# Towards a multiphysics modelling framework for thermosensitive liposomal drug delivery to solid tumour combined with focused ultrasound hyperthermia

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### Abstract

Systemic toxicity and insufficient drug accumulation at the tumour site are main barriers in chemotherapy. Thermosensitive liposomes (TSL) combined with high intensity focused ultrasound (HIFU) has emerged as a potential solution to overcome these barriers through targeted drug delivery and localised release. Owing to the multiple physical and biochemical processes involved in this combination therapy, mathematical modelling becomes an indispensable tool for detailed analysis of the transport processes and prediction of tumour drug uptake. To this end, a multiphysics model has been developed to simulate the transport of chemotherapy drugs delivered through a combined HIFU–TSL system. All key delivery processes are considered in the model; these include interstitial fluid flow, HIFU acoustics, bioheat transfer, drug release and transport, as well as tumour drug uptake. The capability of the model is demonstrated through its application to a 2-D prostate tumour model reconstructed from magnetic resonance images. Our results not only demonstrate the feasibility of the model to simulate this combination therapy, but also confirm the advantage of HIFU–TSL drug delivery system with enhancement of drug accumulation in tumour regions and reduction of drug availability in normal tissue. This multiphysics modelling framework can serve as a useful tool to assist in the design of HIFU–TSL drug delivery systems and treatment regimen for improved anticancer efficacy.

### Graphical abstract

## Keywords

Chemotherapy Drug delivery High intensity focused ultrasound Mathematical model Thermosensitive liposome## Introduction

Numerous anticancer drugs have been developed with significant cytotoxicity observed in preclinical studies, but their clinical applications are limited for various reasons. Among these poor treatment efficacy and systemic toxicity are the main obstacles for effective chemotherapy in patients. This is largely attributed to insufficient drug penetration from microvasculature into tumour interior (Jain 1987a), resulting in inadequate drug accumulation for effective tumour cell killing. On the other hand, undesirable drug deposition in normal tissue may damage healthy cells so as to cause various side effects.

Thermosensitive liposome (TSL) combined with high intensity focused ultrasound (HIFU) offers a promising drug delivery system to overcome the limitations of conventional chemotherapy through actively targeted and triggered release of drugs at the tumour site. With this delivery system, drug-loaded TSL are intravenously administrated and then circulate with bloodstream to reach the tumour site. For well-designed TSL that are stable in blood, encapsulated drugs cannot be released at normal physiological temperature, and mild hyperthermia is required to trigger the release of the encapsulated contents (Gaber *et al.* 1995; Lindner *et al.* 2004; Tagami *et al.* 2011). Localised heating can be achieved by HIFU, which is favoured for its non-invasive nature and accuracy in clinical thermal therapy (Hynynen 2011). The safety and feasibility of the combined HIFU–TSL system have been reported in a very recent clinical trial, demonstrating enhanced intratumoural drug delivery for targeted treatment of liver tumours (Lyon *et al.* 2018). However, there is still a lack of understanding on the drug transport mechanism and tumour drug uptake in response to temperature variation upon HIFU heating.

Given the multiple steps involved in heat transfer and drug transport and the complex interplays among tumour, TSL and anticancer drugs, mathematical modelling plays an increasingly important role in gaining a deep understanding of drug delivery systems and predicting their treatment efficacy. Early mathematical models (Baxter and Jain 1989, 1990, 1991) were developed to study the transport of interstitial fluid and macromolecules in solid tumour. These were subsequently extended to multiple scales for examining the effects of different factors on drug delivery (Eikenberry 2009; El-Kareh and Secomb 2000; Goh *et al.* 2001; Liu *et al.* 2013; Luu and Uchizono 2005; Teo *et al.* 2005; Zhao *et al.* 2007). Results based on a 1-D drug transport model demonstrated the importance of blood temperature in determining the overall delivery outcome using TSL (El-Kareh and Secomb 2003). The performance of TSL was further compared to conventional chemotherapy and stealth liposomes under different conditions (Gasselhuber *et al.* 2012a), whilst a systematic parametric study was performed to identify the most influential factors in determining the peak intracellular concentration using a TSL–HIFU system (Liu and Xu 2015). The drug distribution in a 2-D idealised tumour geometry via TSL-mediated delivery was examined using a transport model that ignored the effect of convection (Zhang *et al.* 2009). Although a bioheat transfer model and macroscopic transport model were integrated in a pioneer study to simulate TSL drug delivery under different heating schedules (Gasselhuber *et al.* 2012b), existing spatially resolved drug transport models do not include HIFU acoustics and temperature-dependent tumour and drug properties.

The aim of the present study is to develop a multiphysics model for TSL-mediated drug delivery combined with HIFU heating. This requires integration and coupling of models for HIFU acoustic field, tissue and blood temperature, interstitial fluid flow, TSL and drug transport/reaction and tumour drug uptake. The integrated model has been applied to real tumour geometry reconstructed from patient-specific images. Heat-induced variations in transport properties of the drug and tumour are considered, which are temporally and spatially dependent. Anticancer efficacy is evaluated based on the percentage of survival tumour cells by solving the pharmacodynamics equation using the predicted intracellular drug concentration.

## Results

A total dose of 50 mg/m^{2} (Gabizon *et al.* 1994) TSL doxorubicin is intravenously administrated (Gasselhuber *et al.* 2012b) at the beginning of the treatment for a 70 kg patient, with HIFU heating being applied simultaneously (Liu and Xu 2015) over a period of 60 min (Grüll and Langereis 2012). The two focus regions are heated sequentially during the treatment, and sonication duration for each region is 1.0 s.

### Interstitial fluid flow

*et al.*2010).

High IFP in tumour is largely attributed to the leaky vasculature and dysfunctional lymphatics. It has been reported that large pores on tumour vasculature could increase the wall hydraulic conductivity by 10-fold compared to normal blood vessels (Baxter and Jain 1989), making fluid exchange across the vessel wall much easier. Additionally, the lack of functional lymphatic vessels results in less fluid being drained out of the extracellular space, causing the build-up of IFP in the tumour.

### HIFU acoustic field

### Temperature distribution in tissue and blood

### Drug transport

Governing equations for drug transport under TSL-mediated delivery

Pharmacokinetics model | |

\(\frac{{{\text d}C_{\text{lp}} }}{{\text d}t} = - CL_{\text{lp}} C_{\text{lp}} - k_{\text{rel}} C_{\text{lp}}\) | (1-1) |

Liposome encapsulated drug concentration in the interstitial fluid ( | |

\(\frac{{\partial C_{\text{le}} }}{\partial t} + \nabla \cdot \left( {C_{\text{le}} \varvec{v}} \right)=D_{\text{l}} \nabla^{2} C_{\text{le}} + F_{\text{v}} \left( {1 - \sigma_{\text{l}} } \right)C_{\text{lp}} + P_{\text{l}} \frac{S}{V}\left( {C_{\text{lp}} - C_{\text{le}} } \right)\frac{{{\text{Pe}}_{\text{l}} }}{{{\text{e}}^{{{\text{Pe}}_{\text{l}} }} - 1}} - F_{\text{ly}} C_{\text{le}} - k_{\text{rel}} C_{\text{le}} , {\text P}{{\text e}_{\text l}}=\frac{{{F}_{v}}\left( 1-{{\sigma }_{l}} \right)}{{{P}_{\text l}}\frac{S}{V}}\) | (1-2) |

Free doxorubicin concentration in blood plasma ( | |

\(\frac{{\partial C_{\text{fp}} }}{\partial t} = k_{\text{rel}} C_{\text{lp}} - \frac{{V_{\text{T}} }}{{V_{\text{B}} }}\left[ {F_{\text{v}} \left( {1 - \sigma_{\text{f}} } \right)C_{\text{fp}} + P_{\text{fe}} \frac{S}{V}\left( {C_{\text{fp}} - C_{\text{fe}} } \right)\frac{{{\text{Pe}}_{\text{f}} }}{{{\text{e}}^{{{\text{Pe}}_{\text{f}} }} - 1}}} \right] - CL_{\text{fp}} C_{\text{fp}} - \left( {k_{\text{a}} C_{\text{fp}} - k_{\text{d}} C_{\text{bp}} } \right)\) | (1-3) |

Bound doxorubicin concentration in blood plasma ( | |

\(\frac{{\partial C_{\text{bp}} }}{\partial t} = \left( {k_{\text{a}} C_{\text{fp}} - k_{\text{d}} C_{\text{bp}} } \right) - \frac{{V_{\text{T}} }}{{V_{\text{B}} }}\left[ {F_{\text{v}} \left( {1 - \sigma_{\text{b}} } \right)C_{\text{bp}} + P_{\text{be}} \frac{S}{V}\left( {C_{\text{bp}} - C_{\text{be}} } \right)\frac{{{\text{Pe}}_{\text{b}} }}{{{\text{e}}^{{{\text{Pe}}_{\text{b}} }} - 1}}} \right] - CL_{\text{bp}} C_{\text{bp}}\) | (1-4) |

Free doxorubicin concentration in interstitial fluid ( | |

\(\frac{{\partial C_{\text{fe}} }}{\partial t} + \nabla \cdot \left( {C_{\text{fe}} \varvec{v}} \right) = D_{\text{fe}} \nabla^{2} C_{\text{fe}} + F_{\text{v}} \left( {1 - \sigma_{\text{f}} } \right)C_{\text{fp}} + P_{\text{fe}} \frac{S}{V}\left( {C_{\text{fp}} - C_{\text{fe}} } \right)\frac{{{\text{Pe}}_{\text{f}} }}{{{\text{e}}^{{{\text{Pe}}_{\text{f}} }} - 1}} - F_{\text{ly}} C_{\text{fe}} + k_{\text{d}} C_{\text{be}} - k_{\text{a}} C_{\text{fe}} + D_{\text{c}} V_{ \hbox{max} } \left( {\frac{{C_{\text{i}} }}{{C_{\text{i}} + k_{\text{i}} }} - \frac{{C_{\text{fe}} }}{{C_{\text{fe}} + k_{\text{e}} \varphi }}} \right) + k_{\text{rel}} C_{\text{le}}\) | (1-5) |

Bound doxorubicin concentration in interstitial fluid ( | |

\(\frac{{\partial C_{\text{be}} }}{\partial t} + \nabla \cdot \left( {C_{\text{be}} \varvec{v}} \right) = D_{\text{be}} \nabla^{2} C_{\text{be}} + F_{\text{v}} \left( {1 - \sigma_{\text{b}} } \right)C_{\text{bp}} + P_{\text{be}} \frac{S}{V}\left( {C_{\text{bp}} - C_{\text{be}} } \right)\frac{{{\text{Pe}}_{\text{b}} }}{{{\text{e}}^{{{\text{Pe}}_{\text{b}} }} - 1}} - k_{\text{d}} C_{\text{be}} + k_{\text{a}} C_{\text{fe}}\) | (1-6) |

Intracellular doxorubicin concentration ( | |

\(\frac{{\partial C_{\text{i}} }}{\partial t} = V_{ \hbox{max} } \left( {\frac{{C_{\text{fe}} }}{{C_{\text{fe}} + k_{\text{e}} \varphi }} - \frac{{C_{\text{i}} }}{{C_{\text{i}} + k_{\text{i}} }}} \right)\) | (1-7) |

### Tumour drug uptake

## Discussion

In this study, a multiphysics model is developed to predict the temporal and spatial profiles of temperature, drug concentrations and tumour cell survival for a TSL doxorubicin delivery system combined with HIFU. This integrated model is applied to a realistic prostate tumour geometry reconstructed from medical images. Our modelling results demonstrate the advantage of the combined HIFU–TSL therapy in achieving enhanced drug release at the targeted tumour site, while reducing drug availability in normal tissues. Simulations of HIFU heating reveals that when the acoustic power is focused at a preselected region, only the targeted region would experience sufficiently high temperature to trigger fast drug release. Since temperature in the normal tissue is below the hyperthermia requirement, majority of the encapsulated doxorubicin remains inside the liposomes. While the primary purpose of HIFU heating is to trigger drug release from TSL, transport properties of the drug and tumour tissue also vary in response to temperature elevation, leading to improved drug transport and uptake during hyperthermia.

Tumour drug distribution and uptake can be highly heterogeneous when a single-element transducer is used, as the coverage of its focus region is very limited. This would be a serious limitation for the treatment of large tumours. In order to achieve a uniform hyperthermia temperature distribution within a tumour, multi-element transducers have been used which can simultaneously generate multiple focus regions, allowing wider coverage of the targeted tumour. Careful design of HIFU scanning trajectory and focus region locations can help homogenise the temperature distribution, so as to achieve enhanced drug release in the entire tumour. Our results also show that incorporating a temperature controller is effective at maintaining the desired tumour temperature during HIFU heating.

It is worth noting that TSL formulations can be unstable at body temperature in human blood (Hossann *et al.* 2010; Tagami *et al.* 2011, 2012). Ideal TSL should be able to release its payload upon hyperthermia only (Zou *et al.* 1993). However, this is difficult to achieve *in vivo*. Leaking at normal body temperature can lead to build-up of free doxorubicin in the circulatory system and normal tissue, increasing the risk of damage to normal cells and systemic side effects, such as cardiotoxicity (Cusack *et al.* 1993; Legha *et al.* 1982). Therefore, further improvements in TSL facilitated delivery combined with HIFU triggered release should aim to minimise drug leakage at body temperature, increase the release rate at the target temperature upon hyperthermia and optimise the heating schedule (Liu and Xu 2015). As a cost effective approach, mathematical modelling can facilitate comprehensive parametric analyses to identify the most influential factors and optimise their ranges, in order to provide guidance for the design of TSL and treatment regimen.

Our mathematical model was developed to describe the key interplays between solid tumour and HIFU–TSL drug delivery system. Although a 2-D model is used in this study for the sake of computational cost, the model can be applied directly to 3-D tumour geometry. Nevertheless, the present model involves a number of assumptions. First, physical complexities including cavitation and non-linear propagation of acoustic waves are ignored, but the effects of these complexities are expected to be minor for the conditions examined here (Sheu *et al.* 2011; Solovchuk *et al.* 2012). Second, the influence of temperature elevation on tumour properties is complex, and not all variations are included in the current model, such as change in interstitial fluid viscosity and temperature-dependent plasma clearance. Third, focus regions are ideally positioned in the tumour, without considering movement trajectory of ultrasound transducer, vasculature distribution and other microenvironment in tumour. Finally, there is a lack of experimental data for direct comparison and validation of our integrated model. Nevertheless, the adopted models have been validated separately by other researchers for HIFU acoustic field (Sheu *et al.* 2011) and HIFU-induced temperature variation (Staruch *et al.* 2011). The macroscopic transport model has been shown to be able to provide qualitative predictions on drug transport and accumulation in solid tumours (Bhandari *et al.* 2017; Zhan and Wang 2018). Given the complexities involved in the drug delivery process, more rigorous validation studies are needed in the future. Since averaged and representative values of model parameters are used in our simulations, the numerical results presented in this study are meant for qualitatively analysis rather than quantitative prediction.

## Conclusions

A multiphysics model has been developed to examine the spatial distribution and temporal variation of TSL drug concentration, temperature upon HIFU heating and drug uptake for a combined HIFU–TSL doxorubicin delivery system. Our results demonstrate the capability of the model to simulate such a complex system, and indicate that this combined drug delivery system is able to achieve a localised treatment with enhanced drug delivery to tumour interior while keeping a relatively low level of drug concentration in normal tissue. The developed modelling framework can serve as a foundation for further comprehensive studies of TSL-mediated drug delivery with HIFU heating.

## Methods

### Mathematical model

Governing equations for interstitial fluid flow

Continuity equation | |

\(\nabla \cdot \varvec{v} = F_{\text{v}} - F_{\text{ly}}\) | (2-1) |

\(F_{\text{v}} = K_{\text{v}} \frac{S}{V}\left[ {p_{\text{v}} - p_{\text{i}} - \sigma \left( {\pi_{\text{v}} - \pi_{\text{i}} } \right)} \right], F_{\text{ly}} = K_{\text{ly}} \frac{{S_{\text{ly}} }}{V}\left[ {p_{\text{i}} - p_{\text{ly}} } \right]\) | (2-2) |

Momentum equation | |

\(\frac{{\partial \left( {\rho \varvec{v}} \right)}}{\partial t} + \nabla \cdot \left( {\rho \varvec{vv}} \right) = - \nabla p_{\text{i}} + \nabla \cdot\varvec{\tau}- \frac{\mu }{\kappa }\varvec{v}\) | (2-3) |

Acoustic model for high intensity focused ultrasound

Acoustic intensity ( | ||

\(I_{\text{a}} = {p_{\text{a}}^{2} } / {2\rho \nu_{\text{a}} }\) | (3-1) | |

Acoustic pressure ( | ||

\(p_{\text{a}} = ik_{\text{ac}} \rho \nu_{\text{a}} \varPsi ,\,\,k_{\text{ac}} = {2\uppi } / {\lambda}\) | (3-2) | |

Acoustic velocity potential ( | ||

\(\varPsi \left( {R,\theta } \right) = \frac{\varvec{u}}{2\uppi }\mathop \smallint \limits_{0}^{2\uppi } \mathop \smallint \limits_{0}^{\text{b}} s^{ - 1} {\text{e}}^{{ - ik_{\text{ac}} s}} R_{1} {\text{d}}R_{1} {\text{d}}\beta ,\) \(\varvec{ u} = u_{0} {\text{e}}^{i\omega t} ,\) \(s = \sqrt {R^{2} - 2Rr_{1} { \sin }\theta { \cos }\beta + zR_{1}^{2} } ,\) \(z=1-2hR\cos \theta /{{b}^{2}},~{{r}_{1}}={{R}_{1}}\sqrt{1-\left( R_{1}^{2}/4{{A}^{2}} \right)}\) | (3-3) |

Bioheat transfer model

Bioheat transfer in tumour and normal tissue | |

\(\rho_{\text{t}} c_{\text{t}} \frac{{\partial T_{\text{t}} }}{\partial t} = k_{\text{t}} \nabla^{2} T_{\text{t}} - \rho_{\text{b}} c_{\text{b}} w_{\text{b}} \left( {T_{\text{t}} - T_{\text{b}} } \right) + 2\alpha_{\text{t}} I_{\text{a}}\) | (4-1) |

Bioheat transfer in blood | |

\(\rho_{\text{b}} c_{\text{b}} \frac{{\partial T_{\text{b}} }}{\partial t} = k_{\text{b}} \nabla^{2} T_{\text{b}} + \rho_{\text{b}} c_{\text{b}} w_{\text{b}} \left( {T_{\text{t}} - T_{\text{b}} } \right) - \rho_{\text{b}} c_{\text{b}} w_{\text{b}} \left( {T_{\text{b}} - T_{\text{n}} } \right) + 2\alpha_{\text{b}} I_{\text{a}}\) | (4-2) |

Pharmacodynamics model

Pharmacodynamics model | |

\(\frac{{{\text d}D_{\text{c}} }}{{\text d}t} = - \frac{{f_{ \hbox{max} } C_{\text{i}} }}{{EC_{50} + C_{\text{i}} }}D_{\text{c}} + k_{\text{p}} D_{\text{c}} - k_{\text{g}} D_{\text{c}}^{2}\) | (5-1) |

#### Interstitial fluid flow

The bulk movement of interstitial fluid in tissue extracellular space and the fluid exchange with systemic blood and lymphatic systems are crucial for drug transport in solid tumour. Given the size of a tumour is typically 2–3 orders of magnitude larger than the inter-capillary distance, tumour and its surrounding normal tissue are treated as rigid porous media, with blood capillaries being represented as a uniformly distributed source term (Baxter and Jain 1989). Table 2 shows the continuity and momentum equations applied to describe the flow of incompressible Newtonian interstitial fluid in a porous medium.

#### HIFU acoustic field

Since acoustic energy radiated from a concave surface tends to be focused near the centre of curvature, such radiators have been used to produce localised heating in clinical applications. The effect of non-linear acoustic wave propagation on the heating of biological tissues by HIFU can be ignored if the applied focal intensity is within the range of 100–1000 W/cm^{2} and the peak negative pressure in the range of 1–4 MPa (Sheu *et al.* 2011; Solovchuk *et al.* 2012). The acoustic field under the aforementioned conditions can be described by the fundamental equations (O’Neil 1949) given in Table 3.

#### Bioheat transfer

Both the tissue and blood are heated by absorbing the acoustic energy induced by HIFU, and heat can be transferred between these two compartments if there is a temperature difference. Because of continuous blood circulation, the heated blood at temperature *T*_{b} will flow out of the heating region while blood at normal temperature *T*_{n} flows in. The corresponding energy conservation equations are summarised in Table 4.

#### Drug transport

Drug transport is determined by drug properties and the *in vivo* environment after administration. The dynamic process of TSL drug delivery includes triggered drug release from liposomes, association/disassociation of free drugs with proteins, plasma clearance, drug circulation within bloodstream, and exchanges among the circulatory system, extracellular space and tumour cells. The full set of mathematical descriptions is given in Table 1.

#### Pharmacodynamics model

Dynamic tumour cell density is determined by drug cytotoxicity, cell proliferation and physiologic degradation, as described in Table 5. In this study, cell proliferation and physiologic degradation are assumed to have reached equilibrium at the start of each treatment.

### Model geometry

The geometry of a prostate tumour is reconstructed from magnetic resonance (MR) images acquired from a patient using a 3.0-Tesla MR scanner (DISCOVERY MR750, GE, Schenectady, New York, USA). Multi-slice anatomical images of the prostate were acquired in three orthogonal planes with echo-planer (EP) sequence, with each image comprising 256 by 256 pixels. In order to build a 2-D model of the tumour and its holding tissue, a representative transverse MR image is selected to cover the maximum tumour dimension, as shown in Fig. 5A. The transverse image is processed using image analysis software MIMICS (Materialise HQ, Leuven, Belgium), and the tumour is segmented from its surrounding normal tissue based on the local signal intensity.

The tumour shown in Fig. 5B is located at the corner of the normal tissue with a dimension of 47 mm (maximum width) by 38 mm (maximum depth) in the 2-D cross-sectional model. Two focus regions are chosen in order to achieve good coverage of the tumour (Mougenot *et al.* 2009). Both regions are elliptical with major and minor axis radii of 8.7 mm and 1.5 mm, respectively. ANSYS ICEM CFD (ANSYS Inc., Canonsburg, USA) is used to generate the computational mesh, which consists of 64,966 triangular elements in total. This is obtained based on mesh independence tests which show that differences in predicted acoustic pressure, temperature and drug concentration between the adopted mesh and a 10-times finer mesh are less than 1%.

### Model parameters

*et al.*2014a, b). Since temperature elevation in response to HIFU heating may influence several properties used in the drug transport model, temperature dependence of the following transport properties is considered: vascular permeability, diffusivity, transmembrane rate, blood perfusion rate and drug release rate from TSL.

Biophysical parameters for tumour and normal tissues

Parameter | Definition | Unit | Tumour tissue | Normal tissue | References |
---|---|---|---|---|---|

| Surface area of blood vessels per unit tissue volume | m | 2.0 × 10 | 7.0 × 10 | |

| Hydraulic conductivity of the micro-vascular wall | m/(Pa·s) | 2.10 × 10 | 2.70 × 10 | |

| Density of interstitial fluid | kg/m | 1000 | 1000 | (Goh |

| Dynamic viscosity of interstitial fluid | kg/(m·s) | 7.8 × 10 | 7.8 × 10 | (Goh |

| Permeability of the interstitial space | m | 4.56 × 10 | 2.21 × 10 | |

| Vascular fluid pressure | Pa | 2080 | 2080 | |

| Osmotic pressure of the plasma | Pa | 2666 | 2666 | |

| Osmotic pressure of interstitial fluid | Pa | 2000 | 1333 | |

| Average osmotic reflection coefficient for plasma proteins | 0.82 | 0.91 | ||

| Hydraulic conductivity of the lymphatic wall times surface area of lymphatic vessels per unit volume of tumour tissue | (Pa·s) | 0 | 4.17 × 10 | (Goh |

| Intra-lymphatic pressure | Pa | 0 | 0 | (Goh |

| Cell density | 10 | 1 × 10 | – | (Eikenberry 2009) |

| Total tumour volume | m | 5 × 10 | 3 × 10 | (El-Kareh and Secomb 2000) |

| Total blood volume in body | m | 5 × 10 | 5 × 10 | (El-Kareh and Secomb 2000) |

Transport parameters for doxorubicin

Parameter | Definition | Unit | Free doxorubicin | Bound doxorubicin | References |
---|---|---|---|---|---|

| Permeability of vasculature wall in tumour | m/s | 3.00 × 10 | 7.80 × 10 | |

| Permeability of vasculature wall in normal tissue | m/s | 3.75 × 10 | 2.50 × 10 | |

| Diffusion coefficient in interstitial fluid of tumour | m | 3.40 × 10 | 8.89 × 10 | (Goh |

| Diffusion coefficient in interstitial fluid of normal tissue | m | 1.58 × 10 | 4.17 × 10 | (Goh |

| Osmotic reflection coefficient | 0.15 | 0.82 | ||

| Doxorubicin-protein binding rate | s | 0.833 | – | (Eikenberry 2009) |

| Doxorubicin-protein dissociation rate | s | – | 0.278 | (Eikenberry 2009) |

| Tumour fraction extracellular space | 0.4 | (Eikenberry 2009) | ||

| Rate of transmembrane transport | kg/10 | 4.67 × 10 | – | |

| Michaelis constant for transmembrane transport | kg/m | 2.19 × 10 | – | (Eikenberry 2009; El-Kareh and Secomb 2000; Kerr |

| Michaelis constant for transmembrane transport | kg/10 | 1.37 × 10 | – | (Eikenberry 2009; El-Kareh and Secomb 2000; Kerr |

| Cell-kill rate constant | s | 1.67 × 10 | – | (Eliaz |

| Drug concentration producing 50% of | kg/10 | 5.00 × 10 | – | (Eliaz |

| Cell proliferation rate | s | 3.00 × 10 | – | (Liu |

| Cell physiologic degradation rate | s | 3.00 × 10 | – | (Liu |

| Plasma clearance in tumour | s | 2.43 × 10 | 0 | (Benet and Zia-Amirhosseini 1995; Robert |

| Plasma clearance in normal tissue | s | 2.43 × 10 | 0 | (Benet and Zia-Amirhosseini 1995; Robert |

Transport parameters for liposome

Parameter | Definition | Unit | Tumour | Normal tissue | References |
---|---|---|---|---|---|

| Liposome permeability of vasculature wall | m/s | 3.42 × 10 | 8.50 × 10 | |

| Liposome diffusion coefficient | m | 9.00 × 10 | 5.80 × 10 | |

| Reflection coefficient for liposome | – | 0.95 | 1.0 | (Zhan |

| Plasma clearance in tumour | s | 2.23 × 10 | 2.23 × 10 | (Gabizon |

Acoustic and thermal properties

Parameter | Definition | Unit | Value | References | ||
---|---|---|---|---|---|---|

Blood | Tumour | Normal tissue | ||||

| Ultrasound speed | m/s | 1540 | 1550 | 1550 | (Sheu |

| Density | kg/m | 1060 | 1000 | 1055 | (Sheu |

| Specific heat | J/(kg·K) | 3770 | 3800 | 3600 | (Sheu |

| Thermal conductivity | W/(m·K) | 0.53 | 0.552 | 0.512 | (Sheu |

| Absorption coefficient | – | 1.5 | 9 | 9 | (Sheu |

| Blood perfusion rate at 37 °C | s | – | 0.002 | 0.018 | (Vaupel |

| Universal gas constant | J/(mol·K) | – | 8.314 | 8.314 | (Schutt and Haemmerich 2008) |

| Activation energy | J/mol | – | 6.67 × 10 | 6.67 × 10 | (Schutt and Haemmerich 2008) |

| Frequency factor | s | – | 1.98 × 10 | 1.98 × 10 | (Schutt and Haemmerich 2008) |

#### HIFU transducer

^{2}and 1.98 MPa, respectively.

HIFU transducer parameters

Inside diameter (mm) | Outside diameter (mm) | Focal length (mm) | Frequency (MHz) |
---|---|---|---|

20.0 | 70.0 | 62.64 | 1.10 |

#### Temperature controller

*et al.*2004). The phase transition temperature of TSL strongly depends on its formulation, and a typical value for TSL doxorubicin is 42 °C based on experimental studies (Tagami

*et al.*2012). In order to achieve the target temperature range, a feedback temperature controller is incorporated, which is designed to adjust the applied ultrasound power according to the maximum temperature in the heated region (

*T*

_{max}). The transducer power is adjusted according to the controller parameter

*P*(

*t*), which is determined by

*k*

_{cp}and

*k*

_{ci}are proportional-integral (PI) control parameters, which are set as 5.0 and 6.0 × 10

^{−3}respectively, based on values in the literature (Staruch

*et al.*2011) and our previous studies.

#### Vascular permeability

*et al.*1995). The logarithmic values of free doxorubicin permeability were found to be proportional to temperature (Dalmark and Storm 1981) as shown in Fig. 13A. The permeability of bound doxorubicin is assumed to follow the same relation. Compared to baseline values at 34 °C, extracellular concentrations of TSL encapsulated doxorubicin increased by 38-fold and 76-fold upon heating to 42 °C and 45 °C, respectively (Gaber

*et al.*1996), based on which the relationship between fold increase in TSL permeability and temperature can be obtained as shown in Fig. 13B.

#### Diffusivity

*D*strands for diffusivity,

*T**is the thermodynamic temperature and

*μ*is the dynamic viscosity of solvent. Subscripts 1 and 2 correspond to conditions 1 and 2, respectively. Owing to the lack of relevant data, dynamic viscosity values in Eq. 2 are assumed to be those of water (Keenan and Keyes 1936), whose dependence on temperature is given in Fig. 13C.

#### Transmembrane rate

*in vitro*experiments (El-Kareh and Secomb 2000; Kerr

*et al.*1986). It has been suggested that increased cellular uptake of doxorubicin with heating can lead to improved outcome when the drug is administrated simultaneously with hyperthermia (El-Kareh and Secomb 2000). Based on data in the literature (Nagaoka

*et al.*1986), there is a 2.2-fold increase at 42 °C. Here, the fold increase (

*k*

_{tv}) at temperature

*T*is obtained by linear interpolation as

#### Blood perfusion rate

*w*) varies with temperature (Schutt and Haemmerich 2008) according to the following relation

*w*

_{0}represents the perfusion rate at 37 °C, and the thermodynamic temperature

*T**is a function of time

*t*.

*R*is the universal gas constant, and

*A*

_{f}and

*ΔE*represent the frequency factor and the activation energy, respectively.

#### Drug release rate from TSL

*et al.*1993). The exact release rate varies according to the composition of liposome, its preparation procedure and heating temperature (Tagami

*et al.*2011). The relation between percentage release and exposure time is found to follow the first-order kinetics expressed as (Afadzi

*et al.*2010)

*%R(t)*is the percentage of drug released at exposure time

*t*,

*R*

_{c}is the total percentage of drug released at a given heating temperature. This equation is used to fit experimental data (Tagami

*et al.*2012) and an example for 37 °C is given in Fig. 13D. Drug release rates at different temperatures in the range of 37–42 °C are summarised in Table 11.

Release rates at various temperatures

| 37.0 | 38.0 | 39.0 | 40.0 | 41.0 | 42.0 |
---|---|---|---|---|---|---|

| 4.17 × 10 | 5.45 × 10 | 1.49 × 10 | 2.82 × 10 | 4.25 × 10 | 5.41 × 10 |

Note that the TSL formulation used in Tagami *et al.*’s study (2012) still allows some drugs to be slowly released at 37 °C, although release at 42 °C is much faster. Linear interpolation is performed to obtain release rate at temperatures between the available temperature points, and the release rate is assumed to be constant if temperature exceeds 42 °C.

### Numerical methods

The mathematical models are implemented into a finite-volume method based computational fluid dynamics code, ANSYS FLUENT (ANSYS Inc., Canonsburg, USA). The user defined scalar function is used to code bioheat transfer model, pharmacokinetics, mass transfer equations for drug transport and pharmacodynamics model. The partial differential equations are spatially discretised by using the 2nd order UPWIND scheme, and the SIMPLEC algorithm is employed for pressure–velocity coupling. The convergence is controlled by setting residual tolerances of the momentum equation, mass transfer equations and bioheat transfer equations to be 1 × 10^{−5}, 1 × 10^{−8} and 1 × 10^{−10}, respectively. Interstitial fluid equations are solved first to obtain a steady-state solution for the entire computational domain. The obtained pressure and velocity fields are adopted as initial values for the simulation of bioheat transfer, drug transport, drug uptake and cell density simultaneously. The 2nd order implicit backward Euler scheme is employed for temporal discretization. A fixed time-step size of 0.1 s is chosen after time-step sensitivity tests.

### Boundary conditions

For a single chemotherapy treatment, the simulated time window is much shorter than that required for tumour growth. Consequently, all boundaries of the tumour and normal tissue are assumed to be fixed. The interface between the tumour and normal tissue is treated as an internal boundary where all variables are continuous. At the outer surface of normal tissue, a relative pressure of 0 Pa and temperature of 37 °C are specified, together with zero flux of drugs.

## Notes

### Compliance with Ethical Standards

### Conflict of interest

Wenbo Zhan, Wladyslaw Gedroyc and Xiao Yun Xu declare that they have no conflict of interest.

### Human and animal rights and informed consent

All imaging data was analysed anonymously and patient information was de-identified prior to this analysis. The anonymized images were saved from the P ACS system by the radiologist who made the image acquisition. Given the archiving of the images for this study was done with an anonymous patient number, the patient could not be identified away from the P ACS system. A prior agreement was made to undertake computational modelling work using totally anonymised images without requiring further specific ethics committee approval for individual patients. Therefore, formal ethical approval was not required for this retrospective study. For this reason, written consent was not obtained from the patient to use the data in this specific study.

## References

- Afadzi M, de L Davies C, Hansen YH, Johansen TF, Standal O-V, Masoy S-E, Angelsen B (2010) Ultrasound stimulated release of liposomal calcein. In: Ultrasonics symposium (IUS), 2010 IEEE. IEEE, 2107–2110Google Scholar
- Baxter LT, Jain RK (1989) Transport of fluid and macromolecules in tumors. I. Role of interstitial pressure and convection. Microvasc Res 37:77–104CrossRefGoogle Scholar
- Baxter LT, Jain RK (1990) Transport of fluid and macromolecules in tumors. II. Role of heterogeneous perfusion and lymphatics. Microvasc Res 40:246–263CrossRefGoogle Scholar
- Baxter LT, Jain RK (1991) Transport of fluid and macromolecules in tumors. III. Role of binding and metabolism. Microvasc Res 41:5–23CrossRefGoogle Scholar
- Benet LZ, Zia-Amirhosseini P (1995) Basic principles of pharmacokinetics. Toxicol Pathol 23:115–123CrossRefGoogle Scholar
- Bhandari A, Bansal A, Singh A, Sinha N (2017) Transport of liposome encapsulated drugs in voxelized computational model of human brain tumors. IEEE Trans Nanobiosci 16:634–644CrossRefGoogle Scholar
- Curiel L, Chavrier F, Gignoux B, Pichardo S, Chesnais S, Chapelon J (2004) Experimental evaluation of lesion prediction modelling in the presence of cavitation bubbles: intended for high-intensity focused ultrasound prostate treatment. Med Biol Eng Comput 42:44–54CrossRefGoogle Scholar
- Cusack BJ, Young SP, Driskell J, Olson RD (1993) Doxorubicin and doxorubicinol pharmacokinetics and tissue concentrations following bolus injection and continuous infusion of doxorubicin in the rabbit. Cancer Chemother Pharmacol 32:53–58CrossRefGoogle Scholar
- Dalmark M, Storm HH (1981) A Fickian diffusion transport process with features of transport catalysis. Doxorubicin transport in human red blood cells. J Gen Physiol 78:349–364CrossRefGoogle Scholar
- Eikenberry S (2009) A tumor cord model for doxorubicin delivery and dose optimization in solid tumors. Theor Biol Med Model 6:16–36CrossRefGoogle Scholar
- Eliaz RE, Nir S, Marty C, Szoka FC (2004) Determination and modeling of kinetics of cancer cell killing by doxorubicin and doxorubicin encapsulated in targeted liposomes. Cancer Res 64:711–718CrossRefGoogle Scholar
- El-Kareh AW, Secomb TW (2000) A mathematical model for comparison of bolus injection, continuous infusion, and liposomal delivery of doxorubicin to tumor cells. Neoplasia 2:325–338CrossRefGoogle Scholar
- El-Kareh AW, Secomb TW (2003) A mathematical model for cisplatin cellular pharmacodynamics. Neoplasia 5:161–169CrossRefGoogle Scholar
- Gaber MH, Hong K, Huang SK, Papahadjopoulos D (1995) Thermosensitive sterically stabilized liposomes: formulation and
*in vitro*studies on mechanism of doxorubicin release by bovine serum and human plasma. Pharm Res 12:1407–1416CrossRefGoogle Scholar - Gaber MH, Wu NZ, Hong K, Huang SK, Dewhirst MW, Papahadjopoulos D (1996) Thermosensitive liposomes: extravasation and release of contents in tumor microvascular networks. Int J Radiat Oncol Biol Phys 36:1177–1187CrossRefGoogle Scholar
- Gabizon A, Catane R, Uziely B, Kaufman B, Safra T, Cohen R, Martin F, Huang A, Barenholz Y (1994) Prolonged circulation time and enhanced accumulation in malignant exudates of doxorubicin encapsulated in polyethylene-glycol coated liposomes. Cancer Res 54:987–992PubMedGoogle Scholar
- Gasselhuber A, Dreher MR, Rattay F, Wood BJ, Haemmerich D (2012a) Comparison of conventional chemotherapy, stealth liposomes and temperature-sensitive liposomes in a mathematical model. PLoS One 7:e47453CrossRefGoogle Scholar
- Gasselhuber A, Dreher MR, Partanen A, Yarmolenko PS, Woods D, Wood BJ, Haemmerich D (2012b) Targeted drug delivery by high intensity focused ultrasound mediated hyperthermia combined with temperature-sensitive liposomes: computational modelling and preliminary
*in vivo*validation. Int J Hyperth 28:337–348CrossRefGoogle Scholar - Goh YM, Kong HL, Wang CH (2001) Simulation of the delivery of doxorubicin to hepatoma. Pharm Res 18:761–770CrossRefGoogle Scholar
- Granath KA, Kvist BE (1967) Molecular weight distribution analysis by gel chromatography on Sephadex. J Chromatogr A 28:69–81CrossRefGoogle Scholar
- Grüll H, Langereis S (2012) Hyperthermia-triggered drug delivery from temperature-sensitive liposomes using MRI-guided high intensity focused ultrasound. J Controll Release 161:317–327CrossRefGoogle Scholar
- Hossann M, Wang T, Wiggenhorn M, Schmidt R, Zengerle A, Winter G, Eibl H, Peller M, Reiser M, Issels RD, Lindner LH (2010) Size of thermosensitive liposomes influences content release. J Controll Release 147:436–443CrossRefGoogle Scholar
- Hynynen K (2011) MRIgHIFU: a tool for image-guided therapeutics. J Magn Reson Imaging 34:482–493CrossRefGoogle Scholar
- Jain RK (1987a) Transport of molecules across tumor vasculature. Cancer Metastasis Rev 6:559–593CrossRefGoogle Scholar
- Jain RK (1987b) Transport of molecules in the tumor interstitium: a review. Cancer Res 47:3039–3051PubMedGoogle Scholar
- Keenan JH, Keyes FG (1936) Thermodynamic properties of steam: including data for the liquid and solid phases. Wiley, New YorkGoogle Scholar
- Kerr DJ, Kerr AM, Freshney RI, Kaye SB (1986) Comparative intracellular uptake of adriamycin and 4′-deoxydoxorubicin by non-small cell lung tumor cells in culture and its relationship to cell survival. Biochem Pharmacol 35:2817–2823CrossRefGoogle Scholar
- Legha SS, Benjamin RS, Mackay B, Ewer M, Wallace S, Valdivieso M, Rasmussen SL, Blumenschein GR, Freireich EJ (1982) Reduction of doxorubicin cardiotoxicity by prolonged continuous intravenous infusion. Ann Intern Med 96:133–139CrossRefGoogle Scholar
- Lindner LH, Eichhorn ME, Eibl H, Teichert N, Schmitt-Sody M, Issels RD, Dellian M (2004) Novel temperature-sensitive liposomes with prolonged circulation time. Clin Cancer Res 10:2168–2178CrossRefGoogle Scholar
- Liu C, Xu XY (2015) A systematic study of temperature sensitive liposomal delivery of doxorubicin using a mathematical model. Comput Biol Med 60:107–116CrossRefGoogle Scholar
- Liu C, Krishnan J, Xu XY (2013) Investigating the effects of ABC transporter-based acquired drug resistance mechanisms at the cellular and tissue scale. Integr Biol 5:555–568CrossRefGoogle Scholar
- Luu KT, Uchizono JA (2005) P-glycoprotein induction and tumor cell-kill dynamics in response to differential doxorubicin dosing strategies: a theoretical pharmacodynamic model. Pharm Res 22:710–715CrossRefGoogle Scholar
- Lyon PC, Gray MD, Mannaris C, Folkes LK, Stratford M, Campo L, Chung DYF, Scott S, Anderson M, Goldin R, Carlisle R, Wu F, Middleton MR, Gleeson FV, Coussios CC (2018) Safety and feasibility of ultrasound-triggered targeted drug delivery of doxorubicin from thermosensitive liposomes in liver tumours (TARDOX): a single-centre, open-label, phase 1 trial. Lancet Oncol 19:1027–1039CrossRefGoogle Scholar
- Mougenot C, Quesson B, de Senneville BD, de Oliveira PL, Sprinkhuizen S, Palussière J, Grenier N, Moonen CT (2009) Three-dimensional spatial and temporal temperature control with MR thermometry-guided focused ultrasound (MRgHIFU). Magn Reson Med 61:603–614CrossRefGoogle Scholar
- Nagaoka S, Kawasaki S, Sasaki K, Nakanishi T (1986) Intracellular uptake, retention and cytotoxic effect of adriamycin combined with hyperthermia
*in vitro*. Jpn J Cancer Res 77:205–211PubMedGoogle Scholar - Nugent LJ, Jain RK (1984) Extravascular diffusion in normal and neoplastic tissues. Cancer Res 44:238–244PubMedGoogle Scholar
- O’Neil H (1949) Theory of focusing radiators. J Acoust Soc Am 21:516–526CrossRefGoogle Scholar
- Raghunathan S, Evans D, Sparks JL (2010) Poroviscoelastic modeling of liver biomechanical response in unconfined compression. Ann Biomed Eng 38:1789–1800CrossRefGoogle Scholar
- Robert J, Illiadis A, Hoerni B, Cano J-P, Durand M, Lagarde C (1982) Pharmacokinetics of adriamycin in patients with breast cancer: correlation between pharmacokinetic parameters and clinical short-term response. Eur J Cancer Clin Oncol 18:739–745CrossRefGoogle Scholar
- Rodvold KA, Rushing DA, Tewksbury DA (1988) Doxorubicin clearance in the obese. J Clin Oncol 6:1321–1327CrossRefGoogle Scholar
- Saltzman WM, Radomsky ML (1991) Drugs released from polymers: diffusion and elimination in brain tissue. Chem Eng Sci 46:2429–2444CrossRefGoogle Scholar
- Schutt DJ, Haemmerich D (2008) Effects of variation in perfusion rates and of perfusion models in computational models of radio frequency tumor ablation. Med Phys 35:3462–3470CrossRefGoogle Scholar
- Sheu TWH, Solovchuk MA, Chen AWJ, Thiriet M (2011) On an acoustics–thermal–fluid coupling model for the prediction of temperature elevation in liver tumor. Int J Heat Mass Transf 54:4117–4126CrossRefGoogle Scholar
- Solovchuk MA, Sheu TW, Lin W-L, Kuo I, Thiriet M (2012) Simulation study on acoustic streaming and convective cooling in blood vessels during a high-intensity focused ultrasound thermal ablation. Int J Heat Mass Transf 55:1261–1270CrossRefGoogle Scholar
- Staruch R, Chopra R, Hynynen K (2011) Localised drug release using MRI-controlled focused ultrasound hyperthermia. Int J Hyperth 27:156–171CrossRefGoogle Scholar
- Swabb EA, Wei J, Gullino PM (1974) Diffusion and convection in normal and neoplastic tissues. Cancer Res 34:2814–2822PubMedGoogle Scholar
- Tagami T, Ernsting MJ, Li SD (2011) Optimization of a novel and improved thermosensitive liposome formulated with DPPC and a Brij surfactant using a robust
*in vitro*system. J Controll Release 154:290–297CrossRefGoogle Scholar - Tagami T, May JP, Ernsting MJ, Li S-D (2012) A thermosensitive liposome prepared with a Cu
^{2+}gradient demonstrates improved pharmacokinetics, drug delivery and antitumor efficacy. J Controll Release 161:142–149CrossRefGoogle Scholar - Teo CS, Hor Keong Tan W, Lee T, Wang C-H (2005) Transient interstitial fluid flow in brain tumors: effect on drug delivery. Chem Eng Sci 60:4803–4821CrossRefGoogle Scholar
- Vaupel P, Kallinowski F, Okunieff P (1989) Blood flow, oxygen and nutrient supply, and metabolic microenvironment of human tumors: a review. Cancer Res 49:6449–6465PubMedGoogle Scholar
- Wolf MB, Watson PD, Scott DR 2nd (1987) Integral-mass balance method for determination of solvent drag reflection coefficient. Am J Physiol 253:H194–H204PubMedGoogle Scholar
- Wu NZ, Da D, Rudoll TL, Needham D, Whorton AR, Dewhirst MW (1993a) Increased microvascular permeability contributes to preferential accumulation of stealth liposomes in tumor tissue. Cancer Res 53:3765–3770PubMedGoogle Scholar
- Wu NZ, Klitzman B, Rosner G, Needham D, Dewhirst MW (1993b) Measurement of material extravasation in microvascular networks using fluorescence video-microscopy. Microvasc Res 46:231–253CrossRefGoogle Scholar
- Yuan F, Leunig M, Huang SK, Berk DA, Papahadjopoulos D, Jain RK (1994) Microvascular permeability and interstitial penetration of sterically stabilized (stealth) liposomes in a human tumor xenograft. Cancer Res 54:3352–3356PubMedGoogle Scholar
- Yuan F, Dellian M, Fukumura D, Leunig M, Berk DA, Torchilin VP, Jain RK (1995) Vascular permeability in a human tumor xenograft: molecular size dependence and cutoff size. Cancer Res 55:3752–3756PubMedGoogle Scholar
- Zhan W, Wang C-H (2018) Convection enhanced delivery of liposome encapsulated doxorubicin for brain tumour therapy. J Controll Release 285:212–229CrossRefGoogle Scholar
- Zhan W, Gedroyc W, Xu XY (2014a) Effect of heterogeneous microvasculature distribution on drug delivery to solid tumour. J Phys D 47:475401CrossRefGoogle Scholar
- Zhan W, Gedroyc W, Yun XuX (2014b) Mathematical modelling of drug transport and uptake in a realistic model of solid tumour. Protein Pept Lett 21:1146–1156CrossRefGoogle Scholar
- Zhang A, Mi X, Yang G, Xu LX (2009) Numerical study of thermally targeted liposomal drug delivery in tumor. J Heat Transf 131:043209CrossRefGoogle Scholar
- Zhao J, Salmon H, Sarntinoranont M (2007) Effect of heterogeneous vasculature on interstitial transport within a solid tumor. Microvasc Res 73:224–236CrossRefGoogle Scholar
- Zou Y, Yamagishi M, Horikoshi I, Ueno M, Gu X, Perez-Soler R (1993) Enhanced therapeutic effect against liver W256 carcinosarcoma with temperature-sensitive liposomal adriamycin administered into the hepatic artery. Cancer Res 53:3046–3051PubMedGoogle Scholar

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