A Biophysics-Based Mathematical Model of Shear-Induced Platelet Activation and Receptor Shedding: Re-examining Previous Experimental Data

Abstract

The power-law model, originally developed for shear-induced hemolysis, has been employed to predict shear-induced platelet activation and receptor shedding. However, its empirical nature lacks mechanistic explanations and violates physical reality by not imposing an upper limit, often leading to inaccuracies. Recent studies suggest that mechanical pulling of platelet GPIb-IX complex triggers the unfolding of its mechanosensitive domain, a crucial process to platelet activation, which can be explained by Bell’s model of bond unbinding under force. Motivated by these findings, we propose a novel mathematical model for shear-induced platelet activation (P-selectin) and shear-induced platelet receptor (GPIbα, GPVI, and GPIIb/IIIa) shedding based on the principle of bond unbinding. The model was examined using experimental data from previous studies in which blood samples were exposed to different combinations of constant shear stress and exposure time. The new model demonstrated an excellent fit with experimental data with an overall coefficient of determination R² > 0.8, mapping the trends in platelet activation and receptor shedding (except for GPIIb/IIIa) across a range of shear conditions. This new model not only addresses the intrinsic upper bound error in the power-law model, but also provides a theoretical foundation into blood damage under shear stress.

1. Introduction

Severe postoperative complications, such as hemolysis, hemostasis disorder, and infections remain a critical concern in mechanical circulatory support (MCS) and a major obstacle to its broader application.^1,2 Device-induced blood damage to various blood components, including red blood cell (RBC), platelet, leukocyte and von Willebrand factor (VWF), is one of the main factors contributing to these complications in patients.^3–5 Shear stress and exposure time act as pivotal factors in device-induced blood damage, especially in high shear blood-contacting medical devices, such as rotary blood pumps in extracorporeal membrane oxygenation systems, ventricular assistant devices, and heart-lung machines for cardiopulmonary bypass.^6,7 To understand blood damage in MCS and improve the biocompatibility of these devices during the design and development stages, the conventional power-law model was initially developed to relate the hemolysis index (HI) to shear stress (τ) and exposure time (t), i.e., $\text{HI}=\text{A}{\tau }^{\alpha }{\text{t}}^{\beta }$.⁸ While useful as an empirical tool, this model lacks a fundamental physical reasoning necessary to explain the underlying mechanisms of hemolysis, which has often been criticized.⁹ Another inherent flaw in the power-law model lies in its mathematical structure. When the shear stress or exposure time exceeds a certain threshold, the HI value can exceed 1. This implies that the extent of hemolysis can go beyond complete damage, which is a clear violation of physical reality and can lead to problematic predications compared to experimental data.

Recent studies have shifted the focus toward platelet defects in these devices, including platelet activation and receptor shedding, due to increasing concern about hemostasis disorders, as platelets play a critical role in physiological hemostasis and pathological thrombosis.^10–12 Shear-induced platelet activation is closely associated with device-associated thrombosis while shear-induced receptor shedding is closely linked to bleeding in MCS patients.^5,13 The power-law models have, therefore, been adapted to estimate these shear-induced platelet defects.^11,14,15 However, predicting of shear-induced platelet injury and related hemostatic complications is more challenging due to the behavior of platelets undergoing complex biochemical and mechanical processes. The prediction often overestimates the experimental data.^16,17 In addition, unlike the stable membrane of the RBC, platelets would have a drastic disk-to-sphere morphological change upon activation,^18,19 limiting the development of an accurate structural model for platelets to implement the conventional strain-based Lagrangian approach to evaluate platelet activation.^20,21 While multiscale platelet models based on molecular dynamics or other particle-based methods have been proposed that relate platelet deformation under shear stress to activation and adhesion,^22,23 this approach is computationally expensive and challenging to integrate into conventional computational fluid dynamics (CFD) simulations at the device scale. In addition, apart from the power-law model, no models, to the authors’ knowledge, have been proposed to specifically address shear-induced platelet receptor shedding in the literature.

In this work, we propose a new mathematical model of shear-induced platelet activation and receptor shedding based on recent findings characterizing platelet behavior that highly likely to contribute their activation. The rationale is built on the hypothesis that shear stress, acts on the platelet key receptor and causes the unbinding of its protein bond, which triggers either platelet activation or receptor shedding. The probability of such unbinding may thus further represent the overall platelet activation or receptor shedding. We examined the model by using experimental data from previous tests conducted under various combinations of constant shear stress and exposure time.

2. Materials & Methods

2.1. Bell’s model for molecular bond binding and unbinding

Bell’s model is a foundational theoretical framework initially used to analyze cellular adhesion or the attachment of cells to surfaces, where adhesion is mediated by reversible bonds between specific molecules.²⁴ It has since been widely extended to describe the rates of bond formation (binding or attachment) and bond dissociation (unbinding or detachment) between two molecules under applied force,²⁵ see the schematic illustration in Figure 1 (a). Unlike the deterministic approach of macroscopic classical Newtonian mechanics, Bell’s law implies for the probabilistic behavior of molecular bonds under force over time. The rates of bond attachment and detachment between two molecules under force, as described by Bell’s law, are,

$$k(F)=\left\{\begin{array}{ll}{k}_0\hfill & \text{attach}\hfill \\ k_0\text{exp}\left(F/F_0\right)\hfill & \text{detach}\hfill \end{array}\right.,$$ (1)

where $k_0$ (unit in s⁻¹) is the unbinding rate under zero external force, which may be related to the characteristics of the two molecules. $F_0=k_{B}T/\lambda$ is the characteristic bond force, where $k_B$ is the Boltzmann constant, T is the temperature, and $\lambda$ is the mechanosensitivity of the characteristic bond separation distance. Each unbinding is modeled stochastically using Bell’s model, and the unbinding probability (P) for each bond under a constant force (F) within a time interval (t) equals to,²⁶

$$P\left(F,t\right)=1-exp\left[-k_{0}exp\left(F/F_0\right)t\right]$$ (2)

Figure 1.

Schematic illustration of the foundational mechanisms assumed in the new model. (a) Bell’s model of molecular bond binding and unbinding under force. (b) Sequence of events in shear-induced platelet activation. (c) Mechanism of shear-induced platelet receptor shedding. MSD-mechanosensitive domain; VWF- von Willebrand factor.

2.2. Hypothesis of Shear-induced Platelet Activation and Receptor Shedding

2.2.1. Platelet activation

a). VWF interaction with GPIba.

Recent studies have increasingly supported interaction between VWF and the glycoprotein Ibα (GPIbα) receptor as a key factor in shear-induced platelet activation. Studies demonstrated that blocking the interaction between the VWF A1 domain and platelet GPIbα significantly suppresses the level of platelet activation, with only minimal activation observed even under extreme shear conditions^27,28.

b). GPIb-IX-V mechanosensation.

In 2015, Zhang et al. identified a juxtamembrane mechanosensitive domain (MSD) within the GPIb-IX complex.²⁹ Their research showed that the pulling of the VWF, when its A1 domain is engaged with GPIb-IX, induces the unfolding of the MSD in GPIbα, potentially contributing to platelet mechanosensing and the shear resistance of the VWF-platelet interaction. Of note, in their study, the most probable MSD unfolding force as a function of loading rate was well-fitted to the Bell-Evans’ model. Since they used a constant loading rate (i.e., $F=R_{f}t$) of the applied force instead of a constant force.²⁵

c). Platelet activation and express of P-selectin.

Zhang et al. further suggested that MSD unfolding may induce a conformational change in the adjacent extracellular domains of GPIbβ and GPIX, transmitting a signal across the platelet membrane that eventually leads to intracellular events and final platelet activation, which results in its morphological transition from a discoid to a spherical shape and the expression of key activation biomarkers, e.g., P-selectin and PAC-1. Studies have showed that the platelet marginal band, composed of a bundle of microtubules, plays a significant role in this process by the action of motor proteins along the microtubules, leading to the coiling of the marginal band.^30,31 This coiling or collapse of the marginal band corresponds with platelet activation and its shape transition from a discoid to a sphere.

d). Shear-induced platelet activation.

We hypothesize that shear-induced platelet activation and the expression of key biomarkers, such as P-selectin, is resulted from a series of events (Figure 1 (b)): VWF unfolding under shear, VWF A1 domain anchoring to GPIbα, GPIb-IX MSD unfolding, transmembrane signaling, marginal band coiling, and ultimately platelet activation. While all steps undoubtedly contribute more or less to the platelet activation in this chain reaction, if one step, for example, MSD unfolding, is the most critical one, then it is reasonable to assume that the probability of a single MSD unfolding represents the likelihood of all GPIb-IX MSDs unfolding within a platelet, assuming that each GPIb-IX MSD unfolding is an independent event. Furthermore, this probability can also be mathematically equated to the overall platelet activation level in the total population.

2.2.2. Platelet receptor shedding

With the hypothesis that the unfolding of critical domains of MSD in GPIb-IX is governed by the unbinding kinetics described by Bell’s law in shear-induced platelet activation, it is straightforward to propose the same kinetics on shear induced receptor shedding. Instead of unfolding, receptor shedding can be directly influenced by shear forces under extreme flow conditions. This suggests that shear forces act on receptors, e.g., GPIbα, directly break the critical protein bonds (i.e, unbinding) that maintain its integrity, leading to receptor shedding, as shown in the schematic illustration in Figure 1 (c). Consequently, shear-induced platelet receptor shedding could be understood through the same probabilistic framework under Bell’s model, as described in Eq. (2).

2.3. New model of shear induced platelet activation and receptor shedding

According to the hypothesis described above, the macroscale levels of shear-induced platelet activation and receptor shedding are critically governed by the unbinding of key molecular bonds under microscale; therefore, their levels can be described by the function in Eq. (2). In fact, this unbinding probability in Eq. (2) is directly related to external force and time interval, which are the same two essential factors—shear stress and exposure time—that are well known to cause platelet activation and receptor shedding under shear stress. However, it is important to clarify that shear stress is not directly equivalent to the tensile force applied to the receptors. Conventionally, shear stress employed in power-law model is defined as a Von Mises-like scalar stress, representing the overall magnitude of the applied forces.⁹

$$\tau ={\left[\frac{1}{6}\sum {\left({\tau }_{ii}-{\tau }_{jj}\right)}^2+\sum {\tau }_{ij}^2\right]}^{1/2}$$ (3)

${\tau }_{ij}$ is the shear stress tensor that is the product of shear rate (${\gamma }_{ij}$) and blood viscosity ($\mu$). To utilize the probability function in Bell’s model, we need to relate the shear stress to the exerted tensile force on the bonds, i.e., $F=\varnothing \left(\tau \right)$. For simplicity, we can assume a linear relationship $F=a\tau$. By defining a new coefficient ${\tau }_0=F_0/a$, the new model for shear-induced platelet defects (activation or receptor shedding) can be expressed as

$$PD\left(\tau ,t\right)=1-exp\left[-k_{0}exp\left(\tau /{\tau }_0\right)t\right]$$ (4)

3. Results

In our previous work, we performed experimental studies to derive the coefficients of the power-law models for hemolysis, platelet activation, and receptor shedding in human blood, using our novel shearing devices (hemolyzer).^14,15 A schematic illustration of the experiment is shown in Figure 2. These devices feature a rotating rotor within a narrow gap, allowing for a uniform Couette flow characterized by constant high shear stresses and short exposure times.³² The original experimental data were collected to fit the power-law model.^14,15 The experiment included shear stress ranges from 25 to 350 Pa and exposure times from 0.039 to 1.5 s, covering the typical shearing conditions observed in clinical devices such as axial and centrifugal pumps in ventricular assist devices. Reusing those data, we performed a preliminary examination of the new mathematical models for shear-induced platelet activation, focusing on P-selectin expression, and shear-induced platelet receptor shedding for GPIbα, GPVI, and GPIIb/IIIa. In addition to the non-robust method, we also included two robust methods, least absolute residuals (LAR) and Bisquare, to assess potential outlier data. Table 1 presents the coefficient of determination (R²) and root mean squared error (RMSE) for each method, along with the coefficient of determination for the power-law models from previous publications for reference. Figure 3 shows the experimental data points and the fitted surfaces generated by the new model for platelet activation, indicated by P-selectin (Figure 3 (a)), and receptor shedding, including GPIbα (Figure 3 (b)), GPVI (Figure 3 (c)), and GPIIb/IIIa (Figure 3 (d)).

Figure 2.

Schematic setup of the hemolyzer experiment used to assess blood damage under constant shear stress and exposure time.

Table 1.

Parameters obtained from fitting the new model to experimental data for shear-induced platelet activation and receptor shedding.

PDI	$k_0$	${\tau }_0$	Robust method	R2	RMSE	Power law R2
P-selectin	0.04318	148.61	Off	0.7742	0.0590	0.7963
	0.02856	129.47	LAR	0.9909	0.0119
	0.03205	134.57	Bisquare	0.8535	0.0475
GPIbα	0.05546	126.74	Off	0.8154	0.0554	0.77
	0.04506	116.04	LAR	0.8087	0.0564
	0.05319	126.06	Bisquare	0.8153	0.0554
GPVI	0.05226	110.77	Off	0.8043	0.0716	0.73
	0.04360	102.44	LAR	0.8005	0.0723
	0.04057	99.60	Bisquare	0.8323	0.0662
GPIIb/IIIa	0.05172	135.23	Off	0.6408	0.0693	0.79
	0.04887	143.66	LAR	0.6037	0.0728
	0.04457	127.58	Bisquare	0.5970	0.0734

PDI-platelet damage index; R²-Coefficient of determination; RMSE-root mean squared error.

Figure 3.

Fitting of the new model to experimental data without applying any robust method. (a) Platelet activation (P-selectin). (b) GPIbα receptor shedding. (c) GPVI receptor shedding. (d) GPIIb/IIIa receptor shedding.

For P-selectin, fitting with robust methods, such as LAR and Bisquare, significantly improves the fit due to the larger data set, which is reflected in a higher coefficient of determination and a lower RMSE. For the receptor shedding, the fits are close regardless of the robust method used. All fits for P-selectin, GPIbα and GPVI using the new model show excellent results, with an overall R² > 0.8, and show improved fit compared to power-law model. GPIIb/IIIa, on the other hand, does not fit well by all methods, with R² < 0.65. This discrepancy is likely due to the fact that, unlike the GPIbα and GPVI, which exist in a fixed number of copies on the platelet surface that can only be shed, the GPIIb/IIIa is also affected by its increased expression during (shear-induced) platelet activation due to externalization.^33,34 While we reference the coefficient of determination from the power-law model, it is important to note that comparing the fit of the new model to the power-law model may be unnecessary, as one model possesses a physical basis while the other does not.

4. Discussion

In this work, a new biophysics-based mathematical model for shear-induced platelet activation and receptor shedding has been developed. We hypothesized that the shear stress acts on the platelet, either directly or indirectly, leading to the unbinding of a key protein molecular bond that subsequently governs platelet activation or receptor shedding. This model offers a comprehensive explanation for shear-induced platelet activation and receptor shedding and effectively fits data from previous experiments. Unlike the conventional power-law models, the parameters in the new model possess physical significance. It also addresses the upper bound issue present in conventional power-law models, which suggests that the platelet activation level, for example, would potentially exceed beyond fully activation, when shear stress or exposure time reaches a certain value. When applied to conventional CFD in the device scale, the new model can be transformed into an implicit form by taking the derivative with respect to time in Eq. (4), yielding the expression of $dPD\left(\tau ,t\right)/dt=k_0\mathit{\text{exp}}\left(\tau /{\tau }_0\right)\left(1-PD\left(\tau ,t\right)\right)$. Notably, this form has a strong resemblance to the expression proposed by Soares et al., where they proposed a formulation to account for a sensitization response of platelets during shear-induced activation.³⁵ However, their model was derived from the observation that the platelet activation rate tapers off as activation progresses and did not explicitly address the underlying biophysical mechanisms.

While we propose that shear-induced platelet activation is primarily due to the pulling of VWF bound to GPIbα receptors, leading to the unfolding the MSD and subsequent activation processes, we shall also mention other historically proposed alternative mechanisms for shear-induced platelet activation. For example, high shear forces, acting on platelets, are likely to expose and activate the membrane GPIIb/IIIa complex, which is initially hidden by other surface molecules in unstimulated platelets.³⁶ Some emerging mechanisms, including mechano-destruction, mechano-activation, and mechano-transduction, offer alternative explanations independent of VWF-GPIbα binding.³⁷ In these theories, shear stress activates platelets by modulating the material properties of the platelet, such as overall stiffness and membrane fluidity, or recognizing platelet-activating biochemical signals. However, to the best of the authors’ knowledge, no specific quantitative formula for platelet activation has been developed based on these alternative mechanisms. Moreover, even fewer models exist for shear-induced platelet receptor shedding.

Several aspects of the new model remain open for further investigation. For example, (1) there is currently no direct evidence to support our assumption that receptor detachment conforms to the Bell’s model. (2) The exact relationship between the shear stress and the resulting tensile force needs to be clarified. In this study, we have simplistically assumed a linear correlation to facilitate the analysis. (3) It is important to note that the tensile force is not necessarily constant. In the hemolyzer experiment, the exposure time was long enough for the shear force to be considered constant. In medical devices, however, platelets are exposed to constantly fluctuating shear forces. Whether they can respond immediately to such transient force should be investigated also in future studies. (4) The new model has intrinsic limitations. Once it reaches full activation, for instance, it remains fully activated indefinitely. However, platelets can undergo apoptosis synchronously, leading to exhaustion and consequently a reduction in activation levels over time.³⁸ Furthermore, the new model suggests that the platelet defects may inevitably increase with time even in the absence of shear stress, which may not be true in low shear scenarios, e.g., physiological conditions. This discrepancy could result from the implicit premise that the probability of unbinding dominates over that of binding under high shear conditions. It may need to consider the probability of re-binding at low shear as described in Eq. (1), which could play an important role in preventing platelet defects under low shear conditions and in modulating platelet function throughout their life span.

While there is no doubt that the above aspects necessitate further classification, we provide a physically realistic framework to quantify shear-induced platelet activation and receptor shedding in this work. This model is adequate for current purpose.

5. Conclusion

In this paper, we proposed a new mathematical model for shear-induced platelet activation and receptor shedding based on the biophysical rationale that these phenomena at the macroscale are the result of the probability of unbinding of the key protein bonds of platelet receptors under tensile force at the microscale. The proposed model fits well to the previously collected data on platelet activation (P-selectin) and receptor shedding of GPIbα and GPVI in blood shearing experiment with constant shear stress and exposure time.