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Numerical and experimental evaluation of platelet deposition to collagen coated surface at low shear rates
Journal of Biomechanics 46 (2013) 430–436
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Numerical and experimental evaluation of platelet deposition to collagen coated surface at low shear rates Klaus Affeld n, Leonid Goubergrits, Nobuo Watanabe, Ulrich Kertzscher Biofluid Mechanics Laboratory, Charite´—Universit¨ atsmedizin Berlin, Thielallee 73, 14195 Berlin, Germany
a r t i c l e i n f o
Keywords: Platelet deposition Shear rate Couette flow Monte Carlo method
a b s t r a c t Platelet deposition to collagen-coated surface under low shear conditions was investigated using an experimental model. The flow chamber was created by combining a stationary and a rotational glass plates spaced 50 mm apart. Blood filled into this space was subjected to a simple Couette flow. Both glass plates were covered with albumin to render them anti-thrombogenic. However, one spot 1 1 mm in size was covered with collagen. This spot was where the platelets deposited. The device was mounted on an inverted microscope and the platelet deposition was recorded. Platelets were dyed to render them fluorescent. The blood used was human blood from healthy volunteers. It was subjected to a range of low shear rates (below 700 1/s) to find out how they act on platelet deposition. The results show a characteristic curve with elevated platelet deposition in the range of 150 1/s. For the interpretation of these results a numerical model was developed. It applies the Monte Carlo method to model a random walk of platelets. This diffusive motion was superimposed on the convective motion by the Couette flow. A satisfactory match to the experimental data was achieved. & 2012 Elsevier Ltd. All rights reserved.
1. Introduction Thrombo-embolical events in cardiovascular devices start with the deposition of platelets onto the material of the implant. This is only the beginning of a complex sequence of events, finally leading to a thrombus, which can block the flow through the device or can dislodge. This process has been subject to a large number of publications. Some authors of recent papers model the complex chain of reactions, including as many reactions as possible. Flamm and Diamond (2012) review such models spanning multi-scale phenomena, from the scale of a calcium ion to the diameter of the aorta. Such models may contain 77 reactions with 132 kinetic rate constants and 70 species, or with 28 fluid phase species and complexes and 44 lipid-bound factors and complexes. For the engineer designing a cardiovascular device, a much simpler model is needed. It should work with assumptions that can be checked experimentally. In the search for such a model, the literature was reviewed. The first experimental models did not include numerical analysis; they were devised to observe the generation of a thrombus (Petschek et al., 1968). Ex-vivo blood was directed to flow onto a glass plate. This flow chamber generates a stagnation point flow and a radial outflow. Later Baldauf et al. (1978) and Reininger et al. (1992) used this flow
n
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type for further studies. The deposition of platelets could be observed with a dark field reflected light microscope. This model was soon replaced by a parallel flow model, which was more suitable for mathematical analysis (Turitto and Baumgartner, 1975a), and a numerical model was proposed. More models were developed by various authors and reviewed (Stubley et al., 1987). The basic equation for all models is D @c=@x wall ¼ k c ð1Þ At the wall, no convective motion is assumed, so the platelet transport is caused by diffusion only. The platelet flux – the left side of Eq. (1) – depends on the platelet diffusivity D and the concentration gradient at the wall. This flux must be equal with the deposition rate of platelets on the surface—the right side of Eq. (1). The deposition rate is assumed to depend on the concentration c of available free platelets and a reaction constant k, a quantity reflecting the likelihood of deposition. The diffusivity of platelets in platelet-rich plasma DPRP is about 1.5 10 13 m2/s. If, however, red blood cells are also present, the diffusivity is much larger, to the order of two magnitudes (Turitto and Baumgartner, 1975b). This is caused by the mixing action of red blood cells as they turn and tumble in a shear flow (Goldsmith, 1972; Goldsmith et al., 1995; Tokarev et al., 2011). Zydney and Colton (1988) have reviewed experiments and models and proposed a diffusivity model for platelets in full blood (Eq. 2) which is still used today. This model assumes a linear dependence of the shear rate g on the diffusivity. Furthermore, the diffusivity also
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depends on the hematocrit H and the diameter of red blood cells dRBC: 2 D ¼ DPRP ð1HÞ þ 0:15 dRBC H=4 g ð1HÞ1:8 ð2Þ At a distance from the surface, basic laws of fluid mechanics determine the transport of platelets. Platelet deposition at the wall is thought to be limited by only one surface property: the surface availability. The concept of surface availability (Turitto and Baumgartner, 1975a) is that already-adhered platelets form a monolayer on the surface. They occupy a site and exclude oncoming platelets from adhering. It is assumed that this surface availability attains zero when the surface is completely covered. This is reflected in Eq. (3) for the reaction constant. The surface coverage is defined by the surface density of already-adhered platelets M(x, t) in relation to the surface density of full coverage MN. In this way, a time dependence of platelet deposition is achieved. k ¼ k0 1 M ðx,t Þ=M 1 ÞÞ
ð3Þ
The reaction constant k0 has been a subject of discussion by various authors. Some state that it is constant. However, comparison with experiments lead other authors to the assumption that the reaction constant increases with the shear rate (David et al., 2001), either linearly or in a more complex way (Sorensen, 2002; Weller, 2008). The many discussions and open questions have lead to the work presented in this paper, which consists of an experimental model and a numerical model. The most elementary viscous flow has been selected – the plane Couette flow – and, in addition, the model is limited only to the initial phase of platelet deposition. In this way, the question of surface availability is excluded. The complex cascade of thrombus generation is truncated to its first phase.
2. Materials and methods 2.1. Experimental model The experimental model is designed to simulate the initial phase of thrombus generation in the human body. If in a blood vessel the endothelial layer is locally disrupted, collagen fibers are exposed. Platelets carried by the bloodstream become activated upon contact with collagen and deposit on its surface. This is considered the initial phase of the coagulation system. The experimental model is designed to create the simplest conditions: stationary blood flow, constant low (o1000 1/s) shear rate which does not activate platelets (Hellums, 1994), anti-thrombogenic surface throughout the device, collagen exposed in a small area, freshly-drawn citrated human blood with non-activated platelets. A flow with a constant shear rate is approached by a parallel plate design (Watanabe et al., 2011). Fig. 1 shows the experimental model: a circular stationary glass plate 40 mm in diameter and an upper rotational plate 35 mm in diameter are spaced 50 mm apart. A set of 50 mm gap for each experiment is done with a precision of the depth of sharpness of the used microscope. The resulting error of the gap set was 4.6 mm. This space is filled with blood and the Couette flow is created when the upper plate is rotated. Both plates are microscopic circular glass coverslips of 175 mm thickness for best optical conditions. The model is positioned in an inverted microscope (Fluovert FU, Leica AG, Germany). The coverslip glass plates are covered with albumin (Carl Roth GmbH & Co KG, Karlsruhe, Germany). This albumin layer makes the glass anti-thrombogenic and thus non-activated platelets will not deposit on the glass. To induce platelet deposition, a spot of collagen (Horm, Nycomed Austria GmbH, Austria) with a diameter of 1000 mm is applied at a radius of 7.5 mm on the lower stationary coverslip. The field of view of the microscope is directed onto this spot. A shear rate between zero and 700 1/s can be generated at this spot. Blood was drawn from the cubital vein of voluntary healthy male donors, which gave informed consent on the work. For each run, two samples each of 9 ml of blood were drawn into vacuum tubes (‘‘Vacutainer’’) containing 1 ml of anticoagulant (0.106 mol/L sodium citrate) (Sarstedt AG, ¨ Nurmbrecht, Germany). The first sample was discarded to exclude platelets that were activated by venipuncture. For fluorescent labeling of the platelets, 0.01 g of Quinacrine dihydrochloride (Sigma Aldrich Chemie GmbH, Steinheim, Germany) was dissolved into 50 g of phosphate-buffered solution (Invitrogen, UK). Then 100 mL of that solution was added to the blood sample. After gentle shaking, the sample was protected from light and kept at room temperature. The samples were used within three hours after venipuncture. Each sample was divided into six portions allowing performing up to six runs. Prior to each run about 100 mL of blood was put onto the
Fig. 1. Couette flow model with two parallel plates (coverslips). Coverslips were used because of their superior optical qualities. The lower coverslip is fixed, while the upper rotates. Blood fills the 50 mm gap between the coverslips. A small spot 1 mm in diameter on the lower coverslip is covered with collagen. The platelets are dyed fluorescent and glow when illuminated by excitatory light. lower coverslip. The upper coverslip was carefully lowered avoiding the generation of air bubbles. Through the lower stationary plate light with 355 to 425 nm wavelength illuminated the blood. Illuminated platelets emit light at 496 nm. The upper coverslip was set in motion, and after focusing on the stationary coverslip plane the recording began. The process of deposition was recorded with a video camera (UI-2230-M, Imaging Development Systems GmbH, Germany) with a resolution of 1024 768 pixels and a frame rate of 20 fps and covering a field of view of 480 360 mm2. Once deposited on the plate, platelets remained fixed and were easily distinguishable from the moving platelets. Camera images were processed and resulted in surface coverage by the platelets as a function of time and shear rate. The duration of each run was 60 s. After this time, platelets start to form large aggregates that create wakes affecting the probability of adhesion, a phenomenon which is not a part of this study.
2.2. Numerical model The objective of the numerical model is to reproduce and to interpret the results of the experiments. The numerical model follows the concept of Virchow (1856), who concluded that three parameters were responsible for thrombus generation: the quality of the blood, the quality of the vessel wall and the quality of the flow. He stated in hypothesis that these three players interact and this became known as Virchow’s triad. In this numerical model, only the initial short phase of platelet deposition is considered. The highly complex system of coagulation with its many stages of platelet activation and the fibrinogen–fibrin system are left out. The numerical model consists of the following elements: a) Shear flow field: In the experimental model, the flow field is the simplest and is characterized by an approximately homogeneous shear rate in the field of view. For the numerical model, a rectangular box is considered, which represents a small part of the Couette flow field. This rectangular box has a height of 50 mm, a width of 1000 mm and a length of 1200 mm (Fig. 2). Its base plane is defined as a wall at which the platelets deposit. The flow enters the box through the front plane. At the inflow side of the box new platelets are inserted at each time step. The number of platelets inserted at each time step is proportional to the flow in the respective velocity. It is thus proportional to time, distance z to the base plane and shear rate to achieve a homogeneous concentration. Platelets which leave the box through its lateral or upper sides during a time step are reflected inwards. For this simple flow and geometry the mirror reflection does not make notable difference. For more complicated flows, however, more appropriate periodic boundary conditions should be incorporated. Platelets close to the bottom can deposit if certain conditions are met according to our model. The flow has no vertical velocity component and
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b)
c)
d)
e)
f)
g)
K. Affeld et al. / Journal of Biomechanics 46 (2013) 430–436 platelets can approach the wall by diffusion only. A linear velocity profile is specified according to the shear rate. Blood model: The model fluid is assumed to be a single-phase fluid with the Newtonian viscosity of blood. Red blood cells are not modeled. However, their mixing effect on platelets is taken into account. This mixing augments the diffusivity of platelets. This is computed with the help of an experimentally derived diffusivity function described by Eq. (2). Platelets: Platelets are assumed to be virtual particles having no influence on the flow field. These particles are considered the center of the platelets, having a virtual diameter of 2 mm. This plays a role when the platelet approaches the wall. The platelets are distributed evenly and are inserted at the inflow plane. The model simulates the finite number of platelets according to the platelet concentration of the donor. In an initial step, particles were distributed evenly in the domain. In the following steps, new particles enter at the inlet plane or by dispersion through the upper face of the deposition domain. Movement of platelets: The movements of platelets are simulated with the Monte Carlo random walk method. This allows the observer to follow the path and the fate of each individual platelet and as such is very close to microscopic observation as performed in the experimental method. Platelets move with the flow field from their initial position after the start. They move stepwise according to the flow field. The flow field step vector is computed individually for each platelet position. This approach guaranties that grid dependence is avoided. Additionally at each time step, a diffusion step vector is computed and added to the flow field step vector of each platelet. A time step tstep of 10 ms is allotted to one numerical cycle. After each time step, every platelet attains a new position. If a platelet penetrates the vertical side planes, its position is horizontally reflected. If a platelet approaches the wall by this motion, it enters the deposition domain. Diffusivity of platelets: The diffusivity is modeled with diffusion steps based on the random walk model. The diffusion steps are computed with a Gaussian distribution and a mean square distance of 2 Ddim D tstep, where Ddim ¼ 3 for three-dimensional space, D being the effective diffusivity and tstep the time step length. The directions of the diffusion step vectors are computed in such a manner that all directions are evenly distributed. For the platelet effective diffusivity D, the equation of Zydney and Colton is used (see Eq. 2). Deposition domain: The deposition domain is the rectangular box just above the fixed base plane. The height of this box is h and is subject to variation. Once a platelet enters the deposition domain it is close enough to interact with a wall. The first panel of 200 mm length in the flow direction is defined to be antithrombogenic . Platelets touching the wall there will not be deposited. The following panel of 1000 mm length represents the collagen spot and is thrombogenic. A platelet touching here can deposit according to the probability that it will do so. Probability of deposition: The probability of deposition is defined as the probability of a platelet at a certain distance h to the wall to deposit within the next time step tstep. It can be experimentally assessed by observation of
moving platelets (Machin et al., 2005). In the numerical model, the distance h is a parameter of the model. The probability of deposition pdeposition is the product of two individual probabilities and one flow-related factor. One probability pplatelet characterizes the reactivity of platelets; another probability pwall characterizes the thrombogenicity of the wall. A third factor fflow characterizes the influence of the shear rate at the wall on the deposition and is a property of the flow. Each of those three numbers is a number between zero and one. All three act together and define the probability of deposition for each individual platelet: pdeposition ¼ pplatelet pwall f flow
ð4Þ
If for instance pdeposition is one, the platelet will deposit in the current time step. h) Probability–reactivity of the platelet: This probability characterizing the reactivity of the platelet models the activation state of the platelet. It is set to one if the platelet is fully activated, but is always a number larger than zero. If this were not the case, the deposition of a non-activated platelet contacting a maximally activating surface, such as collagen, could not occur 0 o pplatelet r 1
ð5Þ
i) Probability–thrombogenicity of the wall: Unlike the previous probability, it is possible that this probability characterizing the thrombogenicity of the wall can assume a zero value. This is because there exist certain materials onto which platelets do not deposit. At the other extreme, collagen is a very thrombogenic material. In this case, the probability is set to one. The encounter between non-activated platelets and collagen can be seen as the design point of the coagulation system. 0 r pwall r 1
ð6Þ
j) Influence of the shear rate: The shear rate has a twofold influence on platelet deposition: First, through convection and diffusivity it determines the number of platelets which are available to interact with the wall. Second, the platelet requires a certain amount of time (called residence time) within a certain distance from the wall in order to deposit. A residence time has been proposed by Reininger et al. (1995) and Machin et al. (2005). The platelet travels with a certain velocity along the wall and reacts only if its distance from the wall is small enough. In this way the residence time may be expressed as a function of the platelets velocity and a length—the distance to the wall at which a reaction can take place. This can be visualized as a handshake between two people: one at rest, the other sliding by on a conveyor belt. There is a certain distance between the two people within which the handshake may take place – at most the combined length of the two outstretched arms – and a certain amount of time required for the event to take place. This time is the residence time. It is calculated by dividing the velocity of the conveyor belt by the double arm length. Length divided by velocity, however, has the dimension of shear rate. This residence time is thus defined as proportional to 1/(shear rate). Taking this into account, one derives. n 0 o f flow ¼ min 1, C shearrate =g ð7Þ r 1 with C shearrate 40 1=s
The exponent n is a number close to one. It is assumed to include other influences than the shear rate alone. It should be noted that the flow factor fflow takes the viscous flow into account and acts in a homogenous shear field on every platelet in the same way. k) Sequence of computation: The simulation of platelet deposition is based on discrete time steps and three-dimensional continuous space. All quantities are calculated separately for each platelet. There is no interaction between platelets included in the model.
The following sequence is performed in one computational cycle for each time step:
insertion of new platelets at the inlet (z position) according to inlet velocity
Fig. 2. Geometry of the numerical model for the simulation of platelet deposition in the experimental model. The numbers indicate the dimensions. The height of the rectangular box–the z-axis–is scaled up about 20 times. Flow is shown by red arrows. The white area on the bottom is given the deposition probability of zero, while the shaded part has the probability of one. This is the area that models the collagen surface. The rectangular box of height h ¼ 1 mm marks the deposition domain. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
profile and platelet concentration, while the y position of each inserted platelet is computed with a random number. In this way, an even concentration of platelets at the inlet is achieved. calculation of the next platelet position with the Monte Carlo random walk: calculation of the displacement due to convection, calculation of the diffusivity due to shear rate, calculation of the displacement due to diffusivity, calculation of the new platelet position in space. reflection of platelets at upper, lower, and side walls. detection and removal of platelets that exited the rectangular box through the outlet. calculation of the deposition probability for each platelet present in the deposition domain (see Eq. 4).
K. Affeld et al. / Journal of Biomechanics 46 (2013) 430–436
generation of a random number for each platelet in the deposition domain. If it is random number (0y1) is smaller than its deposition probability pdeposition, the platelet deposits. The results of the simulations are translated to experimental quantities by conversion of deposited platelets to surface coverage rate. A large number of platelets – approximately 9000 – are observed in the rectangular box at all time steps. One run simulates two seconds, which results in the computation of about 2 million individual positions. As platelets enter the deposition space, the calculation of the deposition probability begins. Platelets that deposit at the wall plane are counted, and their rate is computed as a function of the flow—the shear rate.
3. Results 3.1. Experimental results Experiments were performed with blood from three different donors who gave voluntary informed consent. Fig. 3 shows the results of an experiment and its subsequent evaluation. It shows the surface coverage by platelets over time. The curves do not start out with a surface coverage of zero. This indicates that platelets do deposit before the run and the creation of a shear field begins. This can be explained from the first contact of blood with the collagen surface during filling. The surface coverage increases almost linearly over time. The inclines of the curves represent the growth of surface coverage. This is proportional to the deposition rate per area. Fig. 4 shows the data from all three donors. Donors’ blood differs with regards to platelet concentration (donors 1/2/3: 193 000/163 000/ 263 000 1/mL) and hematocrit (donors 1/2/3: 46.3/43.0/40.6%). The data scatter, as is often found in experiments with platelets. However, a peak in the deposition rate is obvious. Fig. 4 at the lower right shows a regression curve of all data points. This curve requires further interpretation. Why is there a higher rise in platelet deposition at low shear rates? Why is there a peak with a subsequent slope? What is causing the peculiar shape of the curve? 3.2. Numerical results The numerical model was designed to quantitatively reproduce and to interpret the experimental results described above. This includes the platelet deposition rate at zero shear rate, the value of the initial incline of the platelet deposition curve, the peak deposition rate and the value of the shear rate at the peak and downward slope. Fig. 5 shows the platelets’ path lines during a time of 100 ms and a shear rate of 100 1/s. The Z-axis is scaled up tenfold. Of the 9000 platelets considered in the volume, only a fraction is depicted here for reasons of clarity. The first feature to be modeled is the deposition rate at shear rate zero, which
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represents no flow condition. There is no convective transport of the platelets and only few platelets enter the deposition space to deposit. The second feature is the rise in the deposition rate accompanying a rising shear rate. Fig. 6a shows the deposition rate as a function of the rising shear rate and of the parameter Cshearrate. All curves show a nearly linear function at low shear rates. After the linear increase, each curve reaches a peak. This third feature occurs at a specific shear rate, which is accounted for in the numerical model with the parameter Cshearrate. The parameter Cshearrate is chosen to match the experimental results. Apparently, this hypothetical approach models the physical– chemical process of the adhering platelet sufficiently well. The fourth feature is the magnitude of the deposition rate. It depends mainly on the concentration of platelets and their reactivity. In Fig. 6b again the deposition rate is plotted over the shear rate and parameters are chosen for a best fit to the experimental results of donor 1. The best fit with coefficient of determination of R2 ¼0.97 was obtained by modifying the parameters with a genetic algorithm. The square root mean error was minimized. For the height of the deposition domain h¼1 mm and the donor’s platelet concentration of 193,000 1/mL, this results in the parameters Cshearrate ¼137 1/s, the exponent n¼0.8, platelet reactivity pplatelet ¼0.023 and (Eq. 7). pwall is set to one. For donors 2 and 3 the same comparison is made as shown in Fig. 6c and d. The match in these cases is less exact with R2 of 0.88 and 0.84. The reason for this remains open and is a subject for further research—along with the experimental method.
4. Discussion Experiments involving platelets often result in data that scatter, as do the data from the experimental model presented here. This is caused both by the excitable nature of the platelet and by the imperfection of the experimental model. Improvements to the model can be achieved with several modifications. The plates of the Couette device are coverslips. They were chosen for their good optical quality, but are very thin and fragile and do not hold their shape well if stressed. The stationary plate on which the platelet deposition is observed will be partially reinforced with a thicker glass plate. The rotating coverslip will be replaced by a solid cylinder. The latter will be equipped with a central pivot resting on the stationary plate. This pivot will define the gap with more precision. Variation of gap size is around 10% and is one cause of the observed data scatter. Step motors for better speed control will replace the electric brush motors. Apart from a better mechanical design, blood needs to be classified more precisely with the method used in the hematological laboratory. Finally, the microscopic methods need to be improved.
Fig. 3. (a) The 190 130 mm2 image of deposited platelet recorded after 60 s of shear flow. Flow is from left to right. (b) Examples of platelet deposition as a function of time representing results (donor 2) from runs on different days and different blood samples at one shear rate of 175 1/s. During the initial 60 s there is little interaction between a single deposited platelets and an incoming one. The inclines of curves are the platelet deposition rates.
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Also required is the assessment of other parameters that are part of the numerical model and are required to validate the numerical model. Such a parameter is the probability of depositing, which is the product pplatelet pwall. It can be accessed from the observation of platelets moving near the wall (Machin et al., 2005). Counting the platelets, measuring their distance from the wall and comparing them to the deposited platelets will enable the computation of a deposition probability. The results of the numerical model show its ability to model the dependency of deposition rate over shear rate. The model contains parameters that can be quantified directly: shear rate, platelet concentration and hematocrit. Other parameters such as the height of the deposition domain, two probabilities pplatelet, pwall and the factor of flow influence fflow have to be estimated or derived by matching to the experimental data. The match obtained thus far can only be achieved with the assumption of a flow factor fflow ¼min(1, (Cshearrate/g)n). This factor is one for a shear rate lower than 137 1/s. In excess of this shear rate, as defined in the model by Cshearrate, the residence time is shorter than the reaction time. This means that the probability to react decreases and fewer platelets deposit. The time to adhere and to form tethers is discussed by Nesbitt et al. (2009). Machin et al. (2005) have discussed such a reaction time and have presented one method of measuring it. The numerical method as it is presented here uses the Monte Carlo method to simulate directly the diffusion of a platelet (Affeld et al., 2004). This results in translocation, which then is superposed on a convective flow field. In this way, diffusion and convection are decoupled in regard to computation. In most of the models, (Sorensen, 2002; David et al., 2001; Weller, 2008) and (Behbahani, 2011), diffusion and convection are not treated separately. This leads to equations that do not converge easily (Sorensen, 2002). There is also another parameter not treated separately in these models: the reaction constant k0. This comprises the reactivity of a platelet and the reactivity of the wall. In the model presented here, both reactivities are separate parameters and can be tested separately, for example with methods by Machin et al. (2005).
The numerical model does provide an explication of the peculiar curve from the experiment. However, the model is applied only to the most elementary flow, the uniform shear flow. In implantable devices, the flow is more complex. The model is presently being applied to such a three-dimensional flow with widely varying shear rates. This is the stagnation point flow, for which many experimental data, that of the authors (Affeld et al., 1995) and of others, are available. Another suitable flow for validation is the sudden tubular expansion by Karino and Goldsmith (1979). The objective of these applications would be a further validation of the model.
5. Conclusions The experimental work of the present study shows a range of shear rates with an elevated deposition rate, in other words: a range of shear rates favorable for the deposition of platelets. The
Fig. 5. Path lines of platelets in the numerical model during a time of 100 ms and a shear rate of 100 1/s. Z-axis is scaled up ten times. There are about 9000 platelets in the volume, however, for reasons of clarity only a fraction of those platelets is shown. A space above the base plane 2 mm thick is the deposition domain. Once a platelet enters this domain, its probability of depositing is calculated.
Fig. 4. Platelet deposition rate plotted over shear rate for three different donors, but identical device. Each donor curve is based on approximately 55 single experiments. The data scatter—a major cause of this thought to be the insufficient control of gap and blood variability. The lower right regression curve is from the data for all three donors. All curves start near zero (1), have an initial incline (2), reach a peak of deposition rate at a certain shear rate (3) and then slope down.
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Fig. 6. (a) Deposition rate plotted over shear rate calculated for three different values of Cshearrate with constant pplatelet ¼0.023 and exponent n ¼0.8. This simulates the influence of a platelet’s residence time in the vicinity of the wall. (b) Deposition rate plotted over shear rate. Match of experimental and numerical results for donor 1: The respective parameters are pplatelet ¼0.023, Cshearrate ¼ 137 1/s and exponent n¼0.8. The resulting coefficient of determination is R2 ¼ 0.97. (c) Match for donor 2: pplatelet ¼ 0.0175, Cshearrate ¼ 225 1/s, and exponent n¼1.5. The R2 is 0.88. (d) Match for donor 3: pplatelet ¼0.0124, Cshearrate ¼ 175 1/s, and, exponent n¼ 1.55. The R2 is 0.84.
numerical model uses the Monte Carlo method of direct simulation of the path of platelets in a flow field. The model can be applied to flow fields obtained from a flow solver. In this case, the flow field is the most elementary case of viscous flow—the shear field is homogeneous throughout. The numerical model replicates the flow between the two Couette plates and the results model the experimental relationship of platelet deposition rate over shear rate. The numerical model, however, needs to be tested in a more complex flow with a variety of shear rates at the wall. Such a flow is known as the stagnation point flow, which is used for further validation of the model.
Conflict of interest statement The authors do not have any financial or personal conflicts of interest.
Acknowledgements We would like to thank Dipl.-Ing. Jens Schaller and Dipl.-Ing. Torsten Schneider for their helpful technical contributions for numerical modeling. Furthermore, we thank F. Probst, K. Hoell, and K. Loewenhoff for their technical contributions for experiments. The study was supported by the German Research Foundation (Grant number: AF 3/37-1) and by the Alexander von Humboldt Foundation. References Affeld, K., Goubergrits, L., Kertzscher, U., Gadischke, J., Reininger, A., 2004. Mathematical model of platelet deposition under flow conditions. The International Journal of Artificial Organs 27, 699–708.
Affeld, K., Reininger, A.J., Gadischke, J., Grunert, K., Schmidt, S., Thiele, F., 1995. Fluid-mechanics of the stagnation point flow chamber and its platelet deposition. Artificial Organs 19, 597–602. Baldauf, W., Wurzinger, L.J., Kinder, J., 1978. Role of stagnation point flow in formation of platelet thrombi on glass surfaces in tubes with various geometry. Pathology, Research & Practice 163, 9–33. Behbahani, M.J., 2011. Modeling and Simulation of Shear-Dependent Platelet Reactions in Blood Vessels and Blood-Contacting Medical Devices. Dr. Hut, Aachen. David, T., Thomas, S., Walker, P.G., 2001. Platelet deposition in stagnation point flow: an analytical and computational simulation. Medical Engineering & Physics 23, 299–312. Flamm, M., Diamond, S., 2012. Multiscale systems biology and physics of thrombosis under flow. Annals of Biomedical Engineering 40 (11), 2356–2364. Goldsmith, H.L., 1972. The flow of model particles and blood cells and its relation to thrombogenesis. Progress in Hemostasis and Thrombosis 1, 97–127. Goldsmith, H.L., Bell, D.N., Braovac, S., Steinberg, A., Mcintosh, F., 1995. Physical and chemical effects of red-cells in the shear-induced aggregation of human platelets. Biophysical Journal 69, 1584–1595. Hellums, J.D., 1994. 1993 Whitaker lecture—biorheology in thrombosis research. Annals of Biomedical Engineering 22 (5), 445–455. Karino, T., Goldsmith, H.L., 1979. Adhesion of human-platelets to collagen on the walls distal to a tubular expansion. Microvascular Research 17, 238–262. Machin, M., Santomaso, A., Cozzi, M.R., Battiston, M., Mazzucato, M., De Marco, L., Canu, P., 2005. Characterization of platelet adhesion under flow using microscopic image sequence analysis. The International Journal of Artificial Organs 28, 678–685. Nesbitt, W.S., Westein, E., Tovar-Lopez, F.J., Tolouei, E., Mitchell, A., Fu, J., Carberry, J., Fouras, A., Jackson, S.P., 2009. A shear gradient-dependent platelet aggregation mechanism drives thrombus formation. Nature Medicine 15 665-U146. Petschek, H., Adamis, D., Kantrowi, Ar, 1968. Stagnation flow thrombus formation. Transactions American Society for Artificial Internal Organs 14, 256–260. Reininger, A.J., Reininger, C.B., Heinzmann, U., Wurzinger, L.J., 1995. Residence time in niches of stagnant flow determines fibrin clot formation in an arterial branching model—detailed flow-analysis and experimental results. Thrombosis and Haemostasis 74, 916–922. Reininger, C.B., Reininger, A.J., Hormann, A., Steckmeier, B., Schweiberer, L., 1992. Quantitative-analysis of platelet-function using stagnation point flow aggregometry—1st clinical-results. International Angiology 11, 247–255. Sorensen, E.N., 2002. Computational Simulation of Platelet Transport, Activation, and Deposition. University of Pittsburgh.
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Stubley, G.D., Strong, A.B., Hale, W.E., Absolom, D.R., 1987. A review of mathematical-models for the prediction of blood-cell adhesion. Physicochemical Hydrodynamics 8, 221–235. Tokarev, A.A., Butylin, A.A., Ataullakhanov, F.I., 2011. Platelet adhesion from shear blood flow is controlled by near-wall rebounding collisions with erythrocytes. Biophysical Journal 100, 799–808. Turitto, V.T., Baumgartner, H.R., 1975a. Platelet deposition on subendothelium exposed to flowing blood—mathematical-analysis of physical parameters. Transactions American Society for Artificial Internal Organs 21, 593–601. Turitto, V.T., Baumgartner, H.R., 1975b. Platelet interaction with subendothelium in a perfusion system—physical role of red blood-cells. Microvascular Research 9, 335–344.
¨ ¨ Virchow, R., 1856. Uber den Faserstoff: Phlogose und Thrombose im Gefaßsystem, Gesammelte Abhandlungen zur wissenschaftlichen Medicin. Meidinger & Sohn Corp, Frankfurt am Main. Watanabe, N., Affeld, K., Schaller, J., Schmitmeier, S., Reininger, A., Goubergrits, L., Kertzscher, U., 2011. Investigation of human platelet adhesion under low shear conditions in a rotational flow chamber. Journal of Biorheology 25, 64–70. Weller, F.F., 2008. Platelet deposition in non-parallel flow. Journal of Mathematical Biology 57, 333–359. Zydney, A.L., Colton, C.K., 1988. Augmented solute transport in the shear-flow of a concentrated suspension. Physicochemical Hydrodynamics 10, 77–96.
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Numerical and experimental evaluation of platelet deposition to collagen coated surface at low shear rates Klaus Affeld n, Leonid Goubergrits, Nobuo Watanabe, Ulrich Kertzscher Biofluid Mechanics Laboratory, Charite´—Universit¨ atsmedizin Berlin, Thielallee 73, 14195 Berlin, Germany
a r t i c l e i n f o
Keywords: Platelet deposition Shear rate Couette flow Monte Carlo method
a b s t r a c t Platelet deposition to collagen-coated surface under low shear conditions was investigated using an experimental model. The flow chamber was created by combining a stationary and a rotational glass plates spaced 50 mm apart. Blood filled into this space was subjected to a simple Couette flow. Both glass plates were covered with albumin to render them anti-thrombogenic. However, one spot 1 1 mm in size was covered with collagen. This spot was where the platelets deposited. The device was mounted on an inverted microscope and the platelet deposition was recorded. Platelets were dyed to render them fluorescent. The blood used was human blood from healthy volunteers. It was subjected to a range of low shear rates (below 700 1/s) to find out how they act on platelet deposition. The results show a characteristic curve with elevated platelet deposition in the range of 150 1/s. For the interpretation of these results a numerical model was developed. It applies the Monte Carlo method to model a random walk of platelets. This diffusive motion was superimposed on the convective motion by the Couette flow. A satisfactory match to the experimental data was achieved. & 2012 Elsevier Ltd. All rights reserved.
1. Introduction Thrombo-embolical events in cardiovascular devices start with the deposition of platelets onto the material of the implant. This is only the beginning of a complex sequence of events, finally leading to a thrombus, which can block the flow through the device or can dislodge. This process has been subject to a large number of publications. Some authors of recent papers model the complex chain of reactions, including as many reactions as possible. Flamm and Diamond (2012) review such models spanning multi-scale phenomena, from the scale of a calcium ion to the diameter of the aorta. Such models may contain 77 reactions with 132 kinetic rate constants and 70 species, or with 28 fluid phase species and complexes and 44 lipid-bound factors and complexes. For the engineer designing a cardiovascular device, a much simpler model is needed. It should work with assumptions that can be checked experimentally. In the search for such a model, the literature was reviewed. The first experimental models did not include numerical analysis; they were devised to observe the generation of a thrombus (Petschek et al., 1968). Ex-vivo blood was directed to flow onto a glass plate. This flow chamber generates a stagnation point flow and a radial outflow. Later Baldauf et al. (1978) and Reininger et al. (1992) used this flow
n
Corresponding author. Tel.: þ49 30 450 553801; fax: þ49 30 450 553938. E-mail address: [email protected] (K. Affeld).
0021-9290/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jbiomech.2012.10.030
type for further studies. The deposition of platelets could be observed with a dark field reflected light microscope. This model was soon replaced by a parallel flow model, which was more suitable for mathematical analysis (Turitto and Baumgartner, 1975a), and a numerical model was proposed. More models were developed by various authors and reviewed (Stubley et al., 1987). The basic equation for all models is D @c=@x wall ¼ k c ð1Þ At the wall, no convective motion is assumed, so the platelet transport is caused by diffusion only. The platelet flux – the left side of Eq. (1) – depends on the platelet diffusivity D and the concentration gradient at the wall. This flux must be equal with the deposition rate of platelets on the surface—the right side of Eq. (1). The deposition rate is assumed to depend on the concentration c of available free platelets and a reaction constant k, a quantity reflecting the likelihood of deposition. The diffusivity of platelets in platelet-rich plasma DPRP is about 1.5 10 13 m2/s. If, however, red blood cells are also present, the diffusivity is much larger, to the order of two magnitudes (Turitto and Baumgartner, 1975b). This is caused by the mixing action of red blood cells as they turn and tumble in a shear flow (Goldsmith, 1972; Goldsmith et al., 1995; Tokarev et al., 2011). Zydney and Colton (1988) have reviewed experiments and models and proposed a diffusivity model for platelets in full blood (Eq. 2) which is still used today. This model assumes a linear dependence of the shear rate g on the diffusivity. Furthermore, the diffusivity also
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depends on the hematocrit H and the diameter of red blood cells dRBC: 2 D ¼ DPRP ð1HÞ þ 0:15 dRBC H=4 g ð1HÞ1:8 ð2Þ At a distance from the surface, basic laws of fluid mechanics determine the transport of platelets. Platelet deposition at the wall is thought to be limited by only one surface property: the surface availability. The concept of surface availability (Turitto and Baumgartner, 1975a) is that already-adhered platelets form a monolayer on the surface. They occupy a site and exclude oncoming platelets from adhering. It is assumed that this surface availability attains zero when the surface is completely covered. This is reflected in Eq. (3) for the reaction constant. The surface coverage is defined by the surface density of already-adhered platelets M(x, t) in relation to the surface density of full coverage MN. In this way, a time dependence of platelet deposition is achieved. k ¼ k0 1 M ðx,t Þ=M 1 ÞÞ
ð3Þ
The reaction constant k0 has been a subject of discussion by various authors. Some state that it is constant. However, comparison with experiments lead other authors to the assumption that the reaction constant increases with the shear rate (David et al., 2001), either linearly or in a more complex way (Sorensen, 2002; Weller, 2008). The many discussions and open questions have lead to the work presented in this paper, which consists of an experimental model and a numerical model. The most elementary viscous flow has been selected – the plane Couette flow – and, in addition, the model is limited only to the initial phase of platelet deposition. In this way, the question of surface availability is excluded. The complex cascade of thrombus generation is truncated to its first phase.
2. Materials and methods 2.1. Experimental model The experimental model is designed to simulate the initial phase of thrombus generation in the human body. If in a blood vessel the endothelial layer is locally disrupted, collagen fibers are exposed. Platelets carried by the bloodstream become activated upon contact with collagen and deposit on its surface. This is considered the initial phase of the coagulation system. The experimental model is designed to create the simplest conditions: stationary blood flow, constant low (o1000 1/s) shear rate which does not activate platelets (Hellums, 1994), anti-thrombogenic surface throughout the device, collagen exposed in a small area, freshly-drawn citrated human blood with non-activated platelets. A flow with a constant shear rate is approached by a parallel plate design (Watanabe et al., 2011). Fig. 1 shows the experimental model: a circular stationary glass plate 40 mm in diameter and an upper rotational plate 35 mm in diameter are spaced 50 mm apart. A set of 50 mm gap for each experiment is done with a precision of the depth of sharpness of the used microscope. The resulting error of the gap set was 4.6 mm. This space is filled with blood and the Couette flow is created when the upper plate is rotated. Both plates are microscopic circular glass coverslips of 175 mm thickness for best optical conditions. The model is positioned in an inverted microscope (Fluovert FU, Leica AG, Germany). The coverslip glass plates are covered with albumin (Carl Roth GmbH & Co KG, Karlsruhe, Germany). This albumin layer makes the glass anti-thrombogenic and thus non-activated platelets will not deposit on the glass. To induce platelet deposition, a spot of collagen (Horm, Nycomed Austria GmbH, Austria) with a diameter of 1000 mm is applied at a radius of 7.5 mm on the lower stationary coverslip. The field of view of the microscope is directed onto this spot. A shear rate between zero and 700 1/s can be generated at this spot. Blood was drawn from the cubital vein of voluntary healthy male donors, which gave informed consent on the work. For each run, two samples each of 9 ml of blood were drawn into vacuum tubes (‘‘Vacutainer’’) containing 1 ml of anticoagulant (0.106 mol/L sodium citrate) (Sarstedt AG, ¨ Nurmbrecht, Germany). The first sample was discarded to exclude platelets that were activated by venipuncture. For fluorescent labeling of the platelets, 0.01 g of Quinacrine dihydrochloride (Sigma Aldrich Chemie GmbH, Steinheim, Germany) was dissolved into 50 g of phosphate-buffered solution (Invitrogen, UK). Then 100 mL of that solution was added to the blood sample. After gentle shaking, the sample was protected from light and kept at room temperature. The samples were used within three hours after venipuncture. Each sample was divided into six portions allowing performing up to six runs. Prior to each run about 100 mL of blood was put onto the
Fig. 1. Couette flow model with two parallel plates (coverslips). Coverslips were used because of their superior optical qualities. The lower coverslip is fixed, while the upper rotates. Blood fills the 50 mm gap between the coverslips. A small spot 1 mm in diameter on the lower coverslip is covered with collagen. The platelets are dyed fluorescent and glow when illuminated by excitatory light. lower coverslip. The upper coverslip was carefully lowered avoiding the generation of air bubbles. Through the lower stationary plate light with 355 to 425 nm wavelength illuminated the blood. Illuminated platelets emit light at 496 nm. The upper coverslip was set in motion, and after focusing on the stationary coverslip plane the recording began. The process of deposition was recorded with a video camera (UI-2230-M, Imaging Development Systems GmbH, Germany) with a resolution of 1024 768 pixels and a frame rate of 20 fps and covering a field of view of 480 360 mm2. Once deposited on the plate, platelets remained fixed and were easily distinguishable from the moving platelets. Camera images were processed and resulted in surface coverage by the platelets as a function of time and shear rate. The duration of each run was 60 s. After this time, platelets start to form large aggregates that create wakes affecting the probability of adhesion, a phenomenon which is not a part of this study.
2.2. Numerical model The objective of the numerical model is to reproduce and to interpret the results of the experiments. The numerical model follows the concept of Virchow (1856), who concluded that three parameters were responsible for thrombus generation: the quality of the blood, the quality of the vessel wall and the quality of the flow. He stated in hypothesis that these three players interact and this became known as Virchow’s triad. In this numerical model, only the initial short phase of platelet deposition is considered. The highly complex system of coagulation with its many stages of platelet activation and the fibrinogen–fibrin system are left out. The numerical model consists of the following elements: a) Shear flow field: In the experimental model, the flow field is the simplest and is characterized by an approximately homogeneous shear rate in the field of view. For the numerical model, a rectangular box is considered, which represents a small part of the Couette flow field. This rectangular box has a height of 50 mm, a width of 1000 mm and a length of 1200 mm (Fig. 2). Its base plane is defined as a wall at which the platelets deposit. The flow enters the box through the front plane. At the inflow side of the box new platelets are inserted at each time step. The number of platelets inserted at each time step is proportional to the flow in the respective velocity. It is thus proportional to time, distance z to the base plane and shear rate to achieve a homogeneous concentration. Platelets which leave the box through its lateral or upper sides during a time step are reflected inwards. For this simple flow and geometry the mirror reflection does not make notable difference. For more complicated flows, however, more appropriate periodic boundary conditions should be incorporated. Platelets close to the bottom can deposit if certain conditions are met according to our model. The flow has no vertical velocity component and
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b)
c)
d)
e)
f)
g)
K. Affeld et al. / Journal of Biomechanics 46 (2013) 430–436 platelets can approach the wall by diffusion only. A linear velocity profile is specified according to the shear rate. Blood model: The model fluid is assumed to be a single-phase fluid with the Newtonian viscosity of blood. Red blood cells are not modeled. However, their mixing effect on platelets is taken into account. This mixing augments the diffusivity of platelets. This is computed with the help of an experimentally derived diffusivity function described by Eq. (2). Platelets: Platelets are assumed to be virtual particles having no influence on the flow field. These particles are considered the center of the platelets, having a virtual diameter of 2 mm. This plays a role when the platelet approaches the wall. The platelets are distributed evenly and are inserted at the inflow plane. The model simulates the finite number of platelets according to the platelet concentration of the donor. In an initial step, particles were distributed evenly in the domain. In the following steps, new particles enter at the inlet plane or by dispersion through the upper face of the deposition domain. Movement of platelets: The movements of platelets are simulated with the Monte Carlo random walk method. This allows the observer to follow the path and the fate of each individual platelet and as such is very close to microscopic observation as performed in the experimental method. Platelets move with the flow field from their initial position after the start. They move stepwise according to the flow field. The flow field step vector is computed individually for each platelet position. This approach guaranties that grid dependence is avoided. Additionally at each time step, a diffusion step vector is computed and added to the flow field step vector of each platelet. A time step tstep of 10 ms is allotted to one numerical cycle. After each time step, every platelet attains a new position. If a platelet penetrates the vertical side planes, its position is horizontally reflected. If a platelet approaches the wall by this motion, it enters the deposition domain. Diffusivity of platelets: The diffusivity is modeled with diffusion steps based on the random walk model. The diffusion steps are computed with a Gaussian distribution and a mean square distance of 2 Ddim D tstep, where Ddim ¼ 3 for three-dimensional space, D being the effective diffusivity and tstep the time step length. The directions of the diffusion step vectors are computed in such a manner that all directions are evenly distributed. For the platelet effective diffusivity D, the equation of Zydney and Colton is used (see Eq. 2). Deposition domain: The deposition domain is the rectangular box just above the fixed base plane. The height of this box is h and is subject to variation. Once a platelet enters the deposition domain it is close enough to interact with a wall. The first panel of 200 mm length in the flow direction is defined to be antithrombogenic . Platelets touching the wall there will not be deposited. The following panel of 1000 mm length represents the collagen spot and is thrombogenic. A platelet touching here can deposit according to the probability that it will do so. Probability of deposition: The probability of deposition is defined as the probability of a platelet at a certain distance h to the wall to deposit within the next time step tstep. It can be experimentally assessed by observation of
moving platelets (Machin et al., 2005). In the numerical model, the distance h is a parameter of the model. The probability of deposition pdeposition is the product of two individual probabilities and one flow-related factor. One probability pplatelet characterizes the reactivity of platelets; another probability pwall characterizes the thrombogenicity of the wall. A third factor fflow characterizes the influence of the shear rate at the wall on the deposition and is a property of the flow. Each of those three numbers is a number between zero and one. All three act together and define the probability of deposition for each individual platelet: pdeposition ¼ pplatelet pwall f flow
ð4Þ
If for instance pdeposition is one, the platelet will deposit in the current time step. h) Probability–reactivity of the platelet: This probability characterizing the reactivity of the platelet models the activation state of the platelet. It is set to one if the platelet is fully activated, but is always a number larger than zero. If this were not the case, the deposition of a non-activated platelet contacting a maximally activating surface, such as collagen, could not occur 0 o pplatelet r 1
ð5Þ
i) Probability–thrombogenicity of the wall: Unlike the previous probability, it is possible that this probability characterizing the thrombogenicity of the wall can assume a zero value. This is because there exist certain materials onto which platelets do not deposit. At the other extreme, collagen is a very thrombogenic material. In this case, the probability is set to one. The encounter between non-activated platelets and collagen can be seen as the design point of the coagulation system. 0 r pwall r 1
ð6Þ
j) Influence of the shear rate: The shear rate has a twofold influence on platelet deposition: First, through convection and diffusivity it determines the number of platelets which are available to interact with the wall. Second, the platelet requires a certain amount of time (called residence time) within a certain distance from the wall in order to deposit. A residence time has been proposed by Reininger et al. (1995) and Machin et al. (2005). The platelet travels with a certain velocity along the wall and reacts only if its distance from the wall is small enough. In this way the residence time may be expressed as a function of the platelets velocity and a length—the distance to the wall at which a reaction can take place. This can be visualized as a handshake between two people: one at rest, the other sliding by on a conveyor belt. There is a certain distance between the two people within which the handshake may take place – at most the combined length of the two outstretched arms – and a certain amount of time required for the event to take place. This time is the residence time. It is calculated by dividing the velocity of the conveyor belt by the double arm length. Length divided by velocity, however, has the dimension of shear rate. This residence time is thus defined as proportional to 1/(shear rate). Taking this into account, one derives. n 0 o f flow ¼ min 1, C shearrate =g ð7Þ r 1 with C shearrate 40 1=s
The exponent n is a number close to one. It is assumed to include other influences than the shear rate alone. It should be noted that the flow factor fflow takes the viscous flow into account and acts in a homogenous shear field on every platelet in the same way. k) Sequence of computation: The simulation of platelet deposition is based on discrete time steps and three-dimensional continuous space. All quantities are calculated separately for each platelet. There is no interaction between platelets included in the model.
The following sequence is performed in one computational cycle for each time step:
insertion of new platelets at the inlet (z position) according to inlet velocity
Fig. 2. Geometry of the numerical model for the simulation of platelet deposition in the experimental model. The numbers indicate the dimensions. The height of the rectangular box–the z-axis–is scaled up about 20 times. Flow is shown by red arrows. The white area on the bottom is given the deposition probability of zero, while the shaded part has the probability of one. This is the area that models the collagen surface. The rectangular box of height h ¼ 1 mm marks the deposition domain. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
profile and platelet concentration, while the y position of each inserted platelet is computed with a random number. In this way, an even concentration of platelets at the inlet is achieved. calculation of the next platelet position with the Monte Carlo random walk: calculation of the displacement due to convection, calculation of the diffusivity due to shear rate, calculation of the displacement due to diffusivity, calculation of the new platelet position in space. reflection of platelets at upper, lower, and side walls. detection and removal of platelets that exited the rectangular box through the outlet. calculation of the deposition probability for each platelet present in the deposition domain (see Eq. 4).
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generation of a random number for each platelet in the deposition domain. If it is random number (0y1) is smaller than its deposition probability pdeposition, the platelet deposits. The results of the simulations are translated to experimental quantities by conversion of deposited platelets to surface coverage rate. A large number of platelets – approximately 9000 – are observed in the rectangular box at all time steps. One run simulates two seconds, which results in the computation of about 2 million individual positions. As platelets enter the deposition space, the calculation of the deposition probability begins. Platelets that deposit at the wall plane are counted, and their rate is computed as a function of the flow—the shear rate.
3. Results 3.1. Experimental results Experiments were performed with blood from three different donors who gave voluntary informed consent. Fig. 3 shows the results of an experiment and its subsequent evaluation. It shows the surface coverage by platelets over time. The curves do not start out with a surface coverage of zero. This indicates that platelets do deposit before the run and the creation of a shear field begins. This can be explained from the first contact of blood with the collagen surface during filling. The surface coverage increases almost linearly over time. The inclines of the curves represent the growth of surface coverage. This is proportional to the deposition rate per area. Fig. 4 shows the data from all three donors. Donors’ blood differs with regards to platelet concentration (donors 1/2/3: 193 000/163 000/ 263 000 1/mL) and hematocrit (donors 1/2/3: 46.3/43.0/40.6%). The data scatter, as is often found in experiments with platelets. However, a peak in the deposition rate is obvious. Fig. 4 at the lower right shows a regression curve of all data points. This curve requires further interpretation. Why is there a higher rise in platelet deposition at low shear rates? Why is there a peak with a subsequent slope? What is causing the peculiar shape of the curve? 3.2. Numerical results The numerical model was designed to quantitatively reproduce and to interpret the experimental results described above. This includes the platelet deposition rate at zero shear rate, the value of the initial incline of the platelet deposition curve, the peak deposition rate and the value of the shear rate at the peak and downward slope. Fig. 5 shows the platelets’ path lines during a time of 100 ms and a shear rate of 100 1/s. The Z-axis is scaled up tenfold. Of the 9000 platelets considered in the volume, only a fraction is depicted here for reasons of clarity. The first feature to be modeled is the deposition rate at shear rate zero, which
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represents no flow condition. There is no convective transport of the platelets and only few platelets enter the deposition space to deposit. The second feature is the rise in the deposition rate accompanying a rising shear rate. Fig. 6a shows the deposition rate as a function of the rising shear rate and of the parameter Cshearrate. All curves show a nearly linear function at low shear rates. After the linear increase, each curve reaches a peak. This third feature occurs at a specific shear rate, which is accounted for in the numerical model with the parameter Cshearrate. The parameter Cshearrate is chosen to match the experimental results. Apparently, this hypothetical approach models the physical– chemical process of the adhering platelet sufficiently well. The fourth feature is the magnitude of the deposition rate. It depends mainly on the concentration of platelets and their reactivity. In Fig. 6b again the deposition rate is plotted over the shear rate and parameters are chosen for a best fit to the experimental results of donor 1. The best fit with coefficient of determination of R2 ¼0.97 was obtained by modifying the parameters with a genetic algorithm. The square root mean error was minimized. For the height of the deposition domain h¼1 mm and the donor’s platelet concentration of 193,000 1/mL, this results in the parameters Cshearrate ¼137 1/s, the exponent n¼0.8, platelet reactivity pplatelet ¼0.023 and (Eq. 7). pwall is set to one. For donors 2 and 3 the same comparison is made as shown in Fig. 6c and d. The match in these cases is less exact with R2 of 0.88 and 0.84. The reason for this remains open and is a subject for further research—along with the experimental method.
4. Discussion Experiments involving platelets often result in data that scatter, as do the data from the experimental model presented here. This is caused both by the excitable nature of the platelet and by the imperfection of the experimental model. Improvements to the model can be achieved with several modifications. The plates of the Couette device are coverslips. They were chosen for their good optical quality, but are very thin and fragile and do not hold their shape well if stressed. The stationary plate on which the platelet deposition is observed will be partially reinforced with a thicker glass plate. The rotating coverslip will be replaced by a solid cylinder. The latter will be equipped with a central pivot resting on the stationary plate. This pivot will define the gap with more precision. Variation of gap size is around 10% and is one cause of the observed data scatter. Step motors for better speed control will replace the electric brush motors. Apart from a better mechanical design, blood needs to be classified more precisely with the method used in the hematological laboratory. Finally, the microscopic methods need to be improved.
Fig. 3. (a) The 190 130 mm2 image of deposited platelet recorded after 60 s of shear flow. Flow is from left to right. (b) Examples of platelet deposition as a function of time representing results (donor 2) from runs on different days and different blood samples at one shear rate of 175 1/s. During the initial 60 s there is little interaction between a single deposited platelets and an incoming one. The inclines of curves are the platelet deposition rates.
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Also required is the assessment of other parameters that are part of the numerical model and are required to validate the numerical model. Such a parameter is the probability of depositing, which is the product pplatelet pwall. It can be accessed from the observation of platelets moving near the wall (Machin et al., 2005). Counting the platelets, measuring their distance from the wall and comparing them to the deposited platelets will enable the computation of a deposition probability. The results of the numerical model show its ability to model the dependency of deposition rate over shear rate. The model contains parameters that can be quantified directly: shear rate, platelet concentration and hematocrit. Other parameters such as the height of the deposition domain, two probabilities pplatelet, pwall and the factor of flow influence fflow have to be estimated or derived by matching to the experimental data. The match obtained thus far can only be achieved with the assumption of a flow factor fflow ¼min(1, (Cshearrate/g)n). This factor is one for a shear rate lower than 137 1/s. In excess of this shear rate, as defined in the model by Cshearrate, the residence time is shorter than the reaction time. This means that the probability to react decreases and fewer platelets deposit. The time to adhere and to form tethers is discussed by Nesbitt et al. (2009). Machin et al. (2005) have discussed such a reaction time and have presented one method of measuring it. The numerical method as it is presented here uses the Monte Carlo method to simulate directly the diffusion of a platelet (Affeld et al., 2004). This results in translocation, which then is superposed on a convective flow field. In this way, diffusion and convection are decoupled in regard to computation. In most of the models, (Sorensen, 2002; David et al., 2001; Weller, 2008) and (Behbahani, 2011), diffusion and convection are not treated separately. This leads to equations that do not converge easily (Sorensen, 2002). There is also another parameter not treated separately in these models: the reaction constant k0. This comprises the reactivity of a platelet and the reactivity of the wall. In the model presented here, both reactivities are separate parameters and can be tested separately, for example with methods by Machin et al. (2005).
The numerical model does provide an explication of the peculiar curve from the experiment. However, the model is applied only to the most elementary flow, the uniform shear flow. In implantable devices, the flow is more complex. The model is presently being applied to such a three-dimensional flow with widely varying shear rates. This is the stagnation point flow, for which many experimental data, that of the authors (Affeld et al., 1995) and of others, are available. Another suitable flow for validation is the sudden tubular expansion by Karino and Goldsmith (1979). The objective of these applications would be a further validation of the model.
5. Conclusions The experimental work of the present study shows a range of shear rates with an elevated deposition rate, in other words: a range of shear rates favorable for the deposition of platelets. The
Fig. 5. Path lines of platelets in the numerical model during a time of 100 ms and a shear rate of 100 1/s. Z-axis is scaled up ten times. There are about 9000 platelets in the volume, however, for reasons of clarity only a fraction of those platelets is shown. A space above the base plane 2 mm thick is the deposition domain. Once a platelet enters this domain, its probability of depositing is calculated.
Fig. 4. Platelet deposition rate plotted over shear rate for three different donors, but identical device. Each donor curve is based on approximately 55 single experiments. The data scatter—a major cause of this thought to be the insufficient control of gap and blood variability. The lower right regression curve is from the data for all three donors. All curves start near zero (1), have an initial incline (2), reach a peak of deposition rate at a certain shear rate (3) and then slope down.
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Fig. 6. (a) Deposition rate plotted over shear rate calculated for three different values of Cshearrate with constant pplatelet ¼0.023 and exponent n ¼0.8. This simulates the influence of a platelet’s residence time in the vicinity of the wall. (b) Deposition rate plotted over shear rate. Match of experimental and numerical results for donor 1: The respective parameters are pplatelet ¼0.023, Cshearrate ¼ 137 1/s and exponent n¼0.8. The resulting coefficient of determination is R2 ¼ 0.97. (c) Match for donor 2: pplatelet ¼ 0.0175, Cshearrate ¼ 225 1/s, and exponent n¼1.5. The R2 is 0.88. (d) Match for donor 3: pplatelet ¼0.0124, Cshearrate ¼ 175 1/s, and, exponent n¼ 1.55. The R2 is 0.84.
numerical model uses the Monte Carlo method of direct simulation of the path of platelets in a flow field. The model can be applied to flow fields obtained from a flow solver. In this case, the flow field is the most elementary case of viscous flow—the shear field is homogeneous throughout. The numerical model replicates the flow between the two Couette plates and the results model the experimental relationship of platelet deposition rate over shear rate. The numerical model, however, needs to be tested in a more complex flow with a variety of shear rates at the wall. Such a flow is known as the stagnation point flow, which is used for further validation of the model.
Conflict of interest statement The authors do not have any financial or personal conflicts of interest.
Acknowledgements We would like to thank Dipl.-Ing. Jens Schaller and Dipl.-Ing. Torsten Schneider for their helpful technical contributions for numerical modeling. Furthermore, we thank F. Probst, K. Hoell, and K. Loewenhoff for their technical contributions for experiments. The study was supported by the German Research Foundation (Grant number: AF 3/37-1) and by the Alexander von Humboldt Foundation. References Affeld, K., Goubergrits, L., Kertzscher, U., Gadischke, J., Reininger, A., 2004. Mathematical model of platelet deposition under flow conditions. The International Journal of Artificial Organs 27, 699–708.
Affeld, K., Reininger, A.J., Gadischke, J., Grunert, K., Schmidt, S., Thiele, F., 1995. Fluid-mechanics of the stagnation point flow chamber and its platelet deposition. Artificial Organs 19, 597–602. Baldauf, W., Wurzinger, L.J., Kinder, J., 1978. Role of stagnation point flow in formation of platelet thrombi on glass surfaces in tubes with various geometry. Pathology, Research & Practice 163, 9–33. Behbahani, M.J., 2011. Modeling and Simulation of Shear-Dependent Platelet Reactions in Blood Vessels and Blood-Contacting Medical Devices. Dr. Hut, Aachen. David, T., Thomas, S., Walker, P.G., 2001. Platelet deposition in stagnation point flow: an analytical and computational simulation. Medical Engineering & Physics 23, 299–312. Flamm, M., Diamond, S., 2012. Multiscale systems biology and physics of thrombosis under flow. Annals of Biomedical Engineering 40 (11), 2356–2364. Goldsmith, H.L., 1972. The flow of model particles and blood cells and its relation to thrombogenesis. Progress in Hemostasis and Thrombosis 1, 97–127. Goldsmith, H.L., Bell, D.N., Braovac, S., Steinberg, A., Mcintosh, F., 1995. Physical and chemical effects of red-cells in the shear-induced aggregation of human platelets. Biophysical Journal 69, 1584–1595. Hellums, J.D., 1994. 1993 Whitaker lecture—biorheology in thrombosis research. Annals of Biomedical Engineering 22 (5), 445–455. Karino, T., Goldsmith, H.L., 1979. Adhesion of human-platelets to collagen on the walls distal to a tubular expansion. Microvascular Research 17, 238–262. Machin, M., Santomaso, A., Cozzi, M.R., Battiston, M., Mazzucato, M., De Marco, L., Canu, P., 2005. Characterization of platelet adhesion under flow using microscopic image sequence analysis. The International Journal of Artificial Organs 28, 678–685. Nesbitt, W.S., Westein, E., Tovar-Lopez, F.J., Tolouei, E., Mitchell, A., Fu, J., Carberry, J., Fouras, A., Jackson, S.P., 2009. A shear gradient-dependent platelet aggregation mechanism drives thrombus formation. Nature Medicine 15 665-U146. Petschek, H., Adamis, D., Kantrowi, Ar, 1968. Stagnation flow thrombus formation. Transactions American Society for Artificial Internal Organs 14, 256–260. Reininger, A.J., Reininger, C.B., Heinzmann, U., Wurzinger, L.J., 1995. Residence time in niches of stagnant flow determines fibrin clot formation in an arterial branching model—detailed flow-analysis and experimental results. Thrombosis and Haemostasis 74, 916–922. Reininger, C.B., Reininger, A.J., Hormann, A., Steckmeier, B., Schweiberer, L., 1992. Quantitative-analysis of platelet-function using stagnation point flow aggregometry—1st clinical-results. International Angiology 11, 247–255. Sorensen, E.N., 2002. Computational Simulation of Platelet Transport, Activation, and Deposition. University of Pittsburgh.
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K. Affeld et al. / Journal of Biomechanics 46 (2013) 430–436
Stubley, G.D., Strong, A.B., Hale, W.E., Absolom, D.R., 1987. A review of mathematical-models for the prediction of blood-cell adhesion. Physicochemical Hydrodynamics 8, 221–235. Tokarev, A.A., Butylin, A.A., Ataullakhanov, F.I., 2011. Platelet adhesion from shear blood flow is controlled by near-wall rebounding collisions with erythrocytes. Biophysical Journal 100, 799–808. Turitto, V.T., Baumgartner, H.R., 1975a. Platelet deposition on subendothelium exposed to flowing blood—mathematical-analysis of physical parameters. Transactions American Society for Artificial Internal Organs 21, 593–601. Turitto, V.T., Baumgartner, H.R., 1975b. Platelet interaction with subendothelium in a perfusion system—physical role of red blood-cells. Microvascular Research 9, 335–344.
¨ ¨ Virchow, R., 1856. Uber den Faserstoff: Phlogose und Thrombose im Gefaßsystem, Gesammelte Abhandlungen zur wissenschaftlichen Medicin. Meidinger & Sohn Corp, Frankfurt am Main. Watanabe, N., Affeld, K., Schaller, J., Schmitmeier, S., Reininger, A., Goubergrits, L., Kertzscher, U., 2011. Investigation of human platelet adhesion under low shear conditions in a rotational flow chamber. Journal of Biorheology 25, 64–70. Weller, F.F., 2008. Platelet deposition in non-parallel flow. Journal of Mathematical Biology 57, 333–359. Zydney, A.L., Colton, C.K., 1988. Augmented solute transport in the shear-flow of a concentrated suspension. Physicochemical Hydrodynamics 10, 77–96.