Overview of the Targeted Thrombolytic System and Modelling Strategy
An overview of the targeted thrombolytic system (24) is illustrated in Fig. 1. Briefly, tPA-loaded nanovesicles (tPA-NV) bind with αIIbβ3 integrins (INT) expressed on the surface of activated platelets (PLT) in the clot. Upon binding, tPA is released and rapidly binds to fibrin binding sites in the fibrin fibre network. In addition, there is a small amount of tPA leakage from unbound NV into the plasma phase due to the concentration gradient between the tPA level inside the NV and its surroundings. Once tPA reaches its therapeutic level in the plasma, fibrinolysis occurs. Full details of fibrinolytic kinetics can be found in our previous work (24, 28,29,30). From here onwards, NV is used interchangeably with tPA-NV for brevity and NV following tPA release and leakage is referred to as NVemp.
The computational model comprises three parts, as shown in Fig. 1. Firstly, the systemic PKPD model predicts plasma levels of NV, tPA and other fibrinolytic proteins described by a compartmental model. Secondly, the local PD model includes a number of biochemical reaction kinetics, i.e., tPA leakage, binding/unbinding between NV and INT and triggered tPA release in addition to the fibrinolytic reaction kinetics. Finally, the convective and diffusive transport of species in a clotted vessel is coupled with the aforementioned systemic PKPD and local PD models.
Model Equations
The modelling strategy adopted in this work is to combine two models developed in our previous work: systemic PKPD-local PD modelling in (28) and targeted thrombolysis kinetics in (24). Here we only present the fundamental model equations describing each part shown in Fig. 1. A full list of equations and model details can be found either in Section A of the Supporting Information or our previous publications (24, 28).
Systemic PKPD Model
A single compartmental model is used to resolve temporal systemic concentrations of tPA and plasma proteins by accounting for tPA administration, hepatic clearance, systemic secretion and plasma reactions.
$$ \frac{d{C}_{tPA, sys}(t)}{dt}=\frac{I_{tPA}(t)}{V_cM{w}_{tPA}}-{k}_{el, tPA}{C}_{tPA, sys}(t)+{S}_{tPA}(t)+{r}_{tPA}^{plasma}(t) $$
(1)
$$ \frac{d{C}_{i, sys}(t)}{dt}=-{k}_{el,i}{C}_{i, sys}(t)+{r}_i^{plasma}(t)+{S}_i(t),\kern0.5em \mathrm{at}\ i=\mathrm{PLG},\mathrm{PLS},\mathrm{AP},\mathrm{AP}\hbox{-} \mathrm{PLS},\mathrm{FBG},\mathrm{MG}\ \mathrm{and}\ \mathrm{PAI} $$
(2)
where t is time, C the systemic concentration in μM, I the infusion rate in mg/s, Vc the volume of plasma, Mw the molecular weight, kel the elimination rate constant, S the generation rate and r the reaction source term and the subscript i,sys denotes systemic species i. For NV administration, temporal concentrations of NV (CNV) and NV after tPA leakage (CNVemp) are resolved by:
$$ \frac{d{C}_{NV, sys}(t)}{dt}=\frac{I_{NV}(t)}{V_cM{w}_{NV}}-{k}_{el, NV}{C}_{NV, sys}(t)+{r}_{NV}^{plasma}(t) $$
(3)
$$ \frac{d{C}_{NVemp, sys}(t)}{dt}=-{k}_{el, NV}{C}_{NVemp, sys}(t)+{r}_{NVemp}^{plasma}(t) $$
(4)
It is assumed that tPA-loaded NV (NV) and empty NV (NVemp) have the same clearance rate. Depending on the type of drugs, either ItPA(t) or INV(t) is set to 0. The reaction source terms for each component in Eqs. (1) to (4) can be calculated by combining relevant individual reaction terms presented in Section A.2 of the Supporting Information.
Coupled Transport of Species and Local PD Model
Flow in the thrombosed artery is assumed to be governed by the continuity equation for incompressible flow in Eq. (5) and Darcy’s law for flow through a porous medium in Eq. (6),
$$ \frac{\partial Q(t)}{\partial x}=0 $$
(5)
$$ Q(t)=\frac{\varDelta {P}_x{L}_{clot, rem}(t)A}{\mu {\int}_0^L\frac{R_{clot, tot}\left(t,x\right)}{\varepsilon \left(t,x\right)}\; dx} $$
(6)
where x is the axial coordinate, Q the volumetric flowrate along the occluded artery, ΔPx the pressure drop per unit length across the clot, Lclot,rem the remaining fibrin clot length, A the cross-sectional area of the artery, μ the blood viscosity, Rclot,tot the total clot resistance, ε the porosity and L the length of the clot. Equations to calculate variables dependent on the progression of clot lysis can be found in Section A of the Supporting Information, which includes: the total resistance of the clot (Rclot,tot), the resistance of fibrin fibre network (Rclot,FBR) and activated platelets (Rclot,PLT) present in the clot, the volume fraction of fibrin fibre network (ϕFBR) and activated platelets (ϕPLT) in the clot and the local porosity (ε) in the occluded artery.
The transport of unbound species is governed by:
$$ {\displaystyle \begin{array}{c}\frac{\partial {C}_j\left(t,x\right)}{\partial t}=-\frac{\partial u(t){C}_j\left(t,x\right)}{\partial x}+{D}_j\frac{\partial^2{C}_j\left(t,x\right)}{\partial {x}^2}+{r}_j^{tot}\left(t,x\right),\\ {}\kern4.199998em \mathrm{for}\;j=\mathrm{tPA},\mathrm{PLG},\mathrm{PLS},\mathrm{FBG},\mathrm{AP},\mathrm{AP}\hbox{-} \mathrm{PLS},\mathrm{MG},\mathrm{PAI},\mathrm{NV},{\mathrm{NV}}_{\mathrm{emp}}\end{array}} $$
(7)
where u is the Darcy velocity (see Section A.3 in the Supporting Information for details on how to calculate this parameter) and D is the diffusion coefficient. The superscript tot denotes the reaction source term, i.e., the sum of both the plasma phase and clot-bound phase. The activated platelets within the clot can move out of the clot depending on the extent of fibrin degradation, i.e., the extent of lysis, EL. Total activated platelets, NV-bound activated platelets and empty NV-bound activated platelets are obtained by the modified convection-diffusion-reaction equation in Eq. (8).
$$ {\displaystyle \begin{array}{c}\frac{\partial {C}_k\left(t,x\right)}{\partial t}=-M(t)\frac{\partial u(t){C}_k\left(t,x\right)}{\partial x}+M(t)\cdot {D}_{PLT}\frac{\partial^2u(t){C}_k\left(t,x\right)}{\partial {x}^2}+{r}_k^{tot}\left(t,x\right),\\ {}\kern10.67999em \mathrm{for}\ k=\mathrm{PLT}\hbox{-} \mathrm{tot},\mathrm{PLT}\hbox{-} \mathrm{NV},\mathrm{PLT}\hbox{-} {\mathrm{NV}}_{\mathrm{emp}}\end{array}} $$
(8)
$$ M(t)=\left\{\begin{array}{c}1,\kern0.75em \mathrm{for}\kern0.5em {E}_L>{E}_{L, crit}\\ {}1-\tanh \left(m\left(1-\frac{E_L}{E_{L, crit}}\right)\right),\mathrm{for}\ {E}_L\le {E}_{L, crit}\end{array}\right. $$
(9)
where M is the mobility coefficient determined by EL as in Eq. (9) and m is an arbitrary constant that determines the slope of the mobility curve at the critical extent of lysis EL,crit, shown in Fig. 2. All simulations presented here were performed with m = 10.
The concentration of activated platelets and free INT sites in the clot is calculated as:
$$ {C}_{PLT\hbox{-} free}\left(t,x\right)={C}_{PLT, tot}\left(t,x\right)-{C}_{PLT\hbox{-} NV}\left(t,x\right)-{C}_{PLT\hbox{-} N{V}_{emp}}\left(t,x\right) $$
(10)
$$ {C}_{INT,k}\left(t,x\right)={N}_{INT}{C}_k\left(t,x\right),\mathrm{for}\ k=\mathrm{PLT}\hbox{-} \mathrm{free},\mathrm{PLT}\hbox{-} \mathrm{NV},\mathrm{PLT}\hbox{-} {\mathrm{NV}}_{\mathrm{emp}} $$
(11)
where NINT is the average number of integrins expressed on an activated platelet. The content of activated platelets in the clot that varies over time is defined using the extent of activated platelets, EPLT:
$$ {E}_{PLT}\left(t,x\right)=\frac{C_{PLT, tot}\left(t,x\right)}{C_{PLT,0}} $$
(12)
where CPLT,0 is the initial concentration of activated platelets in the clot. The concentrations of bound tPA, PLG and PLS with the fibrin fibre (tPA-F, PLG-F and PLS-F, respectively) in the clot and lysed fibrin binding sites with PLS still present (PLS-Flysed) are obtained by solving the following equations:
$$ \frac{\partial {n}_h\left(t,x\right)}{\partial t}={r}_h^{clot}\left(t,x\right)\ \mathrm{for}\ \mathrm{h}=\mathrm{tPA}\hbox{-} \mathrm{F},\mathrm{PLG}\hbox{-} \mathrm{F},\mathrm{PLS}\hbox{-} \mathrm{F},\mathrm{PLS}\hbox{-} {\mathrm{F}}_{\mathrm{lysed}} $$
(13)
$$ \frac{\partial {n}_{FBR}\left(t,x\right)}{\partial t}=-{r}_{deg}\left(t,x\right) $$
(14)
where nh is the concentration of bound phase species and nFBR is the total fibrin binding sites. Using the calculated nFBR by Eq. (14), the extent of lysis EL is calculated by Eq. (15), which is used to determine the mobility of activated platelets in the clot, as in Eq. (9),
$$ {E}_L=1-\frac{n_{tot}}{n_{tot,0}} $$
(15)
where ntot,0 is the initial concentration of fibrin binding sites.
Solution Procedure
Initial and Boundary Conditions
In order to solve the ordinary and partial differential equations in Eqs. (1)–(4), (7)–(8) and (13)–(14), initial and inlet concentrations of species should be provided. Temporal systemic concentrations are obtained by solving Eqs. (1) to (4) using the following initial conditions.
For tPA, NV, NVemp, PLG, PLS, AP, AP-PLS, FBG, MG and PAI:
$$ {C}_i(t)={C}_{i,0}\ \mathrm{at}\ t=0,\mathrm{for}\ i=\mathrm{species} $$
(16)
$$ {C}_i\left(t,x\right)={C}_{i,0}\ \mathrm{at}\ t=0\ \mathrm{and}\ x\ge 0,\mathrm{for}\ i=\mathrm{species} $$
(17)
$$ {C}_i\left(t,x\right)={C}_{i, sys}(t)\ \mathrm{at}\ t>0\ \mathrm{and}\ x=0,\mathrm{for}\ i=\mathrm{species} $$
(18)
$$ \frac{{\partial C}_i\left(t,x\right)}{\partial x}=0\ \mathrm{at}\ t>0\ \mathrm{and}\ x={d}_{clot}+{L}_{clot},\mathrm{for}\ i=\mathrm{species} $$
(19)
where dclot is the distance between the bifurcation and clot front and Lclot is the clot length.
For NV administration, an additional condition at the clot front, described by Eq. (20), is applied to deal with the sudden elevation of tPA level due to the triggered release upon the contact of NV with PLT.
$$ \frac{\partial {C}_{tPA}\left(t,x\right)}{\partial x}=0\ \mathrm{at}\ t>0\ \mathrm{and}\ x=\mathrm{clot}\kern0.17em \mathrm{front}\ \left(\varepsilon <1\right) $$
(20)
Imposing Eq. (20) at the clot front ensures that the spatial profile of tPA concentration is continuous and differentiable throughout the domain for numerical robustness.
For PLT, PLT-NV and PLT-NVemp:
$$ {C}_{PLT\hbox{-} tot}\left(t,x\right)\left\{\begin{array}{c}{C}_{PLT,0},\kern0.5em \mathrm{for}\ {d}_{clot}\le x\le {d}_{clot}+{L}_{clot}\\ {}0,\kern0.5em \mathrm{otherwise}\end{array}\right.\ \mathrm{at}\ t=0 $$
(21)
$$ {C}_{PLT\hbox{-} NV}\left(t,x\right)=0\kern0.5em \mathrm{at}\ t=0\ \mathrm{and}\ x\ge 0 $$
(22)
$$ {C_{PLT\hbox{-} NV}}_{emp}\left(t,x\right)=0\kern0.5em \mathrm{at}\ t=0\ \mathrm{and}\ x\ge 0 $$
(23)
The initial concentration of activated platelets in the clot CPLT,0 is calculated from the initial volume fraction of activated platelets (ϕf,0) in the clot. It is assumed that activated platelets are homogeneously distributed within the clot.
For tPA-F, PLG-F, PLS-F and FBR:
$$ {n}_m\left(t,x\right)=0\kern0.5em \mathrm{at}\ t=0\ \mathrm{and}\ x\ge 0,\mathrm{for}\ m=\mathrm{tPA}-\mathrm{F},\mathrm{PLG}-\mathrm{F},\mathrm{PLS}-\mathrm{F} $$
(24)
$$ {n}_{FBR}\left(t,x\right)=\left\{\begin{array}{c}{n}_{FBR,0},\kern0.5em \mathrm{for}\ {d}_{clot}\le x\le {d}_{clot}+{L}_{clot}\\ {}\begin{array}{cc}0,& \mathrm{otherwise}\end{array}\end{array}\ \mathrm{at}\ t=0\right. $$
(25)
where nFBR,0 is the initial number of fibrin binding sites, which is estimated using the radius of fibrin fibres. The initial concentration of fibrin binding sites is assumed to be uniform over the clot region.
Model Integration and Numerical Details
Model solutions are obtained by an in-house code programmed in MATLAB R2019b (The MathWorks, Inc., Natick, MA, United States). A total of 10 ordinary differential equations shown in Eqs. (1)–(4) are solved for the temporal systemic concentrations of tPA, NV, NVemp, PLG, PLS, AP, AP-PLS, FBG, MG and PAI and a total of 18 partial differential equations shown in Eqs. (7)–(8) and (13)–(14) are solved for temporal and spatial concentrations of tPA, NV, NVemp, PLG, PLS, AP, AP-PLS, FBG, MG, PAI, PLT-free, PLT-NV, PLT-NVemp, tPA-F, PLG-F, PLS-F, PLS-Flysed and FBR. Eqs (1)–(4) are solved first and the obtained solutions are used as inlet conditions for Eqs. (7)–(8) and (13)–(14).The same numerical procedure for model integration is employed as in our previous work and can be found in (28).
Simulation Scenarios and Model Parameters
Sixteen simulations are performed in this study (Table 1): the first 12 scenarios are designed to compare the effectiveness of free tPA and NV under different clot conditions (i.e,. the clot position and composition). The selected scenarios are based on the values reported in clinical studies: composition and porosity in (31), clot position (32) and size (33, 34). Scenarios 1 and 2 represent more distal clots, while Scenarios 5 and 6 correspond to platelet-rich clots compared to the other simulated scenarios. Scenarios 7 to 12 have a clot of higher overall porosity (ε0 = 0.98) with different platelet and fibrin contents than Scenarios 1 to 6 (ε0 = 0.95). For these scenarios, only the coupled species transport with the local PD model is solved with constant inlet concentrations of plasma proteins, i.e. without the systemic PKPD model. The inlet concentration of tPA is chosen to be 0.035 μM, the therapeutic level typically achieved during a continuous infusion in the recommended dosage regimen for the treatment of ischaemic stroke (28, 30). For the simulation scenarios with NV, the inlet NV concentration equivalent to 0.035 μM of tPA (0.035 μM/vrel, vrel the ratio of encapsulated tPA to NV) is chosen for a fair comparison.
Table 1 List of simulation scenarios. Here NV stands for tPA-loaded nanovesicle Scenarios 13 to 16 are designed to study the effects of different dosing regimens compared to the standard treatment protocol with free tPA. For these scenarios, the systemic PKPD model is solved, and its solutions are used as inlet conditions for the species transport model with the local PD model. The recommended regimen with free tPA for the treatment of ischaemic stroke is selected as a reference case: a total tPA dose of 0.9 mg/kg is intravenously administered with 10% of the total dose as a bolus and the remaining 90% as a continuous infusion over 1 h (Scenario 13). For Scenario 14, the equivalent NV amount is given with the same dosing regimen as Scenario 13. Different dosing regimens are simulated for NV in Scenarios 15 and 16; a total dose of 0.09 mg/kg (10% of the recommended dose) as a bolus in Scenario 15 and a total dose of 0.45 mg/kg (50% of the recommended dose) with the standard dosing protocol (a combination of bolus and continuous infusion) in Scenario 16.
Values for the model parameters are the same as in our previous studies (24) with a modification of NV and PLT diffusivity. Diffusivity values for the NV and platelets should be carefully chosen since diffusion-dominated transport is anticipated in an occlusive clot, unlike in our previous work where non-occlusive clots were simulated (24). Lipid nanovesicles are reported to have diffusivity in the range of 0.126 × 10−12 to 1.689 × 10−12 m2/s depending on their size (mean diameter of 184 nm to 216 nm, a similar range to our NV) (35). Therefore, a diffusivity value of 1.6 × 10−12 m2/s is chosen for the simulations. PLT diffusivity was estimated by the Stokes-Einstein equation (36) and is chosen to be 3.1 × 10−14 m2/s. The half-life time of the developed NV is a new parameter needed for the systemic PKPD model and is chosen to be 133 min, as found in the literature for tPA-loaded PEGylated liposomes (25).
Simulation conditions for all 16 scenarios are summarised in Table 1, and other conditions used in common for all the scenarios are listed in Section B of the Supporting Information, along with kinetics parameters, initial concentrations and other model parameters.