Jana Kasparkova

Hana Kostrhunova

Vojtech Novohradsky

Lili Ma

Guangyu Zhu

Elena R Milaeva

Alexender A Shtill

Robin Vinck

Gilles Gasser

Viktor Brabec
email: brabec@ibp.cz

Alexey A Nazarov
email: nazarov@med.chem.msu.ru

Is antitumor Pt(IV) complex containing two axial lonidamine ligands a true dual-or multi-action prodrug?

Keywords: Bioinorganic Chemistry, Lonidamine, Medicinal Inorganic Chemistry Metals in Medicine Pt(IV) complexes

This work studied the mechanism of action of a Pt(IV) complex 2 bearing two axial lonidamine ligands, which are selective inhibitors of aerobic glycolysis. The presence of two lonidamine ligands in 2 compared to the parent Pt(II) complex increased its antiproliferative activity, cellular accumulation, and changed its cell cycle profile and mechanism of cell death. In 3D cell culture, 2 showed exceptional antiproliferative activity with IC50 values as low as 1.6µM in MCF7 cells. The study on the influence of the lonidamine ligands in the Pt complex on glycolysis showed only low potency of ligands to affect metabolic processes in cancer cells, making the investigated complex, not a dual-or multi-action prodrug. However, the Pt(IV) prodrug effectively delivers cytotoxic Pt(II) complex into cancer cells.

Introduction

Platinum(II) drugs are the current gold standard in cancer chemotherapy. [START_REF] Wang | Cellular processing of platinum anticancer drugs[END_REF][START_REF] Kelland | The resurgence of platinum-based cancer chemotherapy[END_REF][START_REF] Keefe | Aptamers as therapeutics[END_REF][START_REF] Florea | Cisplatin as an anti-tumor drug: Cellular mechanisms of activity, drug resistance and induced side effects[END_REF] Currently, we have a good understanding of cell transport and mechanism of action of platinum drugs, possible side effects, and resistant phenomena. [START_REF] Fuertes | Biochemical modulation of cisplatin mechanisms of action: Enhancement of antitumor activity and circumvention of drug resistance[END_REF][START_REF] Brabec | Role of DNA repair in antitumor effects of platinum drugs[END_REF][7][START_REF] Wu | Specific DNA structural attributes modulate platinum anticancer drug site selection and cross-link generation[END_REF][START_REF] Brabec | Cytotoxic platinum coordination compounds. DNA binding agents[END_REF][START_REF] Johansson | Characterization of New Potential Anticancer Drugs Designed To Overcome Glutathione Transferase Mediated Resistance[END_REF] To overcome some limitations and side effects of conventional Pt(II) antitumor drugs, Pt(IV) prodrugs are attracting considerable interest, mostly due to reduced general toxicity, the possibility for oral administration, and red-ox kinetics as part of the mechanism of action. [START_REF] Gibson | Platinum(IV) anticancer prodrugs -hypotheses and facts[END_REF][START_REF] Johnstone | The Next Generation of Platinum Drugs: Targeted Pt(II) Agents, Nanoparticle Delivery, and Pt(IV) Prodrugs[END_REF][START_REF] Gibson | Multi-action Pt(IV) anticancer agents; do we understand how they work?[END_REF][START_REF] Lee | Development of an Efficient Dual-Action GST-Inhibiting Anticancer Platinum(IV) Prodrug[END_REF] Additionally, Pt(IV)-based prodrugs have the potential of combining several bioactive moieties in a single molecule releasing the FDA-approved Pt(II) drugs, cisplatin, carboplatin, or oxaliplatin, together with other antiproliferative agents. Several Pt(IV) drug candidates have been investigated in the clinical trials; however, none has been approved as an anticancer drug. The current shift of interest in anticancer Pt(IV) complexes is to modify axial position(s) with innocent spectators, cancer-targeting agents, photodynamic therapy photosensitizers, or bioactive moieties to increase antitumor efficacy and/or overcome resistance. [START_REF] Ang | Rational Design of Platinum(IV) Compounds to Overcome Glutathione-S-Transferase Mediated Drug Resistance[END_REF][START_REF] Dhar | Polyvalent oligonucleotide gold nanoparticle conjugates as delivery vehicles for platinum(IV) warheads[END_REF][START_REF] Pathak | The Prodrug Platin-A: Simultaneous Release of Cisplatin and Aspirin[END_REF][START_REF] Kenny | Toward multi-targeted platinum and ruthenium drugs-A new paradigm in cancer drug treatment regimens?[END_REF][START_REF] Wang | Emerging platinum(iv) prodrugs to combat cisplatin resistance: from isolated cancer cells to tumor microenvironment[END_REF][START_REF] Tang | Oxaliplatin-Based Platinum(IV) Prodrug Bearing Toll-like Receptor 7 Agonist for Enhanced Immunochemotherapy[END_REF][START_REF] Babu | Multiaction Pt(IV) Carbamate Complexes Can Codeliver Pt(II) Drugs and Amine Containing Bioactive Molecules[END_REF][START_REF] Muhammad | Mitochondria-targeting monofunctional platinum(II)-lonidamine conjugates for cancer cell deenergization[END_REF][START_REF] Karges | A Multi-action and Multi-target RuII-PtIV Conjugate Combining Cancer-Activated Chemotherapy and Photodynamic Therapy to Overcome Drug Resistant Cancers[END_REF][START_REF] Fronik | Structure-Activity Relationships of Triple-Action Platinum(IV) Prodrugs with Albumin-Binding Properties and Immunomodulating Ligands[END_REF] Upon entering the cell, Pt(IV) prodrugs can be reduced by intracellular reductants to generate DNA-damaging Pt(II) drugs. Simultaneously, bioactive axial moieties are released inside the cancer cells, resulting in a combined effect that triggers several processes, which determine the cell's fate. Thus, Pt(IV) complexes are often dual or multi-action drugs that can increase the efficacy and/or overcome resistance to conventional Pt(II) anticancer drugs. On the other hand, markedly enhanced accumulation of Pt(IV) prodrugs due to their hydrophobic character has been shown to play a key role in the markedly enhanced antitumor activity of several Pt(IV) derivatives of conventional antitumor Pt(II) drugs with bioactive ligands. [START_REF] Alessio | Antiproliferative activity of Pt(IV)-bis(carboxylato) conjugates on malignant pleural mesothelioma cells[END_REF][START_REF] Gabano | Pros and cons of bifunctional platinum(IV) antitumor prodrugs: two are (not always) better than one[END_REF][27] Lonidamine (1, Figure 1) is a selective inhibitor of aerobic glycolysis in cancer cells [START_REF] Cervantes-Madrid | Reviving lonidamine and 6-diazo-5oxo-L-norleucine to be used in combination for metabolic cancer therapy[END_REF][START_REF] Nath | Mechanism of antineoplastic activity of lonidamine[END_REF][START_REF] Shang | The Anticancer Activity of Indazole Compounds: A Mini Review[END_REF] and was evaluated in clinical trials as monotherapy or combination therapy against lung, prostate, breast, and non-small cell lung cancer. [START_REF] Cervantes-Madrid | Reviving lonidamine and 6-diazo-5oxo-L-norleucine to be used in combination for metabolic cancer therapy[END_REF][START_REF] Di Cosimo | Lonidamine: efficacy and safety in clinical trials for the treatment of solid tumors[END_REF][START_REF] Gridelli | A phase II study of VM-26 plus lonidamine in pretreated small cell lung cancer[END_REF][START_REF] Pacini | FEC (5-fluorouracil, epidoxorubicin and cyclophosphamide) versus EM (epidoxorubicin and mitomycin-C) with or without lonidamine as first-line treatment for advanced breast cancer. A multicentric randomized study. Final results[END_REF] Additionally, lonidamine is commonly used as a chemosensitizer to enhance the antitumor effects of chemotherapeutic drugs based on its disruption of energy metabolism. [START_REF] Huang | The potential of lonidamine in combination with chemotherapy and physical therapy in cancer treatment[END_REF][START_REF] Shutkov | Ru(III) Complexes with Lonidamine-Modified Ligands[END_REF] Several Pt(IV) complexes with lonidamine were prepared to target mitochondrial pathways. [START_REF] Muhammad | Mitochondria-targeting monofunctional platinum(II)-lonidamine conjugates for cancer cell deenergization[END_REF][START_REF] Chen | Effective platinum(IV) prodrugs conjugated with lonidamine as a functional group working on the mitochondria[END_REF][START_REF] Muhammad | Multiaction Platinum(IV) Prodrug Containing Thymidylate Synthase Inhibitor and Metabolic Modifier against Triple-Negative Breast Cancer[END_REF] Moreover, tethering of lonidamine to Ru(II)-arene moiety yields compounds with highly relevant cytotoxicity in human glioblastoma cell lines. [START_REF] Nazarov | Organometallic anticancer agents that interfere with cellular energy processes: a subtle approach to inducing cancer cell death[END_REF] Recently, we reported on the new highly active Pt(IV) complexes with lonidamine as axial ligands. [START_REF] Nosova | Enhancing the Cytotoxic Activity of Anticancer PtIV Complexes by Introduction of Lonidamine as an Axial Ligand[END_REF][START_REF] Okulova | Antiproliferative activity of Pt(IV) complexes with lonidamine and bexarotene ligands attached via succinate-ethylenediamine linker[END_REF] We found that conjugation with lonidamine resulted in compounds with superior activity compared to the original organic drug, but also cisplatin. Out of all prepared complexes, we selected the most active one, namely (OC-6-33)-dichloridobis[1-(2,4dichlorobenzyl)-1H-indazole-3-carboxylato](ethane-1,2-diamine)platinum(IV) (2 Figure 1), i.e., Pt(IV) derivative of direct and simple analog of cisplatin dichlorido(ethane-1,2diamine)platinum(II)with lonidamine at both axial positions, [START_REF] Nosova | Enhancing the Cytotoxic Activity of Anticancer PtIV Complexes by Introduction of Lonidamine as an Axial Ligand[END_REF] to investigate the mechanism underlying the antiproliferative effects of this prodrug in cancer cells. We were particularly interested in investigating the contribution of platinum and lonidamine to killing cancer cells cultured in traditional two-dimensional (2D) monolayers and three-dimensional (3D) spheroids. 1) and (OC-6-33)-dichloridobis [1-(2,4-dichlorobenzyl)-1Hindazole-3-carboxylato](ethane-1,2-diamine)platinum(IV) [START_REF] Kelland | The resurgence of platinum-based cancer chemotherapy[END_REF].

Results and Discussion

Synthesis 1-(2,4-Dichlorobenzyl)-1H-indazole-3-carboxylic acid (Lonidamine (1)) [START_REF] Duan | Method for synthesis of lonidamine and related indazole derivatives[END_REF] and (OC-6-33)dichloridobis [1-(2,4-dichlorobenzyl)-1H-indazole-3-carboxylato](ethane-1,2diamine)platinum(IV) (2) [START_REF] Nosova | Enhancing the Cytotoxic Activity of Anticancer PtIV Complexes by Introduction of Lonidamine as an Axial Ligand[END_REF] were prepared using published procedures. Compounds were characterized using 1 H NMR and mass spectrometry and were fully identical to the described in the literature. [START_REF] Nosova | Enhancing the Cytotoxic Activity of Anticancer PtIV Complexes by Introduction of Lonidamine as an Axial Ligand[END_REF] The purity of prepared compounds was established by combustion analysis.

Antiproliferative activity

The first prerequisite for using the investigated potential antitumor substances as effective chemotherapeutics is their sufficient antiproliferative activity in cancer cell lines. Successful antitumor chemotherapeutics should eliminate the expansion of solid tumor mass and/or indispose cancer cells in cell division.

Antiproliferative activity of complex 2 was studied by the commonly used MTT assay after 72 h of incubation on three human cancer cell lines of different origin cultured in 2D monolayers; breast cancer MCF-7 cells, highly invasive, triple-negative breast carcinoma MDA-MB-231 cells, and colorectal carcinoma cells HCT-116 (Table 1). Moreover, the human noncancerous lung fibroblasts MRC-5pd30 were also included in the study. For comparative purposes, identical tests were performed with the clinically approved platinum(II) anticancer drug, cisplatin.

The data revealed that complex 2 showed potency in the submicromolar scale (Table 1). Notably, 2 displayed 20 -to 78-fold greater potency than clinically approved cisplatin. In addition, the experiments also confirm significantly (ca. 5-6-fold) lower activity in noncancerous skin fibroblasts, thus indicating selectivity for cancer over noncancerous human cells. Antiproliferative activity of free lonidamine was considerably lower than the activity of 2 or cisplatin (328 -379fold or 20 -78-fold, respectively).

The MTT assay is dependent on mitochondrial respiration, specifically on the reduction of MTT to formazan catalyzed by mitochondrial succinate dehydrogenase. [START_REF] Berridge | Characterization of the Cellular Reduction of 3-(4,5dimethylthiazol-2-yl)-2,5-diphenyltetrazolium bromide (MTT): Subcellular Localization, Substrate Dependence, and Involvement of Mitochondrial Electron Transport in MTT Reduction[END_REF] Thus, some test compounds capable of affecting mitochondrial function may interfere with the conversion of the dye to colored formazan so that the MTT assay may yield false results. Platinum(IV) complex 2 contains lonidamine which has been known to inhibit aerobic glycolysis in cancer cells by inhibiting the mitochondrially bound hexokinase. [START_REF] Brawer | Lonidamine: basic science and rationale for treatment of prostatic proliferative disorders[END_REF] Thus, it is a different enzyme than the one exploited in the MTT assay. However, to make sure that the results obtained using the MTT assay are not affected by the effect of lonidamine on mitochondria, we also used another assay for quantitative estimation of the number of viable cells in a culture based on the mechanism not associated with mitochondria, namely the neutral red (NR) uptake assay. This assay is based on the ability of viable cells to incorporate and bind the supravital dye neutral red in the lysosomes. [START_REF] Repetto | Neutral red uptake assay for the estimation of cell viability/cytotoxicity[END_REF] The results obtained using this assay showed that the IC50 values determined for 2 and cisplatin in MCF-7 cells did not differ significantly from those obtained by the MTT assay (Table 2).

To emphasize the possibility that the new antitumor compounds might be promising candidates for clinical testing and to improve the relevance of our in vitro results, we used 3D cell cultures (spheroids), which are much better at replicating in vivo environment than traditional 2D cultures. Cells grow in complex 3D cultures with heterogeneous regions, nutrient and oxygen gradients, intercellular and cell-extracellular matrix interactions, closely reflecting the tumor microenvironment.

[45] Hence, 3D growth of immortalized established cell lines or primary cell cultures is regarded as a more stringent and representative model for performing in vitro drug screening.https://www.nature.com/articles/srep19103 -ref6[45-47] 3D cell cultures possess several important in vivo features of tumors such as cell-cell interaction, [START_REF] Baker | Deconstructing the third dimension -how 3D culture microenvironments alter cellular cues[END_REF] hypoxia, [START_REF] Wartenberg | Regulation of the multidrug resistance transporter P-glycoprotein in multicellular tumor spheroids by hypoxia-inducible factor (HIF-1) and reactive oxygen species[END_REF] drug penetration, [START_REF] Minchinton | Drug penetration in solid tumours[END_REF] response and resistance,https://www.nature.com/articles/srep19103 -ref9 [START_REF] Wartenberg | Regulation of the multidrug resistance transporter P-glycoprotein in multicellular tumor spheroids by hypoxia-inducible factor (HIF-1) and reactive oxygen species[END_REF] and production/deposition of the extracellular matrix. [START_REF] Kimlin | In vitro three-dimensional (3D) models in cancer research: An update[END_REF] Last but not least, authentic 3D cell culture models using human cells can circumvent the drawbacks of animal models that, aside from the high cost and ethical considerations, are not always able to recapitulate human diseases or capture the side effect of drugs accurately. Also, several features of the in vivo solid tumors have been exhibited by tumor spheroids. The similarities in the drug responsiveness among the tumor spheroids and the animal models might largely be due to their similarities in enhanced cellular interactions via adhesion and secretion of soluble factors of tumors that lead to the low pH and hypoxia. [START_REF] Ravi | 3D cell culture systems: Advantages and applications[END_REF] Therefore, we also assessed the ability of complex 2 to inhibit 3D spheroid formation from MCF-7 cells known to reproducibly form spheroids, [START_REF] Pulze | MCF7 Spheroid Development: New Insight about Spatio/Temporal Arrangements of TNTs, Amyloid Fibrils, Cell Connections, and Cellular Bridges[END_REF] under 3D cell culture conditions (Table 3). The 3D spheroids were cultured for 96 h to grow up to a tissue mass of around 200 μm in diameter, as described in the Experimental Section. The cells were then treated with various concentrations of 2 for an additional 72 h. The CellTiter-Glo 3D cell viability assay was used to determine IC50 values. The results (Table 3) confirmed that 2, which is a very active drug in conventional 2D cell cultures (Table 1), exhibited exceptional activity also in the 3D spheroids formed from MCF-cells, being ~11-fold more effective than clinically used cisplatin.

The effect on the morphology of 3D spheroids formed from the suspension of MCF-7 single cells is shown in Figure 2. Spheroids formed by the control, untreated cells were of round-shape morphology with a well-defined surrounding edge (Figure 2, panel A). However, spheroids generated after the treatment with 2 (Figure 2, panels D-F) displayed heterogeneous morphology with a high number of dissociated cell clumps. Furthermore, these dissociated cell clumps were found to a greater extent than in the samples treated with cisplatin (Figure 2, panels B,C). Therefore, this microscopic study showed superior activity of 2 in the 3D spheroids formed from breast cancer MCF-7 cells.

Cellular accumulation

Cellular accumulation plays a fundamental role in the antiproliferative activity of low-molecularmass compounds, including platinum complexes. It is so because cellular uptake or efflux belong to the initial steps in the mechanism of their antitumor effects. [START_REF] Brabec | Cytotoxic platinum coordination compounds. DNA binding agents[END_REF][START_REF] Johnstone | The Next Generation of Platinum Drugs: Targeted Pt(II) Agents, Nanoparticle Delivery, and Pt(IV) Prodrugs[END_REF] Therefore, we examined first how 2 accumulates in cancer cells. MCF-7 cells were treated with 10 µM 2 or cisplatin for 6 h, and the amount of platinum in cells was determined by ICP-MS. Platinum from 2 was taken up by the MCF-7 cells ca. 11-fold more efficiently than platinum from cisplatin (Table 4).

It was shown for several Pt(IV) complexes that they are transported into cells by passive diffusion dependent on their lipophilicity. [START_REF] Göschl | The role of the equatorial ligands for the redox behavior, mode of cellular accumulation and cytotoxicity of platinum(IV) prodrugs[END_REF][START_REF] Kostrhunova | A new anticancer Pt(IV) prodrug that acts by mechanisms involving DNA damage and different epigenetic effects[END_REF] However, it cannot be excluded that the involvement of active and facilitated transport mechanisms in the cellular accumulation of the investigated complex 2 can also affect their final cellular accumulation. [START_REF] Göschl | The role of the equatorial ligands for the redox behavior, mode of cellular accumulation and cytotoxicity of platinum(IV) prodrugs[END_REF] Therefore, we also examined whether the cellular uptake of 2 correlates with its lipophilic properties (log P). The log P value for 2 determined previously [START_REF] Nosova | Enhancing the Cytotoxic Activity of Anticancer PtIV Complexes by Introduction of Lonidamine as an Axial Ligand[END_REF] was positive (the compounds are lipophilic). Contrastingly, cisplatin itself is hydrophilic (log P = -2.3), [START_REF] Karmakar | Platinum(IV)-estramustine multiaction prodrugs are effective antiproliferative agents against prostate cancer cells[END_REF] A clear correlation between the complex lipophilicity and cellular accumulation (Table 4) confirmed an assumption that hydrophobic rather than hydrophilic platinum compound penetrates the cytoplasmic membrane more effectively. This supports the hypothesis that 2 is transported into cells by passive diffusion, although the involvement of active and facilitated transport mechanisms cannot be excluded. The data also revealed that the enhanced cellular accumulation of platinum from the more hydrophobic 2 significantly contributes to its increased antiproliferative activity in MCF-7 cells compared with hydrophilic cisplatin. Moreover, an important implication of the marked differences between the cellular accumulation of platinum from 2 and cisplatin is that a significant part of the molecules of 2 enters the cells intact without being reduced already in the extracellular milieu of cultured cells.

Cell cycle profiles of MCF-7 cells

To further examine the mechanism(s) of action (MoA) of 2, we investigated the cell-cycle profiles of MCF-7 cells treated with the investigated complex 2. The cells were treated with various concentrations of 2 for 24 h (Figure 3). The cycle distribution of MCF-7 cells treated with 2 was analyzed by flow cytometry with untreated cells and cisplatin-treated cells as controls. As shown in Figure 3, exposure to a sublethal dose of cisplatin resulted in the accumulation of cells mainly in the S phase and slightly also in the G2/M phase. After treatment with 2, the cells were also primarily arrested in S-phase, similarly to cisplatin. However, the percentage of cells accumulated in the S phase increased significantly compared to the effect of cisplatin. In contrast, the number of cells arrested in the G2/M phase remained unchanged compared to the control. Thus, the impact of 2 on the cell cycle resembles, to some extent, the parental cisplatin, supporting the view that the 2 acts via an overall mechanism similar to that of cisplatin.

Nature of cell death

Cell-based assays, such as the MTT assay, cannot distinguish between the cytostatic and cytotoxic effects of the investigated agents. [START_REF] Eastman | Improving anticancer drug development begins with cell culture: misinformation perpetrated by the misuse of cytotoxicity assays[END_REF] Therefore, we also investigated whether the treatment of MCF-7 cells with 2 can induce cell death and studied the mechanism of cell death. The levels of apoptosis and necrosis induced in MCF-7 cells by 2 or cisplatin at various concentrations over 12 h exposure time were quantified by the assay employing FITC Annexin V with 7-aminoactinomycin D (7-AAD) for the identification of apoptotic and necrotic cells by flow cytometry. The results demonstrated that 2 induced a 7.46 % fraction of early apoptosis at 0.25 μM, and the ratio increased to 10.64% at 0.5 μM treatment. In addition, 2 showed a comparable fraction of late necrosis and late apoptosis compared to cisplatin. The results indicated apoptosis as a predominant mode of cell death; however, the effect of 2 was much more pronounced than that of cisplatin (Figure 4). These findings suggest that the nature of cell death induced in cancer cells by cisplatin and 2 was very similar or even identical. Thus, one can envision scenarios in which the mechanism of the antiproliferative effect of complex 2 is determined mainly by the impact of its platinum moiety.

DNA platination and effect of DNA repair on the toxic effects of 2 in cancer cells in vitro

The important pharmacological target of cisplatin and its antitumor derivatives, including Pt(IV) prodrugs such as that examined in the present work, is nuclear DNA. [START_REF] Brabec | Cytotoxic platinum coordination compounds. DNA binding agents[END_REF] Cisplatin and its antitumor derivatives function in part through coordinative binding to DNA to form adducts that induce unique specific cellular responses, mainly culminating in apoptosis induction in cancer cells. Therefore, we further investigated whether the treatment of MCF-7 cells with 2 can result in the coordinative platination of nuclear DNA. The platination of DNA in MCF-7 cells treated with 2 was examined under experimental conditions identical to those used in the cellular uptake experiments (Table 4) using ICP-MS. The results (Table 5) show that under these experimental conditions, 15 picograms of platinum were associated with one microgram of DNA. This result insinuates that at least a part of 2 was reduced, releasing Pt(II) complex capable of coordinative binding to DNA because Pt(IV) complexes are substitution inert.

Nonetheless, to further explore whether DNA damage caused by the binding of platinum from 2 is involved in the mechanism of its antiproliferative activity, we performed an experiment to determine if DNA is an important target also for 2, as is the case with other Pt(IV) complexes derived from cisplatin-containing bioactive axial ligands. [START_REF] Kostrhunova | A new anticancer Pt(IV) prodrug that acts by mechanisms involving DNA damage and different epigenetic effects[END_REF][START_REF] Kasparkova | A photoactivatable platinum(IV) complex targeting genomic DNA and histone deacetylases[END_REF][START_REF] Kasparkova | Molecular aspects of antitumor effects of a new platinum(IV) drug[END_REF][START_REF] Novohradsky | Antitumor platinum(IV) derivatives of oxaliplatin with axial valproato ligands[END_REF][START_REF] Novohradsky | Epigenetic and antitumor effects of platinum(IV)-octanoato conjugates[END_REF][START_REF] Zajac | Platinum(IV) derivatives with cinnamate axial ligands are potent agents against both differentiated and tumorigenic cancer stem Rhabdomyosarcoma cells[END_REF][START_REF] Kostrhunova | A multi-action Pt IV conjugate with oleate and cinnamate ligands targets human epithelial growth factor receptor HER2 in aggressive breast cancer cells[END_REF] The target may be defined as a molecule whose alteration leads to cell death. One of the criteria used to suggest that DNA is the target is based on the observation that the drug exhibits higher toxicity in cells that are deficient in DNA repair [START_REF] Brabec | Cytotoxic platinum coordination compounds. DNA binding agents[END_REF] because the persistence and toxicity of DNA lesions depend on the capacity of cells to repair the damage. Therefore, the pair of Chinese hamster ovary CHO-K1 cell line (wildtype) and its mutant line MMC-2 [nucleotide excision repair (NER)-deficient, carrying the ERCC3/XPB mutation] was used to distinguish whether NER-reparable DNA damage induced by 2 is involved in the mechanism of action.

Cisplatin is believed to manifest its activity at least partially by damaging DNA, [START_REF] Brabec | Cytotoxic platinum coordination compounds. DNA binding agents[END_REF][START_REF] Johnstone | The Next Generation of Platinum Drugs: Targeted Pt(II) Agents, Nanoparticle Delivery, and Pt(IV) Prodrugs[END_REF] so it was not surprising that also in accord with the previously published data, [START_REF] Kostrhunova | A new anticancer Pt(IV) prodrug that acts by mechanisms involving DNA damage and different epigenetic effects[END_REF] we found (Table 5) that cisplatin was ca. 11-fold more efficient in inhibiting proliferation of MMC-2 than CHO-K1 cells. Similarly, 2 was 9.4-fold more efficient in inhibiting proliferation of MMC-2 than CHO-K1 cells (Table 5), indicating that reparable DNA damage by platinum moiety released from 2 contributes significantly to the resulting biological activity.

Effect of 2 on glycolysis in cancer cells

To further explore the role of lonidamine ligands in the platinum(IV) prodrug 2 in the mechanism of its antiproliferative activity, we examined the effectiveness of 2 in inhibiting aerobic glycolysis in MCF-7 cancer cells. We opted for this study because 2 contains lonidamine, which is known to inhibit aerobic glycolysis in cancer cells by inhibiting the mitochondrially bound hexokinase. [START_REF] Brawer | Lonidamine: basic science and rationale for treatment of prostatic proliferative disorders[END_REF] Aerobic glycolysis is less efficient from an energetic standpoint than oxidative phosphorylation; therefore, cancer cells need to uptake more glucose to meet energy requirements. Moreover, lactate is produced and secreted to the microenvironment to maintain the cellular environment homeostasis.[64] Therefore, the effectiveness of 2 in inhibiting aerobic glycolysis in MCF-7 cancer cells was assessed by measuring the level of cellular glucose metabolism and lactate production using Amplex® Red Glucose/Glucose Oxidase Assay Kit (Life Technologies).

As expected, treatment of MCF-7 cells with free lonidamine for 24 h resulted in significantly (P ˂ 0.02) lower glucose consumption compared with the untreated control or cells treated with cisplatin (Figure 5A). Interestingly, the effect of 2 was markedly less pronounced than that of free lonidamine at equitoxic concentrations. The slight decrease in glucose consumption was observed only at a relatively high concentration of 2, corresponding to 3 times the IC50; however, this effect was above the level of statistical significance (p = 0.07). Similarly, the effect of 2 on the level of lactate excreted after 6 h of incubation with 2 was low and insignificant (Figure 5B), even at very high concentrations (corresponding to 5 IC50). In contrast, free lonidamine decreased lactate excretion significantly under the same conditions.

The data indicate that lonidamine ligands in axial positions of 2 were much less sufficient in shifting the cellular metabolism from glycolysis than free lonidamine, although they both induced the same antiproliferative effects (equitoxic concentrations were used). Thus, the observations that the effectiveness of 2 in inhibiting aerobic glycolysis in MCF-7 cancer cells is very low demonstrate that lonidamine ligands in 2 are functionally of only minor importance in the mechanism of its antiproliferative activity in cancer cells. These results were further confirmed by a glycostress test performed on MCF-7 cells using a Seahorse XF Analyzer. In this assay, subtle variations of the extracellular acidification rate (ECAR), which results from glucose consumption, can be followed in real-time while cells are subjected to different inhibitors injected in the medium. Each inhibitor shuts down or activates a metabolic pathway, allowing to probe how the metabolism of cells is affected by test compounds. For example, oligomycin inhibits mitochondrial respiration resulting in a metabolic shift towards glycolysis. In normal conditions, the oligomycin injection is therefore expected to be followed by a sharp increase in the ECAR (Figure 6B). While this was the case with untreated cells or cells treated with 50 μM of 2 or cisplatin, no increase in the ECAR was observed when cells were pretreated with 100 μM of lonidamine, indicating a severely impaired glycolytic capacity and glycolytic reserve (Figure 6A,C). These results, therefore, confirm that the lonidamine ligands in 2 do not inhibit glycolysis to the same extent as free lonidamine.

Effect of 2 on the level of glutathione in cancer cells.

In addition to the glycolysis inhibition, lonidamine was shown to affect many other metabolic processes in cancer cells. [START_REF] Nath | Mechanism of antineoplastic activity of lonidamine[END_REF] Among them, reducing cellular levels of glutathione (GSH) is particularly important, as GSH interactions of platinum-based drugs lead to the elimination of Ptdrugs from cells and contribute to the resistance of cancer cells. [START_REF] Han | Intracellular glutathione-depleting polymeric micelles for cisplatin prodrug delivery to overcome cisplatin resistance of cancers[END_REF] Thus, a decrease of intracellular concentration of GSH by lonidamine ligands from 2 should potentiate the activity of Pt moiety of 2, resulting in a synergistic increase of activity. The level of GSH was then measured in MCF-7 cells treated with 2, cisplatin, and free lonidamine at various concentrations and incubation periods. Monochlorobimane, a cell-permeant probe for quantifying glutathione levels in cells, was utilized for this experiment. To eliminate the possible impact of the presence of dead or damaged cells on the resulting data, we took advantage of flow cytometry, and the signals arising from potentially damaged cells were gated out.

The results shown in Figure 6 revealed that the level of GSH in cells was significantly elevated as a result of treatment with free lonidamine (except for the lowest concentration at 24 and 48 h of treatment). In contrast, 2 was markedly less effective in increasing glutathione concentration, with provable effect only after a relatively long time of incubation (72 h) The low efficiency of 2 in increasing GSH concentration in cancer cells (Figure 7) is consistent with the data demonstrating the low effectiveness of 2 in inhibiting aerobic glycolysis in MCF-7 cancer cells (Figures 5A,B). These results indicate that two of the important functionalities of lonidamine, which affect metabolic processes in cancer cells, are unlikely to substantially contribute to the overall antiproliferative activity of 2 via their inherent biological activity; the effect of lonidamine ligands in 2 rather consists in the increase of accumulation of DNA damaging Pt moiety in cancer cells due to the elevated lipophilicity of 2.

Tolerance of 2 in vivo and organ distribution

Finally, we tested the acute toxicity of complex 2 in Balb/c mice after a single bolus i.p. injection of the compound dissolved in saline. The doses were 70-125 mg/kg; higher doses were not achievable due to limited water solubility. Animals were monitored for 21 days after injection. No deaths were detected after injections of 125 mg/kg or 70 mg/kg. Mice in these cohorts had normal hair cover; no changes in nutritional behavior were registered over the entire period of observation. Biodistribution of 2 was studied in Balb/c mice using ICP-MS analysis of the platinum content in individual tissues. Compound 2 was injected i.p. at 75 mg/kg single bolus administration. The liver, kidney, lung, brain, and blood were collected after 24 h. This period is sufficient for a pilot judgment about drug distribution. Blood samples were separated into the plasma and cellular fractions by centrifugation. As shown in Table 6, accumulation of 2 mostly in the liver and in the kidney, followed by lung and blood. As expected, complex 2 weakly penetrated the blood-brain barrier. Conclusions A Pt(IV) derivative of a direct and simple analog of cisplatin [cis-ethane-1,2diaminedichloridoplatinum(II)] with lonidamine at both axial positions (complex 2) was prepared and characterized by physicochemical methods. The investigated Pt(IV) complex showed exceptional antiproliferative activity in cancer cells cultured in 2D monolayers or contained in 3D spheroids (Tables 1 and3). Complex 2 also displayed selectivity for cancer over noncancerous human cells. Additionally, our data indicate that platinum from 2 was taken up by the cancer cells markedly more efficiently than platinum from cisplatin, apparently due to the higher hydrophobicity of 2. An important implication of the marked differences between the cellular accumulation of platinum from 2 and cisplatin is that a significant part of the 2 enters the cells intact without being reduced already in the extracellular milieu of cultured cells. Our findings also demonstrate that the impact of 2 on the cell cycle and the nature of cell death resembles that of the parental cisplatin, supporting the view that the 2 acts via an overall mechanism similar to that of cisplatin. We also explored whether DNA damage caused by the binding of platinum from 2 is involved in the mechanism of its antiproliferative activity and found out that reparable DNA damage by platinum moiety released from 2 significantly contributes to its overall biological activity consistent with the reduction of the substitution-inert 2 by intracellular reductants.

Further studies were aimed at demonstrating whether the overall mechanism of antiproliferative activity of 2 also involved the effects of two important functionalities of lonidamine, which affect metabolic processes in cancer cells, namely the inhibition of glycolysis and the reduction of cellular levels of GSH. The results indicated that these important functionalities of lonidamine were unlikely to substantially contribute to the overall antiproliferative activity of 2 via their inherent biological activity. Thus, it is reasonable to conclude that the effect of lonidamine ligands in 2 consists predominantly in the enhancement of lipophilicity of 2, resulting in the increase of accumulation of DNA damaging Pt moiety in cancer cells.

The low potency of lonidamine ligands in 2 to affect metabolic processes in cancer cells, which free lonidamine is able to affect very efficiently, deserves further discussion. Anticancer Pt(IV) prodrugs containing biologically active axial ligands are known to be reduced by intracellular reductants upon entry into cancer cells. This reduction leads to the simultaneous release of the antitumor Pt(II) complex and molecules of the biologically active substance, which synergistically potentiate the antiproliferative effects of the Pt(II) complex. Another factor that significantly contributes to the higher antiproliferative efficacy of Pt(IV) prodrugs compared to parental Pt(II) complexes is that biologically active axial ligands increase substantially the hydrophobicity of the Pt(IV) complex and thus its accumulation in cancer cells. Thus, it is possible that, in the case of treatment of cancer cells with 2, the concentration of antitumor Pt (II) complex in the treated cells is markedly increased compared to the situation where cancer cells are subjected to the significantly more hydrophilic parental Pt(II) complex under the same conditions. At the same time, however, the concentration of lonidamine molecules released in cancer cells may be insufficient to manifest the effect of lonidamine on metabolic processes in cancer cells.

Conclusively, the results of this study show that the task of Pt(IV) complex 2 used as a prodrug is to deliver the cytotoxic Pt(II) drug to the cancer cells predominantly and should consider 2 not as a dual-or multi-action prodrug but as the prodrug that effectively delivers cytotoxic Pt(II) complex, rather than lonidamine, to the tumor.

Experimental

The chemicals and instruments Reagents were purchased from Aldrich unless specified otherwise. All solvents were purified and degassed prior to use. NMR spectra were recorded on a Bruker FT-NMR Avance III 500 MHz instrument at 500.32 (1H). Chemical shifts were referenced relative to the solvent signal for 1H. ESI mass spectra were recorded on an LC/MSn ion trap mass spectrometer amaZon SL (Bruker, Bremen, Germany) with MeOH as a solvent. Elemental analysis was performed at Moscow State University with MicroCube Elementar analyzer.

Material and cell lines for biological studies

Cisplatin, N,N-dimethylformamide (DMF), and dimethylsulfoxide (DMSO) were from Sigma-Aldrich (Prague, Czech Republic). MTT and neutral red (NR) were from Calbiochem (Darmstadt, Germany) and Sigma Aldrich, Prague, Czech Republic), respectively. Stock solutions for cellular studies were prepared by dissolving Pt compounds or lonidamine in DMF to a final concentration of 5 or 20 mM, respectively, and serially diluted before testing in cell culture media. To avoid DMF toxicity, the final DMF concentration in the cell culture medium did not exceed 1% v/v.

The human breast cancer MCF-7 cells and human colorectal carcinoma cells HCT 116 were kindly supplied by Professor B. Keppler, University of Vienna (Austria). Highly invasive breast carcinoma MDA-MB-231 cells and human MRC-5 pd30 cells derived from normal lung tissue were purchased from the European collection of authenticated cell cultures (ECACC; Salisbury, UK). Chinese hamster ovary CHO-K1 cell line (wild-type) and its derivative MMC-2 carrying the ERCC3/XPB mutation (NER-deficient) cell line were kindly supplied by Dr. M. Pirsel, Cancer Research Institute, Slovak Academy of Sciences, Bratislava (Slovakia). The MCF-7, HCT-116, MDA-MB-231, MRC-5 pd30, CHO-K1, and MMC-2 cells were grown in DMEM medium (Dulbecco's Modified Eagle's Medium, high glucose (4.5 gL -1 , PAA) supplemented with gentamycin (50 mgmL -1 , Serva) and 10 % heat-inactivated fetal bovine serum (PAA, Pasching, Austria). The medium for MRC-5 pd30 was further fortified by 1% non-essential amino acids (Sigma-Aldrich, Prague, CzechRepublic). The cells were cultured in an incubator at 37 °C in a humidified 5 % CO2 atmosphere and subcultured 2-3 times a week.

In vitro antiproliferative assays

The effect of the investigated compounds on cell proliferation was evaluated using a common MTT assay or, alternatively, by neutral red (NR) assay. The cells were seeded in 96-well tissue culture plates at a density of 2x10 3 (MCF-7), 4x10 3 (MDA-MB-231 and HCT-116), 1x10 4 (MRC-5pd30), and 3x10 3 (CHO-K1 and MMC-2) cells/well in 100 µL of the medium. After overnight incubation at 37 °C, the cells were treated with the tested compounds in the concentration range of 0 to 50 µM or 0 to 100 µM for 2 or cisplatin, respectively, in a final volume of 200 µL/well for 72 h. As for the MTT assay, 20 µL of a freshly diluted MTT solution (1.25 mgmL -1 in PBS) was added to each well after the treatment period, and the plates were incubated at 37 °C in a humidified atmosphere for 3 h. Afterward, the medium was removed, and the resulting formazan product was dissolved in 100 μL of DMSO. The number of living cells in each well was evaluated by measuring the absorbance at 570 nm (reference 620 nm) using the Spark® multimode microplate reader (Tecan, Raleigh, US). For the NR assay, 20 μL of a 0.33% solution of NR in PBS was added to each well after the treatment period, and the cells were incubated at 37 °C for 3 h. Then, the dyecontaining medium was thoroughly removed, and the cells were quickly rinsed twice with PBS. Then, the incorporated dye was solubilized in 200 μL of 1% acetic acid in 50% ethanol. The absorbance was measured at 540 nm (reference wavelength at 690 nm). In both assays, the reading values were converted to the percentage of control. Antiproliferative effects were expressed as IC50 values calculated from the curves constructed by plotting cell survival (%) versus drug concentration. All experiments were independently repeated at least three times, each repetition made in duplicate.

Toxic effects of the investigated compounds were also determined in 3D cultures of MCF-7 cells. The cells were seeded in ultra-low attachment 96-well plates at a density of 600 cells/well in the growth medium [DMEM; B-27 supplement (1V), EGF (epidermal growth factor, 20 ngmL - 1 ) and BSA (1.5 mgmL -1 )] and allowed to grow for four days. The resulting 96 h-old spheroids were treated with the investigated compounds for an additional 72 h. The CellTiter-Glo 3D cell viability assay (Promega, Prague, Czech Republic) was used to determine IC50 values according to the manufacturer's protocol.

Cellular accumulation

Cellular accumulation of platinum from 2 and cisplatin was determined as already described. [START_REF] Kostrhunova | A subset of new platinum antitumor agents kills cells by a multimodal mechanism of action also involving changes in the organization of the microtubule cytoskeleton[END_REF] In brief, MCF-7 cells were seeded in 10 cm Petri dishes and kept in a drug-free medium at 37 °C in a 5 % CO2 humidified atmosphere overnight. Then, the cells were treated with compounds at their 10 μM concentrations in the cell growth medium when confluency reached about 80% and allowed drug exposure for 6 h. The cells were subsequently detached using 0.25 % trypsin, washed 3-times carefully with ice-cold PBS, collected by centrifugation, and counted. The cell pellets were digested with concentrated HNO3, and the quantity of metals taken up by the cells was determined by ICP-OES.

Cell cycle

MCF-7 cells were seeded in 6-well plates at a density of 2×10 5 cells per well. After attaching for 12 h, cells were treated with cisplatin and 2 at different concentrations for 24 h. Cells were harvested by trypsinization, followed by washing three times with ice-cold PBS and fixed in 5 mL of cold 70% ethanol at -20 °C overnight. Before analysis, cells were spun down, washed twice again with PBS, and stained with PI solution (20 μgmL -1 , containing 0.1% Triton X-100 and 200 μgmL -1 RNase A, pH 7.4) for 15 min at 37 ºC. Samples were finally analyzed by flow cytometer (BD FACS Calibur).

Cell death MCF-7 cells were seeded in 6-well plates at a density of 2.5×10 5 cells per well and allowed to attach for 12 h. Medium containing cisplatin and 2 was added to the cells at different concentrations. After 12 h treatment, cells were collected by trypsinization and washed twice in PBS. Then the cell pellets were washed once and resuspended in Annexin-V binding buffer at the density of 1 ×10 6 cells per mL for staining. A volume of 5 µL of FITC Annexin-V and 1 µL of the PI solution was added according to the manufacturer's protocol. The samples were analyzed by a flow cytometer (BD FACS Calibur) after 15 min staining at r.t.

Intracellular DNA platination

Cellular accumulation of the investigated complexes was determined in MCF-7 cells. Briefly, the cells were seeded on 100 mm Petri dishes with a plating density of 7x10 5 cells/dish. After overnight pre-incubation in a drug-free medium at 37 °C in 5% CO2, the cells were treated with platinum compounds (0.25 µM) and allowed 24 h of drug exposure. After the treatment, the cells were extensively washed with PBS (37 °C), detached by 0.25% trypsin, and washed twice with ice-cold PBS. The cell pellets were then lysed in DNAzol (DNAzol, MRC) supplemented with RNase A (100 μgmL -1 ). Genomic DNA was precipitated from the lysate with 100%ethanol, washed twice with 75% ethanol, and resuspended in water. The DNA content in each sample was determined by UV spectrophotometry. To avoid the affection of high DNA concentration and related viscosity on detection of platinum in the samples, the DNA samples were digested in the presence of 30% hydrochloric acid (Suprapur, Merck Millipore). The amount of metal bound to DNA was quantified by ICP-MS. All experiments were performed in triplicate.

Glucose consumption by cells

MCF-7 cells were seeded on 24-well plates (8×10 4 cells/600 μL/well) and grown overnight. The medium was then changed, and the cells were treated with 2, lonidamine, or cisplatin diluted in the media (400 µL) or left untreated and grown for an additional 24 h. Subsequently, an aliquot of the medium was withdrawn from the wells, and the cells were trypsinized, harvested, and counted. The glucose concentration in the medium was determined using the Amplex® Red Glucose/Glucose Oxidase Assay Kit (Life Technologies) following the manufacturer's protocol.

The glucose accumulation in cells was calculated as the difference between the glucose concentration in the samples and glucose concentration in the cell-free medium and normalized to the number of cells in each sample. It was verified that under the experimental conditions, the viability of cells after the treatment was at least 93%.

Lactate production in cells MCF-7 cells were seeded in 24-well culture plates at a density of 2.5×10 4 /well. After overnight incubation in the complete growth medium, the medium was removed, and cells were washed twice with PBS. Afterward, 400 µL of a serum-free medium without Phenol Red was added to the cells, either drug-free or containing 2, lonidamine, or cisplatin at a final, roughly equitoxic concentrations corresponding to 2-fold, 3-fold, or 5-fold of the IC50. After 6-h treatment, the plates were spun down at 1000 rpm for 5 min to pellet the insoluble material. The cell culture medium from each sample (150 μL) was withdrawn and loaded into a 10 kDa cutoff spin filter column and spun down at 12 000 rpm for 10 min to remove lactate dehydrogenase. Eluents (4 μL) were added to a 96-well culture plate, and lactate concentration was measured using an enzymatic colorimetric Lactate Assay Kit (Sigma-Aldrich) according to the manufacturer's protocol. The amount of lactate present in the samples was determined from a standard curve plotted for the lactate standards supplied in the kit. The experiments were performed in triplicate. It was verified that under the experimental conditions, the viability of cells after the treatment was at least 95%.

Glycostress test

MCF-7 cells were seeded at a density of 20,000 cells/well and incubated for 24 h at 37 °C, 5% CO2 in 100 µL DMEM supplemented with 10% FBS in a Seahorse XF 96-well cell culture microplate (Agilent). Cells were then washed with Seahorse XF DMEM medium (Agilent) and incubated for 1 h at 37 °C, 5% CO2 prior to the assay. 18 min after the start of the assay, 2 (final concentration: 50 µM), cisplatin (final concentration: 50 µM), lonidamine (final concentration: 100 µM), or the vehicle (DMSO) for untreated cells were injected into the wells (8 replicate wells). The assay was then performed according to the manufacturer's instructions using a Seahorse XFe96 Analyzer (Agilent).

Quantification of intracellular glutathione

The intracellular level of GSH was quantified by using GSH-selective cell-permeant probe monochlorobimane. This probe is essentially non-fluorescent until it reacts with glutathione. For this assay, MCF7 cells were seeded on the 6w plates at the density of 2·10 5 cells/well. After the overnight incubation, the cells were treated with tested compounds at their equitoxic concentrations corresponding to 1, 3, 5 xIC50,72h and cultured for a further 24, 48, and 72 h. Then, samples were harvested and stained with 100 μM of monochlorobimane for 30 min at 37°C. Samples were washed with DPBS, and the fluorescence associated with the cells was analyzed using the flow cytometer CellStream (Luminex, USA). Samples were excited at 405 nm, and the emission was detected on the channel 456/51 nm. The only population of single cells was included in the analysis; debris and other particles were gated out of the analysis. The data were processed by using Luminex analysis software.

In vivo acute toxicity

The Balb/c female mice (8-10 weeks old, weight 20-22 g) were bred and hosted at the animal facility of Blokhin Cancer Center. Mice were kept at 21±1ºC, 50-60% humidity; food and water were added ad libitum. All manipulations were performed in accordance with the European Convention for the Protection of Vertebrate Animals used for Experimental and other Scientific Purposes (ETS 123). Compound 2 was injected i.p.; 70-125 mg/kg single bolus administration in 200 µl saline. Each cohort contained six mice. Animals were monitored for 21 days after injection.

General behavioural activity, nutritional habits, and hair cover integrity were the criteria of acute toxicity.

Biodistribution experiments

Compound 2 was injected i.p. (75 mg/kg) into three 10-week-old male Balb/c mice (weight ~20 g). Animals were sacrificed at 24 h after injection; blood and organ samples were collected for quantitative measurements of Pt content.

ICP-MS sample preparation and measurement

Sample digestion was performed using batches of 8 samples, 100 mg each. Each batch contained 2 control samples. Specimens were placed in one-chamber autoclaves to which 1.0 mL of concentrated HNO3 (65%, max. 0.0000005% Hg, GR, ISO, Merck) and 0.1 mL of HCl (37%; max. 0.0000005 % Hg; PA-ACS-ISO; Panreac, Spain) were added. The chambers were closed with caps and evacuated in the autoclave titanium housings. The autoclaves were placed in an electric furnace and incubated at 160 °C for 1 h and 180 °C for another 1 h. After cooling, the solutions were diluted with water to 10 mL and transferred to plastic bottles. The same procedure was performed in autoclaves without samples, and the resulting solutions were used as control samples. The Pt content was determined by inductively coupled plasma mass spectrometry (ICP-MS) using an Х-II spectrometer (Thermo Scientific, USA) and the following parameters: RF generator power 1400 W, nebulizer PolyCon, spray chamber cooling 3 о С, plasma gas flow rate 12 L/min, auxiliary flow rate 0.9 L/min, nebulizer flow rate 0.9 L/min, sample update 0.8 mL/min, resolution 0.8M. The main parameters of MS measurements were: detector mode double (pulse counting and analogous) and scanning mode Survey Scan and Peak Jumping. The settings for the Survey Scan were: number of runs 10, dwell time 0.6 ms, channels per mass 10, acquisition 13.2 sec. The settings for the Peak Jumping were: sweeps 25, dwell time 10 ms, channels per mass 1, acquisition 34 sec. The drug-treatment period was 72 h. The results are expressed as mean values ± SD from three independent experiments performed in duplicate. The results are expressed as mean values ± SD from three independent experiments performed in a quadruplicate; the drug-treatment period was 72 h. The cells were treated with platinum compounds at the 0.25 µM concentration for 24 h. Results are expressed as the mean values ± SD from at least three independent experiments in triplicate for each run. b The drug-treatment period was 72 h. The results are expressed as mean values ± SD from three independent experiments, each performed in triplicate.

2 Figure 1 .

 21 Figure 1. Structures of lonidamine (1) and (OC-6-33)-dichloridobis[1-(2,4-dichlorobenzyl)-1Hindazole-3-carboxylato](ethane-1,2-diamine)platinum(IV) (2).

Figure 2 .

 2 Figure 2. Representative bright-field images of 3D spheroids formed from MCF-7 cells untreated or treated with 2 or cisplatin. The 3D spheroids were cultured for 96 h to grow up to a tissue mass of around 200 μm in diameter, as described in the Experimental Section. The cells were further treated with the Pt complexes, and shots were taken after 72 h. Panels: A, control, untreated spheroids; B and C, the spheroids treated with cisplatin at 4 and 30 µM concentrations, respectively; D, E, and F, the spheroids treated with 2 at its 1, 4, and 7 µM concentration, respectively. Scale bars represent 200 μm.

Figure 3 .

 3 Figure 3. Cell cycle distribution of MCF-7 cells untreated or treated for 24 h at concentrations indicated in the Figure. The experiment was performed three times, and the results are expressed as mean.

Figure 4 .

 4 Figure 4. Effects of 2 and cisplatin on activation of the apoptotic and necrotic pathways in MCF-7 cells determined by the assay employing FITC Annexin V with 7-AAD by flow cytometry. Cells were treated for 12 h with 2 or cisplatin at the concentrations indicated in the figure. The results are expressed as the mean of two independent experiments with duplicate runs.

Figure 5 :

 5 Figure 5: (A) Glucose consumption by MCF-7 cells as affected by 2, cisplatin, and free lonidamime after 24 h of treatment at the indicated equitoxic concentrations. The amount of glucose taken up to the control, untreated cells was taken as 100%. (B) Modulation of extracellular lactate level in MCF-7cells after 6-h treatment with 2, cisplatin, or free lonidamine at the concentrations indicated in the graph. The lactate level in the medium containing untreated control cells was taken as 100%. Figures A and B represent a mean ± SEM from three independent experiments. Stars indicate a difference from control, untreated cells (t-test): * p ˂ 0.05.

Figure 6 :

 6 Figure 6: (A) Real-time Extracellular Acidification Rate (ECAR) as affected by the test compounds (2, cisplatin and lonidamine), followed by the injection of glucose, the mitochondrial respiration inhibitor oligomycin, and the glycolysis inhibitor 2-deoxyglucose in the medium as indicated by dashed lines. Only lonidamine prevented cells from switching their metabolism toward glycolysis when treated with oligomycin. (B) Effect of test compounds on the glycolytic parameters measured by the seahorse XFe Analyzer. Results presented as mean ± SD obtained using at least 8 replicate wells. Stars indicate a difference from untreated cells (t-test): * p < 0.001, ** p < 0.0001.

Figure 7 :

 7 Figure 7: Levels of GSH determined in MCF-7 cells by flow cytometry. The cells were treated with equitoxic concentrations of the investigated compounds corresponding to 1-, 3-, or 5-fold IC50,72h and incubated for 24, 48, or 72 h. The median of the fluorescence of the treated samples was normalized to the untreated control level. Results are expressed as the mean ± SEM; stars indicate the difference from the untreated control: * p˂0.1; ** p˂ 0.05

Table 1 .

 1 IC50 mean values (μM) of 2 and cisplatin evaluated using MTT assay in cancer cells aThe drug-treatment period was 72 h. The results are expressed as mean values ± SD from three independent experiments performed in duplicate.

		MCF-7	MDA-MB-231	HCT 116	MRC-5 pd30
	2	0.36 ± 0.01	0.281 ± 0.003	0.29 ± 0.01	1.9 ± 0.1
	cisplatin	7.2 ± 0.8	22 ± 2	11 ± 2	11 ± 4
	lonidamine	118 ± 9	133 ± 7	110 ± 4	>150

a

Table 2 .

 2 IC50 mean values (μM) of 2, lonidamine, and cisplatin evaluated using MTT and NR assays in MCF-7 cancer cells a

		MTT	NR
	2	0.36 ± 0.01	0.372 ± 0.006
	cisplatin	7.2 ± 0.8	7.8 ± 0.1
	lonidamine	118 ± 9	123 ± 8
	a

Table 3 .

 3 IC50 values [mean values, μM] a for 2 and cisplatin in 3D MCF-7 cell culture for a 72 h incubation period determined by using CellTiter-Glo 3D cell viability assay a

	2	1.6 ± 0.4
	cisplatin 18 ± 3

Table 4 .

 4 Cellular platinum accumulation in MCF-7 cells and log P (octanol/water) values obtained by the shake flask method MCF-7 cells were treated with 10 µM compounds for 6 h. Results are expressed as the mean values ± SD from four independent experiments in triplicate for each run.

		2	cisplatin
	Cellular uptake	208 ± 11	20 ± 9
	(µg Pt/10 6 cells) a
	log P b	1.8 c	-2.3 d
	a

b

Partition coefficient log P (octanol/water). c Taken from Ref.

[START_REF] Nosova | Enhancing the Cytotoxic Activity of Anticancer PtIV Complexes by Introduction of Lonidamine as an Axial Ligand[END_REF]

d Taken from Ref.

[START_REF] Karmakar | Platinum(IV)-estramustine multiaction prodrugs are effective antiproliferative agents against prostate cancer cells[END_REF]

Table 5 .

 5 Platinum content of genomic DNA (pg Pt/µg DNA) isolated from MCF-7 cells treated with 2 or cisplatin. IC50 values [mean values, µM] for 2 and cisplatin in Chinese hamster ovary CHO-K1 cell line (wild-type) and its mutant cell line MMC-2 deficient in DNA repair as determined by MTT assay.

		DNA platination a CHO-K1 b	MMC-2 b
	2	15.3 ± 0.8	4.2 ± 0.2	0.447 ± 0.005
	cisplatin	1.4 ± 0.2	22 ± 3	2.0 ± 0.1
	a

Table 6

 6 Distribution of compound 2 in mice tissues in ng/organ .

	Organ	24 h
	Liver	394.21
	Kidney	69.97
	Lung	21.48
	Brain	0.37
	Blood cell fraction	8.15
	Plasma	4.93

Acknowledgments

This work was supported by the Czech Science Foundation (Grant no. 20-14082J) and Russian Foundation for Basic Research (Grant No. 19-53-26002). This work was also financially supported by an ERC Consolidator Grant (PhotoMedMet) to G.G. (GA 681679) and by a Qlife prématuration funding (G.G. and R.V.).

Data availability statement

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


Sándor Bilicz
email: biliczs@sch.bme.hu

Emmanuel Vazquez

József Pávó

Marc Lambert

Szabolcs Gyimóthy

Eddy-current testing with the Expected Improvement optimization algorithm

Keywords: eddy current testing, inversion, global optimization, Gaussian processes, Expected Improvement algorithm

This paper presents an inverse problem methodology in the domain of non-destructive testing, and more precisely eddy-current testing. Our objective is to use a precise but expensive-to-evaluate model of the electromagnetic induction phenomenon in a conductive material and to estimate the characteristics of a flaw by minimization of a regularized criterion with the Expected Improvement (EI) global optimization algorithm. The EI algorithm is designed to estimate a global optimum of a function with a restricted budget of function evaluations. Thus, we expect to be able to estimate the characteristics of a flaw with a relatively low cost despite resorting to an expensive model of the induction phenomenon. The efficiency of the approach is discussed in the light of preliminary numerical examples obtained using synthetic data.

INTRODUCTION

This paper presents an inverse problem methodology in the domain of non-destructive testing, and more precisely eddycurrent testing (ECT). As illustrated by a large number of contributions in the last ten years, the estimation of the characteristics of scattering objects remains quite challenging from a theoretical, computational and/or experimental perspective, even in the case of the simplest configurations (e.g., a homogeneous embedding space). In this paper, our objective is to use a precise but expensive-to-evaluate model of the electromagnetic induction phenomenon in a conductive material.

Then, to estimate the characteristics of a flaw, a regularized criterion is minimized using the Expected Improvement (EI) global optimization algorithm, the latter being designed to estimate a global optimum of a function with a restricted budget of function evaluations. The EI algorithm is well-known in the domain of global optimization, but to the best of our knowledge, using the EI algorithm for the ECT problem remains largely unexplored. Preliminary results for the characterization of a single volumetric defect have been presented by the authors in [START_REF] Bilicz | Characterization of a 3D defect using the Expected Improvement algorithm[END_REF]. In this paper, a more challenging case is studied. We consider two infinitesimally thin material defects close to each other. This kind of problem turned out to be more complicated than the case of single defect (see, e.g. [START_REF] Pávó | Reconstruction of group of cracks in plate specimens using ECT impedance data[END_REF], or a little bit different application by [START_REF] Bréard | Low-Frequency Modeling of 3-D Coupled Obstacles and Inversion by Differential Evolution[END_REF]).

Our contribution is twofold: (i) from the point of view of ECT, we would like to show that the methodology we propose is promising; (ii) from the point of view of the inverse problem community, we would like to bring attention to an efficient optimization algorithm that renders practical the use of precise but expensive direct models.

The paper is organized as follows. First, we present the theoretical formulation of the direct problem and some aspects of its numerical implementation ( § 2). Then the inverse problem -along with our regularizing assumptions -will be presented ( § 3). Next the new stochastic approach and the optimization task will be in the focus ( § 4). Finally, numerical examples will illustrate the proposed method and the conclusions of the experiments will be drawn ( § 5).

THE FORWARD PROBLEM

In this subsection the theoretical formulation of the direct problem and some aspects of its numerical implementation are discussed.

Assume the arrangement shown in Fig. 1 where two infinitesimally thin cracks located in the planar surfaces S 1 and S 2 inside a non-ferromagnetic plate are searched by a pancake type ECT coil. The normal vectors of the S 1 and S 2 surfaces are n1 and n2 , respectively. In this Subsection the method used for the solution of the forward problem is discussed. In other words, the method used for the calculation of the change of the impedance of the ECT coil due to the presence of the cracks (∆Z) is described when the location and the orientation of S 1 and S 2 are arbitrary. During the solution of the inverse problem, however it will be assumed that the cracks are outer defects (OD) and they are parallel to each other, consequently n1 = n2 = x ( x is the unit vector of the x co-ordinate axis). Fig. 1 shows the particular arrangement that is actually assumed for the inverse problem. The electromagnetic interaction of the exciting coil with nonconducting cracks whose thickness are considerably smaller than their other dimensions might be calculated based on the so-called infinitesimally thin crack approximation described by [START_REF] Bowler | Eddy-current interaction with an ideal crack. I. The forward problem[END_REF]. This model assumes the cracks in the otherwise homogeneous conductor as mathematical surfaces that block the flow of the eddy currents across their surfaces, in other words the normal component of the electric field must be zero on the surfaces representing the cracks. In the same time, the magnetic field is assumed to be continuous on the two sides of these surfaces.

x x y z A B D 1 D 2 L 1 L 2 σ 0 , µ 0 h l d S 1 S 2 r 1 r 2
The above boundary conditions might be satisfied by replacing the S 1 and S 2 mathematical surfaces with current dipole distributions, p 1 (r) n1 and p 2 (r) n2 , having the same effect as the presence of the crack (the function p 1 (r) and p 2 (r) are non-zero only on the surfaces S 1 and S 2 , respectively). Thus, the sum of the electromagnetic field generated by the exciting coil and the secondary sources in the homogeneous conductor will provide the field generated by the crack-probe interaction. Consequently this field must satisfy the above boundary conditions. In the case of cracks with planar surfaces, the described consideration leads to the following integral equation reported by [START_REF] Pávó | Reconstruction of group of cracks in plate specimens using ECT impedance data[END_REF]:

0 = E i n1 (r 1 ) -jωµ 0 S 2 g 21 (r 1 |r ′ )p 2 (r ′ ) dr ′ - -jωµ 0 lim r→r 1± S 1 g 11 (r|r ′ )p 1 (r ′ ) dr ′ , r 1 ∈ S 1 , (1) 0 = E i n2 (r 2 ) -jωµ 0 S 1 g 12 (r 2 |r ′ )p 1 (r ′ ) dr ′ - -jωµ 0 lim r→r 2± S 2 g 22 (r|r ′ )p 2 (r ′ ) dr ′ , r 2 ∈ S 2 , ( 2
)
where r k± (k = 1, 2) denotes the limiting values of the approaches towards the point r k from the nk ornk directions, ω is the angular frequency of the sinusoidal excitation of the probe coil and µ 0 is the permeability of the vacuum.

E i nk = nk • E i (k = 1, 2)
, where E i is the so-called incident field that is the electric field generated by the current of the exciting coil in the defect-free plate. g kl (r|r ′ ) = nl • G(r|r ′ ) • nk (k, l = 1, 2) where G(r|r ′ ) is the Green's dyad transforming the current excitation into the electric field.

The surfaces of the cracks are discretized on a regular lattice where the unknown dipole density function is approximated by a piecewise linear function. The integral equation is tested by first order testing functions and the kernel of the integral equation is evaluated by the formulae published by [START_REF] Pávó | Reconstruction of crack shape by optimization using eddy current field measurement[END_REF]. After solving the linear system of equations obtained by the described discretization of the integral equations (1)-( 2), the dipole density function can be calculated. Knowing p 1 (r) and p 2 (r), the impedance variation of the probe, ∆Z, due to the presence of the cracks can be calculated based on the reciprocity theorem [START_REF] Bowler | Eddy-current interaction with an ideal crack. I. The forward problem[END_REF]:

∆Z = - 1 I 2 2 k=1 S k E i nk (r)p k (r) dr, ( 3
)
where I is the current of the exciting coil.

The results of the described solution of the forward problem are tested against measurements available for single artificial cracks (unfortunately measured results on two parallel cracks are not available in the literature). The comparison verified the validity of the results.

For rectangular shaped cracks being parallel to each other and being perpendicular to the surface of the plate, the integral equations ( 1)-( 2) can be solved by approximating p 1 (r) and p 2 (r) with the function series introduced in [START_REF] Pávó | Calculation of eddy current probe signal with global approximation[END_REF]. Using this method, the results on several test configurations are very close to those obtained by the method described above. The good agreement supports the validity of both solution methods.

THE INVERSE PROBLEM

When the probe coil scans the zone containing the cracks, the variations of the coil impedance are measured at each point of a regular rectangular grid, above the supposedly damaged zone of the plate. In other words, a so-called surface scan is performed. The scan consists of N coil locations where the measured impedance variations are

∆Z k , k = 1, 2, . . . , N.
The problem is to characterize the flaw from this set of measured impedance variations, which will be viewed as the input data of the inverse problem.

Regularizing assumptions

It is customary to introduce some regularizing assumptions to reduce the ill-posedness of the inverse problem. In this work, two rectangular shaped thin cracks are assumed in the plate specimen, under (or near) the surface scan measurement (see Fig. 1). Both cracks are of OD type. Their orientation is supposedly known, together with the "geometrical center" of the two-cracks-system (i.e. the distances of the crack midpoints from the origin are the same for both cracks in the x (A/2) and in the y (B/2) directions as well). As a result, only 6 geometrical parameters are needed to describe the configuration: the length of the cracks (L 1 , L 2 ), their depth (D 1 , D 2 ) and the x-y distances of their centers (A, B), respectively (see Fig. 1). To simplify the notations, the unknown defect parameters are collected in a vector t:

t = [A, B, L 1 , L 2 , D 1 , D 2 ].
The definition domain of t will be denoted by T, and will be referred to as the parameter space.

The forward solver ( § 2) is used to compute the impedance variation due to a fault characterized by its parameter vector t.

A simulated surface scan consists of such computed impedance variations, denoted as ∆Z k (t), k = 1, 2, . . . , N. Obviously, the simulation is performed at exactly the same coil positions as the real measurement, thus both the real and the simulated surface scan consists of N coil locations.

Optimization task

The next step for solving the inverse problem is to achieve the strongest resemblance between the impedance signal obtained by simulation, {∆Z k (t), k = 1, . . . , N}, and the measurement, {∆Z k , k = 1, . . . , N}, by tuning the parameter vector t.

The applied inversion method requires a finite parameter space T, meaning that one has to set lower and upper bounds for each parameter and, moreover, the parameter axes are needed to be discretized. Consequently, the space T has to consist of a finite number of parameter points t. The bounds and discretization of each parameter are summarized in Table 1. The parameters are independent from each other, thus, the parameter space T can be imagined as a 6 dimensional hypercube, spanned by the 6 parameter axes. The hypercube includes 5 2 • 7 4 = 60025 parameter points. To give a mathematical form to the resemblance between two surface-scan impedance signals, we define the similarity function

Q(t) = N k=1 |∆Z k (t) -∆Z k | 2 N k=1 |∆Z k | 2 . ( 4
)
Our objective is thus to minimize Q(t), i.e. to find t = arg min t∈T Q(t) .

STOCHASTIC MODELING AND OPTIMIZATION OF THE SIMILARITY FUNCTION

Our objective is to implement the Expected Improvement algorithm to minimize Q(t). One iteration of such an algorithm involves mainly two steps:

(1) the construction of an approximation of the similarity function from a set of past evaluations of the function obtained at previous iterations. To this end, a random process is chosen as a model of the similarity function and an interpolation by kriging is performed. (2) the search of the maximum of the Expected Improvement over the parameter space T then the computation of the similarity function at that parameter point.

This two-step procedure is repeated iteratively until a stopping criterion is met, i.e. a sequential optimization algorithm is obtained.

Kriging interpolation

Let us assume that we have already observed the multivariate scalar function Q(t) at n points t 1 , t 2 , . . . , t n of T. In this way,

n function values Q 1 = Q(t 1 ), Q 2 = Q(t 2 ), . . . , Q n = Q(t n )
are known. We would like to predict the function value at unobserved sites. One method to achieve this goal is to use kriging, a random process approach developed in the 60s in geostatistics [START_REF] Chiles | Geostatistics, Modeling Spatial Uncertainty[END_REF]. The method is also well-known for modeling computer simulations [START_REF] Sacks | Design and analysis of computer experiments[END_REF].

Let ξ(t) be a Gaussian random process that models the function Q(t). Thus, each observation Q k is considered as the realization of the Gaussian random variable ξ(t k ) (k = 1, 2, . . . , n). Kriging computes the best linear unbiased predictor (BLUP) of ξ(t). Let us denote this prediction by ξ(t). The predictor is linear in the sense that it is a linear combination of the observed random variables ξ(t k ), k = 1, 2, . . . , n, which can therefore be written as

ξ(t) = n k=1 λ k (t)ξ(t k ). ( 5
)
Unbiasedness relates to the fact that the mean of ξ(t) is equal to the mean of ξ(t), i.e. the mean prediction error is zero:

E[ε(t)] = E[ ξ(t) -ξ(t)] = 0. (6)
The term "best" means that the prediction error ε(t) of the kriging predictor has the smallest variance among all unbiased predictors. This variance (also called kriging error) may be written as

σ2 (t) = var[ ξ(t) -ξ(t)] = E[( ξ(t) -ξ(t)) 2 ] , (7)
using the unbiasedness condition [START_REF] Bowler | Eddy-current interaction with an ideal crack. I. The forward problem[END_REF].

The objective is to find the coefficients λ k (t) in (5) that provide the BLUP. The kriging error can be written using the covariance function, which describes the dependence between two random variables of the process at different points. Let us denote the covariance function by

k(t a , t b ) = cov[ξ(t a ), ξ(t b )],
where t a and t b are two points in T. Let us denote by K, the covariance matrix whose entries correspond to the covariances of the random process between the observation points t 1 , t 2 , . . . , t n :

K =                 k(t 1 , t 1 ) k(t 1 , t 2 ) . . . k(t 1 , t n ) k(t 2 , t 1 ) k(t 2 , t 2 ) . . . k(t 2 , t n ) . . . . . . . . . . . . k(t n , t 1 ) k(t n , t 2 ) . . . k(t n , t n )                 (8)
If one has some prior knowledge on the function to be modeled it can be reflected by giving a prior mean to the predictor. Since in our case no information is available but the observed function values, a constant (but unknown) mean E[ξ(t)] = C is assumed.

To simplify the notations let us collect the coefficients λ k (t) into a vector λ λ λ(t) = [λ 1 (t) λ 2 (t) . . . λ n (t)] T , and denote by k(t) the vector whose elements are the values of the covariance between t and the observation points:

k(t) = [k(t, t 1 ) k(t, t 2 ) . . . k(t, t n )] T .
It can be shown that the determination of λ λ λ(t) boils down to computing the solution of the linear system of equations (see, e.g., [START_REF] Villemonteix | An informational approach to the global optimization of expensive-to-evaluate functions[END_REF]):

             1 K : 1 1 . . . 1 0                           λ λ λ(t) µ(t)              =              k(t) 1              (9)
where µ(t) is the Lagrange multiplier, which corresponds to the enforcement of the unbiasedness condition.

Once the vector λ λ λ(t) has been computed, a predicted value of the similarity function Q(t) can be written as

Q(t) = n k=1 λ k (t)Q k . ( 10
)
It is easy to show that the function t → Q(t) interpolates Q(t) at observed points. An interesting property of kriging, is that an estimate of the uncertainty of the prediction is available via the kriging error, which can be written as σ2 (t) = k(t, t)λ λ λ(t) T k(t)µ(t).

(11) This feature will be essential in the adaptive sampling strategy of the Expected Improvement algorithm.

Covariance model

Before focusing on the optimization algorithm, we mention very briefly how the covariance function is chosen in practice (see also [START_REF] Villemonteix | An informational approach to the global optimization of expensive-to-evaluate functions[END_REF]).

First, a simplifying assumption is made -as usually done in geostatistics (see, e.g., [START_REF] Chiles | Geostatistics, Modeling Spatial Uncertainty[END_REF]) -namely that the random process is stationary. Then, the covariance function is a univariate function k(h) where h is a distance between two points t a , t b ∈ T. This distance needs not to be the classical Euclidean distance. In our case, the components of t ∈ T are of different kinds. Thus, it is reasonable to use some anisotropic distance, which may be written as

h = D d=1 t a,d -t b,d ρ d 2 . ( 12
)
where t a,d and t b,d are the d th components of the vectors t a and t b , respectively, and D the dimension of T. The parameters ρ d , d = 1, 2, . . . , D, represent the range of the covariance, or the typical correlation distance, in the direction of the d th component.

Second, a parameterized covariance function is chosen and its parameters are estimated using the data with a maximum likelihood method (see, e.g, [START_REF] Villemonteix | An informational approach to the global optimization of expensive-to-evaluate functions[END_REF]). We use the Matérn covariance function, which can be written as

k(h) = σ 2 2 ν-1 Γ(ν) 2 √ ν h ν K ν 2 √ ν h , (13)
where K ν is the modified Bessel function of the second kind of order ν. The parameter ν controls the regularity of the random process -the higher the ν, the more regular the process is. The parameter σ 2 is the variance of the process (k(0) = σ 2 ). The parameters ρ d in ( 12) are also estimated by maximum likelihood (the covariance function has indeed D + 2 parameters).

Expected Improvement

Since the similarity function Q (to be minimized) requires to compute the solution of a forward problem, the computational cost of evaluating Q is non-negligible. Moreover, for an inversion method to be useful in practice, the computational burden

mm

Step in the y dir. 0.5 mm of the method must not be too high. Thus, we wish to limit the number of evaluations of Q in the inversion procedure, which means that the optimization method used to minimize Q must be efficient.

The expected improvement (EI) algorithm is an iterative method to find the global minimizers of an expensive-to-evaluate function [START_REF] Jones | A taxonomy of global optimization methods based on response surfaces[END_REF]. The method is based on interpolation by kriging of the function to be optimized. Let us assume that Q has been evaluated at n points

Q 1 = Q(t 1 ), Q 2 = Q(t 2 ), . . . , Q n = Q(t n
). An iteration of the EI algorithm provides the location of the next evaluation.

First, an interpolation Q of Q is computed by kriging from the set of past evaluations, along with the variance of the kriging error σ2 (t). Denote the current minimal value by

Q min = min k=1,...,n Q k . Define the improvement over Q min at a point t ∈ T by I(t) = max(0, Q min -Q(t)
). However, Q(t) is unknown except at the n evaluation points. Since Q(t) is modeled by the Gaussian random process ξ, a natural idea is to express the expected value of I(t), which is called the expected improvement and appears to have a very convenient analytical form:

EI(t) ≡ E[I(t)] = σ(t) uΦ(u) + ϕ(u) , (
u = Q min -Q(t) σ(t) .
The next evaluation point is chosen according to the highest value of the expected improvement. Since ( 14) is straightforward to compute, the maximization of the EI over T is not a problem in practice. A natural stopping criterion is when the EI is smaller than a small positive number.

Note that the EI algorithm is known to be consistent, i.e. the algorithm converges to the location of the global minimizer (under some assumptions) [Locatelli, 1997, Vazquez and[START_REF] Vazquez | On the convergence of the expected improvement algorithm[END_REF]. In theory, the convergence rate is unknown but it appears in practice that this type of algorithm converges very rapidly, as illustrated in our numerical studies (see next section).

NUMERICAL EXAMPLES

In this section a couple of numerical examples are selected to illustrate the presented inversion method. The parameters of the ECT configuration are shown in Table 2 (see also Fig. 1).

The surface scan consists of 451 measured impedance valuesobserved at the nodes of a rectangular grid characterized by the parameters given in Table 2. The center of the surface scan is at the origin of the x-y coordinate system and the edges of the grid are parallel to the related coordinate axes. Since no experimental measurements are available at that time, the so-called "measured impedance variation" (∆Z k , k = 1, . . . , N in (4)) of the defects in the assumed test cases are obtained by numerical simulation.

The necessary initialization of our iterative process implies the evaluation of the similarity function at some well-chosen points of the parameters space. There is no "best" way to choose such initial points without complementary a priori informations, therefore it seems reasonable to spread them more or less uniformly on the parameter space. Too few points can give misleading information on the similarity function whereas too many points may be unnecessary (thus "uneconomical" in the sense of computation time). Two types of initialization are presented in the paper:

• Initialization "A": all the vertices of the 6D parameter hypercube (2 6 = 64 points) and the midpoint of the hypercube (1 point): Σ65 points; • Initialization "B": the same points as in Init. "A", and all the midpoints of the sides of the hypercube (2 • 6 = 12 points): Σ77 points.

The parameters of the cracks to be retrieved and the obtained results are reported in Table 3 andTable 4. The depths (D 1 , D 2 ) are given in percents of the plate thickness d. After the initialization (65 or 77 similarity function evaluations) a fixed number of iteration cycles were performed (120 in all the cases). In the column "It." the number of cycle, in which the final best function value (Q min ) was found, is presented; that minimum value (related to the obtained solution) is also reported in the tables.

Table 3 and Table 4 illustrate the efficiency of EI algorithm with which an "acceptable" solution -in the sense of a low similarity function -is obtained in a few number of iterations.

The performance of the algorithm is illustrated in Fig. 3. On the bottom diagrams in Fig. 3(a) and Fig. 3(b), the log 10 of the maximal expected improvement (i.e. over the total discrete parameter space) at the related iteration cycle is presented.

One can see that it is decreasing as the iterations are being performed. However, the decrease is not monotonous. As it was mentioned, in our experiments a fixed number of iterations were performed. However, it would be better to construct a stopping criterion which stops the iterations automatically right after finding the solution -to reduce the number of "unnecessary" function calls at the end. By now, no really good stopping rule has been developed. One can rely on the maximal EI value, Line scan along the x = 0 line. Solid line: real crack configuration, dashed line: retrieved configuration (case Initialization "B"). The measured data is almost the same in both cases, thus it is extremely difficult to make difference between the configurations.

stating that if it is lower than a small limit, the iterations can be stopped. But the choice of this limit is not obvious -as it is illustrated in Fig. 3. In Fig. 3(b), one can see that a small Q value (quite close to the "final" solution in iteration No. 109) was obtained in iteration No. 60. The parameter vector related to the observation in iteration No. 60 is t = [1.0, 0, [START_REF] Bowler | Eddy-current interaction with an ideal crack. I. The forward problem[END_REF]4,60,40] (in the same units as in Table 4). This parameter point is close to the real solution and to the reconstructed one (see Table 4) as well. However, the maximal EI value in iteration No. 60 is almost 100 times higher than the one in iteration No. 109.

Note that the two presented performances are related to the same physical problem, only the initialization of the inversion procedure differs. In spite of this, significant differences can be found between the behaviour of the algorithm in the two cases. Thus, one can see that the choice of initial points strongly influences the latter performance. As it was mentioned above, there is no general best choice of initial points, moreover, it may also depend on the problem to be solved.

It is also important to note that the similarity function must be complicated. This is illustrated in Fig. 5: the function Q(t) is probably quite flat around its global minimizer, or physically, different configurations can cause very similar measurable output signal.

The computations were performed on a PC with 16 Gb RAM and a 64 bits CPU at 3 GHz. The computation of the Green's function and of the incident field was made in advance, separately. One average iteration cycle of the optimization loop took approximately 106 sec. Almost all of that time is needed to perform the kriging prediction and EI computation, the computation of the objective function takes just a few seconds. (At first sight, one can think that the applied inversion method is more expensive-to-evaluate than the objective function itself -but note that by using this inversion method, at most some hundred function calls are needed, whereas the parameter space T consists of more than 60 thousand points!) 3 andTable 4). Top: current best objective function value (the minimum of all the past observations). The found minimum is marked by a circle. Bottom: log 10 of the maximal EI in the current cycle.

CONCLUSION

A method for ECT inversion was presented. The stochasticbased optimization algorithm was coupled with a surface integral-based forward solver -the developed inversion method seems to be efficient in the light of the presented test cases.

The kriging interpolation of the similarity function provides a cheap surrogate model of the electromagnetic "black-box". Since this "black-box" includes electromagnetic field computation tasks, it is usually expensive-to-evaluate. This is why the use of such surrogate models seems to be fruitful. This kind of surrogate modeling of electromagnetic phenomena has been a popular method for years and now seems to be having its "second honeymoon" [START_REF] Sykulski | New trends in optimization in electromagnetics[END_REF]. Several methods exist for the solution of electromagnetic optimization problems. The main novelty of this paper is the application of such an existing (and in other domains widely used) optimization method for the inverse problem of double-crack characterization in ECT.

A couple of numerical examples illustrate the performance of the developed inversion method. The most important conclusion to be drawn is that quite few number of similarity function evaluations were enough to find an acceptable solution -even if the similarity function is "badly behaved" from the point of view of optimization. It is also important to emphasize that the performance of the algorithm is strongly influenced by the choice of initial observation points -as it was pointed out via our numerical examples.

In this paper, only a brief introduction to the stochastic tools and the first results were presented -the research is far from being finished. The next step can be carrying out a precise comparison of the performances of the proposed EI method and other methods. It is also an interesting question how the parameterization of the defects influences the performance. The choice of the applied similarity function can take effect on the behaviour of the algorithm as well.

Fig. 1 .

 1 Fig. 1. Schematic drawing of the geometry of the investigated forward and inverse problems

  14) where Φ(•) is the normal cumulative distribution function, ϕ(•) is the normal density function, and u is defined by

Fig. 2 .

 2 Fig. 2. Magnitude of the measured impedances in case No. #4.Line scan along the x = 0 line. Solid line: real crack configuration, dashed line: retrieved configuration (case Initialization "B"). The measured data is almost the same in both cases, thus it is extremely difficult to make difference between the configurations.

Fig. 3 .

 3 Fig. 3. Typical performance of the algorithm. The curves are realted to Case No. #4, (see Table3 and Table 4). Top: current best objective function value (the minimum of all the past observations). The found minimum is marked by a circle. Bottom: log 10 of the maximal EI in the current cycle.

Table 1 .

 1 Discretization of the parameter axes.

	Parameter	A	B	L 1	L 2	D 1	D 2
	Minimum value (mm)	0.25	0	2	2	0.1250	0.1250
	Maximum value (mm)	1.25	4	8	8	0.8750	0.8750
	Step size (mm)	0.25	1	1	1	0.0125	0.0125
	No. of points	5	5	7	7	7	7

Table 2 .

 2 Parameters of the ECT configuration.

		Metal plate
	Thickness (d)	1.25 mm Conductivity (σ 0 )	10 6 S/m
		Probe coil
	Inner radius (r 1 )	0.6 mm	Outer radius (r 2 )	1.6 mm
	Height (l)	0.8 mm	Lift-off (h)	0.5 mm
	No. of turns	140	Frequency	150 kHz
		Surface scan
	Points in the x dir.	11	Points in the y dir. 41
	Step in the x dir.

Table 3 .

 3 Defect parameters, in case of Initialization "A". (exact/reconstructed)

	No.	A mm	B mm	L 1 mm	L 2 mm	D 1 %	D 2 %	It.	Q min 10 -3
	#1 0.50/0.50 3/3 3/3 5/5 40/40 60/60 57	0
	#2 0.75/0.75 1/1 4/4 4/3 30/40 50/50 112 21.6
	#3 0.75/0.75 2/3 7/5 3/3 50/50 30/50 95	28.6
	#4 1.00/0.75 0/0 6/7 6/3 60/60 40/40 60	8.56
	Table 4. Defect parameters, in case of Initialization
			"B". (exact/reconstructed)
	No.	A mm	B mm	L 1 mm	L 2 mm	D 1 %	D 2 %	It.	Q min 10 -3
	#1 0.50/0.50 3/3 3/3 5/5 40/40 60/60 81	0
	#2 0.75/0.75 1/1 4/2 4/4 30/40 50/50 118 6.98
	#3 0.75/0.75 2/2 7/7 3/3 50/50 30/30 79	0
	#4 1.00/0.75 0/0 6/6 6/5 60/60 40/40 109 4.73

All in all, in the light of the presented preliminary results, the authors surmise that the use of surrogate models in ECT inversion opens new perspectives and it is worthwhile to continue the research in this domain.