Probing vasoocclusion phenomena in sickle cell anemia via mesoscopic simulations

Shear Flow System.

Experiments in ref. 16 have used an in vitro shear flow system to show that the light density cells exhibit larger adhesion, whereas the densest ISCs correspond to the least adhesive group. However, the following fundamental question remains unresolved: Are the different adhesive responses a consequence of the different adhesive proteins on the cell membranes, or rather of the biomechanical properties of the different cell groups? Kaul et al. (6) explored this issue by examining the alteration of SS-RBC adhesivity after a dehydration/rehydration treatment on individual cell groups and found that the cell adhesivity of the deformable SS2 and the dense SS4 cells can be reversed after controlled treatment. They hypothesized that both light- and high-density cells have similar “adhesion potential,” whereas the heterogeneous cell adhesivity is mainly due to the differences in cell deformability and shape peculiarities among the multiple cell groups. To investigate the validity of this hypothesis, we simulated the cell adhesive dynamics of SS-RBCs with different cell rigidity and morphologic characteristics.

We consider three distinct SS-RBCs under shear flow, as shown in Fig. 1. Following ref. 6, we assume that the three cells share similar “adhesive potential,” and hence we set identical adhesive parameters in all three simulations. Cell I represents a SS2 deformable discocyte of cell rigidity similar to the healthy RBC. We set the shear modulus and bending rigidity , i.e., similar to healthy cell rigidity (17⇓–19). Cell III represents an ISC of the SS4 type. Experimental measurements (20, 21) of cell rigidity report scattered values, which further depend on the deoxygenation–oxygenation cycle. For example, an ektacytometer study (21) showed that the cell rigidity of oxygenated ISC is ∼12 times larger than a healthy cell, whereas a micropipette study (20) showed that the cell rigidity of the SS4 group is about 2–3 times larger than the healthy cell; however, this value increases to 20 times the healthy value after one deoxygenation–reoxygenation cycle. In the present work, we set the shear modulus , because an ISC undergoes at least one deoxygenation–reoxygenation cycle. The modification to cell bending rigidity by the deoxygenation–reoxygenation process is unknown; here we set and also conduct sensitivity studies. Cell II represents a rigid discocyte cell (22) of SS3 type with medium MCHC value; we set and for the comparative study.

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Fig. 1.

Sickle cells in shear flow: (A) Successive snapshots of SS-RBCs in shear flow, Movie S1. Labels I, II, and III correspond to a deformable SS2 cell, rigid SS3 cell, and ISC, respectively. Arrow indicates the flow direction. (B and C) Instantaneous contact area and velocity for SS-RBC in shear flow conditions.

Fig. 1A shows successive snapshots of the three cells. For the same shear rate and same membrane adhesive characteristics, the three cells exhibit substantially different adhesive dynamics (Movie S1). Cell I shows firm adhesion to the lower plate after the bond formation stage; the corresponding contact area (see SI Text for definition) is about , as shown in Fig. 1B. Cell II shows transient adhesion to the lower plate initially, but then undergoes a periodic flip movement along the flow direction, characterized by the peak values of the instantaneous cell velocity at 0.30, 0.53 and s. This indicates a weaker cell adhesivity than cell I. Accordingly, the contact area achieves minimum values at those times. The cell eventually detaches from the plate after two to three flips. Differently from cells I and II, cell III does not show firm/transient adhesion to the plate; instead, it directly detaches from the lower plate and moves freely without adhesive bonds established thereafter. Given the similar adhesive potential, the present results validate the hypothesis of heterogeneous cell adhesive dynamics under shear flow conditions. To further quantify the effect of the cell rigidity and morphology on adhesive properties, next we investigate cell adhesion in static conditions by using free-energy analysis.

Static Conditions.

Fig. 2 shows the instantaneous contact area between the cell and the plate for the three SS-RBCs. As in the shear flow conditions, the contact area shows an inverse relationship with the cell rigidity. Whereas the contact area for all of the three cells increases to within the initial stage, the contact areas between the plate and cells I and II further increase to and , respectively, at the final stage. These results can be understood by analyzing the change of free energy due to the adhesive interaction, i.e.,where represents the free-energy change due to the cell deformation, and represents the free-energy change due to the adhesive bond formation between the cell and adhesive ligands. In the present system can be approximated by , where represents the change of contact area and β is the energy coefficient determined by the adhesive affinity. Fig. 2B shows for different adhesive affinities (therefore yielding different contact areas). The numerical results are fitted by , where α depends on the cell rigidity of individual cells. The equilibrium state is determined by the balance between the two energy terms, e.g., , leading to different contact areas. For SS2 cells (corresponding to small α), the adhesive interaction plays a dominant role in the initial stage, resulting in cell deformation and further increase of the contact area. In contrast, a rigid SS4 cell exhibits “solid”-like properties with a larger energy barrier for cell deformation. After the initial stage, the adhesive interaction driven by the bond formation cannot overcome the free-energy increase induced by the cell deformation and therefore prohibits further increase of contact area.

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Fig. 2.

Adhesion in static conditions: (A) Instantaneous contact area between the SS-RBC and the plate coated with adhesive ligands. Black curve represents the contact area of a discocyte with cell rigidity similar to the ISC (cell III). (B) Increase of cell free energy as a function of the contact area. (C) Adhesive force between the cell and the plate as a function of the membrane rigidities for cell morphologies. Error bar represents the adhesive force computed from four independent simulations. (Inset) Sketch of the simulation setup, where a uniform lift force is applied on the upper part of an ISC.

Compared with the shear flow system, we note that the equilibrium contact area obtained in the static condition is smaller. This discrepancy is mainly due to the greater cell deformation under the shear flow condition, i.e., part of the deformation free energy is balanced by the hydrodynamic force exerted on the cell membrane under the shear flow condition. The extended cell membrane facilitates the bond formation and results in a larger contact area. Moreover, compared with the ISC, a discocyte with similar cell rigidity yields a larger contact area in equilibrium state, as shown in Fig. 2A. This result demonstrates that the peculiar elongated and curved characteristics of ISC may further prevent the membrane receptors from interacting with the ligands.

To quantify the effects of shape and rigidity on cell adhesion, we compute the adhesive force between the cell and plate for different cases. Starting from the steady state obtained from the static condition, we apply uniformly a lift force to the upper part of the cell membrane (Fig. 2C, Inset). The adhesive force is determined as the lift force that drives the cell detaching from the plate in a quasistatic way. The deformable SS2 cells exhibit the largest adhesive force of , whereas this value decreases to 34 pN in the case of rigid cells with and . Compared with the discocyte, the ISC with similar cell rigidity yields an adhesive force of 17 pN. For smaller cell rigidity (corresponding to the rehydration treatment of ISC in ref. 6) with and , the adhesive force is approximately 24 pN, compared with 43 pN for a discocyte with similar cell rigidity. In a recent experimental study (23), force spectroscopy was used to quantify the cytoadherence of RBC invaded by a Plasmodium falciparum parasite. It was reported that the rupture force between the invaded RBC and the CHO cells is within the range . This result is similar to the adhesive force magnitude for the RBC–endothelium interaction reported in the present work.

SS-RBC Suspension in Postcapillary Flow.

Single ISCs may occasionally result in occlusion at precapillary junctions (1, 5, 24); however, most of the occlusion sites are in postcapillaries and exhibit a specific cell pattern, where the adherent deformable cells form a sieve-like configuration and selectively trap the ISCs. We model blood circulation by SS-RBC suspensions first in a tube of diameter , as shown in Fig. 3; small green particles represent the adhesive ligands coated on the tube wall. To quantify the distinct role of different cell groups, we infuse different cell groups into the tube and apply a pressure gradient .

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Fig. 3.

Vasoocclusion in postcapillaries: instantaneous mean velocity of blood flow in a cylindrical tube of infused with different SS-RBC suspensions. Red curve represents the resultant velocity infused with SS2 and ISC cell groups. (Inset) Instantaneous snapshots where SS2 cells adhere to vessel wall, consequently trapping the ISCs and resulting in cell blockage. Green curve represents the blood velocity infused with SS2 and ISC cell groups, where adhesive interaction is only applied to the ISC group. The time axis is scaled by 0.5 for better visualization. The Inset represents a snapshot where transient adhesion is established between ISC and the tube wall. Steady flow is recovered as the cell detaches from the tube wall. Blue curve represents the instantaneous velocity of blood flow infused with SS2 and healthy cell groups. Blood flow exhibits a slowdown but not a full occlusion; Movies S2, S3, and S4.

First, we consider suspensions composed of SS2 cells (labeled by blue) and ISC cells (labeled by red) with hematocrit , similar to experiments (5, 6). Steady flow is achieved by turning off the adhesive interaction during the initial stage, yielding a mean flow velocity of . Starting from the steady state , we turn on the adhesive interaction between the SS-RBCs and the ligand particles and compute the instantaneous flow velocity across the tube, as represented by the red curve in Fig. 3. Steady flow is maintained until one of the SS2 cells adheres to the vessel wall, triggering a sharp decrease of blood flow around . As a positive feedback, the decreased blood flow induces more SS2 cells adherent to the vessel wall, leading to a further decrease of flow rate at and . Moreover, these adherent cells decrease the effective vessel lumen near the adherent region, resulting in a secondary trapping of the ISC groups. The final occlusion state is achieved around with cell patterns similar to the experimental observations (5, 6). We have also performed another three simulations starting from different initial conditions, and in all cases we obtained the same final pattern of full occlusion.

To explore the unique contribution of the deformable cell group, we performed a similar simulation of blood flow where the SS2 cell–ligand interaction is turned off. Starting from the steady flow at , blood flow is simulated for 6 s, as represented by the green line in Fig. 3. Due to the ISC–ligand interaction, transient adhesive bonds can be formed, resulting in the decrease of blood flow at . However, firm adhesion cannot be established: blood flow can recover the initial flow rate when the adherent ISCs detach from the tube wall. Blood occlusion is not observed within the simulation time, in agreement with the experimental observation (6) that the ISCs do not result in consistent microvascular blockage when infused alone. This finding reveals the distinct role of the SS2 cell in the vasoocclusion crisis and is consistent with the positive correlation between the SS-RBC deformability index and disease severity reported in clinical observations (9). The ISC cell group, in contrast, contributes differently to the occlusion crisis. We quantify its unique contribution by simulating the blood suspension mixed with the deformable SS2 and healthy cells. Starting from the steady flow at , we compute the mean velocity with the adhesive interaction between the SS2 and ligands, as represented by the blue line in Fig. 3. The flow velocity decreases due to the cell adhesion at and . Although blood flow exhibits a substantial slowdown, full occlusion is not realized due to the high deformability of the healthy RBCs, which enables the cells to squeeze through the sieve-like region formed by adherent SS2 cells. The present result, in turn, reveals the distinct role of the ISC group in the vasoocclusion process. Although the least adhesive, the ISC group, due to its high cell rigidity and peculiar shape, serves as the particular cell group trapped by adherent cells in the postcapillaries. This finding explains the experimental observation that a large number of dense cells accumulate in the occlusion region but are not observed in the peripheral blood circulation (5).

To quantify the effect of the vascular size on occlusion, we have performed similar simulations but in tubes of different diameters. For , occlusion does not occur with tube diameter larger than (Fig. S5); the sieve-like pattern formed by SS2 cells cannot fully trap the ISCs due to the larger spatial accommodation near the adhesion region. However, for , occlusion is observed with but occlusion is prevented with tube diameter larger than (see Fig. S6 and SI Text for more details).

Effect of Inflammation-Stimulated Leukocytes.

Recent studies (25) have shown that SCA is often accompanied by an inflammatory endothelial phenotype, resulting in elevated leukocyte recruitment in blood circulation. Moreover, studies by Turhan et al. (12) in transgenic-knockout mice have shown that the inflammation-stimulated (by cytokine TNF-α) adherent leukocytes, upon interactions with SS-RBCs, lead to occlusion in venular flow. We use our computational framework to investigate the vasoocclusion induced by inflammation-stimulated leukocytes (Methods). We first consider blood flow of SS-RBC suspension with a single leukocyte in a tube of diameter , as shown in Fig. 4. Due to cell migration, the leukocyte touches the wall at and firm adhesion is established thereafter. The mean flow velocity drops from to during this stage. During the inflammation-stimulated stage, we turn on the cell aggregation interaction between the SS-RBCs and leukocytes (see SI Text for details). Multiple SS-RBCs get trapped by the leukocyte, resulting in full occlusion at . For venular flow with larger diameter, multiple leukocytes may accumulate in the inflammation region. Similar to the experiment (12), we simulate a SS-RBC suspension with three leukocytes in a tube of diameter and compute the mean flow velocity during different time intervals, as shown in Fig. 4. Stage I represents the blood flow during the initial stage of the inflammatory response. Starting from the steady flow at , the mean flow velocity drops to due to the attachment of leukoctyes on the vessel wall at s. Stage II represents the blood flow with moderate RBC–leukocyte interaction; the blood flow exhibits a slowdown due to the adherent leukocytes. Stage III represents the late stage of the inflammatory response, where the RBC–leukoctye interaction is further intensified. Multiple SS-RBCs are trapped on the adherent leukocytes, leading to full occlusion at (Movies S5, S6, S7, S8, S9, and S10).

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Fig. 4.

Effect of leukocytes: instantaneous mean velocity of the blood flow in a tube of , one leukocyte (A) and , , three leukocytes (B). (Inset) Snapshots represent blood cells in free motion, leukocyte adhesion, and blood occlusion states. For , the Inset plot represents the blood flow velocity of the present study (blue) and the experimental results (red) (12), where measurements are taken on venules with average diameter and before and after inflammation–stimulation; Movies S5, S6, S7, S8, S9, and S10.

Discussion.

Our simulations of individual SS-RBCs in shear flow validated the hypothesis of the importance of cell rigidity and morphology (5, 6, 16), and elucidated the distinct behavior of each individual cell group under adhesive conditions. This behavior was further quantified by computing the adhesive force and free -response for all cell groups. In simulations of postcapillary SS-RBC flow, we identified which combination of cell groups and at what adhesion strengths lead to full or transient occlusion states. Under typical physiological conditions, our sensitivity studies with respect to the tube size and hematocrit show that blood occlusion mainly occurs in postcapillaries with diameter smaller than , a finding consistent with the experimental results reported in ref. 5. Further free energy analysis on cell adhesion (Fig. S7) and discussion on the important dimensionless groups is included in SI Text. Moreover, under inflammation-stimulated conditions, the present work quantified the influence of the adherent leukocytes, which may further trap the SS-RBC and lead to vasoocclusion in venular flow, especially in larger vessels. In particular, our simulations revealed and quantified the multiinteractional and multistage nature of vasoocclusion, which may be useful in developing effective biomarkers (26) and therapeutic treatments for SCA. In general, our computational framework provides a capability, which, in conjunction with microfluidic experiments, can be used in the preliminary screening of proposed drugs and the interpretation of clinical outcomes.