# Thermosensitive Liposome-Mediated Drug Delivery in Chemotherapy: Mathematical Modelling for Spatio–temporal Drug Distribution and Model-Based Optimisation

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## Abstract

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## 1. Introduction

## 2. Methods

_{c}= 10 μm) and its surrounding tumour interstitium (with a radius R

_{t}= 120 μm). Given the assumption of axis-symmetry and homogeneous distribution along tumour microvessels, one-dimensional drug transport in the radial direction is formulated.

#### 2.1. Systemic Drug Transport: Pharmacokinetics of TSLs and Anticancer Drug

_{37}the drug release rate at the body temperature, k

_{e}the elimination rate from the central compartment, and k

_{p}and k

_{t}are the rate constants for drug transfer between the central plasma and tissue compartments. The subscripts P, t, Lip denote the plasma, tissue and liposome, respectively. The superscripts B and T represent the systemic level and the tumour compartment, respectively. ${F}_{PV}^{T}$ is the plasma flow per tumour plasma volume, defined as: ${F}_{PV}^{T}={w}_{blood}\left(1-{H}_{ctt}\right)/{v}_{P}^{T}$ in which ${w}_{blood}$, ${H}_{ctt}$ and ${v}_{P}^{T}$ are the perfusion rate in (mL/s/mL), haematocrit and volume fraction of tumour plasma, respectively.

#### 2.2. Drug Transport in the Tumour Compartment

#### 2.2.1. Transport in Tumour Plasma

_{37}and kr

_{42}at the body temperature and mild hyperthermia upon heating, respectively. The dynamic concentrations of TSLs and anticancer drug are described by:

_{c}and T

_{h}are the initial time and duration of HT, respectively.

#### 2.2.2. Transport in the Tumour Interstitium

_{e}is the extracellular concentration, c

_{i}the intracellular anticancer drug, D the diffusivity, ${v}_{e}^{T}$ the volume fraction of tumour EES, k

_{1ci}the rate constant for passive transport, V

_{max}the maximum rate for transmembrane transport, K

_{e}and K

_{i}the Michaelis constants, and R is the release rate constant defined in Equation (6). The blood flow velocity u

_{i}in the interstitial space is determined by the permeability of vessel walls (K) and pressure difference between the blood vessel and tumour interstitial space (P

_{i}) via Darcy’s law.

_{e}and u

_{i}. Starling’s law and Kedem–Katchalsky equations are used to prescribe the transmural blood flow velocity and drug flux at the capillary walls, i.e., at r = R

_{c}. It is assumed that osmotic pressure is negligible compared to the pressure gradient in determining the transmural velocity in solid tumour [4]. These boundary conditions are expressed as:

_{F}is the transmural velocity, P

_{v}the vessel pressure, c

_{v}is the drug concentration in the vessel side, L

_{p}the hydraulic permeability and σ

_{F}the reflection coefficient. At the outer boundary of the tumour cord (r = R

_{t}), no flux is prescribed for c

_{e}as expressed in Equation (14).

#### 2.3. Parameterisation and Initialisation

#### 2.4. Numerical Methods

#### 2.5. Optimisation

_{37}and kr

_{42}) and HT duration (T

_{h}) for maximised intracellular drug concentration (c

_{i}) and minimised systemic drug concentration (c

_{B}

^{P}) over time, which are indicative of therapeutic efficacy and the risk of side effects, respectively. For this purpose, a multiobjective optimisation problem is formulated, which can be solved by the weighted sum method and described as follows:–

_{1}and J

_{2}are the objective functions to be minimised that represent time-averaged systemic plasma drug concentration and intracellular drug concentrations normalised by their respective baseline values (as defined in Table 1), and w

_{i}the weighting factor for the i-th objective function value. The model equations presented in Equations. (1) to (14) serve as equality constraints and each optimisation variable is bounded by its lower and upper limits defined as: 0.1kr

_{37,ref}≤ kr

_{37,ref}≤ 10kr

_{37}, 0.1kr

_{42,ref}≤ kr

_{42}≤ 10kr

_{42,ref}and 0 ≤ T

_{h}≤ 3600 s. The weighting factor w

_{1}is varied between 0 and 1 and optimisation results for different weighting factors are compared. Also, multiple sets of initial guesses of the optimisation variables are chosen to ensure global optimality. The above optimisation problem is solved via an in-built function in MATLAB (fmincon), designed for constrained nonlinear multivariable problems.

## 3. Results and Discussion

#### 3.1. Simulation Results for the Baseline Scenario

#### 3.2. Comparison of DOX and TOP

#### 3.3. Effects of Drug Release Rates

_{37}would be needed in order to keep the systemic concentration low. On the other hand, a high kr

_{42}would be desirable for enhanced cancer cell killing. Results in Figure 4b show that intracellular drug concentration is highly sensitive to kr

_{42}, but its effect on systemic concentration is rather trivial, especially at moderate and high kr

_{37}.

_{37}[24]. Ideally, this should be kept as low as possible in order to achieve controlled drug release trigged by hyperthermia. As kr

_{37}increases, more encapsulated drugs would be released before hyperthermia is applied, rendering the process less temperature sensitive. The rate of drug release at hyperthermia is controlled by kr

_{42}. A very high kr

_{42}value corresponds to instantaneous drug release, which triggers a sharp rise in free drug concentration in the tumour plasma compartment upon hyperthermia, as shown in Figure 2b and Figure 3b. This would lead to a significant difference in drug concentration between the tumour plasma and extracellular space, providing substantial driving force for diffusive drug transport into the intracellular space.

#### 3.4. Effects of Hyperthermia (HT) Duration

#### 3.5. Optimisation Results

_{37}, kr

_{42}and T

_{h}, are simultaneously manipulated during the computer-based optimisation until the objective function value, the weighted sum of systemic plasma and intracellular concentrations as described in Equation (15), reaches its minimum. The weighting factors are also varied for global optimisation, and optimisation results are compared in Table 2. The higher the weighting factor w

_{1}, the more we value the minimisation of systemic drug concentration, and vice versa. As the weighting factor, w

_{1}, changes from 0 to 1, the optimisation problem turns from the maximisation of intracellular concentration to the minimisation of systemic plasma concentration. Table 2 indicates that regardless of the release constants and weighting factor, the optimal heating duration is found to be at its upper limit, 3600 s. Also, a similar behaviour for k

_{37}can be seen; the optimal k

_{37}value is at its lower bound regardless of the weight factor and k

_{42}, which means that a low release rate at 37 °C is always beneficial. On the other hand, the optimal value for k

_{42}depends strongly on the weight factor. As w

_{2}increases (i.e., only maximising the intracellular drug concentration), kr

_{42}approaches its upper bound. When the weight factors are equal (i.e., w

_{1}= w

_{2}= 0.5), a moderate kr

_{42}would be optimal. However, the impact of kr

_{42}on systemic drug concentration is weak, and the rate of increase in intracellular concentration slows down significantly after kr

_{42}reaches 0.7 s

^{−1}, suggesting that further increase in kr

_{42}would not enhance intracellular concentration any further. It is worth noting that our optimisation results have limitations especially arising from the current formulation of objective function; the average systemic and intracellular drug concentrations are used to indicate the extent of cytotoxicity and cancer cell killing efficacy, respectively. The concentration terms in the objective function can be replaced by more direct and/or accurate predictors of cytotoxicity and cancer cell killing efficacy. In addition, the optimisation is based on the assumption of uniform temperature in tumour interstitium.

## 4. Conclusions and Future Perspectives

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Schematic diagram of multiple compartments used in our computational model: Two compartments of “systemic plasma” and “tissues” for systemic effects of drug and the tumour compartment comprising of “tumour plasma”, “tumour extravascular extracellular space (EES)” and “tumour cells”. TSLs: Thermo-sensitive liposomes.

**Figure 2.**Spatio–temporal distributions of TSL and topotecan (TOP) concentrations in different compartments. (

**a**) TSL concentrations in the systemic and tumour plasma compartments, (

**b**) released TOP concentration in the systemic plasma, tumour plasma and extracellular space, (

**c**) spatial profiles of TOP intracellular concentrations at different time points and (

**d**) temporal profiles of TOP intracellular concentrations at different radial positions.

**Figure 3.**Simulation results for two different drugs, DOX and TOP. (

**a**) Free drug concentrations in systemic plasma, (

**b**) free drug concentrations in tumour plasma, (

**c**) temporal intracellular drug concentrations in the intracellular space at r = 0 µm over the course of treatment and (

**d**) spatial intracellular drug concentrations at t = 0.55 h.

**Figure 4.**Drug concentration averaged over the course of the treatment with different combinations of release rate constants kr

_{37}and kr

_{42}(

**a**) in the systemic plasma and (

**b**) in intracellular space.

**Figure 5.**Temporal concentrations of drug with different hyperthermia schedules. (

**a**) Free drug concentration in systemic plasma, (

**b**) drug concentration in intracellular space at r = 0 µm, (

**c**) average and peak drug concentration in systemic plasma and (

**d**) average and peak topotecan concentration in intracellular space at r = 0 µm.

Symbol | Description | Value | Reference |
---|---|---|---|

${k}_{e,Lip}$ | Rate constant of TSLs clearance | 9.417 × 10^{−6} (1/s) | [9] |

${k}_{e}$ | Rate constant of drug clearance | 1.1 × 10^{−3} (1/s) | [6] |

${k}_{p}$ | Transfer constant from systemic plasma to tissue | 1.6 × 10^{−3} (1/s) | [6] |

${k}_{t}$ | Transfer constant from tissue to systemic plasma | 4.68 × 10^{−5} (1/s) | [6] |

$k{r}_{37}$ | Release rate constant from iTSLs at body temperature | 3 × 10^{−4} (1/s) | [16] |

$k{r}_{42}$ | Release rate constant from iTSLs during HT (at 42 °C) | 0.114 (1/s) | [16] |

${K}_{e}$ | Michaelis constant for transmembrane transport | 2.19 × 10^{−4} (µg/mm^{3}) | [19] |

${K}_{i}$ | Michaelis constant for transmembrane transport | 1.37 (ng/10^{5} cells) | [19] |

${V}_{\mathrm{ma}x}$ | Maximum rate for transmembrane transport | 0.28 (ng/(10^{5} cells·min)) | [19] |

${k}_{1ci}$ | Rate for passive intracellular uptake | 6.33 × 10^{−4} (1/s) | Fit to [6] |

${V}_{p}^{B}$ | Volume of systemic plasma | 3.04 (L) | [9] |

${V}_{t}^{B}$ | Volume of body tissue | 64.47 (L) | [9] |

${V}^{T}$ | Volume of tumour tissue | 8.82 × 10^{−2} (L) | Estimated based on a spherical tumour with a radius of 2.7 cm |

${v}_{p}^{T}$ | Volume fraction of tumour plasma | 0.07452 | [9] |

${v}_{e}^{T}$ | Volume fraction of tumour EES | 0.454 | [9] |

${v}_{i}^{T}$ | Volume fraction of intracellular space | 0.454 | [9] |

${w}_{blood}$ | Blood perfusion rate | 0.018 (1/s) | [9] |

${H}_{ctt}$ | Haematocrit for tumour microvasculature | 0.19 | [9] |

$P{S}_{t}$ | Permeability surface area product for drugs | 2.53 × 10^{−3} (1/s) (TOP)7 × 10 ^{−3} (1/s) (DOX) | Estimated based on molecular size [9] |

$P{S}_{L}$ | Permeability surface area product for TSL | 4.76 × 10^{−6} (1/s) (TOP)2.38 × 10 ^{−5} (1/s) (DOX) | Estimated using the $P{S}_{t}$, $P$ and ${P}_{lip}$ of DOX and TOP respectively |

$P$ | Diffusive permeability for drugs | 3.61 × 10^{−7} (m/s) (TOP)1 × 10 ^{−6} (m/s) (DOX) | Estimated based on molecular size [19] |

${P}_{Lip}$ | Diffusive permeability for TSLs and iTSLs | 3.4 × 10^{−9} (m/s) | [19] |

${D}_{}$ | Diffusion coefficient | 4.123 × 10^{−10} (m^{2}/s) (TOP)1.578 × 10 ^{−10} (m^{2}/s) (DOX) | Estimated based on molecular size [12] |

${D}_{Lip}$ | Diffusion coefficient for TSLs and iTSLs | 9 × 10^{−12} (m^{2}/s) | [20] |

${R}_{c}$ | Tumour capillary radius | 10 ($\mathsf{\mu}\mathrm{m}$) | [12] |

${R}_{t}$ | Tumour cord radius | 120 ($\mathsf{\mu}\mathrm{m}$) | [12] |

${V}_{c}$ | Volume of single tumour cell | 1 × 10^{−6} (mm^{3}/cell) | [19] |

$Dose$ | Total dose | 49 (mg) | Calculated at a dose of 0.7 mg/kg in a 70 kg human |

Weighting Factor, w_{1}, for the Systemic Plasma Conc. | Weighting Factor, w_{2}, for the Intracellular Conc. | kr_{37} (10^{−4} s^{−1}) | kr_{42} (s^{−1}) | Hyperthermia Duration (s) | Systemic Plasma Concentration (μg/mL) | Intracellular Concentration (ng/10^{5} cells) |
---|---|---|---|---|---|---|

0 | 1 | 0.3 | 1.14 | 3600 | 0.26 | 0.56 |

0.25 | 0.75 | 0.3 | 1.1 | 3600 | 0.25 | 0.56 |

0.5 | 0.5 | 0.3 | 0.7 | 3600 | 0.24 | 0.55 |

0.75 | 0.25 | 0.3 | 0.13 | 3600 | 0.21 | 0.48 |

1 | 0 | 0.3 | 0.0114 | 3600 | 0.18 | 0.19 |

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**MDPI and ACS Style**

Huang, Y.; Gu, B.; Liu, C.; Stebbing, J.; Gedroyc, W.; Thanou, M.; Xu, X.Y. Thermosensitive Liposome-Mediated Drug Delivery in Chemotherapy: Mathematical Modelling for Spatio–temporal Drug Distribution and Model-Based Optimisation. *Pharmaceutics* **2019**, *11*, 637.
https://doi.org/10.3390/pharmaceutics11120637

**AMA Style**

Huang Y, Gu B, Liu C, Stebbing J, Gedroyc W, Thanou M, Xu XY. Thermosensitive Liposome-Mediated Drug Delivery in Chemotherapy: Mathematical Modelling for Spatio–temporal Drug Distribution and Model-Based Optimisation. *Pharmaceutics*. 2019; 11(12):637.
https://doi.org/10.3390/pharmaceutics11120637

**Chicago/Turabian Style**

Huang, Yu, Boram Gu, Cong Liu, Justin Stebbing, Wladyslaw Gedroyc, Maya Thanou, and Xiao Yun Xu. 2019. "Thermosensitive Liposome-Mediated Drug Delivery in Chemotherapy: Mathematical Modelling for Spatio–temporal Drug Distribution and Model-Based Optimisation" *Pharmaceutics* 11, no. 12: 637.
https://doi.org/10.3390/pharmaceutics11120637