# Mathematical Modelling of Intravenous Thrombolysis in Acute Ischaemic stroke: Effects of Dose Regimens on Levels of Fibrinolytic Proteins and Clot Lysis Time

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## Abstract

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## 1. Introduction

_{2}-antiplasmin. These data can be used to elucidate the mechanism of fibrinolysis and the impact of dose regimen on therapeutic efficacy and toxicity. On the other hand, the clinical trials for AIS have focused on clinical and neurological outcomes, such as modified Rankin Scale and NIHSS [2,11,12,13,14,15,16]. Pathological differences between AMI and AIS may cause different responses to tPA dosage, making it difficult to directly apply the PKPD results on thrombolysis in AMI to AIS patients. Furthermore, AIS inherently exhibits a higher risk of intracranial haemorrhage (ICH) when compared to AMI due to acute brain infarction, which may be the reason for a lower dose of tPA recommended for the treatment of stroke than AMI [17]. Additionally, the approved dosage for treating AIS was determined based on clinical outcomes rather than PKPD studies [17]. Hence, there is a clear need for PKPD studies of tPA during thrombolytic therapy for AIS. However, clinical PKPD study on stroke patients would be impractical, as the treatment outcome is highly time-dependent [18]. Using a mathematical model can be very advantageous in such situations where access to experimental and clinical data is limited.

## 2. Methods

#### 2.1. Overview of Modelling Strategy

_{2}-antiplasmin (AP), α

_{2}-macroglobulin (MG), and plasminogen activator inhibitor type 1 (PAI), are present in addition to tPA. The drug is administered to the central compartment and it moves between the central and peripheral compartments. It is assumed that movement of other fibrinolytic proteins between the two compartments is negligible, since the levels of these proteins in plasma are not expected to significantly change. Temporal concentrations of seven fibrinolytic proteins, including tPA, can be resolved using the compartmental model (or named as the “systemic PKPD model”) for a given dose regimen.

#### 2.2. Mathematical Models

#### 2.2.1. Systemic Pharmacokinetics and Pharmacodynamics (PKPD) Model

_{c}the volume of central compartment, M

_{w,tPA}is the molecular weight of tPA (= 59,042.3 g/mol for alteplase), R is the reaction term for generation or consumption via fibrinolytic reactions, S is the systemic secretion (by endothelial cells for tPA), k

_{el}is the elimination rate constant, and k is the distribution kinetics constants. Subscripts sys, c, and p denote the systemic, central, and peripheral compartments, respectively. The superscript plasma refers to reactions taking places in the plasma phase. For other fibrinolytic proteins, component balance equations can be written as:

_{el,i}and secretion rates S

_{i}can be obtained while using half-life and the initial concentration of each component [20]. Reactions between the fibrinolytic proteins in the plasma are listed in Table 1. Details on the reaction kinetics equations and their parameters (plasma reactions 1 to 5 in Table 1) can be found in the Supporting Information (Table S1).

#### 2.2.2. Coupled Flow, Transport and Clot Lysis Model (“Local Pharmacodynamics Model”)

_{a}. The clot is treated as a porous medium with a length of L

_{clot}, a porosity of ε

_{clot}, and permeability of k

_{clot}and its front face is located at a distance of d

_{clot}away from the entrance of the occluded artery. Volumetric flowrate in the occluded artery Q is determined by the pressure drop per unit length across the clot ∆P

_{x}, which is dependent on the occlusion site and collateralisation [24].

_{clot}is calculated using Davies’ Equation [25]. The continuity equation for incompressible flow is employed.

_{i}the diffusivity of protein i, and R

_{i}

^{tot}is the sum of the rates of reactions taking place both in the plasma and clot that contribute to a change in C

_{i}, which can be obtained by:

^{plasma}and R

^{clot}are the rates of reactions in the plasma and in the clot, respectively, as listed in Table 1. The same model for clot lysis, as developed in the previous study [20], is employed here, and the reaction kinetics in the clot are summarised in Table 1. Temporal concentrations of bound phase proteins are also resolved using the reaction rates, as:

_{j}is the concentration of bound phase protein j. Variation in the total concentration of binding sites n

_{tot}is determined by the degradation of binding sites by bound PLS [20].

#### 2.3. Model Integration and Numerical Procedure

_{i,sys}, are then fed into the local PD model that is described by Equation (7) to serve as boundary conditions for the free phase proteins C

_{i}. Initial concentrations are assumed to be the same as those in the systemic PKPD model. The initial and boundary conditions for the bound phase proteins described by Equation (10) are set to zero, while the initial condition for the total concentration of binding sites in Equation (11) is:

_{tot,0}is estimated using the average radius of the fibrin fibre [20]. The partial differential equations for all transporting species (Equation (7)) are numerically solved while using the finite difference method (a combination of the second order central, backward, and forward schemes) for spatial discretisation and backward Euler method for time integration. Detailed discretisation and integration procedures are included in Supporting Information, along with the results of grid independence study.

#### 2.4. Simulation Details

#### 2.5. Remarks on Kinetics Parameters and Model Validation

_{3,cat}in Table S1, which was adjusted to fit experimental findings. Prior to performing simulations for the selected dosage regimens, our model predictions were compared with several sets of PK data available in the literature [3,4,5,8]. Detailed comparisons can be found in Supporting Information (Section D).

## 3. Results and Discussion

#### 3.1. Effects of tPA Dose on Systemic Concentrations of Thrombolytic Proteins

#### 3.2. Effects of Delays between Bolus and Continuous IV Infusion on Systemic Concentrations

#### 3.3. Effects of Bolus to Continuous Infusion Ratio on Systemic Concentrations

#### 3.4. Systemic Concentrations of Thrombolytic Proteins for New Dosage Regimens

#### 3.5. Effects of Different Dosage Regimens on Recanalisation Time

#### 3.6. Temporal and Spatial Variations in Protein Concentration

_{tot}> 0. During the initial phase (up to around 12 min), there is a homogenous distribution of the total binding sites within the clot, as shown in Figure 9a. Free tPA in plasma is transported at a constant rate before reaching the clot, as shown in Figure 9b. When the amount of drug at the clot front reaches the therapeutic level, the clot starts to degrade (at around 12 min), until the complete dissolution (at around 30 min). During the period of clot lysis, tPA slowly penetrates into the clot, creating a band of high tPA concentration near the clot front. Diamond and Anand have also obtained similar tPA spatial profiles with high concentration gradient at the lysis front [25].

_{tot}around 30 min in Figure 9a. For the FBG concentration that is shown in Figure 9d, there is no distinct pattern apart from the reduction in FBG concentration from 8 to around 6.1 μM over time due to the systemic action of tPA on FBG.

## 4. Conclusions and Future Perspectives

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Overview of the full mathematical model. (

**a**) Two-compartment model for pharmacokinetics and pharmacodynamics (PKPD) of tissue plasminogen activator (tPA) in the body (or the “systemic PKPD model”), (

**b**) one-dimensional (1D) blood flow and species transport in an occluded artery, and (

**c**) fibrinolysis described by binding of fibrinolytic proteins with fibrin fibres and the cleavage of fibrin network by bound plasmin. Coupled 1D blood flow and transport and fibrinolysis are referred to as the “local PD model”.

**Figure 2.**Schematic of 1D blood flow and species transport model. The entrance of the blocked artery corresponds to the bifurcating point where flow splits into two branches. The shaded area represents the clot with a porosity of ε

_{clot}and the open area is clot-free with ε = 1.

**Figure 3.**Flow chart of the solution procedure for the systemic and local models. Input information required for solving each model is specified in the green boxes.

**Figure 4.**Changes in systemic concentrations of fibrinolytic proteins over time with different drug doses: 0.6 (Regimen 2), 0.9 (Regimen 1), and 1.2 (Regimen 3) mg/kg of patient weight, as low (solid line), medium (dashed line) and high (dotted line) doses, respectively. (

**a**) tPA concentration in the central (blue lines) and peripheral (orange lines) compartments, (

**b**) systemic concentrations of fibrinogen (FBG), (

**c**) α

_{2}-macroglobulin (MG), and (

**d**) plasminogen activator inhibitor type 1 (PAI) (shown for 1 min only).

**Figure 5.**Effects of bolus-continuous IV infusion delay on the concentrations of (

**a**) tPA and (

**b**) FBG. Simulated Regimens 4–6 correspond to delays of 5, 10, and 30 min, respectively. The reference regimen with no delay (Regimen 1) is also included for comparison.

**Figure 6.**Effects of bolus to infusion ratios at a fixed dose of 0.9 mg/kg, on the concentration of (

**a**) tPA, and (

**b**) FBG. The amount of bolus is varied from 0% (Regimen 7) to 50% (Regimen 9) of the total dose. Regimen 1 has a bolus to infusion ratio of 1:9, while Regimens 8 and 9 have a ratio of 1:4 and 1:1, respectively.

**Figure 7.**Concentrations of tPA (

**a**) and FBG (

**b**)over time for the new regimens simulated in this study.

**Figure 8.**Predicted recanalisation times for the simulated dosage regimens. The black bar and dashed line are the reference regimen. Different colour bars indicate the therapeutic parameters that are varied: tPA dose (blue bars), bolus-infusion delay (orange), bolus-infusion ratio (yellow), and new regimens (green).

**Figure 9.**Temporal and spatial concentrations of free phase tPA, PLG, and FBG, and the total concentration of binding sites (n

_{tot}) in the clot for the standard dosage regimen. Each map shows variations of concentration in time (vertical axis) and space (horizontal axis). (

**a**) The total binding site and (

**b**) free phase tPA, (

**c**) PLG, and (

**d**) FBG.

No. | Description | |
---|---|---|

Plasma | 1 | $\mathrm{tPA}+\mathrm{PLG}\stackrel{{K}_{1,M}\text{}\text{}{k}_{1,cat}}{\to}\mathrm{tPA}+\mathrm{PLS}$ |

2 | $\mathrm{PLS}+\mathrm{AP}\underset{{k}_{2,r}}{\overset{{k}_{2,f}}{\rightleftarrows}}\mathrm{PLS}\xb7\mathrm{AP}\stackrel{{k}_{2,cat}}{\to}\mathrm{inactive}$ | |

3 | $\mathrm{PLS}+\mathrm{FBG}\stackrel{{K}_{3,M}\text{}\text{}{k}_{3,cat}}{\to}\mathrm{PLS}+\mathrm{FDP}$ | |

4 | $\mathrm{PLS}+\mathrm{MG}\stackrel{{k}_{4}}{\to}\mathrm{inactive}$ | |

5 | $\mathrm{tPA}+\mathrm{PAI}\stackrel{{k}_{5}}{\to}\mathrm{inactive}$ | |

Clot | 1 | $\mathrm{tPA}+\mathrm{F}\underset{{k}_{d,tPA}}{\overset{{k}_{a,tPA}}{\rightleftarrows}}\mathrm{tPA}\cdot \mathrm{F}$ |

2 | $\mathrm{PLG}+\mathrm{F}\underset{{k}_{d,PLG}}{\overset{{k}_{a,PLG}}{\rightleftarrows}}\mathrm{PLG}\cdot \mathrm{F}$ | |

3 | $\mathrm{PLS}+\mathrm{F}\underset{{k}_{d,PLS}}{\overset{{k}_{a,PLS}}{\rightleftarrows}}\mathrm{PLS}\cdot \mathrm{F}$ | |

4 | $\mathrm{tPA}\cdot \mathrm{F}+\mathrm{PLG}\cdot \mathrm{F}\stackrel{{K}_{M}\text{}\text{}{k}_{M,cat}}{\to}\mathrm{tPA}\cdot \mathrm{F}+\mathrm{PLS}\cdot \mathrm{F}$ | |

5 | $\mathrm{PLS}\cdot \mathrm{F}\stackrel{{k}_{deg}}{\to}\mathrm{PLS}\cdot \tilde{\mathrm{F}}$ | |

6 | $\mathrm{PLS}\cdot \tilde{\mathrm{F}}\stackrel{{k}_{d,PLS}}{\to}\mathrm{PLS}+\tilde{\mathrm{F}}$ |

**Table 2.**List of simulated dosage regimens. Regimen 1 is the standard dose for the treatment of ischaemic stroke.

No. | Total Dose | Regimen Description |
---|---|---|

1 | 0.9 mg/kg | 10% as bolus + 90% as continuous over 1 h (reference case) |

2 | 0.6 mg/kg | 10% as bolus + 90% as continuous over 1 h |

3 | 1.2 mg/kg | 10% as bolus + 90% as continuous over 1 h |

4 | 0.9 mg/kg | 10% as bolus + 5-min delay + 90% as continuous over 1 h |

5 | 0.9 mg/kg | 10% as bolus + 10-min delay + 90% as continuous over 1 h |

6 | 0.9 mg/kg | 10% as bolus + 30-min delay + 90% as continuous over 1 h |

7 | 0.9 mg/kg | the total dose as a continuous infusion over 1 h |

8 | 0.9 mg/kg | 20% as bolus + 80% as continuous over 1h |

9 | 0.9 mg/kg | 50% as bolus + 50% as continuous over 1h |

10 | 0.9 mg/kg | 15% as bolus + 50% as continuous over 30 min + 35% as continuous over 1 h |

11 | 0.9 mg/kg | 15% as bolus + 35% as continuous over 30 min + 50% as continuous over 1 h |

12 | 0.9 mg/kg | 10% as bolus + 60% as continuous over 30 min + 30% as continuous over 1 h |

13 | 0.9 mg/kg | 10% as bolus + 30% as continuous over 30 min + 60% as continuous over 1 h |

Symbol | Definition | Values [unit] | Source |
---|---|---|---|

C_{tPA,0} | Initial tPA concentration | 0.05 [nM] | [24] |

C_{PLG,0} | Initial PLG concentration | 2.2 [μM] | [24] |

C_{PLS,0} | Initial PLS concentration | 0 [μM] | [24] |

C_{FBG,0} | Initial FBG concentration | 8 [μM] | [24] |

C_{AP,0} | Initial AP concentration | 1 [μM] | [24] |

C_{MG,0} | Initial MG concentration | 3 [μM] | [24] |

C_{PAI,0} | Initial PAI concentration | 5.23 × 10^{−4} [μM] | [27] |

ε_{clot,0} | Initial porosity of the clot | 0.95 [-] | [24] |

R_{f} | Radius of fibrin fibre in the clot | 100 [nm] | [24] |

μ | Blood viscosity | 0.0037 [Pa∙s] | [20] |

n_{tot,0} | Initial total concentration of binding sites in the clot | 1.74 [μM] | [20] |

∆P_{x} | Pressure drop per unit length across the clot | 60 [mmHg/cm] | [24] |

d_{clot} | Distance from the bifurcation to the clot front | 5 [mm] | [26] |

L_{clot} | Length of clot | 5 [mm] | [22] |

D_{a} | Diameter of occluded artery | 3 [mm] | [28] |

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## Share and Cite

**MDPI and ACS Style**

Gu, B.; Piebalgs, A.; Huang, Y.; Longstaff, C.; Hughes, A.D.; Chen, R.; Thom, S.A.; Xu, X.Y.
Mathematical Modelling of Intravenous Thrombolysis in Acute Ischaemic stroke: Effects of Dose Regimens on Levels of Fibrinolytic Proteins and Clot Lysis Time. *Pharmaceutics* **2019**, *11*, 111.
https://doi.org/10.3390/pharmaceutics11030111

**AMA Style**

Gu B, Piebalgs A, Huang Y, Longstaff C, Hughes AD, Chen R, Thom SA, Xu XY.
Mathematical Modelling of Intravenous Thrombolysis in Acute Ischaemic stroke: Effects of Dose Regimens on Levels of Fibrinolytic Proteins and Clot Lysis Time. *Pharmaceutics*. 2019; 11(3):111.
https://doi.org/10.3390/pharmaceutics11030111

**Chicago/Turabian Style**

Gu, Boram, Andris Piebalgs, Yu Huang, Colin Longstaff, Alun D. Hughes, Rongjun Chen, Simon A. Thom, and Xiao Yun Xu.
2019. "Mathematical Modelling of Intravenous Thrombolysis in Acute Ischaemic stroke: Effects of Dose Regimens on Levels of Fibrinolytic Proteins and Clot Lysis Time" *Pharmaceutics* 11, no. 3: 111.
https://doi.org/10.3390/pharmaceutics11030111