Correction

Main Text

2010. A multiscale model of venous thrombus formation with surface-mediated control of blood coagulation cascade. Biophys. J. 98:1723–1732.

The following section should be added after the “Coagulation pathway submodel” section of our article:

Clarification of the relation of the biochemical reactions submodel to previous models

The model from Hockin et al. (14) is an extension of one of the first computational models predicting thrombin production of the TF-initiated procoagulant pathway in vitro developed by Jones and Mann (12) by including stoichiometric inhibitor systems such as the tissue factor pathway inhibitor-mediated inactivation of tissue factor and active factor VII complex and its product complexes. The ordinary differential equations (ODEs) model in Hockin et al. (14) includes an intermediate stage in the activation of prothrombin to thrombin: meizothrombin as a part of the reaction network that is not represented in the ODE models in Kuharsky and Fogelson (17) and Fogelson and Tania (26). Meizothrombin is an intermediate product generated by one pathway for the conversion of prothrombin to thrombin. In Edwin et al. (31), experimental data clearly demonstrates the importance of meizothrombin and, therefore, it is important to consider its significance when modeling the coagulation reactions in the blood. The model in Hockin et al. (14) captured most experimentally observable parameters and accurately described pathology that results in enhanced or deficient thrombin generation (14).

The general idea of including membrane-phase reactions in coagulation models was introduced and demonstrated in Kuharsky and Fogelson (17) using the ODEs model (see reference to (17) in “Biological and modeling background” section). In our study, we extended the specific model of the coagulation network from Hockin et al. (14), which is different from the one described in Kuharsky and Fogelson (17), by incorporating both solution-phase and membrane-phase reactions with concentrations of membrane-binding sites as described in Kuharsky and Fogelson (17) and by using new assumptions, components, and submodel implementation to generate a multiscale model predicting thrombus development both in time and space.

As stated previously in our article, the main novelty of our modeling approach is in describing in detail production of thrombin on the membrane of each individual platelet (modeled as an extended object with a finite volume and stochastically fluctuating membrane by the CPM) and in coupling the complete set of model components at different space and time scales into a multiscale model. Each platelet or other blood cell is carried by blood flow modeled by a detailed fluid dynamics PDEs model (Navier-Stokes equation and Darcy's law equation) and may be incorporated into a thrombus. We focus on modeling space structure of thrombi developing under different conditions, both in time and space, using PDEs and ODEs coupled with a cell-based stochastic discrete model (CPM) for blood cells (see Fig. 1) and comparing simulations with experimental data obtained by our group. Most of the components in our model are spatially dependent, and they are described by PDEs and stochastic discrete dynamical systems (CPMs) compared to ODEs in the models from Hockin et al. (14), Kuharsky and Fogelson (17), and Fogelson and Tania (26). Therefore, behavior of all components in the system of equations is different compared to Hockin et al. (14), Kuharsky and Fogelson (17), and Fogelson and Tania (26) since we take into account both time and space variables and because all components of the model are linked together through stochastic CPM. The ODE model in Hockin et al. (14) includes meizothrombin as a part of the reaction network. We also specifically model its production by introducing two separate equations (4.25) and (4.45), one for meizothrombin in the plasma flow and the other for meizothrombin on platelet membranes. Namely, the volume concentration of thrombin, p₂ is a sum of binding sites for platelet-bound thrombin (IIa), prothrombin (II), and meizothrombin (mIIa), and their complex $p_2=\left[{\text{II}}_{\text{a}}^{\text{m}}\right]+\left[{\text{II}}^{\text{m}}\right]+\left[{\text{m}\text{II}}_{\text{a}}^{\text{m}}\right]+\left[{\text{Xa:Va:II}}^{\text{m}}\right]$ and the volume concentration $e_2^{\text{mtot}}$ is a sum of active platelet-bound thrombin and meizothrombin $e_2^{\text{mtot}}=\left[{\text{II}}_{\text{a}}^{\text{m}}\right]+\left[{\text{m}\text{II}}_{\text{a}}^{\text{m}}\right]$ (see the Supporting Material).

The concept that spatial separation between the site of coagulation factor activation and site of activity affects the dynamics of thrombus development was suggested by Roberts et al. (32) in a cell-based model of coagulation. In this model, coagulation initiation reactions (TF-FVII activation of FX) occurred on monocytes in blood or fibroblasts in the injured vessel wall whereas the propagation reactions generating a large burst of thrombin occurred on activated platelets. The concept of spatial separation was also raised in Geisen et al. (33) and Hathcock et al. (34), which described the importance of blood-borne tissue factor for thrombus development. Geisen et al. (33) suggested that “This phenomenon would clearly favor thrombus propagation because factors IXa and Xa, both initial products of the TF pathway, would be generated on TF-containing microvesicles at the platelet surface, thereby reducing the distance they must diffuse from the source of TF to the platelet surface.” Later results in Hathcock et al. (34) suggest that thrombus growth would be severely limited if activation of FX from flowing blood required diffusion through the thrombus to TF-FVIIa complexes on cells within the vessel wall. This work from the Nemerson laboratory supported the now widely accepted contention of the cell-based model of coagulation that the physical separation between the sites of coagulation zymogen activation and the sites of coagulation factor activity will affect the dynamics of coagulation factor interactions. In addition to the issue of the FVII-TF activation of FIX and FX, this concern also applies to the separation of sites for PC activation, which requires thrombomodulin (Tm) on quiescent endothelialial cells and sites of APC activity (to inactivate FV and and FVIII) in a developing thrombus.

The protein C related coagulation network component in our multistage model, as indicated in the “Protein C pathway component” subsection, is similar to the one described in Fogelson and Tania (26) and incorporates lateral diffusion of PC from sites outside the plane as introduced in Fogelson and Tania (26), which raised the issue of spatial separation of the site of PC activation and APC activity. The model described in our article extends the approach of Fogelson and Tania (26) in several important ways:

• 1.We model thrombin and thrombomodulin concentration on endothelial cells $\left[{\text{II}}_{\text{a}}^{\text{EC}}\right]$, active protein C generated on the endothelial cell surface [APC^EC], and active protein C [APC] in time and space by partial differential equations (4.46), (4.49) and (4.50).
• 2.Moreover, reactions of complexes of thrombomodulin-thrombin and thrombomodulin-thrombin-protein C, which are modeled by ordinary differential equations (4.47) and (4.48), are coupled to the rest of the reaction network through stochastic CPM. As a result, the model in Fogelson and Tania (26) and the one described in our article behave in a different way. All reactions in Fogelson and Tania (26) are modeled by ordinary differential equations, which means that the spatial variation of thrombin on endothelial cells, APC on endothelial cells, and APC are not included in the Fogelson and Tania model (26).
• 3.Transport of a chemical is modeled in a different way. Namely, in the model deacribed in our article, the velocity of chemical convection depends on time and on the size and structure of the growing thrombus.

The first paragraph of “Coagulation pathway submodel” section should read:

We extend the model of Hockin et al. (14) by including in the model both solution-phase and membrane-phase reactions with concentrations of membrane-binding sites being limited and treated as control variables. (For detailed information on composition of our biochemical submodel and its relation to the previous models, see the new section “clarification of the relation of the biochemical reactions submodel to previous models”.)

On page 1726, the fourth paragraph in the 2nd column should start with:

Note that although some notations in our model coincide with those in Kuharsky and Fogelson (17) and Fogelson and Tania (26), our variables p₂ and $e_2^{\text{mtot}}$ include additional terms for meizothrombin that were not considered in Kuharsky and Fogelson (17) (see the “Relation of biochemical reactions submodel to previous models” section and the Supporting Material).