Sickle cells (RBCs)~citenst{pauling1949sickle,ingram1957gene}. In the mutated form

Sickle cell disease (SCD) is a genetic blood disorder affecting millions of people throughout the world. It is caused by the mutation of a single nucleotide in the gene for hemoglobin, a group of proteins that bind oxygen molecules in the red blood cells (RBCs)~citenst{pauling1949sickle,ingram1957gene}. In the mutated form of hemoglobin (HbS), the native charged surface group, glutamic acid, at $eta$6 site is replaced by a hydrophobic group, valine, inducing polymerization of HbS molecules into long and stiff fibers when deoxygenated. These HbS fibers can distort RBCs into heterogeneous shapes, including elongated, granular, oval, holly-leaf,  and crescent (classic sickle) shapes, depending on the intracellular mean corpuscular hemoglobinconcentration (MCHC) deoxygenation rates~citenst{noguchi1981intracellular,eaton1990sickle,ferrone2004polymerization}. In addition to the morphological changes, sickle RBCs are featured with increased stiffness and compromised cell deformability due to the presence of HbS fibers~citenst{Barabino_ARBE_2010, Li_JBE_2017}. HbS inside the RBCs also damages the membrane of RBCs, resulting in enhanced RBC adhesion to vascular endothelium ~citenst{hebbel2008adhesion, kaul2009sickle}. Both the impaired RBCdeformability and enhanced cell adhesion contribute to  the initiation and propagation of the vaso-occlusion (VOC) events, a hallmark of SCD. Recurrent and unpredictable episodes ofVOC in SCD leads to significant morbidity and reduced quality of life as a result of stroke, frequent painful crisis events, and other serious complications, such acute chest syndrome, splenic sequestration and aplastic crises~citenst{brandao2003optical,lande1988incidence,chien1987red,kaul1989microvascular,frenette2007sickle,kaul2004vivo}. Sickling of RBCs is initiated with supersaturation of the deoxy HbS. The HbS nuclei form after a latency period, also known as delay time. The delay time is an inverse function of MCHC from the 30th to the 50th power, meaning that sickle RBCs withhigh MCHC has stronger propensity to HbS polymerization~cite{ferrone2015delay}. Following the formation of the nuclei, HbS moleucles rapidly polymerize into fibers. The polymerization of HbS molecules has been well described by the double nucleation mechanism~citenst{Ferrone1985,Vekilov2007}, which suggests that the nuclei can form either from HbS monomers (homogeneous nucleation), or on the surface of existing fibers (heterogeneous nucleation). When MCHC is high, homogeneous nucleation overwhelms the heterogeneous nucleation, leading to the formation of multiple fiber domains with randomly oriented fibers~citenst{eaton1987hemoglobin}. In contrast, HbS monomers polymerize primarily through the heterogeneous nucleation pathway at low HbS concentrations and thus tend to create single elongated spherulitic domains~citenst{Briehl1995}. Sickle RBCs have been categorized into four groups (fractions I-IV) on the basis of MCHC~cite{kaul1983erythrocytes}. Fraction I mainly consists of reticulocytes withthe least MCHC, which assume multi-spiculated appearance after deoxygenation. Discocytes with MCHC similar to healthy RBCs prevail in fraction II and they exhibit typical sickle shapes once deoxygenated. Fraction III is also primarily comprised of discocytes but the cells are more rigid than those in fraction II due to the high MCHC.Fraction IV mainly contains irreversible sickle RBCs with high MCHC induced by cell dehydration. It was originally considered that the irreversible sickle RBCs play the key role in triggering VOC. However, subsequent studies indicate that each group of sickle RBCs contributes differently to VOC events~cite{kaul1989microvascular,kaul1994adhesion,kaul2004vivo}. Although irreversible sickle RBCs alone may occasionally cause occlusion at precapillary junctions, VOC mainly occurs in postcapillaries where low-MCHC sickle reticulocytes and RBCs form a sieve-like structure to trap the high-MCHC, irreversible sickle RBCs, leading to reduced blood flow and eventual VOC~cite{kaul1989microvascular,kaul2009sickle}. More recent investigations point out that other blood cell components may also contribute to VOC by stimulating inflammation in the microvasculature, which promote adhesion of leukocytesto the endothelium and sickle RBCs~cite{turhan2002primary,madigan2006pathophysiology,wallace2009nkt}.In past decades, significant progresses have been made in elucidating the underlying mechanism of VOC in SCD ~cite{kurantsin1988vaso,frenette2002sickle,conran2009newer,kaul2009sickle,manwani2013vaso}. However, little effort is devoted to probe the critical condition that triggers the VOC as it is pertinent to a complex interplay of patient-specific factors, such as distribution of intracellular HbS concentration, concentrations of HbF and HbA, partial oxygen pressure and other polymerization-independent factors.