Deep vein thrombosis (DVT) is the formation of a blood clot within the deep veins, usually in the leg and sometimes in the arms or cerebral network [1]. In severe cases, the clot can break loose from the vessel walls and become trapped in smaller vessels, leading to thromboembolic diseases such as pulmonary embolism (PE) [2]. Over 200,000 people suffer from DVT (annually) in South Africa and about 60% of these cases lead to embolisms, while 10% are fatal [3–5]. Little research quantifying the impact of DVT on South Africa’s economy has been conducted. Studies from high-income countries have shown its impact to be significant [3]. The most common risk factors are HIV, tuberculosis, smoking, and recent hospitalization [6].

Detecting DVT before embolism is essential. Wells et al. developed a widely used DVT detection model that predicts pre-test probability for DVT [7]. The model calculates a score which takes pre-existing conditions such as cancer and pregnancy, as well as physically noticeable features such as a swollen leg, into account. Due to the risk of potential PE, above-knee cases are treated with anticoagulants such as unfractionated heparin (UFH) and low–molecular-weight heparins (LMWH) [8, 9]. Kesieme et al. explained that the efficacy of heparin is dependent on the ability to administer the right dosage within the first 24 h [9].

Existing detection and treatment procedures are invasive and expensive. The drugs are administered at dosages that are not necessarily patient-specific. It would be beneficial to develop more accurate and cost-effective methods for detecting and treating DVT. Predicting embolism events in cases of early-diagnosed DVT is equally important. With the development of CFD, the flow and biochemical aspects of blood flow in the vein can be simulated to create models with the potential to improve the information available to clinicians for specific patients. Drug testing is another application of CFD model development, where the dosage of anticoagulants can be estimated on a patient-specific level [10]. The model should have the ability to vary biochemical concentrations to simulate the specific patients’ conditions.

Successful attempts to model thrombosis using computational methods (mainly CFD) have been made. Platelet models focus on the platelet adhesion, activation, accumulation, and aggregation to model clot formation [11–13]. Coagulation network models account for the reactions involved in clot formation using mathematical equations [14]. The reaction-mass model couples the blood flow with the biochemical reactions and studies their effects on each other [15–18]. Integrated models are more developed, accounting for the hemodynamic, platelet, and coagulation reactions. Most of these models are developed in idealized geometries and do not account for the complex geometries of the vessels or mechanical agents such as valves.

To reduce the computational cost, many of the models are simplified by making significant assumptions regarding the clotting outcomes. Most in silico DVT models are flow-based, predicting clot formation from stagnation or vortices resulting from the veins’ complex geometry or obstructions [19]. Most models assume the pulsatile blood flow to be steady [14]. Although this assumption has been proven to be acceptable to a degree regarding the clotting outcomes, there is a need to further investigate how this assumption affects the initiation and propagation of clot growth under flow [14, 20–23].

Thrombosis models that capture complex biochemistry have also been presented. Xu et al. developed a multiscale model of thrombosis used to show the relationship between the blood flow rate and the clot size [24]. The model breaks down the coagulation process into sub-models. The Navier–Stokes equations account for the macroscale dynamics of blood flow. The microscale interactions between platelets, fibrinogen, and the vessel wall are described using the cellular Potts model. The Mann model accounted for the biochemical processes which are described using partial differential equations. The interaction between the blood haemodynamic and biochemical reactions and the effects on the thrombus growth in an idealized three-dimensional cylindrical geometry was studied by Bodnar and Sequeira [18]. They applied a network of 23 advection–diffusion–reaction equations to describe the changes in biochemical concentration and transport of enzymes. At higher fluid shear rates, platelet adhesion is facilitated by the von Willebrand factors (vWF) [25]. This led to the development and validation of a predictive model of thrombus growth under the high shear rate by Mehrabadi et al. The model allowed flow under constant pressure or a volumetric flow rate in a cylindrical channel [26]. Xu et al. studied the deformation and embolization of a growing clot under shear flow conditions using a multi-phase model. The Navier–Stokes equations and the Cahn–Hilliard equations were used to describe blood flow and deformation of the clot. It was assumed that the blood clot has two components, namely, the core and the shell. The volume fraction of platelets to plasma was 0.7:0.3 in the core and 0.3:0.7 in the shell. This model provided a quantifiable relationship between clot permeability and likelihood of embolization [27].

While there are numerous computational models of thrombosis, those focusing solely on DVT are limited. One of the earliest models was developed by Ramunigari et al. Their model captures unidirectional blood flow in a vessel with venous thrombosis present, thereby accounting for the haemodynamic effect of the obstructing thrombus [28]. Fortuny et al. developed a patient-specific DVT model. The model simulates blood flow in a realistic popliteal vein geometry generated from in vivo CT images. The effects of anticoagulants on blood flow around the thrombotic area were studied by varying the blood viscosity as a result of the anticoagulants [27, 29].

Venous valves, an important feature of the vessels where DVT develops, enable flow toward the heart but restrict backflow. Malfunction of the valves can cause significant problems to the blood dynamics. Introducing CFD to medical imaging makes non-invasive studies of the operations of valves possible. A model was developed by Ibrahim et al. to study valve openings in the popliteal vein and their effects on blood flow [30]. The valve opening was varied, and the size of the static zone and vorticity were investigated. Their model showed the possibility of clot formation around the valves in an idealized geometry. Nomehrda et al. and Ohashi et al. developed numerical models to study the flow pattern around the valve’s leaflets [31, 32]. The sinus was predicted to regulate flow. The model was validated with a flow experiment. Naracott et al. and Buescher et al. adopted this experiment and studied the effect of varying the sinus depth on the flow [33, 34]. The effect of DVT on venous valves and the implication of valve failure on blood circulation were studied by Simao et al. Their model applied a finite element method (FEM) with particular boundary and element characterizations to solve the fluid–solid interactions’ (FSI) governing equations. Blood is considered a non-Newtonian fluid under laminar flow, and the wall is considered a flexible material. Their model indicates that a clot forms during abnormal valve functioning [35]. A recent model of valve activities was developed by Hajati et al. [36]. The model applied knowledge from previous studies and was used to investigate the blood flow through the venous valve, taking the vein wall and valve leaflet’s flexibility into account. FSI was applied by solving the governing equations using a Galerkin approach for the FEM.

Experimental clot growth is another important method for visualizing and studying thrombosis under different conditions. Lurie et al. carried out an experiment to understand the mechanism of venous valves on blood flow. Duplex ultrasound was used to probe venous valves of patients with no history of venous disease. The time relationship between the valve movement and blood flow was investigated [37]. An in vitro thrombosis model was developed by Narracott et al. using hypercoagulable milk to represent blood to study clot growth in an idealized cerebral aneurysm geometry [38]. Ovanesov et al. achieved clot growth in vitro by introducing human tissue factor expressing cells into non-flowing human plasma [39]. Onasoga-Jarvis et al. improved on this experiment by growing clots under flow in a hydrodynamic focusing microfluidic device. Varying the shear rate, the experiment was used to investigate the haemodynamic conditions under which fibrin formation occurs [40]. Neeves et al. studied the nature of fibrin mesh formed during thrombosis under flow. Thrombin was introduced into a flow of fibrinogen through a microfluidic membrane [41]. Another in vitro model was developed by Prashad et al. to study clot lysis due to thrombolytic drugs. Whole blood from a healthy human was used to grow clots in a pre-weighed sterile tube, and the clot is weighed before and after drug administration. The intended purpose of this method is to provide a quick and efficient way of observing the effects of newly developed drugs on patient-specific thrombosis [42]. All these models and methods contribute to an improved understanding of the clotting process in DVT. However, few models have managed to capture clot formation and the changing biochemical conditions in complex flow, and none of the models have taken the effect of venous valves into account.

This study presents the developmental process of a three-dimensional DVT computational model in idealized and realistic common femoral vein (CFV) geometries. The model accounts for the reaction between thrombin and fibrinogen under pulsatile blood flow, leading to clot formation within the vessel. The difference in mechanical properties between the flow region and the clot region is accounted for by changes in porosity and permeability. The model is verified using a clot growth experiment. The model is developed in three stages. The first stage presents a two-dimensional physiological model of DVT. This model is used to investigate clot growth under different thrombin inlet boundary conditions. The two-dimensional model is also used to carry out a parametric study to determine the effects of varying the inlet velocity, vein diameter, and thrombin concentration on the shape and size of the clot formed. The two-dimensional model is extended to an idealized three-dimensional geometry. This work depicts the development process with increasing model complexity. The computational model is verified using results from an experimental clot growth study. The experiment comprises a steady flow of fibrinogen in a cylindrical pipe and thrombin injected into the flow at the injury site. A qualitative comparison is made between the experimental clot and the in silico clot. The validated three-dimensional model is used to compare clot formation under steady and pulsatile conditions, taking valve activity into account. The model is then applied to a realistic patient-specific geometry, and the clot formed is compared to the clot formed under physiological conditions.

The results for the two-dimensional, three-dimensional, experimental, and patient-specific models are described below.

Our model corroborates the observations of Young et al., where stenotic flow resulted in improved clot propagation and also in regions close to the clot surface experiencing low shear [54]. An increase in velocity increased the rate of thrombin being carried away from the injury zone, thus reducing the size of the clot. This agrees with the findings of Kattula et al. [53]. The results from the model also concur with work done by Wolberg et al. as the clot size increased with an increase in peak thrombin concentration [52]. Results from this model show that changing the velocity profile with a fixed peak velocity does not significantly affect clot initiation. Clotting under steady flow initiated about a second later than under pulsatile flow [11, 55, 56]. The clot shape under steady flow shown in Figure 15E is similar to the clot observed by Kadri et al. [57]. Owing to a higher maximum strain rate in models with the valve activity, the clots formed are smaller than in purely pulsatile cases [31, 32].

The inclusion of fibrin in the model makes fibrin concentration the most important factor since fibrinogen concentration is constant, as explained by Weisel et al. [58]. Biochemical species’ concentrations in the vein changed over time, in a similar manner to the findings of Bodnar and Sequeira [18]. A slower thrombin dissipation rate was noticed when compared to the set profile. This is due to trapped thrombin inside the clot needing more time to be transported away from the injury site, as noted in Bodnar and Sequeira’s model [18]. The limitation of our model is the simplifications made involving clot formation. The complex biochemical reaction is reduced to two reactions, although the spontaneous and accelerated thrombosis in the presence of an “injury” on the vessel is captured. The flexible vein geometry was reduced to a rigid wall with a no-slip condition. This does not correlate with the experimental study. The no-slip condition in the simulation ensured that the velocity at the wall is always zero, allowing the clot to stick to the wall. To prevent the clot from being carried away in the experimental study, the injection needle was inserted into flow to act as a tether for the clot. The three-dimensional models proved that the two-dimensional models can be extended to patient-specific image-derived models. Improvement on the model could enable better understanding of the clot growth process for specific patients.

In this study, the development of a framework that predicts clot formation in a patient-specific three-dimensional femoral vein geometry using CFD techniques and biochemical reactions is outlined. This study aims to understand the effect of changing these factors on the clot size and determining which factor had the most impact on clot initiation and propagation. Increasing the velocity and vein diameter caused a reduction in clot size, and increasing the thrombin peak concentration increased the clot size. Thrombin concentration was found to be the sole factor determining when clot initiation occurs and the driving factor when determining the size of the clot. The work shows that higher thrombin concentration produces denser, larger clots. The model developed in this work was verified using an experimental clot growth study. The clot formed experimentally is compared to the computationally grown clot. After carrying out a visual comparison between both clots, there is a 24% difference between the heights of these clots with the experimental clot being thicker. Previously, most DVT models were mainly flow-based, studying areas of stagnation and recirculation. Most of them included solid valve walls and sinuses. This model avoids this complexity by simulating blood flow coming out of the valve and includes biochemical reactions on the desired injury zone, which allows for further investigation of the clot formation process. This process also helps us study the initiation and propagation of clots, in addition to identifying regions of clot formation. It is evident that the model can predict clot formation under different flow conditions.