## 1 Introduction  
Microalgae perform photosynthesis using only carbon dioxide (CO\({}_{2}\)) and light as carbon and energy sources, respectively [1; 2; 3]. Unlike traditional crops, microalgae do not require arable land and can utilize wastewater and CO\({}_{2}\)-rich exhaust gases, making them an environmentally sustainable and versatile resource for various industries, including functional foods, pharmaceuticals, animal feed, biofuels, biochemicals, and CO\({}_{2}\) mitigation [4; 5]. Astaxanthin is one of the valuable compounds produced by microalgae. Astaxanthin is a secondary ketocarotenoid pigment that is particularly notable for its powerful antioxidant properties, which surpass those of \(\beta\)-carotene and vitamin C [6; 7]. Astaxanthin has wide applications in the aquaculture, cosmetics, pharmaceutical, nutraceutical, and food sectors [8; 9]. Currently, most astaxanthin is produced through chemical synthesis, but concerns about the safety of synthetic astaxanthin for human consumption have driven interest in natural sources [10; 11]. The green microalga _Haematococcus Iacustis_ (formally known as _H. pluvialis_; [12]) is extensively cultivated for its high astaxanthin content, which can reach up to 5% of its dry weight, to meet the growing global demand for natural astaxanthin [7; 13]. The life cycle of _H. lacustris_ is complex, encompassing three distinct stages: cyst germination, vegetative cell growth, and encystment, the latter being associated with astaxanthin production [14; 15]. Astaxanthin biosynthesis is triggered by reactive oxygen species (ROS) signaling, serving as a defense mechanism against unfavorable conditions [7; 16]. However, the photosynthetic cultivation of _H. lacustris_ is time-consuming, often requiring more than two months [17]. Strategies explored to enhance astaxanthin productivity include inducing cyst germination with higher nitrate concentrations [18; 19] or blue light-emitting diode (LED) light [20]. Additionally, conventional photosynthetic parameters, such as CO\({}_{2}\) supply, light intensity, pH, temperature, and media composition, have been extensively optimized [6; 21; 22]. A range of abiotic stresses, including intense light [23], nutrient deficiency [24], high salinity [25], elevated temperatures [26], the addition of aminolay nanoparticles [27], phytohormone supplementation [28], hydrostatic pressure [29], electrical treatment [30], and ultrasonication [31], have also been employed to stimulate encystment and enhance astaxanthin production in _H. lacustris_. However, many of these methods are costly, energy-intensive, or involve chemicals unsuitable for human and animal consumption. Consequently, developing effective and environmentally friendly strategies to enhance microalgal astaxanthin production remains a significant challenge for large-scale applications [16; 32]. Mechanostimulation, which involves applying mechanical or physical stimuli that include pressure, tension, compression, and shear stress, has emerged as a promising approach [33; 34]. These mechanical stresses on the cell membrane can influence various biological processes, including cytoskeleton dynamics, protein conformation, cell division, cell cycle progression, gene expression, and protein synthesis [35; 36]. Notably, mechanostimulation offers environmental benefits for microalgal applications, operating under ambient conditions without the need for chemical additives. For example, [PERSON] et al. [35] reported enhanced cell division in the green microalga _Chlamydomonas reinhardtii_ under microfluidic compression, which led to increased production of biflagellate daughter cells after 96 h. However, the effects of mechanical stress can vary significantly depending on the organism, the intensity of the stress, and the specific methodology employed [37]. For instance, [PERSON] et al. [38] and [PERSON] et al. [39] developed a microfabricated culture platform that physically compressed vegetative biflagellate cells of _H. lacustris_.