Supercritical Fluid Extraction (SFE) is a green process for the recovery of compounds from a solid matrix using supercritical fluids as solvents. Fluids are in supercritical state when their temperature and pressure are above critical point (T_c, P_c). They demonstrate properties such as low viscosity, density comparable to that of liquids, gas-like diffusion and near zero surface tension. Under these conditions, the extraction capacity of many compounds increases, therefore, supercritical fluids become a suitable solvent for a variety of applications ^[1][14]. The most commonly used solvent for SFE is supercritical CO₂ thanks to its low critical temperature (31.1 °C) and lack of toxicity, which allows the extraction of thermolabile compounds. Moreover, Sc-CO₂ is non-flammable, readily available, cost-effective and can be removed from the extracts by expansion to ambient conditions without any further processing, due to its gaseous state under atmospheric temperature and pressure ^[2][3][9,11]. Apart from that, in the supercritical region, solubility increases with the increase in density, which allows the regulation of selectivity by adjusting extraction conditions, such as temperature and pressure. For highly polar compounds, modifiers, such as alcohols, can be used in order to enhance the solubility. Furthermore, the yield and the selectivity of the process can be improved by the use of co-solvents. The above properties generate a highly selective extraction technique, resulting in extracts with high purity ^[3][11].

Apart from γ-linolenic acid, which is the compound extracted in the majority of SFE applications, Arthrospira (Spirulina) can also provide extracts with high concentrations of carotenoids. Specifically, Canela et al. have recovered 2.27 mg/0.8 kg algae per extraction bead, at the optimal extraction conditions, namely a temperature of 30 °C, 18 MPa pressure and 11 hours extraction time ^[4][27]. Temperature, in that study, varied from 20 to 70 °C and pressure from 15 to 18 MPa. Valderrama et al. have achieved 3% phycocyanine yield and more than 97% astaxanthin recovery by extracting A. maxima strains at 60 °C and 30 MPa, both with and without the use of 10% w/w ethanol ^[5][28]. Similarly, experiments at 40–80 °C, 15–35 MPa and 5–15% v/v ethanol led to 48 mg/100 g_biomass zeaxanthin, 7.5 mg/100 g_biomass cryptoxanthin and 118 mg/100 g_biomass β-carotene yield at 35 MPa and 15% v/v ethanol ^[6][29]. Also, in another study, the maximum amount of 283 μg/g_biomass total carotenoids and 5.01 μg/g_biomass total tocopherols have been recovered from A. platensis at 60 °C and 450 bar with 53.22% v/v ethanol ^[7][30]. SFE on pretreated A. platensis, also, resulted in extract composed of approximately 290 ppm zeaxanthin, 73 ppm myxoxanthophyl fucoside, 55 ppm β-carotene and 535 ppm chlorophyll a with antioxidant activity close to 70 μg/mL (EC₅₀) ^[8][34]. Additionally, Wang et al. have extracted at 48 °C, 20 MPa using ethanol as entrainer, 77.8 g β-carotene/kg_biomass, 113.2 g vitamin a /kg_biomass, 3.4 g α-tocopherol /kg_biomass and 85.1 g flavonoids /kg_biomass ^[9][35]. Finally, 6.84 mg/g_biomass chlorophyll a was recovered from A. platensis at 53.4 °C and 48.7 MPa with 40% aq. ethanol ^[10][37].

Chlorella cultures can be used as a source of carotenoids, such as astaxanthin, canthaxanthin, lutein and β-carotene, chlorophylls and phenolic compounds. The extraction conditions, along with the use of co-solvent, can alter the extract’s composition of bioactive compounds and, thus, their antioxidant activity.

Kitada et al. have studied the effect of pressure, temperature and co-solvent on the carotenoid extraction from C. vulgaris ^[11][59]. Specifically, at 70 °C, 2.5 mL/min flow rate and 300 min extraction time, the lutein extracted was 0.13, 0.46, 0.40 and 0.61 mg/g_biomass at 20, 30, 40 and 50 MPa, respectively. The increase in temperature at a constant pressure of 30 MPa, increased the recovered lutein from 0.46 at 60 °C to 0.57 mg/g at 80 °C. The use of ethanol as co-solvent presented generally better results compared to acetone under the same conditions. Namely, 1.54 mg/g_biomass lutein, 0.13 mg/g_biomass β-carotene, 11.43 mg/g_biomass α-chlorophyll and 3.90 mg/g_biomass β-chlorophyll were recovered with ethanol and 0.94 mg/g_biomass lutein, 0.01 mg/g_biomass β-carotene, 3.30 mg/g_biomass α-chlorophyll and 0.59 mg/g_biomass β-chlorophyll were recovered with acetone. Similarly, another study indicated that the increase in pressure at 40 °C led to higher lutein recoveries. More explicitly, at 20 MPa, 1.34% lutein recovery was achieved, at 30 MPa 1.64% and at 40 MPa 1.78% ^[12][64]. Temperature increase seemed to present the opposite effect at 40 MPa, by decreasing lutein recovery to 0.67% at 80 °C ^[12][64]. The flow rate of ethanol as entrainer resulted in 1.78% lutein recovery at 0.3 mL/min, in 1.80% at 0.4 mL/min and in 1.68% at 0.5 mL/min ^[12][64]. Gouveia et al. using extraction conditions of 40 °C, 30.0 MPa and 0.0397 kg/h Sc-CO₂, have reported maximum total carotenoid recovery of 69.1% for completely crushed C. vulgaris cells without the use of co-solvent, while when mixed with oil and with double the flow rate the recovery obtained was 16.6% ^[13][58]. Fairly crushed and slightly crushed cells without the use of entrainers led to a recovery of 37.3% and 17.4%, respectively. Different co-solvents showed little impact on the carotenoid recovery since 19.7% was achieved with oil and 20.2% with ethanol. Safi et al. accomplished better results in overall extract characterization for bead milled C. vulgaris biomass by increasing pressure from 35 MPa to 60 MPa ^[14][52]. In terms of total mass recovered, at 60 MPa pressure 10.64% yield was achieved, in contrast to 9.7% at 35 MPa. Total carotenoids and total chlorophylls reached 60 MPa 1.72 mg/g_{dry biomass} and 1.61 mg/g_{dry biomass}, respectively.

Mendes et al. have investigated the effect of three operational conditions (temperature, pressure and pretreatment) on the carotenoid recovery ^[15][24]. The optimum carotenoid recovery for crude C. vulgaris, almost 500 mg/kg_{dry algae}, was achieved at maximum temperature and pressure, i.e., 55 °C and 35 MPa. From the three degrees of crushing, whole, slightly, and well crushed, the second presented analogous results with the third, approximately 40% total carotenoids yield, but with larger requirements of Sc-CO₂. In a similar study, under the same extraction conditions, best results were derived for the most intense extraction conditions for both crude and pretreated biomass, i.e., 171.1 mg carotenoids per 100 g oil and 0.05% w/w carotenoid yield ^[16][17][61,62]. Hu et al. have carried out an orthogonal experimental design that consisted of 16 experiments, where each factor consisted of four levels, in order to examine the effect of five factors (temperature, pressure, duration, Sc-CO₂ flow rate and co-solvent quantity) on extraction yield and antioxidant capacity ^[18][46]. Yield reached its maximum value, 7.78%, at 32 °C, 40 MPa, 20 kg/h Sc-CO₂ flow rate, 180 min and 1 mL ethanol per gram of C. pyrenoidosa. The inhibition at those conditions was 42.03%, while the optimum was 54.16% with 3.50% yield at 40 °C, 35 MPa, 20 kg/h Sc-CO₂ flow rate, 150 min and 1.5 mL/g ethanol. Consequently, the most effective parameters were pressure for yield and modifier for antioxidant activity. Georgiopoulou et al. studied the SFE of C. vulgaris and specifically the effect of temperature, pressure and solvent flow rate on total extraction yield, antioxidant activity, total phenolic content and target carotenoid compounds, by applying experimental design ^[19][66]. The experiment under the optimum conditions (60 °C, 250 bar and 40 g Sc-CO₂/min) resulted in 3.37% yield, 44.35 mg_extr/mg_DPPH antioxidant activity using an IC₅₀ assay, total phenolic content equal to 18.29 mg gallic acid/g_extract, 35.55 mg/g_extract total chlorophyll content, 21.14 and 10.00 mg/g_extract total and selected carotenoid content, respectively. Furthermore, the addition of 10% w/w ethanol as entrainer enhanced antioxidant activity and yield. Wang et al. investigated the properties of the extract obtained by the SFE of Chlorella at 50 °C, 31 MPa, 6 Nl/min and the use of 50% aqueous ethanol ^[20][65]. The total polyphenol content of the extract was 13.40 mg_GAE/g_extract, while the total flavonoid content was 3.18 mg_QE/g_extract. The inhibition value in the DPPH assay was 47.24% compared to gallic acid’s 100% inhibition. In other research, in which experimental design was employed, the recovery of lutein from superfine pulverized C. pyrenoidosa with the use of ethanol as entrainer, reached its maximum value, 87.0% extraction yield. The conditions of that experiment were 50 °C, 25 MPa, 240 min duration and 50% w/v ethanol ^[21][47].

Haematococcus pluvialis has gained significant research interest due its high content of natural astaxanthin ^[22][158]. Yothipitak et al. have estimated that the recovery of astaxanthin could reach 22.66 mg/g_biomass by SFE at high pressure and temperature (64 MPa and 90 °C) ^[23][80]. SFE, with or without the use of co-solvent, appears to be an adequate technique for astaxanthin extraction, reaching, in certain cases, more than 80% recovery. Extraction of lyophilized H. pluvialis at 45 °C, 48.3 MPa and 2.7 mL/min Sc-CO₂ flow rate, led to almost 85% astaxanthin recovery ^[24][85]. Likewise, 83% recovery, equal to 22.84 mg/g_biomass, was achieved at slightly higher pressure and flow rate (50 MPa and 3 mL/min) and 80 °C ^[25][81]. Moreover, ethanol as co-solvent has been widely investigated. Bustamante et al. recovered 84% of biomass astaxanthin at 40 °C and 55 MPa with the addition of 4.5 v/v ethanol ^[26][82] and, correspondingly, Pan et al. recovered 73.9% by using 9.23 mL/g_biomass of aqueous ethanol under moderate conditions ^[27][83]. Similar studies of SFE at 70 °C and 40 MPa with 5% v/v ethanol led to 80.6% astaxanthin recovery ^[28][87], while at 65 °C, 43.5 MPa with 2.3 mL/g ethanol and at 55 °C, 20 MPa with 13% w/w ethanol, the recovery obtained was 87.4% and 82.3%, respectively ^[29][30][89,90]. SFE of powdered biomass resulted in 61% astaxanthin recovery at 70 °C and 55 MPa ^[31][84], while SFE of lyophilized and crushed H. pluvialis with 9.4% w/w ethanol as co-solvent led to a recovery of 92% of total carotenoids, 76% of β-carotene and 90% of astaxanthin ^[32][86]. Dried H. pluvialis extraction with 10% v/v olive oil as co-solvent under optimum conditions (70 °C, 40 MPa) resulted in 51% recovery of available astaxanthin ^[33][88]. Finally, extraction of red phase Haematococcus at 65 °C and 55 MPa resulted in high astaxanthin and lutein recoveries, 92–98.6% and 52.3–93%, respectively ^[34][35][91,92]

Supercritical fluid extraction of N. gaditana at 60 °C, 40 MPa and 4.5 mmol/min flowrate led to the recovery of 0.343 μg/mg_biomass total carotenoids and 2.238 μg/mg_biomass chlorophyll a ^[36][96] while at 50 MPa, 2.893 μg/mg_biomass total carotenoids, 0.369 μg/mg_biomass chlorophyll a and almost 0.33% total carotenoid yield were obtained ^[37][38][75,76]. Sánchez-Camargo et al. extracted from the same species 0.18 mg/g_biomass (8.3% recovery) violaxanthin at 55 °C and 40 MPa ^[39][97]. Zeaxanthin extraction from N. oculata was, also, carried out leading to 63.2% recovery and 13.7 mg/g_exract ^[40][101]. Lastly, SFE on Nannochloropsis sp. biomass at 40 °C and 30 MPa, with the addition of 20% w/w ethanol resulted in an extract composed of 13.71% astaxanthin, 22.35% lutein, 13.20% violaxantin and neoxanthin, 34.3% vaucheriaxanthin, 4.71% canthaxanthin, 5.08% β-carotene and 3.37% chlorophyll a ^[41][105].

Scenedesmus cells contain both carotenoids and chlorophylls that can be recovered by SFE with or without the use of co-solvent ^[42][159]. A lutein recovery has been reported for S. almeriansis of 0.0466 mg/g_biomass at 60 °C, 400 bar and extraction duration of 300 min ^[43][111]. Also, for the same species, another study reports a recovery of 2.97 mg/ g_biomass of lutein for a shorter extraction time, but increased temperature and pressure, i.e., 65 °C and 550 bar ^[44][112]. The addition of a polar co-solvent in the SFE could affect the extraction of the target compounds by increasing the solvent’s polarity, and therefore, their solubility in the medium ^[45][160]. Indeed, the lutein yield seemed to have been augmented from 0.206 mg/g_biomass to 2.210 mg/g_biomass by adding 30% v/v ethanol maintaining the same temperature, pressure and time ^[46][118]. Similarly, the yield increased from 0.2105 mg/g_biomass lutein to 0.4361 mg/g_biomass with the addition of 10% v/v ethanol ^[47][119]. Remarkably, the extraction conditions which lead to the maximization of the lutein yield does not always match with the most intense ones. The same phenomenon is observed for β-carotene and lutein extraction, which both reach their maximum recovery (1.5 mg/g_biomass and 0.047 mg/g_biomass, respectively) at 60 °C, 400 bar and 300 minutes total extraction ^[43][111]. In this case, co-solvent contribution seems to be not so intense, since the use of 10% v/v ethanol led to the increase in the extracted β-carotene from 0.0547 mg to 0.0599 mg per dry biomass ^[47][119]. As a result, the best total carotenoid recovery does not occur under very intense extraction conditions. For example, SFEat 40 °C, 400 bar, and 2 h duration resulted in a recovery equal to 48.39 mg/g_extract and 0.303 mg/g_biomass at 250 bar, the same temperature and double duration ^[48][49][114,131]. Additionally, more carotenoids were detected, such as astaxanthin, neoxanthin, violaxanthin and zeaxanthin, and the recovery of all of them, except for violaxanthin appeared to increase with the use of co-solvent ^[47][119].

In terms of chlorophylls, they seem to have similar behavior to carotenoids. At 50 °C, 250 bar, and extraction time equal to 120 min, 15.68 mg/g_extract of chlorophylls were recovered ^[48][114]. Chlorophyll a is extracted in larger quantities in contrast to chlorophyll c. For example, Guedes et al. extracted 0.848 mg/g_biomass of chlorophyll a while chlorophyll b and c quantities obtained were 0.356 mg/g_biomass and 0.018 mg/g_biomass, respectively ^[49][131].

The extraction yields reported in the various studies show significant diversity, possibly due to different species, different cultivation and different SFE conditions. The species obliquus presents the lowest yields among them all. The highest cited is 8.3% at 20 °C, 120 bar and 540 min total extraction time ^[50][2]. Also, SFE at 40 °C, 400 bar for 120 min resulted in 1.15% yield as reported by Gilbert-López et al. ^[48][114], while Choi et al. obtained a yield of 4.20% under almost the same conditions ^[51][115]. By the addition of 15% v/v ethanol as co-solvent, the latter yield was increased to 14.51% ^[51][115]. However, other research presented a 0.247% yield with 5% v/v ethanol at 65 °C, 300 bar and for 90 min, which deviates significantly from the results of the other researchers ^[52][44].

The SFE of the species almeriensis at 60 °C, 400 bar and 120 min total extraction time, led to 1.50% yield ^[44][112]. Similarly, SFE at 45 °C, 300 bar and 90 min with the addition of 5% v/v ethanol resulted in 19.4% yield ^[53][93]. The extraction of species of obtusiusculus at 20 °C, 120 bar and 540 min resulted in a yield of 6.4% ^[50][2]. Ultimately, SFE of unspecified Scenedesmus species led to yields up to 6.81% ^[54][120].

In addition to the species mentioned above, Dunaliella salina cultures are also a major carotenoid and chlorophyll source. Specifically, extraction carried out at 40 MPa and 60 °C recovered 12.17 μg/mg_biomass carotenoids and 0.227 μg/mg_biomass chlorophylls ^[55][74]. By using 5% mol ethanol as co-solvent, under the same conditions, the yield altered to 9.629 μg/mg_biomass carotenoids and 0.700 μg/mg_biomass chlorophylls ^[38][76]. Similarly, Pour Hosseini et al., at slightly lower temperature and without co-solvent, obtained 115.44 μg/g_biomass total carotenoids and 32.68 μg/g_biomass chlorophylls ^[56][77]. Under milder conditions, namely 45 °C and 20 MPa with 5% w/w ethanol, Molino et al. recovered 25.5% of β-carotene from D. salina ^[57][78]. Total carotenoid content was also determined at 27.5 °C, 44.2 MPa and 45 °C, 20 MPa and found to be equal to 7.2 mg/100 g_extract and 25 g/kg_biomass, respectively ^[58][59][72,79].

SFE of Chlrococcum littorale recovered 89% of extractable carotenoids and 48% of chlorophylls ^[60][69], while SFE of Isochrysis galbana at 50 °C and 30 MPa led to the recovery of 16.2 mg/g_biomass carotenoids and 4.5 mg/g chlorophylls ^[61][94]. Chatterjee et al. determined that the total carotenoid content of P. valderianum was equal to 13.43 μg β-carotene equivalent/g_biomass at 50 °C and 50 MPa ^[62][110]. Fujii extracted from Monoraphidium sp. 2.46 mg/g_biomass astaxanthin, which is equal to 101% recovery, by using 20 mL ethanol as entrainer at 60 °C and 20 MPa ^[63][95].

Lastly, carotenoids such as β-carotene, β-cryptoxanthin and zeaxanthin were recovered from Synechococcus sp. Explicitly, maximum recovery 71.6%, 90.3% and 36.4%, of β-carotene, β-cryptoxanthin and zeaxanthin, respectively, was achieved ^[64][122]. Additionally, the SFE at 40 °C,40 MPa and 5% mol ethanol performed by Cardoso et al., resulted in 20.35 mg/g_extract β-carotene and 25.96 mg/g_extract zeaxanthin ^[65][17]. The addition of ethanol as co-solvent appears to have a positive effect on the pigment extraction. Macías-Sánchez et al., by using 5% mol ethanol under the same extraction conditions, achieved an increase from 1.51 to 1.86 μg/g_biomass in carotenoid recovery and from 0.078 to 0.286 μg/g_biomass in chlorophyll recovery ^[38][66][76,123].

The most common fatty acid extracted through SFE from Arthrospira cultures is GLA and, in general, an alcohol as co-solvent is used. GLA yield equal to 0.44% was achieved by conducting SFE of A. maxima at 60 °C, 35 MPa, 2 g/min solvent flow rate and 10% v/v ethanol ^[15][67][68][24,25,26]. Sajilata et al. recovered 102% GLA from A. platensis at 40 °C, 40 MPa and 0.7 L/min Sc-CO₂ flow rate ^[69][32], while other research on the same species, presented 24.7% recovery at 40 °C and 30 MPa with 50% v/v ethanol ^[70][31]. Total fatty acid content was, also, determined. Andrich et al. by performing SFE of A. platensis at 55 °C and 70 MPa obtained a total FA content equal to approximately 40% ^[71][39]. At lower pressure, slightly increased temperature and with 53.22% v/v ethanol as co-solvent, Esquivel-Hernandez et al. recovered from the latter species, 34.76 mg/g_biomass fatty acid ^[7][30]. Qiuhui et al. determined the FA composition of A. platensis extract derived from extraction at 40 °C, 35 MPa and 24 kg/h solvent flow rate ^[72][33]. Specifically, the extract consisted of 16.91% oleic acid, 36.51% linolic acid, 16% α-linolenic acid and 19.68% γ-linolenic acid. Similarly, SFE with ethanol under optimum conditions, 48 °C and 20 MPa, led to the following extract composition: 35.32% palmitic acid, 21.66% α-linolenic acid and 20.58% linoleic acid ^[9][35]. Finally, Mendiola et al. examined the effect of temperature, pressure and the use of co-solvent on palmitic and oleic acid recovery from A. platensis ^[73][36].

Solana et al. studied the composition of the extracts derived from SFE of C. protothecoides at 60 °C, 30 MPa and 5% ethanol, which consisted of 25.68% saturated fatty acids, 13.12% monounsaturated fatty acids, 61.77% polyunsaturated fatty acids, 15.13% Ω-3 and 23.53% Ω-6 ^[52][44]. Extraction of C. vulgaris at 40 °C and 37 MPa, with a mixture of hexane and ethanol as co-solvents, led to extracts composed of 30.05% palmitic acid, 30.22% stearic acid, 3.24% lauric acid, 4.82% myristic acid, 3.01% arachidic acid, 2.54% palmitoleic acid, 3.38% oleic acid, 1.63% linoleic acid, 1.71% docosahexaenoic acid and 2.98% eicosapentanoic acid ^[74][67]. Alhattab et al., by performing SFE of C. saccharophila at 73 °C and 24.1 MPa recovered extracts composed of 20.4% total FAME ^[75][48]. Microwave pretreated C. vulgaris, submitted to SFE at 70 °C and 28 MPa, led to 26.589 mg palmitic acid/ 100 mg_oil, 27.296 mg oleic acid /100 mg_oil, 10.403 mg linoleic acid /100 mg_oil and 16.163 mg α-linoleic acid /100 mg_oil ^[76][57].

Lipid recovery from Chlorella by applying SFE was mainly conducted with the use of co-solvent. In detail, SFE of Chlorella sp. with 5% ethanol at 60 °C and 30 MPa led to 79.53% lipid yield ^[77][54]. Also, at lower pressure while using 0.4 mL/min hexane, lipid yield was determined as 63.78% ^[78][53]. Moradi-kheibari et al. recovered from C. vulgaris 6.68% lipids at 45 °C, 35 MPa and 10% v/w ethanol ^[79][60]. For the same species, with 10% v/v ethanol, 97% of neutral lipids, approximately 25% of glycolipids and 35% phospholipids were recovered at 50 °C and 25 MPa ^[80][63]. Finally, Mendes et al. extracted 54.26 mg/g_biomass lipids from C. vulgaris at 55 °C and 35 MPa ^[17][62].

The SFE of fatty acids from N. gaditana’s were also studied. Molino et al. at 65 °C and 25 MPa recovered approximately 7.5 mg/g_biomass SFAs, 8 mg/g_biomass MUFAs, 10.5 mg/g_biomass PUFAs, 11.50 mg/g_biomass EPAs, while lipid yield was 34.15 mg/g_biomass ^[81][98]. SFE of N. oculata at 40 °C and 20.7 MPa resulted in extracts composed of 35% total SFAs, 45.31% MUFAs and 19.69% PUFAs ^[82][103]. FAME yield from N. granulata reached 18.23 mg/g_biomass at 70 °C and 35 MPa ^[83][99], while in another study for the same species and conditions, crude lipid yield reached 256.3 g/kg_biomass ^[84][100]. Crampon et al. at 60 °C and 40 MPa obtained an extract from N. oculata composed of 93.82% triglycerides and 1.80% sterols ^[85][86][38,102]. Finally, fatty acid composition of Nannochloropsis sp. extracts obtained at 40 °C and 30 MPa was found to be as follows: 25.3% SFAs, 20.1% monoenoic acid, 54.6% PUFAs ^[87][104].

The EFA with the highest concentration in the lipid extracts of Scenedesmus by SFE was found to be α-linolenic acid (ALA). Specifically, for the species obliquus, when extracted at 45 °C and 150 bar for 30 minutes, the percentage of ALA in the extracted lipids reached 21.47% ^[52][44], while in other research it was found to be equal to 28.44% by conducting extraction at 20 °C and 120 bar for 540 min total extraction time ^[50][2]. The concentration of LA in the aforementioned cases was 10.33% and 10.21%, respectively. It should be noted that the optimum extraction conditions, regarding the highest concentration of ALA and LA in the extracts, coincide. Contrariwise, an almost four times higher concentration of LA compared to ALA in S. obliquus extracts obtained by SFE at 40 °C and 379 bar is reported ^[51][115]. Moreover, for the species obstusiusculus, less ALA and LA were recovered in comparison with obliquus under the same conditions ^[50][2]. S. almeriensis extracts, in contrary to other species, contain 2.9% LA while no ALA was detected. However, these extracts contained more EPA (7.9%) compared to those of obliquus and obstusiusculus species which had less than 0.59% ^[53][93].

SFE of B. braunii at 50 °C and 25 MPa resulted in an approximately 18% yield ^[88][41]. Halim et al. have extracted from Chlorococcum sp. a 1.4% FAME yield ^[89][68]. Lyophilized C. cohnii, when extracted with SFE, led to extracts composed of 72% DHA ^[90][71]. Molino et al. recovered 8.47 mg/g_biomass FAME (97.07% recovery) from D. salina at 75 °C and 55 MPa ^[57][78]. Additionally, lipid yield of SFE of Ochromonas danica reached 234.2 mg/g_biomass at 40 °C and 17.2 MPa ^[91][107].

The mathematical modeling of SFE in solid matrixes provides valuable information about the course of extraction. Using as independent variables, the operational conditions, such models describe the progress of the extraction over time, making the optimization and the simulation of the process possible ^[92][93][161,162]. The solid particles are usually depicted as spheres or cylinders and the mass transfer phenomena occurring in the biomass can be described by linear driving force models, shrinking core models, broken plus intact cell models and the combination of the latter ^[93][162]. Some hypotheses can be made in order to simplify the kinetic models, such as immobilized cells with constant density and porosity and isothermal and isobaric conditions in the extractor ^[93][162].

This model based on Lack’s plug flow model was proposed by Sovová and co-workers ^[92][94][161,163], and assumes that cell walls function as an additional resistance to the extraction of the solute. Grinding of the biomass results in disrupted and intact cells where the solute transfers to the supercritical phase through convection and molecular diffusion, respectively ^[93][162]. The extract primarily gets exhausted from the broken cells and gradually from the intact, resulting in three mass transfer periods. Initially, the extraction rate increases constantly and then falls progressively, ending up in a diffusion controlled period ^[95][164]. Sovová’s kinetic model was applied successfully in the SFE of various microalgal biomasses. Specifically, Mouahid et al. employed it for the SFE of Arthrospira platensis, Chlorella vulgaris, Cylindrotheca closterium and Nannochloropsis oculata ^[85][38] and Hernández et al. for Isochrysis sp., Nannochloropsis gaditana, Tetraselmis sp. and Scenedesmus almeriansis ^[53][93]. Solana et al. have studied the extraction kinetics of Chlorella protothecoides, Nannochloropsis salina and Scenedesmus obliquus ^[52][44]. Other studies involve Chlorella vulgaris ^[19][96][55,66], Haematococcus pluvialis ^[26][82] and Nannochloropsis gaditana ^[39][97].

Apart from models such as the linear driving force model (LDF) and shrinking core model, desorption, solubility based on Fick’s diffusion law models are often employed for the description of the SFE process on microalga. Examined species are A. maxima and A. platensis ^[4][67][25,27], C. protothecoides ^[97][43], Chlorococcum sp., Synechococcus sp., D. salina, N. gadiatana ^[37][75] and Nannochloropsis sp. ^[87][104].