Céline ; Loubière

Caroline Sion

Natalia De Isla

Loïc Reppel

Emmanuel Guedon

Isabelle Chevalot

Eric Olmos

Olmos

Impact of the type of microcarrier and agitation modes on the expansion performances of mesenchymal stem cells derived from umbilical cord

Keywords: MSC culture, agitation, microcarrier Mesenchymal stem cells, Wharton's jelly, microcarrier, agitation, cell counting method

Medicine, National Institute of Health). This is mainly due to the promising 6 application of MSC in cell therapies with actions as trophic mediators and 7 displaying bioactive factors secretion [START_REF] Arnold | Mesenchymal stem cells as trophic 604 mediators[END_REF]. Their immuno-modulatory prop-8 erties also present a great interest for this medical purpose, allowing either 9 autologous and allogenic treatments. They are notably involved into treate-10 ments for type I and II Diabetes, cancers and heart failures in some others 11 [START_REF] Mimeault | Stem cells: a revolution in therapeu-606 tics: recent advances in stem cell biology and their therapeutic appli-607 cations in regenerative medicine and cancer therapies[END_REF].

12 MSC were firstly isolated from bone marrow by Friedenstein et al. and 13 identified as a population of anchorage cells with a fibroblast behavior and 14 the capacity to regenerate bone cells [START_REF] Friedenstein | The development of 610 fibroblast colonies in monolayer cultures of guinea-pig bone marrow and 611 spleen cells[END_REF]. Since then, other parts of the 15 body were identified as MSC sources, such as adipose-tissues [START_REF] Bénédicte | Im-613 munomodulatory effect of human adipose tissue-derived adult stem cells: 614 comparison with bone marrow mesenchymal stem cells Brit[END_REF], synovium 16 [START_REF] De | Multipotent mesenchymal stem cells from adult human synovial 618 membrane[END_REF], lung [START_REF] Mark | Stem cells of the 620 alveolar epithelium The[END_REF], muscles [START_REF] Jackson | Mesenchymal progenitor 622 cells derived from traumatized human muscle[END_REF], umbilical-cord blood [8] or umbilical-cord matrix 17 (Wharton's jelly) [START_REF] Hwai-Shi | Mesenchymal 628 stem cells in the Wharton's jelly of the human umbilical cord[END_REF]. In order to establish robust criteria to define MSC, ity into osteoblasts, adipocytes and chondrocytes [START_REF] Dominici | Minimal criteria for defin-631 ing multipotent mesenchymal stromal cells. The International Society 632 for Cellular Therapy position statement[END_REF]. [START_REF] Matthew | 676 Hydrodynamic effects on animal cells grown in microcarrier cultures 677[END_REF] These cells are nevertheless present in limited quantities in the body, [START_REF] Timothy | Mechanical stimuli differentially control stem cell behavior: 680 morphology, proliferation, and differentiation[END_REF] which decrease with ageing [START_REF] Alexandra | Age-related 634 changes in human bone marrow-derived mesenchymal stem cells: conse-635 quences for cell therapies Mech[END_REF]. In vitro expansions are thus necessary to 26 reach the high cell dose requirements for the medical purposes, between 10 9 27 and 10 12 cells/lot [START_REF] Rowley | Meeting lot-size challenges of manufacturing adherent cells 638 for therapy[END_REF]. To improve their performance, expansion processes 28 have been investigated from static planar cultures to stirred tank bioreactors, 29 taking into account both the quantity and the quality of the harvested cells 30 [START_REF] Katharina | hMSC production in dis-641 posable bioreactors with regards to GMP and PAT[END_REF][START_REF] Krishna | Bioprocessing strategies for the large-scale production of human 645 mesenchymal stem cells: a review Stem[END_REF]15].

31

Concerning the quality, MSC have the particularity to be the drug itself, 32 directly injected into the patient, meaning that, in addition to the process 33 itself (robust control and monitoring), all components involved in the culture 34 medium must be suitable for clinical applications. This led to the use of 35 chemically defined culture media displaying much higher costs [16], or to 36 the serum replacement by human blood-derived alternatives like the human 37 platelet lysate (hPL) [17]. The use of hPL has notably been validated for 38 MSC cultures, with higher expansion rates than with fetal bovine serum, 39 making it a good substitute at an affordable cost [18]. It also presents the Heparin (Sigma-Aldrich) was added to this basal medium for planar static 126 cultures at a concentration of 2 IU / mL. Concerning dynamic cultures on 127 microcarriers, first cultures led to the formation of a jelly matrix precipi-128 tate, entrapping microcarriers, even with 2 IU / mL heparin in the culture 129 medium. This gel appeared during the first days and led to high variability 130 of cell adhesion and expansion. The heparin was thus found to be not enough 131 effective to avoid gel formation. Hence, it was chosen to use a mechanical 132 fibrinogen-depletion instead of the increase of heparin concentration, which 133 may have deleterious effects on cell expansion at higher concentrations [39].

134 Similar fibrinogen-depletion methods have already been described by other 135 authors [START_REF] Sandra | Mechan-738 ical fibrinogen-depletion supports heparin-free mesenchymal stem cell 739 propagation in human platelet lysate[END_REF][START_REF] Pachler Karin | A Good Manufacturing 741 Practice-grade standard protocol for exclusively human mesenchymal 742 stromal cell-derived extracellular vesicles[END_REF]. For the present study, the culture medium was maintained ) concentration (g L -1

) concentration (g L -1 282

F exp = < N cell/µC > t 0 +5d -< N cell/µC > t 0 +2d < N cell/µC > t 0 +2d (1)
c v = σ < N cell/µC > • 100 (2)
σ = 1 n -1 n i=1 (N cell/µC (i) -< N cell/µC >) 2
(3)

Y Glu = ∆ 2→5 C consumption Glucose ∆ 2→5 < N cell/µC > •C µC/L (4)
Phenotypic analysis

283

After trypsinization, cells were frozen before the flow cytometry analyses.

284

They were then quickly thawed with fresh culture medium, pre-warmed at 387

Glucose consumption and lactate production kinetics showed that Cytodex-388 1, Star-Plus and Hillex II microcarriers led to higher cell expansions (Fig. 4).

389

These microcarriers presented better fractions of colonized particles, which Concerning Hillex II, cell adhesion was found to be very low with only 9

407

% of adhered cells. The percentage of colonized microcarriers x colo , repre-408 sented in Fig. 6, was found to be related to the number of cells, with higher 409

x colo obtained with cultures displaying more attached cells. Fig. 7 showed strate kinetics due to an early confluence.

432

The mean number of cells may also be related to the metabolite / substrate (Fig. 9 (E)), whereas the total number of cells was found to be the low- activity. It may be supposed that a fraction of viable cell remained in sus-450 pension, without attachment possibility on microcarriers. However, cells in 451 suspension would not be taken into account by the in situ counting method.

452

Cell distribution on microcarriers at t 0 + 2 days and t 0 + 5 days are presented 453 in Fig. 7. c v remained similar over time, except for Hillex II. In this case, 454 the deviation became significantly higher and may be related to the decrease 455 of the colonized microcarrier fraction (Fig. 6).

456

Interesting results were obtained with Lac/Glu ratios, presented in Table 4. different metabolic pathways, such as glutamine that may be a second source In addition to the low cell attachment observed on Hillex II, no cell ex- Cytodex-1 (Fig. 13), despite no significant difference was observed on the 488 mean number of cells per microcarriers on Cytodex-1, Star-Plus and Plastic-489 Plus at t 0 + 2 days and t 0 + 5 days. Hence, the metabolic activity, related to 490 viable cells, was higher for cultures carried out with these two microcarriers, 491 but with different metabolic pathways, involving different carbon sources for 492 cell expansion. As for cultures in Erlenmeyer flasks, Lac/Glu ratios, and 493 displaying values over 2, led to the hypothesis that glutamine was used by 494 cells and led to the production of more lactate (Table 5). Here, this assump-

Microcarriers performances comparison 562

Microcarriers were finally compared according to the three criteria con-563 cerning cell attachment, expansion and ease of detachment. Each agitation 564 mode led to the results summarized in This study demonstrated that the detachment step is still a challenge for 592 the development of a robust process, especially when Cytodex-1 were used.

593

Star-Plus and Plastic-Plus were found to be better compromises, respectively 594 for the orbital and the mechanical agitation modes.

Introduction 1 In

 1 recent years, Mesenchymal Stem / Stromal Cells (MSC) are increas-2 ingly used in regenerative medicine. According to the U.S. National In-3 stitute of Health, the number of completed or ongoing MSC-based studies 4 passed from around 500 in 2016 to 854 in July 2018 (US National Library of5

  for Cellular Therapy (ISCT) proposed three mini-19 malist conditions: (i) the capacity to adhere on plastic in standard culture 20 conditions, (ii) the expression or non-expression of specific surface antigens 21 (positive for CD105, CD73 and CD90; negative for CD45 CD34, CD14 or 22 CD11b, CD79a or CD19 and HLA-DR), and the in vitro differentiation abil-23

40 advantage 42 For

 4042 to be rich in proteins and growth factors, avoiding the need of 41 bFGF addition. the large-scale production of MSC, the technology of microcarriers, Wezel et al. in 1967, is today widely used to allow adherent-44 dependent cell cultures. They consist in micro-spheres from 90 to 400 µm 45 diameter with a density slightly higher than water [19]. Cells adhere on used was α-MEM Eagle solution (Lonza), supple-120 mented with 5 % human platelet lysate (MultiPL'30, Macopharma), 4 mM 121 glutamine (Sigma) and 1 % antibiotic-antimicotic (Gibco). According to the 122 manufacturer instructions, the hPL derived from screened normal human 123 donor platelets, was pooled in large batch sizes of 10 L and was sterilized by 124 filtration of 0.2 µm before filling in closed systems.

  125

136 between 4

 4 and 6 hours at room temperature to allow clot formation. It was 137 then stored at 4 • C at least 18 hours. Before fibrin gel removing, the medium 138 was brought to 37 • C and submitted to mechanical shocks by agitation. The 139 biogel was then physically collapsed and the precipitated fibrin was pelleted 140 by centrifugation (670 × g, 10 min, Jouan G 4.11). Finally, the medium was 141 passed through a 0.22 µm filter.

Five

  microcarriers were compared, including Cytodex-1 (GE Healthcare), 144 widely used for adherent dependent cell cultures, and the Pall Starter Kit 145 composed of Plastic, Plastic-Plus, Hillex II and Star-Plus microcarriers (Pall, 146 SoloHill). Microcarriers properties and culture concentrations are presented 147 in Table 1. The different concentrations of microcarriers were calculated in 148 order to display the same available surface for cell expansion, both in static 149 (A = 30 cm 2 ) and dynamic (A = 250 cm 2 ) modes . Preliminary cultures 150 showed that a coating with hPL enhanced cell attachment on microcarriers 151 (data not shown). Only Cytodex-1 presented similar cell attachment with 152 and without hPL coating. In the present work, all microcarriers were thus 153 coated with hPL by incubating them in pure hPL at 37 • C, 80 % humidity 154 and 5 % CO 2 during two hours. Then, microcarriers were washed with fresh 155 culture medium before cell seeding. 156 Cell culture under static conditions 157 After quick thawing, cells were seeded at 2000 cells / cm 2 in T175 flasks 158 filled with 25 mL culture medium. The medium was changed every 2 or 3 159 days by complete medium removing and addition of the same volume of fresh 160 medium. Once 60-70 % of the confluence was reached, the culture medium 161 was removed and cells were washed with DPBS before the cell detachment, 162 using 5 mL TrypLE (Gibco) per flask. TrypLE is composed of recombinant 163 cell-dissociation enzymes that replace trypsin. Incubation did not exceed 164 5 min and the enzymatic digestion was stopped by adding 20 mL of fresh 165 medium. The suspension was then centrifuged at 130 × g during 5 min, 166 the supernatant was removed and the pellet was dispersed in fresh culture 167 filtration step was performed with 40 µm filters, in order to elim-168 inate potential cell aggregates. 1 mL of the suspension was finally collected 169 to determine the cell number and cell viability. 170 Cell culture on microcarriers in static mode 171 Six well plates with ultra-low attachment surface (Corning) were used to 172 perform static cultures on microcarriers. After addition of the microcarriers 173 at the bottom of the well, cells were seeded at a concentration of 3 10 5 cells 174 / well in a working volume of 5 mL. After 5 days, the supernatant was 175 removed and microcarriers were washed with DPBS. Cell detachments were 176 finally performed by using 1 mL of TrypLE. A complementary agitation can 177 be applied to promote cell detachment and to reduce the incubation time, 178 as proposed by Nienow et al. [42]. An orbital agitation of 75 rpm (orbital 179 diameter d o = 10 mm) was applied during 5 min, and 5 mL of fresh medium 180 was then added to stop the enzymatic digestion. Cells were finally separated 181 from microcarriers with a 70 µm filtration step. 182 Cell culture on microcarriers in agitated mode 183 The following protocol was established from preliminary cultures per-184 formed using both Cytodex-1 and Star-Plus microcarriers, in order to deter-185 mine the suitable operating conditions for cell growth comparison on different 186 microcarriers. Two stirred culture systems were used: (i) Erlenmeyer flasks 187 and (ii) Spinner flasks. Properties of the two systems are presented in Table

  188

2 . 9 F

 29 Cells were recovered from planar cultures at around 70 % of the confluence 189 and were passed through a 40 µm filter to trap potential aggregates before 190 seeding. 70 % of the confluence was found to be a good compromise to recover 191

2 * 10 Page

 210 Cytodex-1 concentrations are given according to the particle dry weight. avoiding the formation of aggregates. After counting, cells 192 were inoculated at a concentration of 0.8 10 6 cells / mL, namely 2 10 6 cells 193 per flask. In Erlenmeyer flasks, a gentle agitation was applied to homoge-194 nize cell and microcarrier suspension, and cultures were let without agitation 195 during one hour to allow cell adhesion. Agitation rate was then fixed at the 196 minimal agitation rate ensuring all particle suspension N = N js = 75 rpm 197 (d o = 10 mm). In Spinner flasks, cultures performed at N = N js led to cell 198 lysis. It was thus decided to let cultures without agitation during one hour 199 with gentle agitations at the beginning and at the mid-point, before working 200 at a low agitation rate N = 40 rpm, at which a sufficient number of particles 201 were suspended without major hydromechanical stress [42]. Both cultures 202 performed in Erlenmeyer and spinner flasks were limited to 5 days, with 50 203 % medium exchange performed at t 0 + 3 d and t 0 + 4 d, in order to ensure a 204 sufficient glucose concentration for cell expansion and to reduce the lactate 205 concentration. At the end of the culture, the supernatant was removed and 206 particles were washed with DPBS. 10 mL of TrypLE were then added with a 207 complementary agitation during 5 min. Fresh medium was used to stop the 208 enzymatic digestion and cells were separated from microcarriers with a 70 209 µm filtration step. Cell number and viability were determined before being 210 frozen. Microcarriers were recovered from filters and microscopic analyses 211 were performed to determine the number of cells remaining attached. 212 Detached cells counting 213 After each detachment, cell number and cell viability were determined 214 from cell suspensions, using the Vi-Cell T M Cell Viability Analyzer Counter 215 (Beckman Counter), based on a Trypan Blue Dye Exclusion method.

232 3 261F 14 Page

 314 transparent and were found to allow blue light to pass through the material.233In this way, cells adhered in the whole bead surface were counted. On the 234 contrary, Plastic, Plastic-Plus and Star-Plus microcarriers were found opaque 235 and only cells present in half of the bead surface were counted. Furthermore, 236 a half-face found empty did not necessarily mean that there was no cell on 237 the other side. Consequently, in order to predict the number of cells on the 238 non-visible surface, a Poisson distribution was integrated in the routine. The 239 routine finally provided the following information: (i ) The mean number of 240 cell per microcarrier < N cell/µC >, (ii ) the distribution of the number of cell 241 per microcarrier, (iii ) the standard deviation of this distribution σ, and (iv ) 242 the percentage of colonized microcarriers x colo , namely the fraction of parti-243 cles containing at least one cell. This method based on image analysis was 244 preliminary validated by cell visual counting. 245 Cell morphology investigation 246 As for cell counting, samples of 200 µL were recovered from a homoge-247 neous particle suspension. After removing the supernatant, cells were stained 248 using a Live / Dead Cell double staining kit (Sigma) during 30 min at 37 • C. 249 It was composed of calcein-AM for living cell membrane staining, and ethid-250 ium homodimer for dead cell nuclei staining. Microscopic analyses were then 251 performed in order to qualitatively determine cell morphology. This oper-252 ation was performed at t 0 + 2 days in order to ensure cell attachment and 253 their final morphology, while no doubling population already occurred. The 254 objective was to distinguish round from spreaded cells. , lactate, glutamine, ammonia and lactate deshydro-257 genase (LDH) concentrations were measured from supernatant samples col-258 lected throughout the cultures, by using the Gallery multiparametric analyzer 259 (Thermo Fisher Scientific). Glucose (GOD-POD), L-Lactic Acid and LDH 260 (SCE) kits from Thermo Fisher Scientific and Glutamine V2 Bio and NH Bio kits from Roche were used. 262 Kinetic parameters calculation 263 Based on the previous experimental measurements, different parameters 264 were calculated to characterize cell culture kinetics. An expansion factor 265 exp , expressed in Eq. (1), used data from cell counting by image analysis 266 and compared the number of cell produced between t 0 + 2 days and t 0 + 5 267 days. Also based on image analysis results, the distribution of the number 268 of cell per microcarrier was determined and the corresponding coefficient of 269 variation c v was calculated according to Eq. (2), with σ the standard devia-270 tion presented in Eq. (3). In this last parameter, N cell/µC (i) corresponds to 271 the number of cell counted on the i th microcarrier and n represents the total 272 number of microcarriers treated by image analysis. Two parameters based 273 on metabolite concentrations were finally calculated. In one hand, Y Glu yield 274 was defined according to the Eq. (4), and described the concentration of 275 glucose consumed on the cell concentration production, from t 0 + 2 days to 276 t 0 +5 days, with ∆ 2→5 C consumption Glucose corresponding to the cumulative glucose con-277 sumption between t 0 + 2 days and t 0 + 5 days, ∆ 2→5 < N cell/µC > for the mean 278 number of cell produced between t 0 + 2 days and t 0 + 5 days, and C µC/L the 279 microcarrier concentration in the system. In the other hand, the Lac/Glu 280 as the cumulative lactate production on the cumulative 281 glucose consumption during the whole culture.

285 37 •

 37 . After centrifugation, cells were re-suspended in DPBS and their viability 286 was validated using a Live / Dead kit assay. Cells were then incubated into 287 DPBS-BSA 0.5 % (Gibco, Sigma-Aldrich) during 15 minutes and double 288 marks were realized using antibodies targeting the specific antigen coupled 289 with Phycoerythrin (PE) or Fluorescein isothiocyanate (FITC) during 45 290 minutes, at room temperature and out of direct light.

Fig. 1 .Figure 1 :Figure 2 :

 112 Fig. 1. Cell detection and counting were validated with 5 % mean deviation 344

390

  seemed to globally impact the cell expansion. On the contrary, Plastic mi-391 crocarriers were notably unsuitable for MSC culture in comparison to other 392 microcarriers regarding the low cell expansion after 5 days of culture and the 393 LDH concentration. LDH concentration was indeed higher for the culture 394 using Platic microcarriers than with others, despite a lower number of cells 395 observed on particles. This confirmed a lower cell viability in comparison to 396 other microcarriers. Only Cytodex-1, Star-Plus, Plastic-Plus and Hillex II 397 microcarriers were thus selected for the dynamic cultures. 398 Cell attachment and expansion comparison with an orbital agitation 399 Cultures in Erlenmeyer flasks were performed without agitation during 400 the first two hours. Then, a continuous orbital agitation was applied to en-401 sure particle suspension, but gathering most of particles inward. They led 402 to different cell attachments depending on the type of microcarrier used. 403 As presented in Fig. 5, Cytodex-1 presented the best capability with 59 404 % of inoculated cells adhered on microcarriers surfaces, whereas Star-Plus 405 and Plastic-Plus presented lower values with 20 % and 37 % respectively.

Figure 3 : 23 Page

 323 Figure 3: Cell nuclei DAPI coloration at t 0 + 5 days in static mode: Cytodex-1 (A), Star-Plus (B), Plastic (C), Plastic-Plus (D) and Hillex II (E) microcarriers.

Figure 4 :

 4 Figure 4: Kinetics of metabolite concentrations over time in static mode and according to the type of microcarrier used: Glucose (A), lactate (B), glutamine (C), ammonia (D) and LDH (E) (n = 3).

24 Page

 24

410 1 ,Figure 5 :

 15 Figure 5: Mean number of cells per microcarrier cultivated with an orbital agitation and according to the type of microcarrier at t 0 + 2 days (black bars) and t 0 + 5 days (white bars) (n = 2).

433 kinetics, presented in Fig. 9 .L - 1 , 26 FFigure 6 :

 91266 Figure 6: Fraction of microcarriers containing at least one cell in culture with an orbital agitation and according to the type of microcarrier at t 0 + 2 days (black bars) and t 0 + 5 days (white bars) (n = 2).

441est. 27 Page

 27 It may be thus assumed that cell viability declined during the first days 442 of the cultures. On the other hand, Cytodex-1 presented the highest Y Glu 443 yield of 2.2 10 -10 mol cell -1 in comparison to 1.4 10 -10 mol cell -1 calculated 444 with Star-Plus. Due to a higher number of cells, the LDH concentrations was 445 notably expected to be higher after 5 days of cultures (Fig.9 (E)). Star-Plus 446 presented a higher number of cells than on Plastic-Plus at t 0 +2 days and thus 447 led to a higher metabolic activity during the first days. However, at t 0 + 5 448 days, the mean number of cells on Plastic-Plus declined despite a metabolic 449

Figure 7 :

 7 Figure 7: Coefficient of variation of the distribution characterizing the number of cell per microcarrier with an orbital agitation and according the type of microcarrier at t 0 + 2 days (black bars) and t 0 + 5 days (white bars) (n = 2).

457Figure 8 :

 8 Figure 8: Cell morphology grown on microcarriers at t 0 + 2 days according the type of microcarrier and the agitation mode. Green staining for living cell membranes and red staining for dead cell nuclei.

469

  except a gentle agitation at mid-time to promote cell homogenization on mi-470 crocarriers. Particles were then dispersed by a mechanical agitation. In this 471 system, cell attachment was similar for Cytodex-1, Star-Plus and Plastic-472 Plus microcarriers with 3.6 to 3.7 cells per microcarrier at t 0 + 2 days, as 473 presented in Fig. 10, and cell displayed similar fibroblast-like morphologies, 474 as shown in Fig. 8. On the contrary, only 1.8 cells per microcarrier were 475 counted on Hillex II microcarriers, with 30 % empty microcarriers as pre-476 sented in Fig. 11. As a consequence, cells on Hillex II presented a more 477 heterogeneous distribution than on Cytodex-1 and Plastic-Plus, with c v = 478 105 %, whereas 68 to 80 % were obtained with the other microcarriers (Fig.

  479

30 Page

 30

Figure 9 :

 9 Figure 9: Kinetics of metabolite concentrations during cultures stirred with an orbital agitation and according to the type of microcarrier used: Glucose (A), lactate (B), glutamine (C), ammonia (D) and LDH (E). Vertical arrows correspond to an exchange of 50 % culture medium (n = 2).

31 PageFigure 10 :

 3110 Figure 10: Mean number of cells per microcarrier cultivated with a mechanical agitation and according to the type of microcarrier at t 0 + 2 days (black bars) and t 0 + 5 days (white bars) (n = 2).

481 pansion was noticed at t 0 + 5 32 PageFigure 11 :

 53211 Figure 11: Fraction of microcarriers containing at least one cell in culture with a mechanical agitation and according to the type of microcarrier at t 0 + 2 days (black bars) and t 0 + 5 days (white bars) (n = 2).

495 tion was notably supported by the results on glutamine consumption with 496 33 PageFigure 12 :

 3312 Figure 12: Coefficient of variation of the distribution characterizing the number of cell per microcarrier with a mechanical agitation and according to the type of microcarrier at t 0 + 2 days (black bars) and t 0 + 5 days (white bars) (n = 2).

Figure 13 :

 13 Figure 13: Kinetics of metabolite concentrations during cultures stirred with a mechanical agitation and according to the type of microcarrier used: Glucose (A), lactate (B), glutamine (C), ammonia (D) and LDH (E). Vertical arrows correpond to 50 % culture medium exchange (n = 2).

Figure 14 :

 14 Figure 14: DAPI cell nuclei staining after cell detachment and filtration: Cytodex-1 (A), Star-Plus (B), Plastic-Plus (C) and Hillex II (D).

Figure 15 :Figure 16 :

 1516 Figure 15: Characterization of GW-MSC immunophenotype, expanded under various dynamic conditions, by flow cytometry: Cells cultivated with an orbital agitation on Plastic-Plus (P+ orb.), Hillex II (H II orb.) and Cytodex-1 (Cyto orb.), and with a mechanical agitation on Cytodex-1 (Cyto mec.) and Star-Plus (S+ mec.).

Table 1 :

 1 Microcarrier physical properties (based on manufacturer data) and concentrations used in this study.

	Dynamic culture
	Static culture
	Diameter range Density	(µm) (g mL -1
	Manufacturer
	Name

Table 2 :

 2 Stirred systems and agitation conditions chosen for cultures in dynamic mode.

			Biotechnology Progress	Page 12 of 53
			Bioreactor
			Erlenmeyer flask	Spinner flask
		Manufacturer	Fisherbrand	DWK Life Sciences
			(Fisher Scientific)	Inc. (Wheaton)
		Mixing mode	Orbital	Mechanic
		Material	Plastic (single-use)	Siliconized glass
		Diameter (mm)	66	40
		V working (mL)	25	25
		N (rpm)	75	40
	217	In situ counting of cells adhered on microcarriers
	218	A dedicated post-treatment was developed to dispose of a robust and
	219	quantitative method to directly count MSC cells attached on microcarriers,
	220	without detachment step. To do that, samples of 200 µL were recovered from
	221	homogeneous particle suspensions. After supernatant removing for metabo-
	222	lite analysis, cells were stained using a DAPI-methanol solution of 1 µg / mL
	223	(DAPI from Sigma) during 15 min at room temperature. Cells were then
	224	washed with DPBS and microscopic analyses were performed. At least 5
	225	photos per operating condition were recorded and loaded in a Matlab script
	226	for post-treatment. First, microcarriers were identified by contour recog-
	227	nition and their coordinates were recorded. Secondly, based on the simple
	228	colour detection by the hue function of Matlab, a mask was applied in or-
	229	der to only keep the blue elements concerning cell nuclei after the DAPI
	230	staining. After contrast adjustments, a binarization was performed and cell
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Table 3 summarizes

 3

	293
	294	Gallios (Beckman Coulter, Brea, CA, USA). The number of positive cells
	295	was obtained using the Kaluza software (Beckman Coulter, Brea, CA, USA).
		15

291

the antibodies used in this study. Cells were finally washed in DPBS and 292 suspended in 300 µL DPBS, before the analysis using the flow cytometer

Table 3 :

 3 List of antibodies used for the flow cytometry analyses.

			Biotechnology Progress			Page 16 of 53
	Antigen Fluorophore Manufacturer Reference Species Isotype	Control isotype FITC DAKO X0927 Mouse IgG1 κ	Control isotype PE DAKO X0928 Mouse IgG1 κ	CD 90 FITC Beckman Coulter IM1838U Mouse IgG2a	CD 34 PE BD Pharmingen 555882 Mouse IgG1 κ	HLA-DR FITC Beckman Coulter IM0463U Mouse IgG2b	CD 73 PE BD Pharmingen 550257 Mouse IgG1 κ	CD 45 FITC DakoCytomation F0861 Mouse IgG1 κ	CD 166 PE Beckman Coulter A22361 Mouse IgG1	CD 44 FITC Beckman Coulter IM1219U Mouse IgG1
					16
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301 medium was replaced with pre-warmed Complete Adipogenesis Differentia-302 tion Medium and re-feed every 3 days. After at least 40 days of culture, cells 303 were washed with DPBS and fixed in a 4 % formaldehyde solution during 304 30 min. Thereafter, cells were twice washed with distilled water and incu-305 bated with Oil Red-O solution (0.5 % in isopropanol) at room temperature 306 for 15 min. Cells were finally rinsed with distilled water before microscopic 307 visualization. 308 Osteogenesis differentiation 309 As recommended by the manufacturer, cells were seeded at 5,000 cells 310 / cm 2 on 24-well plates with 1 mL of culture medium. After 24 h, culture 311 medium was replaced with pre-warmed Complete Osteogenesis Differentia-312 tion Medium and re-feed every 3 days. After at least 40 days of culture, cells 313 were washed with DPBS and fixed in a 4 % formaldehyde solution during 314 30 min. Thereafter, cells were twice washed with distilled water and stained 315 with 2 % Alizirin Red S solution at room temperature for 2-3 min. Cells 316 were finally rinsed with distilled water before microscopic visualization.

317 17 324 solution prepared in 0.1 HCl at room temperature for 30 min. Cells were 325 finally rinsed three times with 0.1 HCl and distilled water before microscopic 326 visualization. 327 Data statistical analysis 328 A one-way analysis of variance (ANOVA) test was used with a Fisher's 329 Least Significant Difference (LSD) test admitting significance at p-value < 330 0.05 to determine if the different groups were significantly different. For 331 static cultures, standard deviations were calculated according to data from 332 cultures performed in parallel (n=3). Concerning dynamic cultures, standard 333 deviations were calculated according to data of different cultures ( n=2). 334 Higher variability may be thus expected than with cultures performed in the 335 same time, with the same inoculum.

Table 4 :

 4 Kinetic parameters calculated for cultures involving an orbital agitation and according to the type of microcarrier used (n = 2).

	Page 29 of 53	Biotechnology Progress
	459
	460	of carbon and energy. In static cultures performed without glutamine, cells
	461	exhibited round morphologies and finally detached after a few days, without
	462	proliferation (data not shown). It was thus observed that glutamine was nec-
	463	essary for cell expansion. Hence, glutamine may be used by cells, even in low
		29
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467 468

Cell cultures in spinner flasks started with one hour without agitation,

Table 5 :

 5 Kinetic parameters calculated for cultures involving a mechanical agitation and according to the type of microcarrier used (n = 2).

			Biotechnology Progress	Page 36 of 53
		Microcarrier F exp	Y Glu	Lac/Glu
			(-) × 10 -10 (mol cell -1 ) (g gl -1 )
		Cytodex-1*	0.60	0.4	2.2
		Star-Plus	0.14	1.7	2.1
		Plastic-Plus 0.15	3.7	2.2
		Hillex II	-0.10	N/A	1.9
	506	on microcarrier performances. Static cultures of cells attached on micro-
	507	carriers were found not representative of what could occur during dynamic
	508	cultures, in particular concerning Hillex II microcarriers. In the case of the
	509	orbital agitation, cells presented the best adhesion capability on Cytodex-
	510	1, the highest expansion rate on Star-Plus, and Plastic-Plus and Hillex II
	511	were found unsuitable for MSC cultures in Erlenmeyer flasks. By using a
	512	mechanical agitation, similar number of cells were obtained with Cytodex-1,
	513	Star-Plus and Plastic-Plus, but cell expansion seemed to involve different
	514	carbon sources with a higher glutamine consumption for cells on Cytodex-1
	515	than with others. Hillex II was still found unsuitable with this agitation
	516	mode. As a result, microcarrier selection in static mode for applications in
	517	dynamic mode may be questionable, considering that performances in this
	518	condition could not be necessarily transposed to agitated cultures.
	519	Cell detachment from microcarriers
	520	The last criterion to compare microcarriers adequacy for MSC cultures
	521	is the ease of detachment. Results of the microscopic analysis performed on
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534

Table 6 :

 6 Cell number counted after detachment according to the type of microcarrier (× 10 5 , n = 2).

	Page 37 of 53	Biotechnology Progress
	Microcarrier	Static	Orbital agitation Mechanical agitation
	Cytodex-1	1.3 ± 0.2	3.4 ± 1.8	0.5 ± 0.1
	Star-Plus	1.9 ± 0.9	2.8 ± 0.6	1.2 ± 1.0
	Plastic	1.1 ± 0.3	N/A	N/A
	Plastic-Plus 1.3 ± 0.5	3.5 ± 0.9	2.0 ± 1.3
	Hillex II	1.7 ± 1.1	0.5 ± 0.3	0.6 ± 0.6
			37
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Table 7 :

 7 Cell viability after detachment according to the type of microcarrier (n = 2).

	Page 39 of 53		Biotechnology Progress
		Microcarrier	Static	Orbital agitation Mechanical agitation
		Cytodex-1	82 % ± 7	83 % ± 1	73 % ± 4
		Star-Plus	81 % ± 10	95 % ± 0	87 % ± 5
		Plastic	78 % ± 5	N/A	N/A
		Plastic-Plus 83 % ± 16	94 % ± 1	84 % ± 8
		Hillex II	82 % ± 7	61 % ± 17	62 % ± 13
	539	Plus, Star-Plus and Hillex II) and the two agitation modes (orbital and me-
	540	chanical) studied. Fig. 15 presented the percentage of expression obtained
	541	from MSC-specific cell surface markers. In all cases, less than 5 % expressed
		HLA-DR, CD45 and CD34. Concerning CD44, CD166, CD90 and CD73,
	544
	545	agitation, where CD90 markers were respectively expressed by 78 % and 75
	546	% of the cell populations. Even that the role of CD90 in MSC remains un-
	547	known, its expression has been related to the maintain of the undifferentiated
	548	state [46] and the immunopressive properties of MSC [47]. Moreover, it has
	549	been reported a decrease on CD90 expression on MSC submitted to mechan-
	550	ical stimulation [48]. Consequently, the decrease on CD90 expression found
	551	in this study could indicate a shift in the stemness state of MSC towards a
	552	state more susceptible to differentiation. Nevertheless, the percent of CD90
	553	expression remains higher than 75 %. All these values allow to validate the
	554	cell immonuphenotype for stem cell characterization.
				39
				John Wiley & Sons

542

more than 90 % cells expressed these markers, except in the cases of cultures 543 on Cytodex-1 with an orbital agitation and on Star-Plus with a mechanical

Table 8 .

 8 In the present study, the

	565
	566	growth performance comparison was done relatively to the different micro-
	567	carriers used, for each agitation mode. In static mode, Cytodex-1, Star-Plus,
		41

574

In dynamic mode using mechanical agitation, Cytodex-1, and Plastic-Plus 575 presented similar attachment and expansion performances, but the detach-576 ment step was still difficult for Cytodex-1, making Plastic-Plus more reliable.

577

In both dynamic cases, Hillex II presented the lowest attachment and expan-578 sion performances, despite promising results in static mode. 579 Conclusion 580 Microcarriers were compared for three different agitation modes, namely 581 static mode, orbital and mechanical agitation. Three culture steps were 582 investigated, involving the attachment efficiency, the cell proliferation and 583

the ease of detachment. It was assumed that all these three criteria were 584 equally important for the cultivation process improvement. Results showed

Table 8 :

 8 Microcarriers performances summary according to the agitation mode.

  Bieback Karen, Kern Susanne, Klüter Harald, Eichler Hermann. Critical 625 parameters for the isolation of mesenchymal stem cells from umbilical 626 cord blood Stem Cells. 2004;22:625-634. [15] Martin Céline, Olmos Éric, Collignon Marie-Laure, et al. Revisiting 647 MSC expansion from critical quality attributes to critical culture pro-Santos Francisco dos, Andrade Pedro Z, Abecasis Manuel M, et al. To-650 ward a clinical-grade expansion of mesenchymal stem cells from human 651 sources: a microcarrier-based culture system under xeno-free conditions Ma Teng, Tsai Ang-Chen, Liu Yijun. Biomanufacturing of human mes-666 enchymal stem cells in cell therapy: Influence of microenvironment on 667 scalable expansion in bioreactors Biochem. Eng. J. 2016;108:44-50. [28] Bongso Ariff, Fong Chui-Yee. The therapeutic potential, challenges and 691 future clinical directions of stem cells from the Wharton jelly of the 692 human umbilical cord Stem Cell Rev. Rep. 2013;9:226-240. [34] Smith J Robert, Pfeifer Kyle, Petry Florian, Powell Natalie, Delzeit Jen-713 nifer, Weiss Mark L. Standardizing umbilical cord mesenchymal stromal 714 cells for translation to clinical use: selection of GMP-compliant medium 715 and a simplified isolation method Stem Cells Int. 2016;2016. ner Wolfgang. Heparin concentration is critical for cell culture with hu-in a serum-free batch culture process Biotechnol. Bioeng. 2010;107:143-

			Biotechnology Progress	Page 48 of 53
	Nomenclature cess parameters Process Biochem. 2017;59:231-243. [8] 627 736 648 man platelet lysate Cytotherapy. 2013;15:1174-1181.
	595 716
	717	[35] Mizukami Amanda, Fernandes-Platzgummer Ana, Carmelo Joana G, et
		al. Stirred tank bioreactor culture combined with serum-/xenogeneic-
	720	1059.	F o r F o r F o r F o r F o r F o r F o r F o r F o r
			P e P e P e P e P e P e P e P e P e
			e r e r e r e r e r e r e r e r e r
			R e R e R e R e R e R e R e R e R e
			v v v v v v v v v
			i e i e i e i e i e i e i e i e i e
			w w w w w w w w w
	668
			48
			John Wiley & Sons

649 [16] 652 Tissue Eng. Part C-Me. 2011;17:1201-1210. 653 [17] Tekkatte Chandana, Gunasingh Gency Ponrose, Cherian KM, Sankara-654 narayanan Kavitha. Humanized stem cell culture techniques: the animal 655 serum controversy Stem Cells Int. 2011;2011. 656 [18] Azouna Nesrine Ben, Jenhani Faouzi, Regaya Zohra, et al. Phenotypical 657 and functional characteristics of mesenchymal stem cells from bone mar-658 row: comparison of culture using different media supplemented with hu-659 man platelet lysate or fetal bovine serum Stem Cell Res. Ther. 2012;3:6. 660 [19] Van Wezel AL. Growth of cell-strains and primary cells on micro-carriers 661 in homogeneous culture Nature. 1967;216:64. 662 [20] Simaria Ana S, Hassan Sally, Varadaraju Hemanthram, et al. Allogeneic 663 cell therapy bioprocess economics and optimization: Single-use cell ex-664 pansion technologies Biotechnol. Bioeng. 2014;111:69-83. 665 [21] 718 free culture medium enables an efficient expansion of umbilical cord-719 derived mesenchymal stem/stromal cells Biotechnol. J. 2016;11:1048-759 153.

Acknowledgments

The authors would like to thank the French Agence Nationale de la 597 Recherche (ANR) and the Interreg VA Grande Région program for their fi-598 nancial support. They also acknowledge Fabrice Blanchard (LRGP, Nancy) 599 for his technical contribution to this work.

John Wiley & Sons

Biotechnology Progress

Abbreviations

FBS


Van Duong Dinh

NON-RADIAL FINITE TIME BLOW-UP FOR THE FOURTH-ORDER NONLINEAR SCHR ÖDINGER EQUATIONS

Keywords: 2010 Mathematics Subject Classification. 35Q55, 35B44 . Fourth-order nonlinear Schrödinger equation, Blow-up

We revisit the finite time blow-up for the fourth-order Schrödinger equation with focusing inhomogeneous nonlinearity -|x| -2 |u| 4 n u. By exploiting localized virial estimates and spatial decay of the nonlinearity, we prove the finite time blow-up of non-radial solutions with negative energy. Our result is the first one dealing with the existence of non-radial blow-up solutions to the fourth-order Schrödinger equations.

Introduction

We consider the Cauchy problem for fourth-order nonlinear Schrödinger equations with focusing inhomogeneous nonlinearity

i∂ t u -µ∆ 2 u + ∆u = -|x| -2 |u| 4 n u, (t, x) ∈ R + × R n , u| t=0 = u 0 ∈ H 2 (R n ), (1.1)
where n ≥ 3, u : R + × R n → C, u 0 : R n → C, and µ > 0. Equation (1.1) belongs to a class of general fourth-order nonlinear Schrödinger equations, namely

i∂ t u -µ∆ 2 u + ∆u = F (x, u)u, (t, x) ∈ R + × R n (1.2)
which was introduced by Karpman [START_REF] Baruch | Singular solutions of the biharmonic nonlinear Schrödinger equation[END_REF] and Karpman-Shagalov [START_REF] Bergé | Soliton stability versus collapse[END_REF] in order to take into account the role of small fourth-order dispersion terms in the propagation of intense laser beams in a bulk medium with Kerr nonlinearity. The function F (x, u) can be understood as a nonlinear potential affected by electron density (see e.g., [4]).

The local well-posedness for (1.1) was established by Cho-Ozawa-Wang [8] using a regularizing argument and Strichartz estimates. More precisely, there exist T * > 0 and a unique solution (1.1). The maximal time of existence satisfies the blow-up alternative: either T * = ∞ or if T * < ∞, then lim t T * u(t) H 2 = ∞. In addition, there are conservation laws of mass and energy

u ∈ C([0, T * ), H 2 (R n )) ∩ L 2 loc ([0, T * ), W 3, 2n n-2 (R n )) to
M (u(t)) = ˆ|u(t, x)| 2 dx = M (u 0 ) (Mass) and E(u(t)) = 1 2 ˆ|∇u(t, x)| 2 + µ|∆u(t, x)| 2 dx - n 2n + 4 ˆ|x| -2 |u(t, x)| 2+ 4 n dx = E(u 0 ). (Energy)
In the case of no harmonic term (∆u), equation (1.1) has a scaling invariance

u λ (t, x) = λ n 2 u(λ 4 t, λx), λ > 0.
A direct computation shows that this scaling leaves the L 2 -norm of initial data invariant. Thus due to the mass conservation, equation (1.1) (without the ∆u term) is usually called mass-critical.

After the pioneering work [8], the inhomogeneous biharmonic nonlinear Schrödinger equation has attracted considerable interest in the mathematical community (see e.g., [7,[START_REF] Baruch | Ring-type singular solutions of the biharmonic nonlinear Schrödinger equation[END_REF][START_REF] Boulenger | Blowup for biharmonic NLS[END_REF] and references therein).

The main purpose of this paper is to study the finite time blow-up for (1.1). Before stating our result, let us recall previous works related to the finite time blow-up for (1.1) and more generally for (1.2).

For the pure power-type nonlinearity, i.e., F (x, u) = |u| p-1 , the first blow-up results was given in a series of works of Baruch-Fibich-Mandelbaum [1-3], where they gave numerical simulations showing the existence of finite time blow-up solutions. In [5], Boulenger-Lenzmann provided rigorous analytical proofs for the existence of blow-up solutions. In particular, they proved that radial H 2 -solutions with negative energy blows up in finite time in both mass-critical and mass-supercritical regimes. In [START_REF] Bonheure | Strong instability of ground states to a fourth order Schrödinger equation[END_REF]10], some finite time blow-up results were proved in the mass-supercritical regime for radial H 2 -solutions with non-negative energy. Still in the mass-supercritical regime, the author [START_REF] Dinh | On blowup solutions to the focusing intercritical nonlinear fourth-order Schrödinger equation[END_REF] showed the existence of finite time blow-up solutions with radial Ḣ2 ∩ Ḣγc -data having negative energy, where γ c is the critical regularity exponent.

We now turn to the case of focusing inhomogeneous nonlinearity (1.1) which is the main focus of this paper. In this case, we are only aware of the following blow-up result due to Cho-Ozawa-Wang [8].

Theorem 1.1 ( [8]). Let µ > 0 and u 0 ∈ H 2 (R n ) satisfy E(u 0 ) < 0. If one of the following conditions holds:

• n ≥ 7 and |x|u 0 ∈ L 2 ;

• n ≥ 5 and u 0 is radial; then the corresponding solution to (1.1) blows up in finite time, i.e., T * < ∞.

The proof of Theorem 1.1 is based on a virial argument in the same spirit of Glassey's work [11] on finite time blow-up for the standard nonlinear Schrödinger equation. However, due to the appearance of a biharmonic operator, a careful analysis is needed to derive a virial inequality which guarantees the finite time blow-up. More precisely, they proved that for sufficiently smooth solutions (e.g., u ∈ C([0, T * ), H 3 (R n ))), the virial quantity

V(u(t)) := (1 -2µ∆) -1/2 xu(t), (1 -2µ∆) -1/2 xu(t) satisfies V(u(t)) ≤ 16t 2 E(u 0 ) + 4t Im u 0 , x • ∇u 0 + C(M (u 0 )) 1+2/n + V(u 0 ), ∀t ∈ [0, T * ).
This inequality yields the finite time blow-up, i.e., T * < ∞ provided that E(u 0 ) < 0. As the above inequality only holds for sufficiently smooth solutions and it is hard to expect such regularity for solutions of (1.1) with H 2 -initial data due to the singularity and lower power of nonlinearity, the authors implemented an approximation with regularized solutions to overcome the difficulty.

Note that in the above-mentioned works, the radial symmetry (or finite variance) plays a crucial role. To the best of our knowledge, there are no results concerning the finite time blow-up for general (non-radial and infinite variance) H 2 -solutions to (1.2) as well as (1.1) available in the literature. The main purpose of this note is to prove the following result.

Theorem 1.2. Let n ≥ 3, µ > 0, and u 0 ∈ H 2 (R n ) satisfy E(u 0 ) < 0. Then the corresponding solution to (1.1) blows up in finite time.

Theorem 1.2 improves the one in [8] in several directions. First, we prove the existence of finite time blow-up solutions without radial symmetry and finite variance. Second, we show the finite time blow-up for all dimensions n ≥ 3, where the local well-posedness for (1.1) is available. Our result is the first one dealing with the finite time blow-up for fourth-order nonlinear Schrödinger equations with general data.

The proof of Theorem 1.2 relies on a localized virial identity and a spatial decaying property of the nonlinearity. First, we derive a virial identity related to (1.1) using an idea of [5]. Second, by choosing a suitable cutoff function and using the spatial decay of nonlinearity, we carefully estimate each terms of the virial identity. Finally, an ODE argument allows us to show the finite blow-up.

This note is organized as follows. In Section 2, we derive a localized virial identity related to (1.1). Section 3 is devoted to the proof of the finite time blow-up given in Theorem 1.2.

Localized virial identity

In this section, we derive a localized virial identity related to (1.1) which plays a key element in our proof. To this end, we introduce

φ(r) = ˆr 0 ζ(s)ds with ζ(r) :=        2r if 0 ≤ r ≤ 1, 2r -2(r -1) 6 if 1 < r ≤ 1 + 1/ 5 √ 6, smooth and ζ (r) < 0 if 1 + 1/ 5 √ 6 < r < 10, 0 if r ≥ 10. Let R > 0 and define the radial function φ R (x) = φ R (r) := R 2 φ(r/R), r = |x|.
(2.1)

We collect some properties of φ R in the following lemma.

Lemma 2.1. We have

∇ j φ R L ∞ R 2-j , 0 ≤ j ≤ 6, (2.2) and supp(∇ j φ R ) ⊂ {|x| ≤ 10R} if j = 1, 2, {R ≤ |x| ≤ 10R} if 3 ≤ j ≤ 6,
(2.3)

and φ R (r) r ≤ 2, φ R (r) ≤ 2, ∀r ≥ 0.
(2.4)

In addition, we have

φ R (r) r -φ R (r) ≥ 0, ∀r ≥ 0.
(2.5)

Proof. The properties (2.2)-(2.4) follow directly from the choice of ζ. Let us check (2.5). For 0 ≤ r/R ≤ 1, (2.5) holds trivially as

φ R (r) r = φ R (r) = 2. For 1 < r/R ≤ 1 + 1/ 5 √ 6, we have φ R (r) r = ζ(r/R) r/R = 2 -2 (r/R -1) 6 r/R and φ R (r) = ζ (r/R) = 2 -12(r/R -1) 5 . Thus φ R (r) r -φ R (r) = 12(r/R -1) 5 -2 (r/R -1) 6 r/R = 2(r/R -1) 5 6 - r/R -1 r/R = 2(r/R -1) 5 5r/R + 1 r/R ≥ 0. For r/R > 1 + 1/ 5 √ 6, we have φ R (r) r = ζ(r/R) r/R ≥ 0, φ R (r) = ζ (r/R) ≤ 0
which shows (2.5).

Let u : [0, T * ) × R n → C be a H 2 -solution to (1.1). We define the virial quantity

M φ R (t) := 2 Im ˆ∇φ R • ∇u(t)u(t)dx. (2.6)
Lemma 2.2 (Localized virial identity). Let n ≥ 3 and µ > 0. Then we have for all t ∈ [0, T * ),

d dt M φ R (t) = µ ˆ∆3 φ R |u(t)| 2 dx -2µ ˆ∆2 φ R |∇u(t)| 2 dx + 8µ k,l,m ˆ∂2 lm φ R ∂ 2 kl u(t)∂ 2 mk u(t)dx -4µ k,l ˆ∂2 kl ∆φ R ∂ k u(t)∂ l u(t)dx -ˆ∆2 φ R |u(t)| 2 dx + 4 k,l ˆ∂2 kl φ R ∂ k u(t)∂ l u(t)dx - 4 n + 2 ˆ∆φ R |x| -2 |u(t)| 2+ 4 n dx + 2n n + 2 ˆ∇φ R • ∇(|x| -2 )|u(t)| 2+ 4 n dx.
(2.7)

Proof. We follow an idea of [5, Lemma 3.1]. We only provide formal computations, the desired identity follows from an approximation argument (see [5, Lemma 3.1] for more details). We rewrite M φ R (t) as

M φ R (t) = u(t), Γ φ R u(t) , where Γ φ R := -i(∇φ R • ∇ + ∇ • ∇φ R ). Recall that if u solves the equation i∂ t u = Hu, then d dt u, Au = u, [H, iA]u ,
where [H, A] = HA -AH is the commutator operator. Thanks to this identity, we have

d dt M φ R (t) = u(t), µ[∆ 2 , iΓ φ R ]u(t) -u(t), [∆, iΓ φ R ]u(t) -u(t), [|x| -2 |u| 4 n , iΓ φ R ]u(t) =: (I) -(II) -(III).
We have from [5, (3.9)] that

[∆ 2 , iΓ φ R ] = 8 k,l,m ∂ 2 kl (∂ 2 lm φ R )∂ 2 mk + 4 k,l ∂ k (∂ 2 kl ∆φ R )∂ l + 2 k,l ∂ k (∆ 2 φ R )∂ l + ∆ 3 φ R which implies (I) = 8µ k,l,m ˆ∂2 lm φ R ∂ 2 kl u(t)∂ 2 mk u(t)dx -4µ k,l ˆ∂2 kl ∆φ R ∂ k u(t)∂ l u(t)dx -2µ ˆ∆2 φ R |∇u(t)| 2 dx + µ ˆ∆3 φ R |u(t)| 2 dx.
We also have (see [5, (3.8

)]) that [∆, iΓ φ R ] = 4 k,l ∂ k (∂ 2 kl φ R )∂ l + ∆ 2 φ R which gives (II) = -4 k,l ˆ∂2 kl φ R ∂ k u(t)∂ l u(t)dx + ˆ∆2 φ R |u(t)| 2 dx.
Finally, we have

[|x| -2 |u(t)| 4 n , iΓ φ R ] = [|x| -2 |u(t)| 4 n , ∇φ R • ∇ + ∇ • ∇φ R ] = -2∇φ R • ∇(|x| -2 |u(t)| 4 n ) which yields (III) = -2 ˆ∇φ R • ∇(|x| -2 |u(t)| 4 n )|u(t)| 2 dx.
Using the identity

∇(|x| -2 |u| 2+ 4 n ) = n + 2 2 ∇(|x| -2 |u| 4 n )|u| 2 - n 2 ∇(|x| -2 )|u| 2+ 4 n ,
we get

∇(|x| -2 |u| 4 n )|u| 2 = 2 n + 2 ∇(|x| -2 |u| 2+ 4 n ) + n n + 2 ∇(|x| -2 )|u| 2+ 4 n .
Thus we obtain

(III) = 4 n + 2 ˆ∆φ R |x| -2 |u(t)| 2+ 4 n dx - 2n n + 2 ˆ∇φ R • ∇(|x| -2 )|u(t)| 2+ 4 n dx.
Collecting the above identities, we prove the result.

Lemma 2.3. Let n ≥ 3 and µ > 0. Then we have for all t ∈ [0, T * ),

d dt M φ R (t) = µ ˆ∆3 φ R |u(t)| 2 dx -2µ ˆ∆2 φ R |∇u(t)| 2 dx -4µ k,l ˆ∂2 kl ∆φ R ∂ k u(t)∂ l u(t)dx -ˆ∆2 φ R |u(t)| 2 dx + 8µ ˆφ R r k |∇∂ k u(t)| 2 + φ R r 2 - φ R r 3 k |x • ∇∂ k u(t)| 2 dx + 4 ˆφ R r |∇u(t)| 2 + φ R r 2 - φ R r 3 |x • ∇u(t)| 2 dx - 4 n + 2 ˆ φ R + (2n -1) φ R r |x| -2 |u(t)| 2+ 4 n dx.
(2.8)

Proof. As φ R is radial, we use the fact that

∂ 2 kl φ R = δ kl - x k x l r 2 φ R r + x k x l r 2 φ R to have k,l ∂ 2 kl φ R ∂ k u(t)∂ l u(t) = k,l φ R r δ kl ∂ k u(t)∂ l u(t) + k,l φ R r 2 - φ R r 3 x k x l ∂ k u(t)∂ l u(t) = φ R r |∇u(t)| 2 + φ R r 2 - φ R r 3 |x • ∇u(t)| 2 .
We also have

k,l,m ∂ 2 lm φ R ∂ 2 kl u(t)∂ 2 mk u(t) = k,l,m φ R r δ lm ∂ 2 kl u(t)∂ 2 mk u(t) + k,l,m φ R r 2 - φ R r 3 x l x m ∂ 2 kl u(t)∂ 2 mk u(t) = φ R r k |∇∂ k u(t)| 2 + φ R r 2 - φ R r 3 k |x • ∇∂ k u(t)| 2 . Furthermore, as ∆φ R = φ R + (n -1) φ R r , we have ∆φ R + n 2 ∇φ R • ∇(|x| -2 )|x| 2 = φ R + (2n -1) φ R r .
From (2.7) and the above identities, we obtain (2.8).

Finite time blow-up

We are now able to show our main result given in Theorem 1.2.

Proof of Theorem 1.2. The proof is divided into two steps.

Step 1. A localized virial estimate. We first show that for all t ∈ [0, T * ),

d dt M φ R (t) = 32E(u(t)) -8 ∇u(t) 2 L 2 + CR -2 + CR -2 ∇u(t) 2 L 2 (3.1)
for some constant C > 0 depending only on µ and M (u 0 ). Thanks to (2.8), we have for all t ∈ [0, T * ),

d dt M φ R (t) = 16µ ∆u(t) 2 L 2 + 8 ∇u(t) 2 L 2 - 16n n + 2 ˆ|x| -2 |u(t)| 2+ 4 n dx -8µ ˆ 2 - φ R r k |∇∂ k u(t)| 2 dx + 8µ ˆ φ R r 2 - φ R r 3 k |x • ∇∂ k u(t)| 2 dx -4 ˆ 2 - φ R r |∇u(t)| 2 dx + 4 ˆ φ R r 2 - φ R r 3 |x • ∇u(t)| 2 dx + 4 n + 2 ˆ (2 -φ R ) + (2n -1) 2 - φ R r |x| -2 |u(t)| 2+ 4 n dx + µ ˆ∆3 φ R |u(t)| 2 dx -2µ ˆ∆2 φ R |∇u(t)| 2 dx -4µ k,l ˆ∂2 kl ∆φ R ∂ k u(t)∂ l u(t)dx -ˆ∆2 φ R |u(t)| 2 dx = 32E(u(t)) -8 ∇u(t) 2 L 2 + E 1 (u(t)) + E 2 (u(t)) + E 3 (u(t)) + E 4 (u(t))
, where

E 1 (u(t)) = -8µ ˆ 2 - φ R r k |∇∂ k u(t)| 2 dx + 8µ ˆ φ R r 2 - φ R r 3 k |x • ∇∂ k u(t)| 2 dx -4 ˆ 2 - φ R r |∇u(t)| 2 dx + 4 ˆ φ R r 2 - φ R r 3 |x • ∇u(t)| 2 dx and E 2 (u(t)) = 4 n + 2 ˆ (2 -φ R ) + (2n -1) 2 - φ R r |x| -2 |u(t)| 2+ 4 n dx, E 3 (u(t)) = -2µ ˆ∆2 φ R |∇u(t)| 2 dx -4µ k,l ˆ∂2 kl ∆φ R ∂ k u(t)∂ l u(t)dx, E 4 (u(t)) = µ ˆ∆3 φ R |u(t)| 2 dx -ˆ∆2 φ R |u(t)| 2 dx.
From (2.4) and (2.5), we see that E 1 (u(t)) ≤ 0. From (2.2) and the conservation of mass, we have

E 4 (u(t)) µ ∆ 3 φ R L ∞ + ∆ 2 φ R L ∞ u(t) 2 L 2 R -2 ,
where the implicit constant depends only on µ and M (u 0 ). We also have from (2.2) that

E 3 (u(t)) R -2 ∇u(t) 2 L 2 .
Finally, using (2.3), we have supp (2

-φ R ) + (2n -1) 2 - φ R r
⊂ {|x| ≥ R}. Thus, using (2.2) and the standard Gagliardo-Nirenberg inequality, we get

E 2 (u(t)) R -2 u(t) 2+ 4 n L 2+ 4 n R -2 ∇u(t) 2 L 2 u(t) 4 n L 2 R -2 ∇u(t) 2 L 2 .
Collecting the above estimates, we prove (3.1).

Step 2. An ODE argument. We will show the finite time blow-up under the assumption that E(u 0 ) < 0. Assume by contradiction that T * = ∞. By the energy conservation, we have from (3.1) that

d dt M φ R (t) ≤ 32E(u 0 ) -8 ∇u(t) 2 L 2 + CR -2 + CR -2 ∇u(t) 2 L 2 , ∀t ∈ [0, ∞).
As E(u 0 ) < 0, we take R > 0 sufficiently large so that We see that Z(t) is non-decreasing, non-negative, and satisfies

d dt M φ R (t) ≤ 16E(u 0 ) -4 ∇u(t) 2 L 2 , ∀t ∈ [0, ∞). ( 3
Z (t) = |M φ R (t)| 2 ≥ A 2 Z(t), ∀t ≥ t 0 .
For t 1 > t 0 , we integrate this inequality over [t 1 , t] to get

Z(t) ≥ Z(t 1 ) 1 -A 2 Z(t 1 )(t -t 1 )
, ∀t ≥ t 1 .

In particular, Z(t) → ∞ as t t * := t 1 + 1 A 2 Z(t1) . Thus M φ R (t) ≤ -AZ(t) → -∞ as t t * . Therefore the solution cannot exist for all t ≥ 0, hence it must blow up in finite time.

  .2) Integrating this inequality, we haveM φ R (t) = M φ R (0) + ˆt 0 d ds M φ R (s)ds ≤ |M φ R (0)| + 16E(u 0 )t, ∀t ≥ 0.This shows that M φ R (t) ≤ 0 for all t ≥ t 0 :=|M φ R (0)| 16|E(u0)| . Integrating (3.2) over [t 0 , t], we get M φ R (t) = M φ R (t 0 ) +On the other hand, we have|M φ R (t)| ≤ ∇φ R L ∞ ∇u(t) L 2 u(t) L 2 ≤ C(R, M (u 0 )) ∇u(t) L 2 , ∀t ≥ 0 which implies M φ R (t) ≤ -A ˆt t0 |M φ R (s)| 2 ds, ∀t ≥ t 0 for some constant A = A(R, M (u 0 )) > 0.Now set Z(t) := ˆt t0 |M φ R (s)| 2 ds, t ≥ t 0 .

	≤ -4	ˆt t0	ˆt t0 ∇u(s) 2 d ds L 2 ds, ∀t ≥ t 0 . M φ R (s)ds

Acknowledgments

This work was supported in part by the European Union's Horizon 2020 Research and Innovation Programme (Grant agreement CORFRONMAT No. 758620, PI: Nicolas Rougerie). The author would like to express his deep gratitude to his wife -Uyen Cong for her encouragement and support. The author also would like to thank the reviewers for their helpful comments and suggestions.