David Lagier
email: david.lagier@ap-hm.fr

David Tonon

Philippe Garrigue

Benjamin Guillet

Laura Giacomino

Jean-Charles Martin

Marie-Christine Alessi

Nicolas Bruder

Lionel J Velly

Thromboxane-prostaglandin receptor antagonist, terutroban, prevents neurovascular events after subarachnoid haemorrhage: a nanoSPECT study in rats

Keywords: Subarachnoid haemorrhage, Single-photon emission computerized tomography, Thromboxaneprostaglandin receptor, Cerebral vasospasm, Blood brain barrier, Apoptosis

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Methods:

The rat double subarachnoid haemorrhage model was produced by twice injecting (days 1 and 2) autologous blood into the cisterna magna. Seventy-eight male Sprague-Dawley rats were assigned to experimental groups. Rats exposed to subarachnoid haemorrhage were allocated to no treatment (SAH group) or TBN treatment by gastric gavage during the first 5 days after haemorrhage (SAH+TBN group). Control rats received artificial cerebrospinal fluid injections (CSF group). Sham-operated rats with or without TBN administration were also studied. Body weight and Garcia neurological scores were assessed on day 2 and day 5. We used nanoscale singlephoton emission computed tomography (nanoSPECT) to measure brain uptake of three radiolabelled agents: 99m Technetium-diethylenetriaminepentacetate ( 99m Tc-DTPA), which indicated blood-brain barrier permeability on day 3, 99m Technetium-annexin V-128 ( 99m Tc-Anx-V128), which indicated apoptosis on day 4, and 99m Technetiumhexamethylpropyleneamineoxime ( 99m Tc-HMPAO), which indicated cerebral perfusion on day 5. Basilar artery narrowing was verified histologically, and cerebral TP receptor agonists were quantified. Results: 99m Tc-DTPA uptake unveiled blood-brain barrier disruption in the SAH group. TBN mitigated this disruption in the brainstem area. 99m Tc-Anx-V128 uptake was increased in the SAH group and TBN diminished this effect in the cerebellum. 99m Tc-HMPAO uptake revealed a global decreased perfusion on day 5 in the SAH group that was significantly counteracted by TBN. TBN also mitigated basilar artery vasoconstriction, neurological deficits (on day 2), body weight loss (on day 5) and cerebral production of vasoconstrictors such as Thromboxane B2 and Prostaglandin F2α.

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Background

Most survivors of a cerebral aneurysm rupture experience subarachnoid haemorrhage (SAH)-related morbidity and mortality, due to early brain injury and delayed cerebral ischemia [START_REF] Al-Khindi | Cognitive and functional outcome after aneurysmal subarachnoid hemorrhage[END_REF]. Recent studies have highlighted a complex pathophysiology of post-SAH neurovascular events [START_REF] Macdonald | Delayed neurological deterioration after subarachnoid haemorrhage[END_REF] that extends beyond cerebral vasospasm [START_REF] Rabinstein | Secondary brain injury after aneurysmal subarachnoid haemorrhage: more than vasospasm[END_REF]. First, early brain injury implies microvascular spasms [START_REF] Friedrich | Experimental subarachnoid hemorrhage causes early and long-lasting microarterial constriction and microthrombosis: an in-vivo microscopy study[END_REF], loss of arteriolar autoregulation [START_REF] Jaeger | Continuous monitoring of cerebrovascular autoregulation after subarachnoid hemorrhage by brain tissue oxygen pressure reactivity and its relation to delayed cerebral infarction[END_REF], apoptosis [START_REF] Cahill | Mechanisms of early brain injury after subarachnoid hemorrhage[END_REF] and microthrombosis [START_REF] Pisapia | Microthrombosis after experimental subarachnoid hemorrhage: time course and effect of red blood cell-bound thrombin-activated pro-urokinase and clazosentan[END_REF]. Then, inflammatory response and oxidative stress induced by late degradation of blood products in the subarachnoid space [START_REF] Dreier | Products of hemolysis in the subarachnoid space inducing spreading ischemia in the cortex and focal necrosis in rats: a model for delayed ischemic neurological deficits after subarachnoid hemorrhage?[END_REF] contribute to delayed cerebral ischemia.

Arachidonic acid is metabolized through enzymatic (eicosanoid family: thromboxane A2 and prostaglandins) or non-enzymatic (F2-isoprostanes, 20-hydroxyeicosatetraeonic acid) pathways. Modification of arachidonic acid metabolism has been identified in a rat model of SAH [START_REF] Gaetani | Arachidonic acid metabolism and pathophysiologic aspects of subarachnoid hemorrhage in rats[END_REF]. Then, the CSF level of these metabolites was positively associated with the development of cerebral vasospasm and the neurological outcome in humans [START_REF] Chehrazi | Prostaglandins and vasoactive amines in cerebral vasospasm after aneurysmal subarachnoid hemorrhage[END_REF][START_REF] Corcoran | Are isofurans and neuroprostanes increased after subarachnoid hemorrhage and traumatic brain injury?[END_REF][START_REF] Crago | Cerebrospinal fluid 20-HETE is associated with delayed cerebral ischemia and poor outcomes after aneurysmal subarachnoid hemorrhage[END_REF][START_REF] Lin | Increased levels of F2-isoprostanes following aneurysmal subarachnoid hemorrhage in humans[END_REF]. These lipid metabolites specifically bind the thromboxane and prostaglandin (TP) receptors [START_REF] Feletou | The thromboxane/endoperoxide receptor (TP): the common villain[END_REF]. TP receptor activation leads to platelet or endothelial cell activation, smooth muscle cell contraction [START_REF] Toth | Isolated human and rat cerebral arteries constrict to increases in flow: role of 20-HETE and TP receptors[END_REF], endothelin-1 production and leucocyte recruitment [START_REF] Ting | Platelet function and Isoprostane biology. Should isoprostanes be the newest member of the orphan-ligand family?[END_REF]. At the vascular level, these agonists have shown [START_REF] Hoffman | Isoprostanes: free radical-generated prostaglandins with constrictor effects on cerebral arterioles[END_REF] to promote vasospasm, thrombosis and parietal remodelling [START_REF] Feletou | The thromboxane/endoperoxide receptor (TP): the common villain[END_REF][START_REF] Feletou | Vasoconstrictor prostanoids[END_REF][START_REF] Gelosa | Terutroban, a thromboxane/prostaglandin endoperoxide receptor antagonist, prevents hypertensive vascular hypertrophy and fibrosis[END_REF]. Similarly, the only effective and currently available treatment, nimodipine, reduced the production of thromboxane-prostaglandin (TP) receptor agonists in a rabbit model of SAH [START_REF] Pickard | Oral nimodipine reduces prostaglandin and thromboxane production by arteries chronically exposed to a periarterial haematoma and the antifibrinolytic agent tranexamic acid[END_REF]. These data suggest that TP receptor antagonists represent interesting tools to prevent brain damage following SAH.

Terutroban (TBN), a specific TP receptor antagonist [START_REF] Gaussem | The specific thromboxane receptor antagonist S18886: pharmacokinetic and pharmacodynamic studies[END_REF], was first studied as an antiplatelet and vasoprotective agent. Lesault et al. [START_REF] Lesault | Daily administration of the TP receptor antagonist terutroban improved endothelial function in high-cardiovascularrisk patients with atherosclerosis[END_REF] revealed that TBN improved endothelial function and arterial vasodilation properties in an atherosclerotic population. However, in a large clinical trial with mid-term outcomes, TBN failed to show clinical relevance compared to aspirin for the secondary prevention of cardiovascular events in stroke patients [START_REF] Bousser | Terutroban versus aspirin in patients with cerebral ischaemic events (PERFORM): a randomised, double-blind, parallel-group trial[END_REF]. Subsequently to this negative result, TBN development was stopped by the industry. However, unlike atherosclerosis, SAH is associated with a massive, localized and acute production of TP receptor agonists, mediated by inflammatory and oxidative pathways [START_REF] Chehrazi | Prostaglandins and vasoactive amines in cerebral vasospasm after aneurysmal subarachnoid hemorrhage[END_REF][START_REF] Crago | Cerebrospinal fluid 20-HETE is associated with delayed cerebral ischemia and poor outcomes after aneurysmal subarachnoid hemorrhage[END_REF][START_REF] Lin | Increased levels of F2-isoprostanes following aneurysmal subarachnoid hemorrhage in humans[END_REF][START_REF] Asaeda | A non-enzymatic derived arachidonyl peroxide, 8-isoprostaglandin F2 alpha, in cerebrospinal fluid of patients with aneurysmal subarachnoid hemorrhage participates in the pathogenesis of delayed cerebral vasospasm[END_REF]. Moreover, Ansar et al. have showed, in a rat model, that intracisternal injection of blood profoundly increased neurovascular TP receptor expression [START_REF] Ansar | Subarachnoid hemorrhage induces enhanced expression of thromboxane A2 receptors in rat cerebral arteries[END_REF].

The objective of this study was to evaluate the effectiveness of TBN in preventing post-SAH neurovascular events. For the in vivo study, we used functional outcomes and nanoscale single-photon emission computed tomography (nanoSPECT) to evaluate blood-brain barrier disruption, neuronal apoptosis and cerebral hypoperfusion. For the post mortem study, we evaluated vasospasm of the basilar artery and cerebral production of TP receptor agonists. The double intracisternal injection SAH rat model was used as it is validated for inducing intracranial hypertension, delayed decrease in cerebral blood flow and basilar artery vasospasm [START_REF] Lee | Characterization of an improved double hemorrhage rat model for the study of delayed cerebral vasospasm[END_REF].

Methods

Animals

Adult male Sprague-Dawley rats (n = 78; 280-320 g; Janvier Labs, France) were housed at 21 ± 0.5 °C with food and water ad libitum on a light-dark reverse cycle. All experiments were approved by the Aix-Marseille University Ethics Committee on Animal Experimentation (CE14; n°3-17012013) and were performed in an accredited laboratory (C13-055-20).

SAH rat model

Rat subarachnoid haemorrhage was performed as previously described by Dusick et al. [START_REF] Dusick | A minimally-invasive rat model of subarachnoid hemorrhage and delayed ischemic injury[END_REF]. Briefly, general anaesthesia was delivered with intraperitoneal injections of ketamine (100 mg/kg) and midazolam (10 mg/kg). Autologous arterial blood was collected from the ventral tail artery, and 250 μL was percutaneously injected into the cisterna magna (day 1). A stereotactic frame was used to fix the animal head by both external auditory canals and to maintain a correct anteflexion of the head. Cisterna magna was punctured percutaneously after localization of the puncture site by manual palpation of osseous landmarks. Progression of the needle was operated with a manual depression in the connected syringe. When CSF backflow appeared, the intracisternal localization of the needle tip was confirmed. Then, blood was injected. This procedure was repeated 24 h later (day 2).

Experimental groups

In the first step, rats were randomly assigned, by drawing lots, to one of the three experimental groups (Fig. 1): in the SAH group (n = 20), rats only received a double intracisternal injection with blood. Animals in the SAH +TBN group (n = 19) received a double intracisternal injection with blood and a daily TBN (Servier®, Suresnes, France) oral gavage (30 mg/kg per day dissolved in saline) from day 1 to day 5. TBN treatment was started immediately after the first intracisternal injection. The cerebrospinal fluid (CSF) group (n = 19) received a double intracisternal injection of 250 μL artificial CSF. Secondarily, two types of sham rats were studied: without treatment (sham group; n = 15) or with daily TBN oral gavage (30 mg/kg per day) from day 1 to day 5 (sham+TBN group; n = 5). In the sham groups, the cisterna magna was punctured without injection and any post mortem analysis was performed.

Functional outcome

All investigators were blinded to group assignment. On days 2 and 5, body weight was recorded and rat behaviour was evaluated with the Garcia neurological scoring system [START_REF] Garcia | Neurological deficit and extent of neuronal necrosis attributable to middle cerebral artery occlusion in rats[END_REF]. The following behaviours were evaluated: spontaneous activity, symmetrical movements of limbs, forelimb outstretching, climbing the wall of a wire cage, axillary touch response, and vibrissae touch response. Each subtest was graded on a scale of 0 (worst) to 3 (best). The total score was calculated as the sum of all subtest scores.

Evaluation of blood-brain barrier permeability, apoptosis and cerebral perfusion by NanoSPECT-CT

NanoSPECT-CT technique provides in vivo, longitudinal, quantitative and functional images. Imaging approaches also offer ethical advantages by largely reducing the number of animals needed (3-R rule). Blood-brain barrier permeability was studied on day 3 by injecting 20 MBq/100 μL 99m Tc-diethylenetriaminepentacetate (DTPA), (Pentacis®, France) into the lateral tail vein. 99m Tc-DTPA is a hydrophilic agent that cannot pass through an intact blood-brain barrier. 99m Tc-DTPA was previously used to explore blood-brain barrier disruptions in clinical studies and in other rodent models of acute neurological injuries [START_REF] Yang | Micro-SPECT/ CT-based pharmacokinetic analysis of 99mTc-diethylenetriaminepentaacetic acid in rats with blood-brain barrier disruption induced by focused ultrasound[END_REF]. On day 4, we evaluated the uptake of 99m Tc-Anx-V128 (Atreus Pharmaceu-ticals®, Ottawa, Canada). 99m Tc-Anx-V128 is a novel radiolabelled agent that target the negative phosphatidyl serine exposed on cell membranes. It is used for quantification of apoptosis. Finally, we studied the kinetics of cerebral perfusion by injecting 20 MBq/150 μL 99m Tc-hexamethylpropyleneamineoxime (HMPAO) (Cer-estab®, GE Healthcare, France). 99m Tc-HMPAO, a lipophilic agent that passes through the blood-brain barrier, is metabolized in neurons, and the metabolites are trapped in the brain. This agent is well-validated for exploring cerebral perfusion in animals [START_REF] Fj | Perfusion scanning using 99mTc-HMPAO detects early cerebrovascular changes in the diabetic rat[END_REF] and humans [START_REF] Powsner | SPECT imaging in cerebral vasospasm following subarachnoid hemorrhage[END_REF]. Since animals remain awake during radiotracer uptake and move freely before imaging, this technique prevents the bias of cerebral blood flow disturbances secondary to anaesthetic exposure. 99m Tc-HMPAO NanoSPECT images were acquired at two time points: in the basal state (day 0) and after SAH conditions were established (day 5). Forty minutes after the radiolabelled agent injection, animals were imaged with a nanoS-PECT/CTplus® camera (Mediso, Hungary). Before SPECT imaging, we obtained a 0.8-mm-slice thickness CT scan of the encephalic region. During the scan, the animals were anaesthetized with isoflurane (1.5 vol%), and body temperature was maintained. Images were reconstructed and analysed with the 3D-ROI module provided in InVivoScope® software v2.0p4 (InviCRO). After co-registration between SPECT and CT scan acquisitions, three regions of interest (ROI) were drawn, based on anatomical landmarks identified in the CT images. The brainstem ROI was delimited cranially by the spheno-occipital synchondrosis and caudally by the foramen magnum. A frontal plane, passing through the fourth ventricle, delimited the cerebellum and brainstem ROIs in the antero-posterior axis. The transverse cerebral fissure was used to distinguish the cerebellum ROI from the cerebral hemispheres. Radioactivity inside each ROI was quantified and corrected according to the tissue volume (MBq/mm 3 ), then divided by the total effective injected dose, after correcting for radioactive decay of 99m Tc. For each ROI, the radioactivity was expressed in parts per million of the injected dose (ppmID/mm 3 ). The uptakes of 99m Tc HMPAO on day 5 were expressed as the percentage of basal uptake (day 0).

Changes in brain TP receptor agonists

Cerebral quantification of TP receptor agonists was performed in a subgroup of 18 rats. On day 5, rats were euthanized and harvested fresh brains were frozen with liquid nitrogen. Thromboxane-B2 (TXB-2), Prostaglandins E2 (PGE2), D2 (PGD2), F2α (PGF2α) and isoprostane 8-iso Prostaglandin A-2 (8-isoPGA2) identification and quantification were performed on a half brain homogenate with high-performance liquid chromatography (HPLC; 1290 Infinity and Zorbax column SB-C18; Agilent®, Les Ulis, France) associated with tandem mass spectrometry (MS/MS; G6460 Agilent Triple Quadripole; Agilent®, Les Ulis, France). Further details are provided in [START_REF] Faouder | LC-MS/MS method for rapid and concomitant quantification of pro-inflammatory and pro-resolving polyunsaturated fatty acid metabolites[END_REF]. Results were expressed in picograms per milligram of protein.

Basilar artery lumen measurement

Histological examination was realized in a subgroup of 19 animals. On day 5, perfusion-fixation was performed by cannulating the left ventricle. The ventricle was perfused at 100 cmH 2 O pressure by 50 mL of 4% formaldehyde. Thereafter, the whole brain was removed and immersed in 4% formaldehyde overnight at 4 °C. Next, the brainstem was carefully dissected and embedded in paraffin. To examine the vascular changes in the basilar artery, axial brainstem slices (5 μm) were stained with haematoxylin and eosin. A blinded observer determined the cross-sectional areas of the basilar artery lumen with computerized image analysis (NIS-Elements 4.3, Nikon®, Champigny sur Marne, France).

Statistics

The number of rats per group was assessed in a power analysis performed with online software (https:// www.dssresearch.com/resources/calculators/sample-size-calculator-percentage/). We expected terutroban to increase the 99m Tc-HMPAO uptake by 50% compared to the SAH group. Past studies showed that cerebral blood flow decreased after a double intracisternal injection in the rat model of SAH [START_REF] Lee | Characterization of an improved double hemorrhage rat model for the study of delayed cerebral vasospasm[END_REF]. Based on these data, we estimated that a minimum of 14 animals for both samples would be required, for an empiric α error level of P < 0.05 and a β error level = 0.4. Therefore, we included rats to reach a minimal sample of 14 rats in each radiolabeled study group, taking into account mortality and radiolabeled injection failure. All data are presented as the mean and standard deviation (SD). Between-group comparisons was conducted with a one-or two-way ANOVA (depending on the outcome) followed by Tukey's multiple comparisons test. Adjusted P values < 0.05 were considered significant. Statistical analyses were performed with Prism® 6.0 (GraphPad Software, Inc., La Jolla, CA). The datasets generated and/or analysed during the current study are available in the "figshare" repository, with the identifier https://figshare.com/s/f7f8f15e1d185ad7e7f4.

Results

Of the 78 rats included, 58 were exposed to intracisternal injections. Among them, 45 rats completed the nanoSPECT imaging protocol. Mortality (22.4%) occurred during the intracisternal injections. Three rats were excluded due to radiolabelled agent extravasation during injection (Fig. 1).

SAH-induced synthesis of TP receptor agonists

The model of SAH was validated by the presence of blood clots in harvested brain of all rats in the SAH and SAH +TBN groups but not in the CSF group (Fig. 2a). Post mortem quantification of cerebral TP receptor agonist revealed the following: in the SAH group, cerebral TXB-2 and PGF2α were significantly increased compared to the CSF group (101 ± 28 versus 53 ± 14 pg per mg of protein; P < 0.001 and 126 ± 51 versus 62 ± 20 pg per mg of protein; P < 0.001 respectively). Cerebral TXB-2 and PGF-2α levels were significantly reduced in the SAH+TBN group compared to the SAH group (TXB-2 68 ± 16 versus 101 ± 28 pg per mg of protein; P = 0.02; PGF-2α 85 ± 16 versus 118 ± 50; P = 0.03; Fig. 2b).

Functional outcome

On day 5, SAH rats treated with TBN (SAH+TBN group) displayed significant less weight loss than the non-treated SAH rats (SAH group) (Fig. 3a) showing a beneficial effect of TBN. This difference was not yet visible on day 2. TBN also improved neurological deficits as measured using the Garcia scale. On day 2 the SAH +TBN group displayed a significant higher Garcia score than the SAH group. On day 5, differences in the Garcia score remain significant between the SAH and the sham or CSF groups (Fig. 3b).

99m Tc-DTPA nanoSPECT imaging On day 3, 99m Tc-DTPA uptake was significantly increased in the SAH group compared to the sham group in the brainstem (0.8 ± 0.5 ppmID/mm 3 versus 0.2 ± 0.1 ppmID/mm 3 , P < 0.001; Fig. 4a,c) and in the cerebellum (0.7 ± 0.4 ppmID/mm 3 versus 0.2 ± 0.1 ppmID/mm 3 , P = 0.03). In the SAH group, an inter-ROI analysis showed that brainstem and cerebellar uptakes were significantly higher than the cerebral hemispheres uptake (P = 0.02 versus cerebellum; P = 0.01 versus brainstem; Fig. 4a,b). No difference was found between CSF and SAH groups in all studied ROIs. TBN had no significant effect on 99m Tc-DTPA uptakes of sham-operated animals. In the SAH+TBN group, a significant reduction in brainstem 99m Tc-DTPA uptake was found compared to the SAH group (0.4 ± 0.1 ppmID/mm 3 versus 0.8 ± 0.5 ppmID/ mm 3 ; P = 0.03; Fig. 4c), but no significant effect was found in the cerebellum (SAH+TBN versus SAH; P = 0.46) and the cerebral hemispheres (SAH+TBN versus SAH; P = 0.7).

99m Tc-Annexin V-128 nanoSPECT imaging On day 4, 99m Tc-Anx-V128 uptake was significantly increased in the SAH group compared to the sham group in the brainstem (0.5 ± 0.2 ppmID/mm 3 versus 0.2 ± 0.1 ppmID/mm 3 , P < 0.001), in the cerebellum (0.6 ± 0.4 ppmID/mm 3 versus 0.2 ± 0.1 ppmID/mm 3 , P < 0.001) and in the cerebral hemispheres (0.5 ± 0.2 ppmID/mm 3 versus 0.2 ± 0.1 ppmID/mm 3 , P < 0.001; Fig. 4d). Compared to the CSF group, 99m Tc-Anx-V128 uptake was significantly increased in the cerebellum (0.6 ± 0.4 ppmID/ mm 3 versus 0.3 ± 0.1 ppmID/mm 3 , P = 0.001). TBN had no significant effect on 99m Tc-Anx-V128 uptakes of sham-operated animals. TBN significantly reduced cerebellar 99m Tc-Anx-V128 uptake induced by SAH (0.3 ± 0.1 ppm ID/mm 3 versus 0.6 ± 0.4 ppmID/mm 3 ; P = 0.01; Fig. 4d). However, TBN did not show any significant effect on SAH-induced 99m Tc-Anx-V128 uptake in the brainstem (SAH+TBN versus SAH; P = 0.83) and in the cerebral hemispheres (SAH+TBN versus SAH; P = 0.71).

99m Tc-HMPAO nanoSPECT imaging On day 5, 99m Tc-HMPAO uptakes (Figs. 5a,b) were significantly reduced in the SAH group compared to the CSF group in all studied ROIs: cerebral hemispheres (-18 ± 20 versus 2 ± 13% of basal uptakes; P = 0.003), cerebellum (-17 ± 18 versus 1 ± 12% of basal uptakes; P = 0.01) and brainstem (-18 ± 17 versus 2 ± 9% of basal uptakes; P = 0.004). In the sham+TBN group, we found increased day 5 99m Tc-HMPAO uptakes: + 9 ± 20% of basal uptakes in the cerebral hemispheres (P = 0.8 versus sham group), + 24 ± 16% of basal uptakes in the cerebellum (P = 0.05 versus sham group), and + 26 ± 12% of basal uptakes in the brainstem (P = 0.04 versus sham group). TBN also significantly prevented the SAH-induced reduction in 99m Tc-HMPAO uptake in all examined ROIs: cerebral hemispheres (3 ± 12% of basal uptakes; P = 0.002 versus SAH group), cerebellum (4 ± 21% of basal uptakes; P = 0.002 versus SAH group) and brainstem (5 ± 21% of basal uptakes; P < 0.001 versus SAH group; Fig. 5b).

Evaluation of the basilar artery lumen

Histological evaluation of the basilar artery on day 5 showed that its lumen was significantly narrowed in the SAH group compared to the CSF group (38,382 ± 6528 μm 2 versus 70,933 ± 14,994 μm 2 ; P < 0.001). Terutroban significantly diminished basilar artery narrowing (56,335 ± 15,728 μm 2 ) compared to the SAH group (P = 0.04; Fig. 6a,b).

Discussion

In the current study, we evaluated the neuroprotective effects of the TP receptor antagonist: TBN, in an experimental SAH rat model. The major findings were as follows: prevented disruption of the blood-brain barrier in the brainstem, alleviated apoptosis in the cerebellum and counteracted global delayed cerebral hypoperfusion. To our knowledge, this study is also the first to describe a rodent SAH model with nanoSPECT-CT imaging.

The blood-brain barrier disruption process has been well described after SAH [START_REF] Doczi | The pathogenetic and prognostic significance of blood-brain barrier damage at the acute stage of aneurysmal subarachnoid haemorrhage. Clinical and experimental studies[END_REF]. Interestingly, we found that 99m Tc-DTPA uptakes increased in the CSF and SAH groups, particularly in the infra-tentorial ROIs (brainstem and cerebellum), near the site of injection. With regard to these results, we hypothesize that volume expansion and subsequent global cerebral ischemia might contribute to the blood-brain barrier rupture as a part of the early brain injury. 99m Tc-Anx-V128 has been developed for the in vivo quantification of apoptotic activities. We have already correlated 99m Tc-Anx-V128 nanoSPECT imaging with histological TUNEL staining during focal cerebral ischemia in rats [START_REF] Garrigue | Single photon emission computed tomography imaging of cerebral blood flow, blood-brain barrier disruption, and apoptosis time course after focal cerebral ischemia in rats[END_REF]. In this work, we have found a large and global increase of 99m Tc-Anx-V128 uptakes in the SAH group indicating an activation of apoptosis. SAH-related apoptosis in both neuronal and endothelial cell is recognized as a major mechanism of the brain injury in the first steps of the disease [START_REF] Cahill | Mechanisms of early brain injury after subarachnoid hemorrhage[END_REF]. Finally, 99m Tc-HMPAO is well-validated for exploring cerebral perfusion [START_REF] Powsner | SPECT imaging in cerebral vasospasm following subarachnoid hemorrhage[END_REF]. A significant decrease in HMPAO uptake was found on day 5 in the SAH group, but not in the CSF group. This highlighted the notion that injection of blood in the subarachnoid space may have contributed to cerebral hypoperfusion [START_REF] Macdonald | Delayed neurological deterioration after subarachnoid haemorrhage[END_REF].

The quantification of cerebral TP receptor agonists revealed a dramatic increase of two well-described inflammation-induced vasoconstrictors after the intracisternal blood injection: TXB-2 [START_REF] Wagerle | Effect of thromboxane A2/endoperoxide antagonist SQ29548 on the contractile response to acetylcholine in newborn piglet cerebral arteries[END_REF] and PGF2α [START_REF] Wanderer | Levosimendan reduces prostaglandin F2a-dependent vasoconstriction in physiological vessels and after experimentally induced subarachnoid hemorrhage[END_REF][START_REF] Wanderer | Vasomodulatory effects of the angiotensin II type 1 receptor antagonist losartan on experimentally induced cerebral vasospasm after subarachnoid haemorrhage[END_REF]. This result validates our model regarding the induction of a cerebral inflammatory response. Thromboxane A2, as well as other prostaglandins, has already been targeted by different anti-inflammatory and antiplatelet agents in previous SAH studies [START_REF] Mees | Antiplatelet therapy for aneurysmal subarachnoid haemorrhage[END_REF]. The use of a TXA2-synthetase inhibitor has improved cerebral blood flow and prevented vasospasm in two experimental studies [START_REF] Chan | The role of the prostacyclin-thromboxane system in cerebral vasospasm following induced subarachnoid hemorrhage in the rabbit[END_REF][START_REF] Takeuchi | Effects of thromboxane synthetase inhibitor (RS-5186) on experimentally-induced cerebral vasospasm[END_REF]. Two clinical studies [START_REF] Suzuki | Clinical study of OKY-046, a thromboxane synthetase inhibitor, in prevention of cerebral vasospasms and delayed cerebral ischaemic symptoms after subarachnoid haemorrhage due to aneurysmal rupture: a randomized double-blind study[END_REF][START_REF] Tokiyoshi | Efficacy and toxicity of thromboxane synthetase inhibitor for cerebral vasospasm after subarachnoid hemorrhage[END_REF] have explored the efficacy of OKY-36046 (Cataclot®), a TXA-2 inhibitor. Despite small sample sizes, they both showed a protective effect on cerebral vasospasm. By contrast, TBN pharmacological properties have already been well described in humans and drug has been safely evaluated in a large clinical trial [START_REF] Bousser | Terutroban versus aspirin in patients with cerebral ischaemic events (PERFORM): a randomised, double-blind, parallel-group trial[END_REF]. By antagonizing the TP receptor, it jointly inhibits the activity of enzymatic (eicosanoids) and non-enzymatic arachidonic acid metabolites (isoprostanes, 20-hydroxyeicosatetraeonic acid). In our work, TBN also diminished brain TXB-2 and PGF-2α syntheses. A similar result has been found in the serum of apo E-deficient mice treated by TBN [START_REF] Cayatte | The thromboxane receptor antagonist S18886 but not aspirin inhibits atherogenesis in apo E-deficient mice: evidence that eicosanoids other than thromboxane contribute to atherosclerosis[END_REF], but the mechanism of this effect on agonist synthesis is unknown. As agonists are synthetized by cells expressing TP receptors on their surface (endothelial cells and platelets), a better understanding of TP receptor intracellular signalling could be of interest.

Our results suggest that TBN exerts protection on the brainstem blood-brain barrier. The protective effect of TBN may be secondary to its endothelioprotective effect, as it has been previously described in the animal with reduced endothelial ICAM-1 expression and leukocyte recruitment [START_REF] Cayatte | The thromboxane receptor antagonist S18886 but not aspirin inhibits atherogenesis in apo E-deficient mice: evidence that eicosanoids other than thromboxane contribute to atherosclerosis[END_REF][START_REF] Gelosa | Terutroban, a thromboxane/prostaglandin endoperoxide receptor antagonist, increases survival in stroke-prone rats by preventing systemic inflammation and endothelial dysfunction: comparison with aspirin and rosuvastatin[END_REF][START_REF] Worth | S18886, a selective TP receptor antagonist, inhibits development of atherosclerosis in rabbits[END_REF]. Indeed, endothelial inflammation, expression of adhesion molecule and leukocyte recruitment are well-known mechanisms of blood-brain barrier disruption [START_REF] Chan | The role of the prostacyclin-thromboxane system in cerebral vasospasm following induced subarachnoid hemorrhage in the rabbit[END_REF]. Moreover, Siler et al. have found a decreased expression of VCAM-1 and less cerebral oedema in mice exposed to epoxyeicosatrienoic acids, a recognized TP receptor antagonist [START_REF] Behm | Epoxyeicosatrienoic acids function as selective, endogenous antagonists of native thromboxane receptors: identification of a novel mechanism of vasodilation[END_REF], after SAH [START_REF] Siler | Soluble epoxide hydrolase in hydrocephalus, cerebral edema, and vascular inflammation after subarachnoid hemorrhage[END_REF]. Finally, TP receptor agonists were already found to enhance endothelial apoptosis by inhibiting cell survival properties of TNFα during inflammation and ischemia-reperfusion [START_REF] Ashton | Inhibition of tumor necrosis factor alpha-mediated NFkappaB activation and leukocyte adhesion, with enhanced endothelial apoptosis, by G protein-linked receptor (TP) ligands[END_REF].

Our data demonstrate a general beneficial effect of TBN on brain perfusion on day 5. An increase in 99m Tc-HMPAO uptake was found in sham-operated and in SAH rats exposed to TBN. This action was probably secondary to an intrinsic vasodilatory property. In support of this view, 99m Tc-HMPAO SPECT imaging has been validated for determining cerebral perfusion effects of cerebrovasoactive drugs acting via multiple modes of pharmacological action (such as nimodipine or acetazolamide) in primates [START_REF] Oliver | Non-human primate SPECT model for determining cerebral perfusion effects of cerebrovasoactive drugs acting via multiple modes of pharmacological action[END_REF]. Likewise, our results suggest a beneficial effect of TBN on basilar artery vasomotricity. Finally, a microvascular dysfunction could impair cerebral perfusion after SAH [START_REF] Terpolilli | Are we barking up the wrong vessels? Cerebral microcirculation after subarachnoid hemorrhage[END_REF]. Recent studies showed that TBN could promote cerebral arterioles' vasodilation and preserve regulative properties of endothelial cells, by protecting the function of endothelial nitric oxide synthase and inhibiting the Rho-kinase pathway [START_REF] Liu | Thromboxane prostanoid receptor activation impairs endothelial nitric oxide-dependent vasorelaxations: the role of Rho kinase[END_REF][START_REF] Horvath | Hypersensitivity to thromboxane receptor mediated cerebral vasomotion and CBF oscillations during acute NO-deficiency in rats[END_REF].

Limitations

The double autologous arterial blood intracisternal injection model had been recognized for its validity [START_REF] Lee | Characterization of an improved double hemorrhage rat model for the study of delayed cerebral vasospasm[END_REF] and low mortality rate [START_REF] Dudhani | A low mortality rat model to assess delayed cerebral vasospasm after experimental subarachnoid hemorrhage[END_REF], but this model has its limitations due to the absence of vascular rupture. We have not found an increase in cerebral 8-isoPGA2 level after blood injection revealing the poor activation of the oxidative stress response in this animal model. Nonetheless, this minimally invasive rat model, by avoiding cutaneous incision and burr-hole drilling, reduces procedure-related intracranial artefacts that could be potentially detected by the highly sensitive nanoSPECT technique. Even if same volumes of blood or CSF were injected, after a strict confirmation of the intracisternal position, we do not have any data on intracranial pressure or cerebral blood flow measurement. Then, the Garcia score in this animal model lacks sensibility, and other functional tests would have been needed for a better neurobehavioural assessment of treatment effect [START_REF] Jeon | Neurological and neurobehavioral assessment of experimental subarachnoid hemorrhage[END_REF]. We have found heterogeneous effects of TBN depending on the type of radiotracers and the considered ROI. For example, TBN protects BBB in the brainstem without exerting a significant effect on apoptosis in this ROI. This topographical discrepancy can be related to the site of blood injection with a diverse effect on each ROI and/or to a lack of power because of too small sample sizes. Despite our past work [START_REF] Garrigue | Single photon emission computed tomography imaging of cerebral blood flow, blood-brain barrier disruption, and apoptosis time course after focal cerebral ischemia in rats[END_REF], our study lacks a histologic correlation of apoptosis with 99m Tc-Anx-V128 uptake in this specific model. 99m Tc-Anx-V128 uptake could also be enhanced by blood and subarachnoid clots as annexin V has an affinity for blood components such as activated platelets. This possibility may reduce 99m Tc-Anx-V128 specificity for apoptotic cells in SAH models. As we have found in the cerebellum ROI, TBN may play a role in apoptosis prevention. Regarding the limits of 99m Tc-Anx-V128, confirmation of this result is needed with other type of investigations dedicated to apoptosis quantification.

Conclusions

With in vivo nanoSPECT imaging in a rodent model of SAH, this study shows that TBN administration after SAH protects the blood-brain barrier, improves cerebral perfusion and may prevent apoptosis. Thus, this TP receptor antagonist shows a promising result in treating post-SAH neurovascular events.

Fig. 1

 1 Fig. 1 Protocol sequences. CSF, cerebrospinal fluid; HPLC, high-performance liquid chromatography; SAH, subarachnoid haemorrhage; TBN, terutroban

Fig. 2

 2 Fig. 2 SAH-induced synthesis of TP receptor agonists. a Dissected brains show subarachnoid blood clots preferentially concentrated in the basal cisterns and around the brainstem in the SAH group. No blood was found in the CSF group. b Quantification of cerebral TP receptor agonists on day 5. Bars indicate the mean ± SD measured in rats of CSF group (n = 6), SAH group (n = 7), or SAH+TBN group (n = 5). **P < 0.05 compared to CSF, §P < 0.05 compared to SAH. CSF, cerebrospinal fluid; SAH, subarachnoid haemorrhage; TBN, terutroban; CH, cerebral hemispheres; CB, cerebellum; BS, brainstem

Fig. 3

 3 Fig. 3 Early (day 2) and delayed (day 5) functional outcomes. a Changes in body weights (% of day 0). Bars indicate the mean and SD measured in rats of sham group (n = 13), CSF group (n = 15), SAH group (n = 15) or SAH+TBN group (n = 14). b Garcia scores measured on days 2 (D2, early effects) and 5 (D5, delayed effects). Bars indicate the mean and SD measured in rats of sham group (n = 12), CSF group (n = 15), SAH group (n = 15) or SAH+TBN group (n = 12); *P < 0.05 compared to sham, **P < 0.05 compared to CSF, §P < 0.05 compared to SAH. CSF denotes cerebrospinal fluid; SAH, subarachnoid haemorrhage; TBN, terutroban

Fig. 4

 4 Fig. 4 NanoSPECT study of blood-brain barrier permeability ( 99m Tc-DTPA) on day 3 and apoptosis ( 99m Tc-Anx-V128) on day 4. a Representative image of the differential brainstem 99m Tc-DTPA uptake in the SAH group (yellow arrow). Colour scales indicate signal intensities from low (blue) to high (white). NanoSPECT images (sagittal view) show the three ROIs (CH, cerebral hemispheres; CB, cerebellum; BS, brainstem). Bars indicate the mean ± SD measured in rats of sham group (n = 15), sham+TBN group (n = 5), CSF group (n = 14), SAH group (n = 15) or SAH+TBN group (n = 15). b Inter-ROI analysis of 99m Tc DTPA uptake in the SAH group: predominating uptake in the infra-tentorial ROIs (CB and BS). *P < 0.05 compared to CH. c 99m Tc-DTPA and d 99m Tc-Anx-V128 nanoSPECT study. Results were expressed as parts per million of injected dose (ppm ID)/ mm 3 of tissue. Bars indicate the mean ± SD measured in rats of sham group (n = 14), sham+TBN group (n = 5), CSF group (n = 14), SAH group (n = 14) or SAH+TBN group (n = 15). *P < 0.05 compared to sham, **P < 0.05 compared to CSF, #P < 0.05 compared to sham+TBN, §P < 0.05 compared to SAH. CSF denotes cerebrospinal fluid; SAH, subarachnoid haemorrhage; TBN, terutroban; CH, cerebral hemispheres; CB, cerebellum; BS, brainstem

Fig. 5

 5 Fig. 5 NanoSPECT study of cerebral perfusion ( 99m Tc-HMPAO) on day 5. a 99m Tc-HMPAO nanoSPECT imaging (sagittal view) shows an example of the 99m Tc-HMPAO brain uptake at day 5. Colour scales indicate signal intensities from low (blue) to high (yellow or white). b 99m Tc-HMPAO uptake (% basal uptake) was measured on day 5. Bars indicate the mean ± SD measured in rats of sham group (n = 15), sham+TBN group (n = 5), CSF group (n = 15), SAH group (n = 15) or SAH+TBN group (n = 15). *P < 0.05 compared to sham, **P < 0.05 compared to CSF, #P < 0.05 compared to sham+TBN, §P < 0.05 compared to SAH. CSF denotes cerebrospinal fluid; SAH, subarachnoid haemorrhage; TBN, terutroban; CH, cerebral hemispheres; CB, cerebellum; BS, brainstem

Fig. 6

 6 Fig. 6 Basilar artery measurements. a Representative photomicrographs showing significant vasoconstriction of basilar arteries on day 5 after subarachnoid haemorrhage induction. Vasoconstriction was attenuated with terutroban (SAH+TBN). Tissue samples were stained with haematoxylin and eosin; × 200 magnification; scale bar = 200 μm. b Basilar artery luminal areas (μm 2 ). Bars indicate the mean and SD measured in rats of CSF group (n = 6), SAH group (n = 7), or SAH+TBN group (n = 6). **P < 0.05 compared to CSF, §P < 0.05 compared to SAH. CSF denotes cerebrospinal fluid; SAH, subarachnoid haemorrhage; TBN, terutroban

Acknowledgments

None.

Funding

Support was provided solely from institutional and/or departmental sources. The producer of terutroban at the time of the study (Servier®) did not fund or influence the study.

Availability of data and materials

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

Abbreviations

Authors' contributions DL and DT designed the study, performed the experiments, analysed the data and drafted the manuscript. PG, BG, LG and J-CM performed the experiments and analysed the data and contributed to the final manuscript. NB, LV and MCA designed the study, analysed the data and contributed to the final manuscript. All authors read and approved the final manuscript.

Ethics approval

All experiments were approved by the Aix-Marseille University Ethics Committee on Animal Experimentation (CE14; n°3-17012013) and were performed in an accredited laboratory (C13-055-20).

Consent for publication

Not applicable.

Competing interests

The authors declare that they have no competing interests.

Publisher's Note

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


Alessandro Di Pretoro

Ludovic Montastruc
email: ludovic.montastruc@ensiacet.fr

Flavio Manenti

Xavier Joulia

Flexibility analysis of a distillation column: Indexes comparison and economic assessment

Keywords:

published or not. The documents may come

Introduction: A measure of flexibility

Flexibility analysis is a step of process design procedure that is often skipped. Sometimes a sensitivity analysis is performed with features similar to the flexibility one but in general they don't overlap.

The standard procedure for chemical plants design consists of the assessment of the optimal design according to the economic and operational aspects. Nevertheless this optimal design is strictly related to the operating conditions, i.e. perturbations, when present, can seriously turn the tables. In these cases an a priori flexibility analysis could be of critical importance to assess in which range of operating conditions a system design is effectively better performant than another one.

The word flexibility commonly refers to the ability to change in order to cope with variable circumstances both in a passive and an active sense; to be more detailed, in process engineering, flexibility can be defined as the ability of a process to accommodate a set of uncertain parameters [START_REF] Hoch | Flexibility analysis leads to a sizing strategy in distillation columns[END_REF]. This concept looks easily understandable but actually the definition standalone says nothing about how we can deal with it. Thus we need an opera-tional definition and to know what flexibility means through the way it could be measured.

The very first publication about the definition of a flexibility index is provided by Swaney and Grossmann (1985a) and soon after [START_REF] Saboo | Design of resilient processing plants 0.8. a resilience index for heat-exchanger networks[END_REF] proposed another possible index to quantify the flexibility of a process design (called in this case "resiliency"). Moreover there exists an additional paper by both [START_REF] Grossmann | Operability, Resiliency, and Flexibility: Process Design Objectives for a Changing World[END_REF] where the flexibility related problems and solution are thoroughly analyzed with a pioneering methodology.

The Swaney and Grossmann flexibility index ( F SG ) mathematically states as follow:

Given that:

• θ N : nominal values of uncertain parameters (base point);

• θ + , θ -: expected deviations for each parameter;

• d : design variables associated to the equipment capacities;

• z : control variables.

The flexibility index, for a given design d, is the solution of the problem: (d, z, If F = 1, the deviations θ + and θ -can be accommodated. In order to have a clearer idea of what these statement means we can help ourselves with the plot in Fig. 1 . ( Swaney and Grossmann, 1985a ).

F SG = max δ (1.1) s.t. max θ T (δ) min z max j J f j
The curve ( d, θ ) is the so called feasibility domain defined by the constraints. Constraints can be physical, legal, operational, economic etc. and in general a deeper study about the way this region is outlined and the conditions to be satisfied, as well as the degrees of freedom analysis, is needed. This step precedes the flexibility analysis since the domain shape is strictly related to the nature of the case study.

As shown by the optimization problem, the flexibility index defined as above ( 1.1, 1.2 ), is the maximum fraction of the expected deviation δ that can be accommodated by the system; it graphically represents the minimum among the maximum fractions of the hyperrectangle sides' lengths that is bounded by the feasible zone.

Moreover, for constraints jointly quasi convex in z and 1 D quasi convex in θ the problem can be decomposed into two level optimization problem:

F SG = min k δ k (1.3) δ k = max z δ (1.4) f j (d, z, θ k ) ≤ 0 j J (1.5) θ k = θ N + δ • θ k (1.6)
and the solution lies at a vertex of the hyperrectangle allowing us to solve the optimization problem by evaluating the feasibility of the design at each vertex. In this way, it can be noted that the explicit solution of the min-max problem can be circumvented; on the other hand certain types of non-convex domains may lead to nonvertex solutions [START_REF] Grossmann | Active constraint strategy for flexibility analysis in chemical processes[END_REF]. Before the introduction of the flexibility study, to move from feasibility study towards the design phase, we just need to know whether the project is feasible under the nominal operating conditions or not, but once we deal with flexibility the qualitative answer yes/no is not sufficient anymore. We need to quantify "how much" the project is feasible and the independent variables ranges enclosing the possible operating conditions.

As anticipated at the beginning, the same year Morari et al. proposed a "resiliency index" defined as the capability to easily recover or adjust to misfortune or change, that is more or less the passive alter ego of the capability to adapt to new, different or changing requirements, i.e. flexibility. [START_REF] Saboo | Design of resilient processing plants 0.8. a resilience index for heat-exchanger networks[END_REF].

This index is based on the same premises of the F SG , i.e. even in this case the very first step to perform is the definition of the feasibility domain. Then the resiliency index RI is defined as the largest total disturbance load, independent of the direction of the disturbance, a system is able to withstand without becoming infeasible.

It mathematically stands as: (1.8) this corresponds to inscribing the largest possible polytope inside the feasible region defined by the inequalities here above. The RI is then equal to the distance between a vertex V of the polytope and the nominal operating point 0 as shown in Fig. 2 . The main advantage of the RI compared to the F SG is that it requires a lower computational effort; this can be easily figured out in the case of an n-dimensional problem where a vertex analysis has to be performed: in the hyperrectangle case we have 2 n vertices to calculate while in the polytope case we have just 2 n of them.

RI = min i | l max i | (1.7) s.t. { max j f j (θ ) ≤ 0 , ∀ l : | l i | i ≤ RI}
Moreover, the disturbance region measured in the first case might be practically more interesting because it expresses directly the disturbance load allowed in the direction of each parameter independently on the others.

For the same reason, since the hyperrectangles sides have to be parallel to the axes even if this configuration doesn' t allow the biggest possible rectangle, the F SG index results to be very conservative and significantly underestimate the actual flexibility of the system.

This representation of the feasible and expected possible zones does not reflect thoroughly the real world since not all the possible operating conditions are equally probable; hence this kind of analysis results in rather conservative estimates. In many real world applications, however, data are usually available that allow a better definition of uncertainty in a statistical sense.

For this reason a novel flexibility analysis approach for processes with stochastic parameters was then proposed in 1990 by [START_REF] Pistikopoulos | A novel flexibility analysis approach for processes with stochastic parameters[END_REF] . It' s shown that the flexibility index can correspond to a multivariate cumulative distribution function transforming the original constraint space to the space of stochastically dependent flexibility function by mean of the analytical properties of the flexibility problem. [START_REF] Pistikopoulos | A novel flexibility analysis approach for processes with stochastic parameters[END_REF].

Given the feasibility region constraints inequalities as:

(d, θ ) ≤ 0 (1.9) the equality defines the boundary of the feasible zone.

The stochastic flexibility index SF can be defined as:

SF = P r{ (d, θ ) ≤ 0 } = { θ : (d,θ ) ≤0 } . . . P D (θ ) dθ (1.10)
where P D is the joint distribution function of the uncertain parameters θ .

The comparison between the stochastic flexibility index SF and the Swaney and Grossmann is better shown in Fig. 3 .

With this methodology we can calculate a weighted estimation of flexibility. Thus, if something is unlikely to happen and our system cannot withstand this operating conditions, it only slightly affects the final value of the flexibility index providing a measurement less conservative and more adherent to reality.

Obviously, the other side of the coin is the strict dependency of this index on the availability of probability distribution data, that is if no data are available this methodology is clearly unusable.

An additional problem to be solved was pointed out in 1995 by [START_REF] Dimitriadis | Flexibility analysis of dynamic systems[END_REF] and it consists in the evaluation of flexibility taking into account the dynamics of the studied system. This topic was already pointed out by [START_REF] Grossmann | Operability, Resiliency, and Flexibility: Process Design Objectives for a Changing World[END_REF] few years before but Pistikopoulos is the very first one to define a specific index namely Dynamic Flexibility (DF) that takes into account the evolution of the system. Actually the definition itself of DF is not so different since it is a modification of the previous index F SG . However, its introduction allows the study of a system taking into account its control loops and their tuning, therefore from an operational point of view its introduction is of critical importance.

The Dynamic Flexibility problem is introduced here below for literature coverage reasons. Nevertheless this index will not be taken into account in the flexibility analysis proposed in the next section both because no other dynamic flexibility indexes to compare it with have been found in literature and because the analysis refers to steady state conditions.

The definition of the Dynamic Flexibility follows the path of the F SG considering the uncertain and control parameters namely θ and z as a function of time, therefore the dynamic flexibility index evaluation problem becomes: (d, x (t) , z(t ) , θ (t ) , t ) ≤ 0 (1.12)

DF (d) = max δ (1.11) s.t. χ (d) = max θ (t) T (δ,t) min z(t ) Z(t ) max j J,t [0 ,H] f j
s.t. h (d, x (t) , z(t ) , θ (t) , t ) = 0 (1.13) x (0) = x 0 (1.14) δ ≥ 0 (1.15) T (δ, t ) = { θ (t) | θ N (t) -δ • θ -(t) ≤ θ (t) ≤ θ N (t) + δ • θ + (t) } (1.16) Z(t ) = { z(t ) | z L (t) ≤ z(t ) ≤ z U (t) } (1.17)
Qualitatively, the dynamic flexibility index, DF, represents the largest scaled deviation of the uncertain parameter profile that the design can tolerate while remaining feasible within the horizon considered.

The dynamic flexibility index problem is a two-stage, semiinfinite, dynamic optimization problem with an infinite number of decision variables. Therefore, in order to solve it, an ad hoc methodology to reduce the dimension of the problem has to be implemented.

Finally, the most recent flexibility index found in literature is provided by Lai and Hui (20 07,20 08) . It has the aim to overcome the problems related to previous indexes, i.e. the requirement of nominal point and the consideration of the critical uncertainty only (causing an underestimation) for F SG and RI, as well as the availability of the probability distribution of all uncertain parameters at the design stage for the SF. This new flexibility metric is much easier to use and does not need a lot of computational effort or available data and it is defined as follow.

Let V 0 be the volume defined by the uncertain parameters:

V 0 = N i =1 ( θ iU -θ iL ) (1.18)
and V f the feasible volume defined as the intersection of the constrained volume and V 0 .

Then the flexibility index F V is defined as the ratio between the feasible space and the uncertain space:

F V = V f V 0 (1.19)
However, S f is usually irregular in shape and its volume ( V f ) determination is not straightforward. To estimate V f , a constructed space ( S e , the region outlined by the thick solid line), whose volume determination is less difficult, can be inscribed inside S f . With a careful selection of the shape of S e , its volume ( V e ) can be used as a close estimate of V f .

As illustrated in Fig. 4 , S e can be constructed by first picking a reference point ( P R ), which is not necessarily the nominal point, within S f . Auxiliary vectors v j with selected directions can be radiated from P R . The interception points ( P j ) of v j and the feasible space boundary are obtained. The S e can then be constructed by joining these P j points according to their positions in space. Since different S e can be generated by different auxiliary vector direction selection schemes, estimation accuracy of V f and F V will depend on the selection scheme employed. The general formulation for the auxiliary vectors' positions in a 3 D space is as follows: and0 ≤ x j , y j , z j ≤ 1 (1.21) [START_REF] Lai | Process flexibility for multivariable systems[END_REF].

max x j ,y j ,z j V e (x j , y j , z j ) (1.20) f k (θ i j ) ≤ 0
s.t. θ i, j = v i j i j i j + θ R i , i j = + • θ i (v i j ≥ 0) | -• θ i | (v i j < 0) i = x, y, z (1.22)
However, whether the feasibility domain is well defined and the constraints equations are know, there's no need to approximate anymore since we can get the exact value of the S f volume through a multiple integral at the cost of a higher computational effort.

An additional remark is worth to be done: the volumetric flexibility index takes into account no perturbation outside the uncertain space and treat every point as equally possible. Therefore, given its definition, we can write it as:

F v = V f V 0 = 1 V 0 V f d θ = V f 1 V 0 d θ (1.23)
that is the same definition as the stochastic flexibility index for a probability step function. A practical application of this analogy between SF and F V will be shown in the following chapters. Finally, on one hand we can say that several flexibility indexes have been found in literature but, on the other hand, except the steady state debutanizer case study proposed by [START_REF] Hoch | Evaluation of design flexibility in distillation-columns using rigorous models[END_REF] ( F SG ), no case studies about distillation have been provided (cf. Table 1 ). Most of the systems found in literature subjected to flexibility analysis have linear constraints or at least a quasi convex feasibility domain; whether a higher complexity can be detected, the flexibility analysis has been conducted with the simple F SG that, due to its straightforward application, results to be the index used in the vast majority cases. However, the behaviour of less used and more complex indexes is worth to be studied in deeper and for a wider range of systems as well; therefore, a complete flexibility analysis of a distillation column, inspired by the aforementioned debutanizer example [START_REF] Hoch | Evaluation of design flexibility in distillation-columns using rigorous models[END_REF], is proposed here below with all the four steady state indexes. An accurate results comparison and economic assessment will follow.

The debutanizer case study

In order to better understand the distillation related application of these indexes, the simple debutanizer case study proposed by [START_REF] Hoch | Evaluation of design flexibility in distillation-columns using rigorous models[END_REF] , with few modifications, has been analyzed in detail.

In short, the system is made up of a standard distillation column, i.e. total condenser, no intermediate feeds/withdrawals and partial reboiler ( Fig. 5 ).

The feed stream is defined by the composition and conditions shown in Table 2 while the given design parameters are listed in Table 3 .

Finally the uncertain parameters are reported in Table 4 , their expected variation ranges θ k ± are the same in either positive or negative direction and equal to 10% of their nominal value as suggested in the paper. The feed variations are related to the nature of the upstream process, the performances of the heat exchangers to the tubes fouling, the water temperature to the seasonality and finally the flooding and weeping velocity are function of the trays technology and status.

The specifications are given by the paper as three inequality constraints, namely:

• Maximum molar fraction of butane in the bottom product = 0.01786; • Maximum molar fraction of pentane in the distillate = 0.025;

• Minimum pentane recovery in the bottom = 0.97.

In this paper the two most restrictive equality relationships have been selected to fulfill the two remaining degrees of freedom:

• Molar fraction of butane in the bottom product = 0.01786; • Pentane recovery in the bottom = 0.97.

The controlled variables are respectively the reflux and distillate flowrates.

Flexibility analysis

Flexibility analysis can be performed either during the design phase or for an already existing equipment (or plant). In the next Swaney and Grossmann (1985a) F SG [START_REF] Floudas | Global optimization in design under uncertainty: feasibility test and flexibility index problems[END_REF] F SG [START_REF] Lai | Process flexibility for multivariable systems[END_REF] F V Refrigeration cycle Swaney and Grossmann (1985b) F SG Reactor-recycle Swaney and Grossmann (1985b) F SG Heat exchanger network [START_REF] Grossmann | Operability, Resiliency, and Flexibility: Process Design Objectives for a Changing World[END_REF] F SG Swaney and Grossmann (1985b) F SG Saboo et Pa chapters flexibility will be both assessed for the debutanizer column as shown in the previous paragraph and for a distillation column to be designed; the results will be then compared. Moreover, the analysis will be conducted separately for the deterministic and stochastic indexes in order to highlight analogies and differences.

Deterministic indexes: Swaney Grossmann F SG and resilience index RI

First of all, in order to evaluate the Swaney and Grossmann flexibility index F SG , we need to estimate the variation ranges of the uncertain variables as shown in Table 4 , called from here on out "expected deviations". Since the system is rather simple (i.e. convex feasibility domain), for the moment there's no need of outlining the whole feasible space as discussed in the first section; the vertex analysis, by increasing the parameters variation percentages until the expected values and keeping constant the ratios between them, can be then easily and effectively performed. Since there are 7 changing parameters, the hyperrectangles has 7 dimensions that means 2 7 = 128 vertices should be theoretically calculated for each simulation but, thanks to the possibility to decouple some independent parameters, the computational effort can be substantially reduced.

The complete flexibility analysis results and the corresponding plots are shown respectively in Table 5 and Figs. 6 and7 , the debutanizer related values are indicated.

The resulting Swaney and Grossmann flexibility index for this case study is then given by:

F SG = al l owed de v iation expected de v iation = 5% 10% = 0 . 5 (3.1)
The bottleneck of flexibility is given by the column minimum diameter equal to 0.634 m, i.e. by the flooding conditions. However, the analysis has been carried out for all the other design parameters standalone as well. The condenser results to be the second most constraining variable, while the reboiler the third one; for a value between 12 and 15% of the allowed deviation we can notice that the maximum diameter becomes lower than the minimum one causing the column design to be impossible with a single diameter column. This last phenomenon occurs because flexibility indexes refer both to positive and negative perturbations causing the range of diameter values able to ensure operable conditions to become smaller for higher flexibility requirements. The conditions that cause the hyperrectangle to exceed the feasibility domain boundaries, i.e. which vertex is tangent one, for each design variable are reported in Table 6 ; "+" and "-" indicate respectively a positive or negative deviation of the uncertain parameter while "/" indicates that the parameter does not affect the constraining design variable.

For the "Condenser area", "Reboiler area" and "Minimum column diameter" critical conditions are achieved because of overfed column and underperforming equipment while for the maximum diameter, i.e. weeping conditions, criticalities are present in case of underfeeding as expected according to the physics of the problem.

Beside the design and sizing, an economic assessment has been performed as well. The purpose is to couple flexibility and investment costs in order to make the best decision during the design phase and avoid the plant underperform; operational costs have not been taken into account since they're univoquely determined due to the fact that we're not changing the number of trays. For more detail about equipment costs correlations cf. Appendix B

The capital costs trends (normalized to the lowest value) as function of F SG flexibility are plotted for each equipment in Figure 8 . All of them increase as flexibility increases (according with their size). The most expensive equipment is the Kettle reboiler that has a much more accurate technology than the other heat exchangers and that works under pressure, while the column is relatively cheap because of its small diameter.

In Fig. 9 three series of percentage data as function of flexibility have been plotted; they refer namely to:

• dC C 0

: the additional investment referred to a column designed in nominal operating conditions, i.e. 0% flexibility (Blue trend);

• dC C e f f
: the cost differences with respect to the case study column design (Red trend); • dC C real : the effective cost differences if the case study column is already available (Yellow trend).

We can notice first that the trends are more or less linear; moreover, we can also point out that part of the investment (about 3%) could have been saved if the reboiler was properly designed, from a flexibility point of view, since its overdesign is practically useless considering the bottleneck of 5.00% given by the column diameter.

Differently from the F SG index, the resilience index defines the largest total disturbance load a system is able to withstand independently of the direction of the disturbance.

It was originally defined for heat exchangers networks, therefore some modifications are needed in order to adapt it to every type of system. First of all the Resilience Index has a dimension that usually is an "heat" related value (kW, K etc.); in this case, in order to make the comparison with the other indexes possible and because of the several perturbations we have to cope with, each one with its different dimension, the percentage deviation value will be used. Then, since we're dealing with a simple system (i.e. quasi-convex domain), we're sure we're going to solve a so called "Class 1 problem", i.e. the standard vertex analysis procedure can be performed successfully.

The complete Resilience analysis results are shown in Tables 7 and8 and Figs. 10 and11 .

The resulting Resilience Index (RI) for this case study is then given by:

RI = min i | l max i | = 9 . 64 (3.2)
First of all we can notice that there are no problems even if the intervals are only half-bounded because we're looking for the minimum of the maximum withstood perturbations. The ∞ values don't mean that we really calculated results for infinite percentage values of parameter deviation, but it means that the withstood percentage perturbation is large enough not to affect the flexibility analysis or enough to fulfill all the physically possible (not only expected) deviation range in that direction.

Then we can focus our attention on the limiting design factor that is, as for F SG index, the column diameter whose corresponding parameter is the flooding velocity. The Resilience Index calculated this way has an higher value than the F SG since we perturbate only one parameter at time leaving unchanged the others. Vice versa, in order to attain a given flexibility value, a smaller oversizing than F SG case is needed.

Even the crossover of the minimum and maximum diameter values, i.e. the completely infeasible conditions with a single diameter column, was more conservative in the F SG analysis where it was about 13% (cf. Fig. 7 ), while in the RI this condition is attained for a 24% (cf. Fig. 11 ) flexibility more or less.

Moreover, it is worth noticing that, beside the column diameter, the second most constraining variable is the reboiler while for the F SG analysis was the condenser. This is the most representative difference between the two indexes because the variables acting on the condenser are U cond and T w in both cases but, while in the RI analysis they are perturbated one at a time, in Swaney and Grossmann analysis they change all at once causing the equipment affected by an higher number of uncertain parameters (in this case the condenser) to be more critical.

As well as the Swaney and Grossmann index costs analysis, the capital costs trends (normalized to the lowest value) as function of RI flexibility are plotted for each equipment in Fig. 12 . All of them increase as flexibility increases (according with their size). The most expensive equipment is the Kettle reboiler that has a much more accurate technology than the other heat exchangers and that works under pressure, while the column is relatively cheap because of its small diameter. In Fig. 13 the three series of percentage data as function of Resilience Index analogous to those in Fig. 9 are reported.

Even in this case the trends are more linear, the differences between F SG and RI indexes economical analysis reflects the differences highlighted in the equipment design analysis as well as the analogies, i.e. for a given flexibility value the capital cost is lower if we consider the Resilience Index flexibility. Obviously it doesn't mean that we can save money just by changing the index we use, it only means that by selecting a different index we are measuring different performances; so for each process the more suitable the index is the more accurate the economical analysis will be, where with the word "suitable" refers to the performances demanded to cope with the possible disturbances.

The Swaney and Grossmann flexibility measures the ability of the system to withstand an overall parameters deviation of F SG • θ k ± %, the Resilience Index measures the case of a single variable, it doesn't mind which one, deviation of RI %, therefore, in order to find the most suitable index, we need to evaluate which of the two kinds of deviation our system is more likely to undergo to. This idea of "disturbance likelihood" leads us then to the stochastic characterization of the flexibility indexes discussed in the following section for the same debutanizer case study.

Stochastic indexes: Pistikopoulos and Mazzuchi SF and Lai and

Hui F V When we move from a deterministic point of view to a stochastic one we have to transform the idea of perturbation expected or not, defined by a well bounded deviation range, into a continuous function describing how much expected the deviation is. To do this we need to use a probability distribution function relating a probability value to each condition "x", i.e. the independent variable(s) (cf. Appendix C for more details).

The probability density function are usually parametric functions, the probability of the independent variable falling within a particular range of values is given by the integral of this variable' s PDF over that range. The probability density function is nonnegative everywhere and its integral over the entire space is equal to one.

Before going any further with the application of these principles to our debutanizer case study, a few observations about the chosen PDF and its properties will be remarked here below.

First of all, in order to univocally define the PDFs we need to set the parameters; the selected values are:

• μ= operating conditions for each variable;

• σ or b = 20% of μ for each variable.

For this case study a 20% variance has been selected since it is about the maximum allowed individual deviation for the variables taken into account in the stochastic flexibility analysis. This choice makes the analysis sensible enough to appreciate the features of the coupling of flexibility and economics, that is the main goal of the paper. A very small σ value would result in a useless flexibility analysis since almost all possible perturbations would be withstood; on the other hand a higher σ value has poor reliability since variables uncertainty range would be too wide and a significant stochastic flexibility value would be never attained. Therefore 20% results a good compromise allowing an analysis more sensible and unbiased by excessive optimism, keeping us as conservative as needed.

We have then to choose which type of PDFs would better describe the system under analysis. Probability reflects the state of the information therefore, since we actually have no data about the probability functions of our parameters deviations, the most general PDF possible should be used in order to have the more unbiased possible results. The condition of "general validity" is satisfied by the Gaussian or normal probability distribution. It is symmetrical with respect to its mean and the 99.73% of cumulative probability falls in the range [ -3 σ, +3 σ ].

For the sake of completeness, in order to prove that the results of the stochastic flexibility analysis are not qualitatively PDF dependent, it will be conducted with a different probability distribution function as well, i.e. the Laplace one. This distribution satisfies, in a sense, the same requirements needed by the system description, that is:

• The maximum probability is attained at the operating conditions; • the probability value is not dependent on the deviation direction.

For further details about Normal and Laplace probability distribution functions cf. Appendix C .

In order to have a visual approach with the stochastic flexibility index meaning, a 2D analysis for the condenser perturbation has been performed first ( Figs. 14 and15 ). Then, once dealt with it, the analysis is shifted to higher dimensions; it is nonetheless worth remarking that the analysis methodology is independent on the dimension of the problem. As we can notice, the two selected parameters, i.e. heat transfer coefficient and cooling water temperature, act on the same design variable, i.e. heat exchanger transfer area. Therefore a single constraint representing the heat balance is present.

However, since we don't want to perform the flexibility analysis of a heat exchanger, on one hand we need to increase the dimension of our problem on the other hand we don't want the computational effort to be too high. To match these two purposes we could, for instance, add the most constraining parameter (according to the previous flexibility analysis), i.e. the flooding velocity. This way we have a 3D domain with a variable ( G f ) related to the column design ( D min ) and the other two ( T w and U cond ) acting on the same design variable ( A cond ) ( Fig. 16 ); the independence of the third parameter on the other two can be immediately noticed since the yellow plane, i.e. flooding constraint, is parallel to the T w x U cond plane.

An additional difference between this index and the previous ones we've got to deal with is that, even if a given design defines univocally a stochastic flexibility value, a given stochastic flexibility value does not define univocally a system design. The two constraints can be shifted in several configuration keeping constant the value of the integral function indeed. Therefore we'll refer as the "x" stochastic flexibility value design to the optimal configuration that attains that value, where optimal simply means cheaper. This need of economical optimization directly links flexibility with design and economic aspects whose trends will be anticipated with respect to the sizing ones.

The results of the stochastic flexibility analysis for each 1% cost increase are reported in Figs. 17 and 18 , the optimal design variables in Figs. 19 and20 .

First of all it's worth highlighting that, differently from previous flexibility indexes, the stochastic one has a non-zero value at operating conditions design.

Moreover the optimal design according to flexibility could be different from the operating conditions design or the economical optimal design.

An additional difference between SF and F SG or RI is that its trend is highly non-linear if expressed as function of the equipments' size, that means that for big oversized equipment the flexibility increase due to a further oversizing is only slightly appreciable. The relative optimal oversizing trend between the condenser and the column keeps being nonetheless almost linear and furthermore it follows the same line independently on the starting design condition as expected.

Finally if we compare the two different distributions results we can immediately notice that they reflect the nature of the distributions themselves. Laplace distribution converges more slowly than the Normal one, therefore the related SF approaches the value 1 only for a bigger oversizing. Nonetheless their trends are qualitatively as similar as the PDF are. The same sizing related remarks are valid if we talk about costs since they're directly related.

For two different equipment design the starting points of the dC C v s. SF lines are different but, after a while, the ending branches of the two curves overlap each other approaching the asymptote. Additional costs are higher for the Laplace distribution case than Normal distribution, as expected, because the deviation likehood is slightly higher even far from the operating conditions. Differently from F SG and RI , costs increase as function of stochastic flexibility index shows a non-linear trend, this means that once a certain point of the curve has been passed the ratio between the increase in flexibility and costs starts declining fast.

The main purpose of Process System Engineering is to enhance the decision making capability of the chemical engineer, therefore we'd like to identify the "certain point" after whom it's not worth keeping spending money in overdesign. The range of convenient oversizing is visible to the unaided eye: the first part of the curve gives a high flexibility increase with a small additional costs but the unwithstood deviation probability is still relevant; on the other hand the last part, even withstanding almost the whole possible deviations, needs a consistent oversizing (i.e. additional costs) to be achieved. Finally we can then conclude that the middle range of the curve is a good compromise between high flexibility and affordable additional cost.

The thorough procedure we propose to assess the optimal range is based on the curve properties: tangent lines have almost the same slope (i.e. derivative) at the beginning and at the end of the curve while it considerably changes within the interval we're interested in.

The procedure then consists in plotting the ratio between the derivative calculated at each point and the derivative calculated at the previous one as illustrated in Fig. 21 for the case of NDF and operating conditions. This way, given the dC C v s. SF plot only, we can obtain a new plot whose trend shows a maximum corresponding to the value SF = 0 . 9409 , i.e. 10% of additional cost. However even the other values near there can be considered good conditions for a flexibility based design, more important than the optimal value itself is to avoid the conditions corresponding to very first and very last part of the plot. The chronologically last flexibility index proposed in scientific literature is the volumetric flexibility index F V . It is defined as the feasible fraction of the uncertain space, that is a line for 1D case, a surface for 2D case, a volume for 3D case and a hypervolume for higher dimensions.

The reason why the volumetric flexibility index is included among the stochastic indexes is that it can be also thought as a SF index particular case whose deviation probability function is described by a step function ( Fig. 22 ) defined as:

1 N i =1 ( θ iU -θ iL ) if | x i | < θ ± i 0 if | x i | > θ ± i (3.3)
therefore the same SF analysis rules and remarks can be applied. Thus, in order to perform this kind of analysis, two parameters have to be set: the nominal point to the operating conditions and the expected deviation value to 20%.

For this index as well a 2D analysis for the condenser perturbation has been performed first ( Fig. 23 ) and then, once dealt with it, the analysis is shifted to higher dimensions. The same SF analysis remarks apply, i.e. the analysis methodology is independent on the dimension of the problem, the two selected parameters act on the same design variable, i.e. heat exchanger transfer area, therefore a single constraint representing the heat balance is present.

The results of this 2D case study are F V = 0 . 4714 for operating conditions and F V = 0 . 8584 for real conditions.

For the F V index the analysis has been shifted in the same SF 3D domain ( Fig. 24 ). The flooding velocity has then been included in the analysis; since it is the most constraining parameter, it reduces significantly the feasibility of the system. The triple integral required by the SF index actually becomes a volume integral since the probability distribution is constant on each interval where it is defined; due to the complexity of PDF to be integrated in stochastic flexibility analysis, the computational effort for the F V analysis prove much lower (seconds vs. minutes).

Even in this case a given volumetric flexibility value does not define univocally a system design, therefore the "x" volumetric flexibility value design refers to the optimal, i.e. cheapest, configuration that attains that value. The results of the volumetric flexibility analysis for each 0.5% cost increase are shown in Figs. 25 and26 .

On one hand we have the analogies with the SF that are non-zero values at operating conditions, optimal sizing independent on the starting conditions and asymptotes at F V = 1 . On the other hand we've got to notice that the trends of the F V analysis results are rather peculiar and very distinctive of this kind of index. In Fig. 26 we can mainly distinguish three zones:

1. Linear trend: In the very first overdesign part both the column diameter and condenser area affect the flexibility of the system, therefore the optimal design strategy is to split the investment according to the proportion expressed by the slope of the line. 2. Horizontal (or vertical) trend: After a while the flooding constraint is almost outside the uncertain parallelepiped. The only thing to do is then to shift the condenser constraint until it exits the parallelepiped as well. In the end a little adjustment of both the design variables is observed and the value F V = 1 is finally attained. 3. Overpayment zone: After the F V = 1 value has been achieved the whole uncertain domain is contained within the feasible boundaries therefore any additional oversizing is, considering flexibility, practically useless. The A cond = 50 . 90 m 2 and D col = 0 . 686 m condition corresponds definitely the maximum achievable flexibility. In the end it is worth remarking that the shape of the oversizing curve is strictly related to the system features and to the expected deviations, the flexibility analysis procedure has nonetheless general validity.

The cost related remarks are the direct consequence of the combination of sizes related ones. The increase in investment costs as function of volumetric flexibility shows an almost linear trend. Since the equivalent probability distribution has a constant value, the F V index tendency results more similar to the F SG and RI indexes one, even if the domain considerations, the way we compute its value and the properties of the oversizing curves practically make it analogous to the SF index.

In the end we can conclude that a 5.5% additional cost (w.r.t. operating conditions) is definitely the maximum investment worth to be done according to a flexibility improvement purpose.

Conclusion

After these four flexibility analysis an indexes comparison is necessary to summarize and comment the relationships between all the results.

First of all we can mainly distinguish two typologies of indexes according to their approach towards the feasible domain: the indexes assessing the minimum of the maximum performance and the ones giving a global assessment within the uncertain or feasible domain.

The F SG and RI indexes are generally more conservative, they're based on n-D hyperpoligons that can be inscribed within the feasible domain, where n is the dimension of the uncertain zone (i.e. the number of possibly perturbated parameters). The first one is more suitable if the expected perturbation involves several variables at the same time, the second one for bigger variations referred to single variables. However, they're both very conservative (specially F SG ) since the described geometrical objects can only scale up or down but their structure is not flexible at all, it doesn't coast the domain, it just stops whether one point is tangent. This way two completely different systems could have the same flexibility index value just because they allow the same perturbation intensity for the most constraining parameter only; on the other hand given a certain starting design, a flexibility index defines univocally the system configuration.

On the other hand we have more optimistic indexes with nonnull value at operating conditions that require integration to be calculated: SF and F V . Even in this case one of them refers to an expected deviation while the other one needs an a priori knowledge or estimation of perturbation likelihood. F V results are similar to the linear ones until the index achieves its maximum value of 1 after whom every kind of oversizing is useless; on the other hand the SF index is very sensitive and smart, it takes into account all the possible deviations proportionally to their likelihood, showing an hyperbolic trend in oversizing/cost vs. flexibility plots that reflects well the real capability of the system to better withstand small and likely deviations than big and unlikely ones. Several configurations may have the same SF or F V index value since they assess a global system property, not the most constraining one; therefore during the flexibility assessment we could need to solve an optimization problem according to the analysis purpose.

In the end it's worth remarking that the value of F SG and F V strictly depends on the expected deviations, nevertheless F SG can be generalized and compared to the whole feasible domain as shown while F V cannot.

Then, from a design point of view we can definitely conclude that the most suitable flexibility index for the particular analyzed system is a function of the expected deviation nature.

An advice of general validity is to perform the flexibility analysis using more than one index, combining this way their advantages; for instance the first two indexes are quite easy to compute while the ones requiring an integral calculation have a higher computational effort demand. Thus we can perform the flexibility analysis with F SG or RI first in order to identify the most constraining variables in order to be able to reduce the dimension of the problem and perform the SF or F V analysis on a smaller domain. Obviously this is a good procedure if some parameters perturbations affect variables whose constraints are very loose; if the order of magnitude of all the F SGi or RI i is almost the same, this preliminary analysis is of little help since we cannot be sure that the most constraining variable for one index will be the most constraining for the other indexes as well unless much less relevant deviations are allowed with respect to the others.

In order to compare the four indexes more immediately, the Table 9 resuming all their main features has been outlined.

On the other hand, from an economic perspective, the introduction of the Capital costs vs. Flexibility relationship and its relative plots let the decision maker, i.e. the engineering, take a more informed decision whatever the adopted flexibility index. Moreover, it clearly shows how an a priori flexibility analysis during the

Fig. 1 .

 1 Fig. 1. Feasibility domain -F SG( Swaney and Grossmann, 1985a ).

Fig. 2 .

 2 Fig. 2. Feasibility domain -F SG vs RI[START_REF] Saboo | Design of resilient processing plants 0.8. a resilience index for heat-exchanger networks[END_REF].

Fig. 3 .

 3 Fig. 3. Feasibility domain -F SG vs SF ( Pistikopoulos and Mazzuchi, 1990 ).

Fig. 4 .

 4 Fig. 4. Feasibility domain -F V[START_REF] Lai | Process flexibility for multivariable systems[END_REF].

Fig. 5 .

 5 Fig. 5. Debutanizer column layout.

Fig. 7 .

 7 Fig. 7. Column max & min diameters ( F SG ).

Fig. 9 .

 9 Fig. 9. Capital costs comparison ( F SG ).

Fig. 11 .

 11 Fig. 11. Column max & min diameters ( RI ).

Fig. 12 .

 12 Fig. 12. Equipments bare module cost ( RI ).

Fig. 13 .

 13 Fig. 13. Capital costs comparison ( RI ).

Fig. 14 .

 14 Fig. 14. Condenser SF : Normal PDF.

Fig. 15 .

 15 Fig. 15. Condenser SF : Laplace PDF.

Fig. 16 .

 16 Fig. 16. 3D SF analysis domain.

Fig. 17 .

 17 Fig. 17. SF results normal PDF.

Fig. 18 .

 18 Fig. 18. SF results Laplace PDF.

Fig. 19 .

 19 Fig. 19. SF optimal sizing NDF.

Fig. 21

 21 Fig. 21. dC C v s. SF derivatives ratio.

Fig. 22 .

 22 Fig. 22. F V probability distribution function.

Fig. 23 .

 23 Fig. 23. 2D F V analysis.

Fig. 24 .

 24 Fig. 24. 3D F V analysis domain -operating conditions.

Fig. 25 .

 25 Fig. 25. F V analysis economic results.

Fig. 26 .

 26 Fig. 26. F V analysis optimal sizing.

Table 1

 1 Flexibility studies in literature.

	Case study	Authors	Index
	Pump and pipe	Grossmann and Morari (1983)	F SG

Table 2

 2 Feed conditions and composition.

	Partial molar flowrates	Value	Unit
	Propylene		0.055	mol/s
	Propane		0.053	mol/s
	Butane (lk)		6.863	mol/s
	Pentane (hk)		2.743	mol/s
	Temperature		Bubble	K
	Pressure		15 • 10 5	Pa
	Table 3
	Design parameters.
	Desing variables
	d	Symbol	Value	Unit
	Rectification stages	N r		9	1
	Stripping stages	N s		10	1
	Column diameter	D col		0.634	m
	Condenser area	A cond		40.00	m 2
	Reboiler area	A reb		26.83	m 2
	Top pressure	P top		4 • 10 5

Table 4

 4 Uncertain parameters θ k .

	Unit

Parameter θ Symbol Value θ N Expected deviation θ k ±

Table 5

 5 Flexibility analysis results: F SG .

	Deviation [%]	Condenser area [ m 2 ]	Reboiler area [ m 2 ]	Minimum diameter max [ m ]	Maximum diameter min [ m ]
	0.00	32.91	22.26	0.603	0.782
	5.00	38.08	24.59	0.634	0.744
	6.66	40.00	25.43	0.644	0.732
	9.35	43.35	26.83	0.662	0.712
	15.00	51.53	30.09	0.701	0.673
	21.00	62.30	34.05	0.746	0.632

Fig. 6. Heat transfer areas ( F SG ).

Table 6

 6 Critical vertices.

	Parameter vs design	Condenser area	Reboiler area	Minimum diameter	Maximum diameter
	Butane flowrate	+	+	+	-
	Pentane flowrate	+	+	+	-
	U Cond	-	/	/	/
	T w	+	/	/	/
	U Reb	/	-	/	/
	G f	/	/	-	/
	G w	/	/	/	+

Fig. 8. Equipments bare module cost ( F SG ).

Table 7

 7 Allowed disturbances loads.

	Parameter	Maximum deviation [%] l max + i	Minimum deviation [%] l max -i	Limiting constraint
	Butane flowrate	11.80	∞	D min
	Pentane flowrate	81.00	∞	D min
	Inlet cooling water temperature	18.28	∞	A cond
	Condenser heat transfer coefficient	∞	17.74	A cond
	Reboiler heat transfer coefficient	∞	17.02	A reb
	Max vapor velocity	∞	9.64	D min
	Min vapor velocity	52.18	∞	D max

Table 8

 8 Flexibility analysis results: RI .

	Deviation [%]	Condenser area [ m 2 ]	Reboiler area [ m 2 ]	Minimum diameter max [ m ]	Maximum diameter min [ m ]
	0.00	32.91	22.26	0.603	0.782
	9.64	36.42	24.64	0.634	0.745
	17.02	39.65	26.82	0.662	0,715
	17.74	40.00	27.06	0.665	0.712
	24.00	43.30	29.29	0.691	0.686

Fig. 10. Heat transfer areas ( RI ).

Appendix B. Capital costs estimations

In order to evaluate the investment cost required for the whole system or make any kind of economic consideration and comparison, we need to estimate the costs of every single equipment.

For this purpose the Guthrie-Ulrich-Navarrete correlations described in the next paragraphs will be used [START_REF] Guthrie | Capital cost estimating[END_REF][START_REF] Guthrie | Process Plant Estimating, Evaluation, and Control[END_REF][START_REF] Navarrete | Planning, Estimating, and Control of Chemical Construction Projects[END_REF][START_REF] Ulrich | A Guide to Chemical Engineering Process Design and Economics[END_REF].

B.1. Purchase equipment cost in base conditions

The purchase equipment cost in base conditions is obtained by mean of the following equation:

where A is the characteristic dimension ad the K i coefficients are relative to the equipment typology (cf. Table 10 ).

The provided coefficients refers to the year 2001 and to a M&S index equal to 1110. In order to update the costs value to the year 2016 we'll refer to a M&S index equal to 1245.2 by mean of the correlation:

. Bare module cost

The equipment bare module cost can be calculated according to the following correlation:

where the bare module factor is given by:

The F M and F P factors refers to the actual constructions materials and operating pressure while the B i coefficients refers to the equipment typology (cf. Table 11 ).

The F P,Kettle value is given by:

where P is the relative pressure in 10 5 Pascal.

For column trays bare module cost a slightly different correlation should be used:

where N is the real trays number, F BM = 1 e F q is given by the correlation:

C.1. Normal probability distribution function

As already mentioned, the condition of "general validity" is represented by the gaussian or normal probability distribution. It is symmetrical with respect to its mean and the 99.73% of cumulative probability falls in the range [ -3 σ, +3 σ ].

The single variable Normal PDF ( Fig. 27 ) states as:

where μ is the mean and σ is the variance. The bivariate Normal PDF ( Fig. 28 ), for a correlation between the variables ρ = 0 , states as:

where y is given by:

Finally the most general n-variables normal PDF with can be defined as:

where is the variance-covariance matrix, -1 its inverse and | | its determinant. Moreover we can standardize, i.e. reconduct to a 0 mean value and variance equal to 1 (variance-covariance matrix equal to the identity matrix), the normal distribution by mean of the independent variable substitution:

obtaining:

for a general n variables standard normal probability distribution [START_REF] Severini | Elements of Distribution Theory[END_REF]. This transformation besides making the calculations easier allows to compare variables with different dimensions, e.g. temperature vs. flowrate vs. velocity etc.

The boundaries of the feasibility domain, if analytically available, have then to be rewritten as functions of the new variable z by inverting the Eq. (C.5) .

C.2. Laplace distribution function

The second distribution function used for the stochastic flexibility analysis is the so called Laplace PDF.

This distribution satisfies, in a sense, the same requirements needed by the system description, that is:

• The maximum probability is attained at the operating conditions; • the probability value is not dependent on the deviation direction.

The analytical expression of the single variable Laplace PDF ( Figure 29 ) states as:

where μ is the mean and b is the diversity local parameter.

Even in this case we can standardize the distribution, i.e. reconduct to a 0 mean value and diversity parameter equal to 1 by mean of the independent variable substitution: the trivariate Laplace PDF (for the bivariate cf. Fig. 30 ), for a correlation between the variables ρ = 0 , states as:

The Laplace distribution function converges more slowly than the normal distribution, therefore we expect a lower flexibility increase by increasing the sizing, i.e. the costs.

Supplementary material

Supplementary material associated with this article can be found, in the online version, at doi: 10.1016/j.compchemeng.2019. 02.004 .