Circulation. 1996;94:763-774
(Circulation. 1996;94:763-774.)
© 1996 American Heart Association, Inc.
Computer Simulation of Systemic Circulation and Clot Lysis Dynamics During Thrombolytic Therapy That Accounts for Inner Clot Transport and Reaction
Sriram Anand, PhD;
Scott L. Diamond, PhD
the Bioengineering Laboratory, Department of Chemical Engineering, The State University of New York at Buffalo.
Correspondence to Dr Scott L. Diamond, Bioengineering Laboratory, Department of Chemical Engineering, The State University of New York, Buffalo, NY 14260. E-mail sdiamond@eng.buffalo.edu.
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Abstract
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Background We developed a computer model to predict lysis rates
of thrombi for intravenous thrombolytic regimens based on the
convective/diffusive penetration of reacting and adsorbing fibrinolytic
species from the circulation into the proximal face of a dissolving
clot.
Methods and Results Solution of a one-compartment plasma model provided the dynamic concentrations of fibrinolytic species that served as inlet conditions for simulation of the one-dimensional spatiodynamics within a dissolving fibrin clot of defined composition. The model predicted the circulating levels of tissue plasminogen activator (TPA) and plasminogen levels found in clinical trials for various intravenous therapies. To test the model predictions under in vitro conditions, plasma clots were perfused with TPA (0.1 µmol/L) and plasminogen (1.0 µmol/L) delivered at constant permeation velocity of 0.1 or 0.2 mm/min. The model provided an accurate prediction of the measured lysis front movement. For TPA administration regimens used clinically, simulations predicted clot dissolution rates that were consistent with observed reperfusion times. For unidirectional permeation, the continual accumulation of adsorbing species at the moving lysis front due to prior rounds of solubilization and rebinding was predicted to provide for a marked concentration of TPA and plasmin and the eventual depletion of antiplasmin and macroglobulin in an advancing (
0.25 mm thick) lysis zone.
Conclusions Pressure-driven permeation greatly enhances and is a primary determinant of the overall rate of clot lysis and creates a complex local reaction environment at the plasma/clot interface. With simulation of reaction and transport, it becomes possible to quantitatively link the administration regimen, plasminogena activator properties, and thrombolytic outcome.
Key Words: fibrinolysis • hemodynamics • pharmacokinetics • plasminogen activators • thrombolysis
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Introduction
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Intravenous thrombolytic therapy for the treatment of acute
MI is conducted in accordance with a specific administration
regimen. Depending on the regimen and plasminogen activator(s)
used, a set of events (plasminogen activation, inhibition, clearance)
determines the constituents of the circulation that come in
contact with the occlusive thrombus. Species in the plasma adjacent
to the clot can penetrate into the proximal face of the clot
via diffusion and pressure-driven permeation. The effects of
pressure-driven permeation have been studied in vitro
1 2 3 4 5 and in vivo
6 7 8 and indicate thrombolysis is diffusion
limited unless transport by pressure-driven permeation is present.
Accurate prediction of thrombolytic outcome must account for
both the biochemical kinetics and penetration rates of proteins
in dissolving thrombi.
Arterial thrombi within the coronary artery are 1 cm long.9 With intravenous administration, plasminogen activators within the systemic circulation are carried efficiently to the nearest unobstructed coronary bifurcation upstream of the clot. If the clot is close to this unobstructed bifurcation, the clot is initially in direct contact with the well-mixed systemic circulation. If the clot is centimeters downstream of an unobstructed bifurcation, then recirculating secondary flows are responsible for delivery of plasminogen activators to the clot. In this case, time lags and inefficient transport coupling may occur between the pharmacodynamics of the systemic circulation and lytic events at the clot. At the proximal face of the clot, thrombolytic therapy is a process characterized by the convective/dispersive penetration of enzymes that bind the erodible fibrin fibers of the blood clot.4 5 10 Protease transport involves the binding of proteins to the fibrin fiber network with subsequent solubilization of these enzymes back into the interstitial fluid of the clot.5
In the present study, we coupled the calculation of the dynamics in the circulation with the spatiotemporal transport/reaction phenomena that occur within a dissolving thrombus. In these simulations, clot lysis rate is a direct and quantitative result of (1) the biochemistry of the circulation, (2) the properties and administration regimen of the thrombolytic agent, (3) the physical structure and biochemical composition of the thrombus, and (4) the local pressure environment. Using simulation and in vitro experimentation, we showed that increases in permeation lead to direct increases in the rate of plasma clot lysis. The simulation provided the tool with which to understand reaction and transport in a coupled manner as these two processes occur in vivo. The agreement between simulations and in vitro plasma clot lysis indicates that fibrinolysis is quantifiable despite its reaction complexity. With continually improving measurements of clot composition, permeability, and reaction kinetics, computer simulation can serve as a useful predictive tool for the evaluation of new agents and therapeutic approaches.
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Methods
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Plasma Clot Lysis Under Conditions of Permeation
Human venous blood was collected with anticoagulation (final
concentration of 0.38% wt/vol sodium citrate) through venipuncture
of healthy, nonsmoking volunteers (who gave informed consent)
in accordance with the guidelines and approval of the Internal
Review Board of the State University of New York at Buffalo.
Platelet-rich plasma (obtained through centrifugation of whole
blood at 150
g for 15 minutes) was further centrifuged at 1900
g for 15 minutes to yield PPP. Small aliquots of recalcified PPP
were promptly suctioned to the midpoint of a 1-mL polystyrene
sterile pipette to form individual 2-cm-long PPP clots. At 4
hours after coagulation, a lytic cocktail containing 0.1 µmol/L
recombinant human TPA (Activase, Genentech Inc) and 1.0 µmol/L
human Glu-plasminogen (American Diagnostica, Inc) diluted in
0.05 mol/L Tris-HCl, pH 7.4, 0.1 mol/L NaCl, and 5 mmol/L CaCl
2 was backfilled adjacent to each PPP clot. A Harvard syringe
pump with multiple syringes was then used to perfuse the solutions
into the clots at constant velocity of 0.1 or 0.2 mm/min. The
position of the lysis front was recorded with time over the
course of the experiment. All experiments were conducted on
a microscope stage maintained at 37°C with an airstream
incubator.
Theory
Intravenous thrombolytic administration regimens for acute MI involve a combination of bolus and continuous infusions. In the circulation, changes in blood chemistry occur due to fibrin-independent plasminogen activation. To describe these changes in the circulation, we calculated the concentration of each species (Ci, µmol/L) through the solution of a set of coupled ordinary differential equations. The differential equation used to model the rate of change of concentration of each species (Ci) in plasma is given in Equation 1. The plasma of the circulation was treated as a single, well-mixed compartment that can receive a user-specified infusion regimen for any species, Ii(t) (µmol/s), and any species may be eliminated at a given rate by clearance, Ki(t). This algorithm represents a generalized approach that is consistent with earlier pharmacodynamic models of thrombolytic therapy.11
 | (E1) |
The reaction network treated in the present study is shown in Fig 1, where fibrin is present only in the clot and is not present in the circulation. In the circulation, the fibrin of Fig 1 is replaced by a nonspecific soluble cofactor for fibrin-independent TPA-mediated plasminogen activation. This fibrin-independent activation of plasminogen by TPA is neglected in the clot and is not shown in Fig 1 because that reaction is slow compared with fibrin-dependent activation. Since fibrinogen, degradation products, and other factors in blood can serve as cofactors for this reaction, the rate constants for fibrin-independent TPA-mediated plasminogen activation in the circulation are lumped and apparent. This approach captures to some extent the evolving role of FDP in the circulation in a time-averaged manner. We have used the activation kinetics of Glu-plasminogen by TPA in the absence of fibrin12 as the apparent Michaelis-Menten kinetic constants (Km=28 µmol/L, k2=0.3 s-1) as originally proposed by Tiefenbrunn et al11 for TPA-mediated plasminogen activation in plasma. In the circulation, the rate of generation (or loss) of the ith species Ri(Cj,Ck) was determined by the reaction between the jth and kth species. Reaction rate expressions may be of any form, typically Michaelis-Menten or second-order association kinetics, as shown in Fig 1 (without fibrin) and Table 1. The clearance rate of each species, Ki, is taken as a first-order process in Ci determined from the half-life of the species where Ki=(k·Ci·Vplasma). The circulatory half-lives of TPA and uPA (two-chain) were taken as 5 and 60 minutes, respectively.13 14 Nominal plasma levels of the various fibrinolytic species (Table 2) were used as the initial conditions for the solution of the set of ordinary differential equations for the 12 soluble species shown in Fig 1. The calculated dynamic levels of species within the circulation then served as inlet conditions for prediction of local clot lysis dynamics (Fig 2).
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Figure 1. Summary of the fibrinolytic reaction pathway. Plasminogen (plg) is activated to plasmin (plm) in the fluid phase by urokinase (uPA), whereas fibrin-bound plg is activated TPA in the fibrin phase. The free-phase plm is inhibited by AP and macroglobulin (MG), whereas the bound-phase plm is protected from inhibition. The uPA and TPA are inhibited in the free phase by PAI-1. Inhibition reactions were assumed to be irreversible. Soluble species can reversibly adsorb and desorb with fibrin under kinetically controlled conditions. The plg and plm compete for the same sites on the intact fibrin monomer, whereas TPA binds a unique site. All kinetic constants for reversible adsorption and enzymatic reaction are given in Table 1. Single-chain to two-chain conversions of TPA and uPA as well as conversions of Glu-plasmin(ogen) to Lys-plasmin(ogen) were not considered.
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Figure 2. Numerical algorithm for predicting systemic and clot lysis dynamics for intravenous thrombolytic therapy.
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At the clot, fibrinolysis is a multicomponent reaction network with fluid-phase reaction as well as heterogeneous reactions catalyzed by species within the fibrin fiber phase (Fig 1
). The rates for reaction and adsorption steps are based on in vitro assays4 5 and are shown in Table 1. For transport of species in eroding fibrin, the instantaneous concentration of a soluble species at some position in the fluid phase of the fibrin gel is described with the use of Equation 2. The rate of change of the local volume-averaged concentration of a species in the fluid phase depends on the net adsorption rate to the fibrin fibers, dispersion, permeation, homogeneous reaction in the free phase, and the effects due to change of volume of the solid phase as species are released during solubilization of the solid-phase and free-phase species are diluted to a small extent in the lost fibrin fiber volume.5 The rate of change of a fibrin-bound concentration of the ith species depends on the rate of binding, dissociation, and heterogeneous reaction (Equations 3 and 4), where the jth species may compete for the rth binding site (
rj=one interaction; rj=no interaction).
 | (E2) |
 | (E3) |
 | (E4) |
The
permeation term in Equation 2 reduces to
(
c/
x) for one-dimensional
flow, where the superficial velocity for pressure-driven permeation
through fibrin is described by Darcy's law as
=k/µ(
P/L)
where
k is the specific permeability and µ is the viscosity
of the permeating plasma.
5 Equations 2 through 4 for each species
in Fig 1
are solved for specified initial conditions c
i(
x,0)
and s
i(
x,0) on the clot domain 0<
x<L
clot. Danckwerts inlet
and outlet boundary conditions were used for the solution of
the partial differential equations (Equations 5 and 6), where
C
iinlet(t)=C
i(t) from Equation 1.
 | (E5) |
 | (E6) |
The effective diffusion
(D
ieff) of proteins in the fluid space of the fibrin network
due to brownian motion is not significantly hindered.
15 16 Hydrodynamic dispersion (brownian diffusion plus micromixing)
will occur during pressure-driven permeation, and this process
scales with the velocity in the pore space (v
x) and the pore
diameter (D
pore) as correlated
17 in Equation 7 for the dispersion
coefficient
DL. The specific permeability of fibrin was based
on the fiber diameter and gel porosity according to the Davies
equation
18 (Equation 8).
 | (E7) |
 | (E8) |
In eroding fibrin, it is
necessary to relate the proteolysis rate with evolving material
properties (D
fiber,
,
k). Through evaluation at all times of
the instantaneous rate of cutting within the fiber [
kcat·s
plasmin,
[µmol/L per second]] at some location
x', it is possible
to calculate the historic amount of cutting within the fiber
(µmol/L of cleavages) made at
x' and t'. In defining a
solubilization factor
that relates the number of equivalent
monomers solubilized to the number of cleavages made, the amount
of solubilization or lysis (
L[
x',t'], µmol/L of fibrin
lysed at
x' and t') from the solid phase can be defined as a
function of the historic amount of cuts made by plasmin at position
x' (Equation 9). For fibrin, the solubilization factor has been
estimated to be a constant with a value of 0.1. This corresponds
to the estimate that for every 10 cleavages by plasmin, an equivalent
of one subunit of fibrin is solubilized.
19
 | (E9) |
The local plasmin
concentration in the fiber and the solubilization rate are formulated
with respect to a volume average within the solid fiber volume.
Plasmin gradients within the fiber are not calculated. Details
for the calculation of the fiber properties during lysis are
given in "Appendix 1." The solution of Equations 1 through 4
with initial conditions and boundary conditions is described
in "Appendix 2."
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Results
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Predicting Circulation Dynamics During Intravenous Thrombolytic Therapy
Calculation of systemic concentrations during thrombolytic therapy
was required for predictions of local clot lysis. We simulated
intravenous administration regimens to predict dynamic TPA and
plasminogen levels in the circulating plasma in humans. The
prediction of systemic TPA levels during therapy was quite accurate
because TPA levels were controlled simply through dilution of
the bolus into the plasma and through its half-life, both of
which are well characterized. These simulations demonstrated
accurate prediction of systemic TPA levels (Fig 3
), which can
peak at >0.1 µmol/L after a bolus infusion of 50 mg.
13 Similarly, a double TPA bolus regimen of 35 mg in each bolus
20 was simulated with suitable accuracy. Simulation of the dynamic
plasminogen level in the systemic circulation involved more
biochemical interactions and required a suitable rate equation
for plasminogen activation levels by TPA in plasma. The simulation
provided agreement within
10% of measured plasminogen values
(Fig 4
) during lytic therapy for various clinical trials.
11 21 22 23 These trials involved the use of varying administrative
schemes, from very modest to more aggressive TPA dosing. The
prevailing
2-antiplasmin and
2-macroglobulin as well as free
and inhibited plasmin levels are expected to be accurately predicted
since they result directly from the prevailing plasminogen levels.
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Figure 3. Comparison of simulated circulating levels of TPA (solid lines) with data (solid circles) obtained from clinical trials for various administration regimens. For TPA administration, measured plasma levels of TPA are shown as a function of time for a double 35-mg bolus20 (A) and a 50-mg bolus21 (B).
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Figure 4. Prediction of systemic circulating levels of plasminogen for four separate TPA administration regimens. The plasminogen levels are predicted and compared with measured values.11 21 22 23
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Predicting In Vitro Plasma Clot Lysis Under Conditions of Permeation
The next step was to validate predictions of clot lysis under various transport conditions created in vitro. We have previously shown5 accurate simulation of plasmin-, uPA-, and TPA-mediated fibrinolysis under conditions of diffusional penetration of fibrinolytic mediators into purified fibrin gels (3 mg/mL) when the concentrations of these agents were physiological or within the clinical range (<1 µmol/L). For permeation velocities up to 1 mm/min, we have also accurately simulated the measured lysis front velocity of dissolving fibrin gels in vitro. In the present work, to recreate a transport regimen comparable to the convection-dominated nature of arterial thrombolytic therapy, a syringe pump was used to infuse a fibrinolytic mixture into PPP clots via permeation at a constant superficial velocity. Despite the apparent simplicity of this experimental configuration, a challenging test was to simulate the lysis of a PPP clot under the conditions of permeation given the numerous reactions that occur. A cocktail of 0.1 µmol/L TPA and 1.0 µmol/L Glu-plasminogen was placed adjacent to 2-cm-long PPP clots, and the superficial permeation velocity through the clot was maintained constant at 0.1 or 0.2 mm/min.
To simulate this experiment, the initial levels of the various fibrinolytic mediators in the clot were taken to be at values that were 66% of nominal values in plasma (Table 2) due to a one-third dilution during recalcification. Thus, the concentration of the fibrin was set initially as 5.823 µmol/L, and the fiber radius was set to 250 nm, resulting in an initial clot porosity of 0.9928. The initial total amount of AP in the clot was set at 0.66 µmol/L, and 30% of this value was treated as cross-linked to the fibrin phase (cAP=0.4653 µmol/L, sAP=27.719 µmol/L). The 2-macroglobulin level was set at 2 µmol/L in the clot, all of which was in the fluid phase. A total concentration of 1.452 µmol/L plasminogen in the clot was allowed to equilibrate to the fibrin (Kd=38 µmol/L24 ), resulting in free and bound concentrations (cplg=1.1094 µmol/L, splg=47.618 µmol/L) of plasminogen. Under these conditions, the simulation captured the measured rate of movement of the lytic front accurately (Fig 5). For prevailing inlet conditions of TPA (0.1 µmol/L), the biochemical reactions were occurring quite rapidly; however, the species transport (greatly enhanced through permeation relative to pure diffusion) remained rate limiting. This was seen where a doubling of the permeation velocity enhanced the lysis rate. The lysis front position in the experiment actually moved slightly ahead of the position of the perfusion front (t) because of dispersion processes that were captured in the simulation with Equation 7. In contrast to the permeation results shown in Fig 4, the lysis front proceeded only 1.1 mm in 60 minutes through diffusional transport (=0) of TPA and plasminogen into a PPP clot. Thus, permeation at a velocity of 0.2 mm/min caused a 15-fold increase in the lysis rate compared with diffusional delivery.
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Figure 5. Simulation and measurement of TPA-mediated lysis of PPP clots (n=2) under conditions of constant superficial permeation velocity as a lysis front moves across the 2-cm-long clot. A mixture of TPA (0.1 µmol/L) and Glu-plasminogen (1 µmol/L) was perfused at constant permeation velocities of 0.1 and 0.2 mm/min into plasma clots (formed at 33% dilution after recalcification). Simulation results (solid lines) and measured front position (solid circles) are shown versus time. The simulated concentration profiles within the dissolving clot at 30 minutes for the clot dissolved by infusion at 0.2 mm/min (*) are given in Fig 6.
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In addition to determination of lysis front position with time, the concentration profiles of the species were simulated at all times corresponding to the experiments shown in Fig 5. At 30 minutes after the initiation of permeation at 0.2 mm/min, the front moved 8 mm as indicated by the data point with an asterisk in Fig 5. For the simulation of this experiment at 30 minutes, the free-phase profiles are shown for plasminogen, plasmin, and TPA (Fig 6A). Robust lysis was predicted to occur in the region of the lysis front where TPA was present at low but catalytic levels. The level of plasmin in the free phase was very small due to rapid inhibition by the free-phase antiplasmin and macroglobulin as well as by the cross-linked antiplasmin that was liberated during lysis into the fluid phase. Due to solubilization of bound (nonactivated) plasminogen over the preceding 30 minutes of the simulation, there was a small peak in the plasminogen concentration near the front. In Fig 6B, the bound phase concentrations based on the total volume of the clot, si·(1-), are shown. Behind the lysis front, the bound concentrations of species in the fibers si were the residual levels of species left in vanishingly thin fibers. These concentrations reached very high values but occur in zones of >99.99% lysis and represent negligible amounts of species because si·(1-)0 in these regions. The bound levels of the various proteins displayed a maximum at the lysis front, which was a thin zone where significant plasminogen activation and lysis were occurring. The local rate of lysis was reduced by the clot levels of the fast-acting plasmin inhibitors. These inhibitors and complexes are shown in Fig 6C. In the zone of full lysis behind the front, the inhibitors were depleted through complexation and perfusion of inhibitor-free liquid (a wash-out effect since only TPA and plasminogen were perfused into the clot). At the lysis front, the concentration of plasmin/antiplasmin was substantially higher than that of plasmin/macroglobulin because of the faster inhibition rate of antiplasmin. The slight peak in free antiplasmin levels was due to release of fibrin-bound antiplasmin during lysis at the advancing front over the preceding 30 minutes. Overall, these simulations indicate that substantial lysis had occurred in the regions where TPA had penetrated to even a small extent. Also, the rate of lysis front advancement was controlled by the rate of permeation of the fibrinolytic mediators.
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Figure 6. Simulation of spatial concentrations after 30 minutes when TPA (0.1 µmol/L) and plasminogen (1.0 µmol/L) were perfused into PPP clots at a constant superficial velocity of 0.2 mm/min. The lysis front has moved to a position of 1 cm into the clot after 30 minutes of lysis. Note that some concentrations are multiplied by a factor of 10 or 10 000 to facilitate presentation. The lysis front profile (A) can be multiplied by 100 to give percent lysis.
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Prediction of Arterial Thrombi Dissolution by Intravenous TPA or uPA Therapy
The superficial velocity of permeation through a clot is dictated by the pressure drop across the thrombus and the specific permeability of the clot. Over the cardiac cycle, the time-averaged permeation velocity depends on the time-averaged pressures both proximal and distal to the clot. The upstream pressure is the time-averaged aortic pressure of 90 mm Hg, whereas the downstream pressure depends on the extent of collateralization. In the absence of collateralization, the time-averaged downstream pressure may be only slightly higher than the mean right atrium pressure of 10 to 20 mm Hg. With collateralization, higher mean pressures of 40 to 50 mm Hg may occur distal to the clot and may provide delivery of the agent to the distal face of the clot (not considered in the present study). For a 1-cm-long clot, the pressure drop across an occlusive arterial thrombus may range from 50 to 80 mm Hg/cm (without collateralization) to 40 mm Hg/cm with collateralization. During MI, the precise phasing of the pulsatile aortic pressure with the pressures distal to the occluding thrombus in a blocked coronary artery is not well defined at present. Over the cardiac cycle, the clot will essentially time-average the pulsatile flow, whereas dispersive processes are likely enhanced by the pulsatility.25
Thrombi have structures and compositions that are dependent on their age and the prevailing hemodynamic conditions during their formation. The biochemical compositions of venous and arterial thrombi have been measured ex vivo,26 and these values are given in Table 2, recalculated for the fluid and fibrin phases. There are very few direct measurements of the amount of fibrin within human thrombi. In Table 2, we took as a worst case scenario that the entire mass of insoluble material of urea-extracted and thrice-washed thrombi measured by Potter vann Loon et al26 is hydrated fibrin (this neglects the wet weight of insoluble platelet actin and cell debris). See "Appendix 3" for details.
Using in vitro kinetic data (Table 1) for the reaction network (Fig 1) and ex vivo arterial thrombi properties and initial plasma composition (Table 2), we simulated a prototype accelerated therapeutic regimen for intravenous TPA therapy for coronary thrombolysis27 involving a 15-mg bolus followed by infusion of 85 mg over 90 minutes. In the simulation, this regimen provided for a relatively constant circulating level of TPA of 0.04 µmol/L (Fig 7A) during the first hour of therapy. Plasminogen dropped to less than half of its original level with concomitant consumption of antiplasmin and macroglobulin and appearance of plasmin/antiplasmin and plasmin/macroglobulin complexes in the circulation by 1 hour.
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Figure 7. Lysis of an arterial thrombus by intravenous TPA thrombolytic therapy (100 mg total dose). Changes in systemic species are shown (A) for a therapy initiated by a 15-mg TPA bolus followed by infusion of 85 mg TPA over 90 minutes. Initial clot length is 1 cm, and initial pressure drop was set at 50 mm Hg/cm. Complete reperfusion is achieved at 52 minutes (B). The lysis and concentration profiles of free and bound species (C through E) across the clot are shown after 10 minutes of therapy.
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The lysis front advanced at a nearly constant velocity of 1 mm every 10 minutes for the first 40 minutes of the therapy (Fig 7B), after which lysis proceeded more rapidly with the remaining 6 mm of the thrombi dissolving in the next 10 minutes. This acceleration toward reperfusion that occurred between 40 and 50 minutes after initiation of the therapy was due to the shortening of the clot at constant proximal pressure head (increasing pressure drop and permeation) and accumulation of reaction species at the lysis front. The prediction of clot lysis (with no adjustable parameters) is consistent with observed recanalization times between 45 and 90 minutes.
To indicate the reaction dynamics during the simulation, "snapshots" of the initial 1-cm clot domain after 10 minutes of therapy are shown in Fig 7C through 7E. At 10 minutes, the lysis front advanced over 1.2 mm into the clot, and the most active lysis zone was seen to be a region of 0.25-mm thickness. In this thin reaction zone, dramatic changes in reactant concentrations were seen. Some fibrin-bound plasminogen was released through solubilization, and this plasminogen readsorbed and accumulated at the lysis front to levels in excess of prevailing plasma levels. Under conditions of permeation for 10 minutes in the simulation, the continual binding of TPA at the lysis front resulted in TPA levels that were twofold to threefold higher than circulating levels. Interestingly, antiplasmin and macroglobulin were predicted to become depleted in the "spent" fluid that was moving with the lysis front (Fig 7E), allowing plasmin to exist as a long-lived species. This was because the reacting fluid at the lysis front was continually reaching plasminogen-laden fibrin that was activated (Fig 7E) and lysed with the released plasmin subsequently inhibited. The free plasmin predicted to exist in the antiplasmin- and macroglobulin-depleted zone would be susceptible to the slower C1 inhibitor (not simulated). We predict that some plasmin/C1 inhibitor complex would form under permeation conditions despite the known kinetic disadvantage of C1 inhibitor in well-mixed systems. The inhibited plasmin complexes formed at the lysis front had no fibrin binding capability (Fig 1) and thus moved with the permeating fluid. This was seen in Fig 7E, where the inhibited complexes penetrated 1 mm farther ahead of the lysis front after 10 minutes of lysis. The lysis front was moving at a velocity slower than the permeation velocity but much faster than would be expected in the absence of pressure-driven permeation. We conclude that in Fig 7, the overall thrombolytic rate (ie, lysis front velocity) achieved as a result of the dosing scheme for this clot structure and composition can be improved only up to the point where the lysis front proceeds at a velocity comparable to the permeation velocity. Once this maximal lysis rate is achieved with higher dosing, further increases in TPA dosage or dosage rate would not offer significant gain. In contrast to Fig 7, the more robust fibrinolysis achieved in the in vitro experiment (Figs 5 and 6) resulted in a lysis front with a velocity that was comparable to the permeation front velocity.
The prevailing systemic levels of circulating species at 10 minutes (see Fig 7A) exist at the inlet of the initial reaction domain at x=0 cm (Fig 7C and 7E). In a one-dimensional simulation, mixing eddies proximal to the clot cannot be simulated since the reacting fluid can move only in the forward direction. Thus, changes in systemic concentrations were not immediately and directly having an impact on reaction dynamics at the lysis front. Rather, changes in the systemic concentrations of species were immediately manifest at the inlet of the initial reaction domain and then began to convect forward. In this one-dimensional approach, the biochemistry of the plasma that reaches the proximal face of the clot in the first minutes of the therapy had the greatest impact on subsequent dynamics. This is consistent with the motivation for front-loaded therapies.28 In the simulation, the permeation of fluid through the lysis front was somewhat faster than the movement of the lysis front, and the constituents of this fresh plasma fed the reactions at the lysis front to a significant extent during the first 10 minutes of the therapy but less so thereafter. (See "Discussion" regarding inlet mixing.)
To demonstrate the marked qualitative and quantitative differences between uPA and TPA, we conducted a simulation of intravenous urokinase (two-chain uPA) therapy for acute MI. At the present, we have not included in the kinetic model the conversion by plasmin of single-chain uPA (prourokinase) to the active two-chain uPA. We used a protocol of a bolus of 8 mg uPA (106 IU Abbokinase) followed by a continuous infusion of 1 mg/h (125 000 IU/h), which provided for an approximately constant level of 0.04 µmol/L uPA in the systemic plasma. In the simulation, since uPA is active in the plasma, a rapid and nearly complete activation of circulating plasminogen was observed within 1 hour after initiation of the therapy, with consequent consumption of antiplasmin and macroglobulin. In contrast to TPA, after 30 minutes of intravenous uPA infusion, only a minor amount of lysis took place (Fig 8). With uPA, the marked depletion of systemic plasminogen in combination with generation of plasmin in the free phase, where it was rapidly inhibited, resulted in rather inefficient therapy compared with TPA. By 90 minutes, antiplasmin was predicted to become depleted in the front of the clot due to washout with the antiplasmin free circulating plasma and continual activation of plasminogen released from the dissolving fibrin (simulation not shown). A zone of complete lysis proceeded 1.5 mm across the front of the clot at 90 minutes. Also, lesser degrees of lysis (5% to 30%) were present at 90 minutes in regions deep within the clot (0.2 to 1 cm). This deep zone of lysis was not seen with TPA and indicates that uPA activated an inner clot pool of plasminogen because its penetration into the clot was not slowed by binding. The distinct patterns of TPA and uPA thrombolysis make the deconvolution of additive versus synergistic biochemical effects of combined TPA/uPA intravenous therapies difficult, since low levels of uPA-mediated inner clot lysis may enhance permeation of TPA (a likely beneficial effect that is based in transport).
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Figure 8. Fibrinolysis of an arterial thrombus mediated by two-chain uPA under conditions of intravenous thrombolytic therapy. The percent lysis profiles across a 1-cm arterial thrombus are shown at various times for a therapy consisting of 8-mg bolus followed by infusion at 1 mg/h. Initial pressure drop, clot structure, and clot composition are the same as in Fig 7.
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Discussion
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We developed a computer algorithm to relate the dynamics of
the systemic circulation to the thrombolytic processes that
occur at the site of a dissolving thrombus during thrombolytic
therapy. The model made accurate predictions of circulating
TPA and plasminogen levels measured during clinical trials for
intravenous TPA therapy. Also, the model made accurate predictions
of in vitro plasma clot lysis under conditions of permeation.
Finally, arterial thrombolysis was simulated for TPA and uPA
intravenous administration. The simulations provided rates of
overall lysis as well as spatial distributions of reacting species.
As a one-dimensional model, the simulations describe the rate at which a lysis front proceeds across a given clot structure. In these one-dimensional simulations, fluid can advance only in the forward direction. Thus, the simulation cannot be used to describe radial fluxes within a clot, dissolution fingering,2 29 30 or fluid mixing at the proximal face of the clot. A common approach to account for eddy mixing at the inlet of a reaction domain is to increase the longitudinal dispersion coefficient, DL. We implemented this approach by increasing DL by 20-fold at all positions of the clot that were 100% lysed. This led to modest increases (20% to 30% at 15 minutes) in the rate of lysis. Still, this approach does not fully capture the impact of mixing at the proximal face of the dissolving clot. The one-dimensional model used in the present study would correspond to anatomic situations where the thrombi are a substantial distance from a well-mixed region that exists upstream at a vessel bifurcation. If coagulated red blood cells accumulate on the proximal face of the clot due to thrombin leakage, blood stagnation, and/or plasma infiltration into the thrombus, this loose structure would offer little resistance to permeation but would damp fluid mixing proximal to the clot. The hemodynamics of this situation are captured in one-dimensional simulations.
A two- or three-dimensional model is required to predict the causes and probability of lytic fingering where dissolution channels proceed through the clot to produce early but low-grade reperfusion. When a channel begins to develop, its growth is fortified by the development of larger pressure gradients at the leading tip of the finger, with consequent faster permeation and accumulation of binding species. Using a two-dimensional model,30 we found that large lytic fingers will occur when the initial thrombus contains a spatial distribution of random fluctuations in initial fibrin density as small as ±10% (on average).
Thrombolysis occurs in the presence of continued coagulation. New fibrin deposition by thrombin released or generated in the clot can be taken into account by allowing the porosity at a particular clot location (/t in Equation 2) to be a function of both lysis and coagulation. In such a model, continuing coagulation within a dissolving thrombus would likely dictate a threshold level of plasminogen activator needed to achieve net lysis as well as indicate some benefit to local clot lysis by systemic fibrinogen consumption.
Less is known quantitatively about the range of permeation flows through real human arterial thrombi during MI. The permeation of plasma through a clot may be nonuniform and slow compared with nominal flow rates through the coronary vessels. Superficial permeation velocities across coronary thrombi are estimated to be 
0.001 cm/s.4 5 To obtain reperfusion within 45 to 90 minutes, some permeation of the lytic agent into the clot must occur since diffusion is such a slow transport process.1 2 3 4 5 10 As Blinc et al31 pointed out, permeation must occur when radiolabeled fibrinogen injected into dogs with existing occlusive coronary clots is found to be distributed along the clot length in <1 hour.32 Diffusional transport would require days or weeks to achieve this distribution.
Clinical evidence also indicates an important role for permeation and its impact on the rate at which reperfusion occurs. The role of permeation is apparent in the observation that patients with hypotension or cardiogenic shock are poor candidates for successful thrombolytic treatment of MI.6 33 The impact of pressure-driven permeation of lytic agents into clot structures was demonstrated in a canine model of LAD thrombosis under conditions of severe hypotension (systolic pressure of 75 mm Hg).7 A LAD catheter administration of TPA to the clot was significantly less effective under conditions of hypotension compared with thrombolysis at systolic pressures of 130 mm Hg.7 Under either condition of 75 and 130 mm Hg systolic blood pressure, the delivery of the agent to the proximal face of the clot was the same (by intracoronary catheter); however, the driving pressures for permeation were markedly different. Similar transport mechanisms have been shown for the lysis of pulmonary embolism.34 35 In human cardiogenic shock and MI, the success of thrombolytic therapy can be enhanced with the administration of ionotropic agents (with consequent increase of systolic pressure from 64 to 102 mm Hg).6 In a canine model of cardiogenic shock with LAD thrombosis in which systolic blood pressure had been reduced to 75 mm Hg through phlebotomy, intra-aortic balloon counterpulsation to increase coronary pressure resulted in enhanced thrombolysis with the use of intravenous TPA.8 36
Studies with ultrasound also demonstrate the importance of transport processes. Francis et al37 demonstrated that 1-MHz ultrasound at intensities of 8 W/cm2 significantly enhanced the rate of lysis of a plasma clot submerged in solutions of TPA (1 µg/mL) as well as uPA. Such levels of ultrasound do not cause fragmentation or heating of the thrombus; rather, micromixing and transport facilitation are the causes of enhanced lysis. Through a cavitation-dependent mechanism, ultrasound was shown to facilitate transport of reactants without modulation of enzymatic activity.38 When ultrasound is applied to fibrin clots that are maintained under a hydrostatic pressure head, the permeability also increases through a cavitation-dependent mechanism that alters fibrin fiber structure.39 Enhancement in permeability through ultrasound facilitated pressure-driven permeation of lytic agents into the whole blood clot with consequent reduction of reperfusion times.40
Specific structural aspects of clots have received attention with respect to lytic susceptibility and penetration of lytic agents. For example, coronary artery thrombi formed during unstable angina tend to be grayish white, whereas those present during acute MI are reddish, as determined with angioscopy.41 These grayish-white coronary thrombi tend to be resistant to thrombolytic therapy.42 These structures are particularly rich in platelets, are older, and have a very tight fibrin structure,43 and they are believed to be formed under flow conditions. The reddish thrombi of acute MI are rich in fibrin and red blood cells and tend to be younger and formed under conditions of stasis. It is certainly possible that the tight fibrin (possibly also containing collagen) networks of aged thrombi do not allow sufficient penetration of lytic agents.
The transport limits are difficult to overcome; it is difficult for a plasminogen activator to penetrate the many centimeters into the long clots of the deep veins or peripheral arteries. Pressure drops across the long venous clots are likely to be an order of magnitude less than the pressure drops across arterial thrombi. Aggressive systemic intravenous therapies are not typically used in these instances. For these clots, catheters are introduced either directly into the clot44 or with guide wires to help open a channel in the thrombus for perfusion.45 46
With the advent of new delivery schemes, administration regimens, and agents, computer simulation is an additional tool for the streamlined evaluation of new therapies. Knowledge of the theoretical optimality of a given approach provides a benchmark by which to evaluate the efficacy of the administration regimen and agent. The methods described in the present work also facilitate an understanding of the dynamics and evaluation of the efficacy of intravessel or intrathrombic delivery of thrombolytic agents.
|
Appendix 1: Theory
|
|---|
To quantify the change in fibrin fiber volume on lysis, the
total length of fibers initially in the gel was first evaluated
(for uniform initial radius R
fo). The total length of fibers
is related to the fibrin fiber density (
fiber), the gel density
(
), and the total clot volume, V
T (cm
3). The total length of
fibers per volume of gel is given as
4
 | (E10) |
for L
T and R
fo [=] nm.
The local dynamic change in fibrin gel porosity () was related to the local extent of lysis (L) by allowing the radius of the fibers (Rf) to shrink at constant fibrin fiber density (fiber=0.28 g/mL of fiber47 and so=fiber/MW monomer), whereas the fiber length per unit volume of gel (LT/VT) was assumed to remain constant (Equations 10 and 11).
 | (E11) |
For an initial fiber radius (Rfo) and a total fiber length density, LT/VT, the initial gel porosity is
 | (E12) |
The local instantaneous porosity (x,t) based on the local instantaneous fiber radius, Rf(x,t), is calculated similarly. The percent lysis is then known at any position by
 | (E13) |
The local concentration of each type of binding site (
r) within the fiber is determined by the fibrin monomer concentration within the fiber (so, assumed constant during lysis as the fiber radius shrinks) multiplied by the sites per monomer, qr. We have kept qr constant in the present work. The local specific permeability k(x,t) is calculated from (x,t) and Rf(x,t) with Equation 8. At any given time, t', the overall specific permeability 
can be evaluated as the inverse of the total resistivity by integrating the local resistivity [k(x,t')]-1 over the length of the clot for a one-dimensional system.
|
Appendix 2: Numerical Methods
|
|---|
The concentration (C
i) of each species in the systemic circulation
was obtained through solution of the system of coupled ordinary
differential equations (Equation 1) describing the plasma as
a one-compartment system with time-dependent sources (intravenous
infusion) and sinks (liver clearance). The integration of the
equations was accomplished with a fourth-order Runge-Kutta solver.
48 To simulate realistic infusion regimens of any fibrinolytic
mediator, the Runge-Kutta solver was designed so that bolus
(mg) or continuous infusions (mg/min) could be specified directly
for any infusion regimen, I(t). Prevailing systemic concentrations
were based on the plasma volume of 3.5 L. Bolus infusions were
treated as rapid continuous infusions that occur over a 5-second
interval. The concentrations of the species in the systemic
circulation were calculated for every 1 millisecond of simulation
and served as inlet conditions (C
iinlet) for the Danckwerts
boundary conditions (Equation 5) for each species balance partial
differential equation.
A finite difference algorithm was used to solve the system of coupled and nonlinear PDEs. A two-step procedure was used to advance the solution from the j to the j+1 time level via a differencing formulation at the j+1/2 time level as previously described.5 During lysis, the transport and material properties (Dfiber, DL, k, , ) within the clot change in time and space. To begin, the values are known at every node in the domain at the j time level for the free and bound concentrations of each species (ci, si), the fiber radius, porosity, and percent lysis. The use of a specialized algorithm for improved material property guesses at the j+1/2 level for the porosity and fiber radius provided for a straight forward solution of cj+1. On calculation of the transport properties at the j+1/2 and j+1 levels, the free-phase concentrations at the j+1 time level can be obtained through gaussian elimination of finite difference matrices evaluated at the j+1/2 level. We observed a monotonic rate of convergence (1 to 25 iterations). For simulations with numerous fast reacting and convecting species through erodible fibrin, a simulation of 60 minutes of lysis required <8 hours of CPU (Silicon Graphics Indigo R4000 or SUN SparcCenter 2000) using t=0.01 second and 0.01-cm node spacing. The conservation of mass of each species for the total system (both free and bound phases) has been rigidly implemented to 0.5%.
|
Appendix 3: Clot Composition and Permeability
|
|---|
The total volume of a thrombus can be subdivided into the volume
occupied by cells and the volume occupied by the biphasic fibrin
gel (hydrated fibrin fiber volume plus water volume). The model
(Equations 2 through 9) describes phenomena in the biphasic
fibrin gel that exists between the cells. Since the thrombus
forms under flow conditions, the relationship of clot properties
to those of whole blood is not a direct one because depletions
and enrichments occur. During thrombosis under flow, platelets
accumulate in the growing clot to levels that are 5 to 15 times
higher than the plasma concentration (of 300 000/µL).
This conclusion is based on measurements of PAI-1 levels in
thrombi
26 49 where PAI-1 levels are 200 times plasma levels
and
5 to 15 times higher than serum levels (containing platelet
PAI-1 released from
-granules). Also, the volume occupied by
red blood cells within the thrombus volume is very likely to
be lower than blood hematocrit. This conclusion is based on
(1) the existence of white and pink thrombi that occur under
arterial conditions and (2) the low red blood cell content of
thrombi formed under high flow in vitro. In a thrombus, the
concentration of fibrin in the volume occupied by the biphasic
fibrin gel is expected to greatly exceed its plasma level (8
µmol/L) and likely has a concentration ranging from 80
to
220 µmol/L. This conclusion is based on in vitro and
ex vivo studies. Coagulated platelet-rich plasma can retract
to
10% of its original volume, resulting in a 10-fold enrichment
of the resulting thrombus concentration of fibrin. Clotted whole
blood can retract to
50% of its original volume. Since the original
red blood cell volume within the retracted whole blood clot
is unchanged during retraction, the fibrin gel volume has lost
considerable serum and in effect has been condensed to only
10% of its original volume, resulting in a high fibrin concentration
(
80 µmol/L) in the gel space of the thrombus. During its
formation, a real thrombus is subjected to hemodynamic forces
that may further consolidate the fibrin gel of the thrombus.
Also, the permeation of fresh fibrinogen into the clot may allow
continuing fibrin polymerization due to high levels of inner
clot thrombin. Platelet release of fibrinogen (50 µg/10
9 platelets at
3x10
9/mL of thrombi) represents a relatively minor
contribution to the fibrin content of a platelet-retracted thrombus
(<1%). The estimate of the high density of fibrin (80 to
200 µmol/L) in a thrombus is consistent with direct measurements
of the insoluble, washed wet pellet (wet fibrin fibers plus
insoluble wet cellular debris), which represents 31% of the
total thrombus weight obtained from arterial and venous thrombi.
26 This wet pellet likely contains substantial fibrin, based on
the earlier report by Brommer et al
50 that fibrin represents
85% of the dry weight of fresh human arterial thrombi. The weight
of pelleted wet fibrin fibers from an 80 µmol/L fibrin
gel is calculated to be
10% of the original wet weight of the
gel based on a fibrin fiber density of 280 mg of fibrin/mL of
fiber
47 for fibers that are 80% water. Similarly, the weight
of pelleted wet fibrin fibers from a 220 µmol/L fibrin
gel is calculated to be
27% of the original wet weight of the
gel.
|
Selected Abbreviations and Acronyms
|
|---|
| AP |
= |
2-antiplasmin |
| FDP |
= |
fibrin degradation products |
| LAD |
= |
left anterior descending coronary artery |
| MI |
= |
myocardial infarction |
| PAI-1 |
= |
plasminogen activator inhibitor type I |
| PPP |
= |
platelet-poor plasma |
| TPA |
= |
tissue plasminogen activator |
| uPA |
= |
urokinase plasminogen activator |
|
|
Acknowledgments
|
|---|
This work was supported by an American Heart Association, National
Center, grant (93-8670) and by National Science Foundation grant
BES 9358236. Also, we acknowledge the SUNY School of Engineering
and Applied Sciences for use of the SUN SparcCenter 2000 machine
as well as National Science Foundation Instrumentation Grant
CTS 9212682 to Dr T.J. Mountziaris for use of the SGI R4000
machine.
Received November 28, 1995;
revision received February 1, 1996;
accepted February 3, 1996.
|
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Abstract
Introduction