the uniform discretisation is not 289 necessarily reflective of the best protocol for estimating the parameters. In fact, 290 in most cases, the performance of uniform discretisation is close to the mean 291 information gain observed across all the angle combinations.

  292

303

  angles from φ a and three angles from φ b . It should be noted that such a trade-off 304 between maximising individual parameter gains is still significantly different than 305 a uniform discretisation.

  306

Figure 2 .Figure 3 .

 23 Figure2. The variation of information criterion S = I(K 1 ; Z) across the 6,884 combinations grouped by the number of angles in a protocol. The vertical lines represent the variation around the mean value, which is shown in black circles. Black text shows the combinations that produce maximum and minimum values of S. The red and blue pointers show S for angle combinations that follow a uniform discretisation of the angular space between 0 and 90 degrees. Red and blue texts show the associated angle combinations.

Figure 4 .

 4 Figure 4. The variation of information criterion S = I(K 1 ; Z) + I(K 2 ; Z) -I(K 1 ; K 2 |Z) across the 6,884 combinations grouped by the number of angles in a protocol. The vertical lines represent the variation around the mean value, which is shown in black circles. Black text shows the combinations that produce maximum and minimum values of S. The red and blue pointers show S for angle combinations that follow a uniform discretisation of the angular space between 0 and 90 degrees. Red and blue texts show the associated angle combinations.

1 ;Figure 5 .

 15 Figure5. The variation of information criterion S = I(K 2 ; Z) + I(K 2 ; Z) -I(K 1 ; K 2 |Z) across the 6,884 combinations. The vertical red lines show the groupings with respect to the number of angles in a protocol and S values are sorted in increasing order within each such grouping. The x-axis is represents the index associated with the protocol and is in logarithmic scale to capture the spread between 1 angle in a protocol (16 values) vs 5 angles in a protocol (4,368 values).

  323

Figure 6 showsFigure 7 Figure 7

 677 Figure6shows S for all the 6,884 combinations in increasing order of magnitude and

  338

SFigure 6 . 16 S

 616 Figure 6. The variation of information criterion S = I(K 1 ; Z) + I(K 2 ; Z) -I(K 1 ; K 2 |Z) across the 6,884 combinations sorted in increasing order of S

Figure 7 .SFigure 8 .Figure 9 . 10 20 S

 7891020 Figure 7. Zoomed view into the first 150 protocols from Figure 6. The upper panel shows S and the lower panel shows the angles (by circles) in the corresponding protocol.

Figure 10 .

 10 Figure 10. Information criterion against the angles when the protocols are restricted to only two angles. The upper panel shows S and the lower panel shows the angles (by circles) in the corresponding protocol.

20 SFigure 11 .

 2011 Figure 11. Information criterion S (in blue) and the log of the determinant of the Fisher Information Matrix inverse log 10 |H -1 | (in red) against the angles when the protocols are restricted to a maximum of two angles. The upper panel shows S and log 10 |H -1 |, while the lower panel shows the angles (by circles) in the corresponding protocol. Note that here e 1 is shown in solid lines (left y-axis) and e 2 is shown in dashed lines (right y-axis).

357the

  Fisher Information Matrix inverse, log |H -1 |. It is encouraging that a high corre-358 spondence between the two metrics is observed. In particular, increases in S , implying 359 higher information gains, are accompanied by corresponding decreases in log |H -1 |, 360 implying a smaller volume of the parameter posteriors. A Pearson correlation coefficient 361 of r = -0.76 is observed between S and log |H -1 | implying a high similarity between the 362 two metrics, and validating the information-theoretic approach in part. Let us remark 363 that, when the number of the parameters become large, evaluating the Hessian would 364 imply a non-negligible computational cost. On the contrary, the method used to evaluate 365 the Mutual Information, being a primarily Monte Carlo based estimation, is less severely 366 dependent on the number of parameters. Furthermore, computation of derivatives 367 (either numerically or through adjoint based methods) may be cumbersome for certain 368 type of models. Finally, we note that the effect of noise on information gain, and hence 369 optimal design, can be easily assessed in the proposed framework by adding noise to 370 the samples of Z, see equation (9).

  optimal design based on information-theoretic quantities of mu-373 tual information and conditional mutual information is proposed. The framework treats 374 information gain as the central criterion for inverse problems and proposes several 375 information-theoretic frameworks for desired sense of optimality. The capabilities of 376 this framework are tested on the optimal design problem for biaxial experiments, where 377 the effect of angle combinations along which the strains are measured is assessed on 378 parameter estimation through information gain. Without including any physics based 379 reasoning and purely through the information-theoretic measures, it is found that low 380 angles ≤ 24 • are not very informative about the parameters relative to high angles.

  This approach only concerns with the posterior of the parameter Θ i and

	178
	179	example:
		1.	The mutual information for any single parameter may be maximised, giving S =

I(Θ; Z) quantifies the amount of 163 information that can be gained on average by knowing one random variable, say Z, 164 about the other, say Θ. Indeed, with this interpretation, MI is a good candidate for the 165 statistical criterion S for optimal design. For an individual parameter, Θ i , or indeed 166 for any combination of parameters {Θ i , Θ j }, the corresponding information gains can 167 be similarly computed through I(Θ i ; Z) and I({Θ i , Θ j }; Z), respectively. Thus, while 168 I(Θ i ; Z) quantifies the information gain individually for the parameter Θ i , the quantity 169 I({Θ i , Θ j }; Z) quantifies information gain for the pair {Θ i , Θ j } jointly. A measure of 170 correlation between the parameters Θ i and Θ j is, however, missing, and is provided by 171 conditional mutual information (CMI) defined as

I(Θ i ; Θ j |Z) I = I(Θ i ; {Θ j , Z}) II -I(Θ i ; Z) III . (

15

)

The CMI I(Θ i ; Θ j |Z) represents the additional information gained about the pa-173 rameter Θ i when both Θ j and Z are known (term II) relative to when only the measure-174 ments Θ i alone are known (term III). Note that CMI is symmetric, i.e. I(Θ i ; Θ j |Z) = 175 I(Θ j ; Θ i |Z), and can be interpreted as a measure of dependence between the parameters 176 given the measurements Z. Also note that both MI and CMI are non-negative.

177

With the above background, many statistical measures can be constructed. For 180 I(Θ i ; Z). 181 ignores all other parameters.

  nats more information compared to the worst combination; and ii) 350 generally speaking, higher angles are more informative compared to lower angles, in 351 particular angles below 24 • . While similar analysis for more than two angle combinations 352 can be easily performed, efficient visual representation of such results is cumbersome

When two angles are considered, the S 346 values in increasing order of magnitude and the corresponding angle combinations are 347 shown in Figure 10. This figure re-iterates observations made previously: i) the choice 348 of combinations significantly affects the information gain, the best combination gives approximately 20 353 and avoided in this manuscript.

Version March 31, 2021 submitted to Axioms https://www.mdpi.com/journal/axioms

Funding: This research was funded by the Engineering and Physical Sciences Research Coun-408 cil of the UK (Grant reference EP/R010811/1 to SP) and (Grant reference EP/P018912/1 and 409 EP/P018912/2 to AA).

410

These observations are then found to be consistent based on physics-based reasoning,

Limitations and future work 391

While the proposed framework is shown to perform well on a two-parameter 392 problem, its performance in higher parameter problems is not assessed. This assessment 393 forms the the primary limitation and area of future assessment. In particular, the 394 problems envisaged are largely related to the performance of the MI and CMI estimators 395 in higher dimensions of both parameters and the measurements. While a dimensionality 396 reduction approach was adopted in this study to minimise the adverse effects of the 397 latter, this may not be possible in many forward and inverse problems. Thus, a large area Lastly, a thorough comparison against classical optimal design methods (C-, E-, T-, 400 V-optimal designs, etc.) needs to be performed, along with the construction and analysis 401 of corresponding information-theoretic metrics.

Author Contributions: All authors contributed equally to writing, editing, and reviewing the 403 manuscript. SP and DL conceptualised the information-theoretic framework. AA and SP con-