Effects of pulsating heat source on interstitial fluid transport in tumour tissues

Abstract

Macromolecules and drug delivery to solid tumours is strongly influenced by fluid flow through interstitium, and pressure-induced tissue deformations can have a role in this. Recently, it has been shown that temperature-induced tissue deformation can influence interstitial fluid velocity and pressure fields, too. In this paper, the effect of modulating-heat strategies to influence interstitial fluid transport in tissues is analysed. The whole tumour tissue is modelled as a deformable porous material, where the solid phase is made up of the extracellular matrix and cells, while the fluid phase is the interstitial fluid that moves through the solid matrix driven by the fluid pressure gradient and vascular capillaries that are modelled as a uniformly interspersed fluid point-source. Pulsating-heat generation is modelled with a time-variable cosine function starting from a direct current approach to solve the voltage equation, for different pulsations. From the steady-state solution, a step-variation of vascular pressure included in the model equation as a mass source term via the Starling equation is simulated. Dimensionless 1D radial equations are numerically solved with a finite-element scheme. Results are presented in terms of temperature, volumetric strain, pressure and velocity profiles under different conditions. It is shown that a modulating-heat procedure influences velocity fields, that might have a consequence in terms of mass transport for macromolecules or drug delivery.

1. Introduction

Phenomena like macromolecule transport in applications like drug delivery are strongly influenced by interstitial flow fields. In this context, understanding how they are modulated under different stimuli is of crucial importance. It is now widely accepted that interstitial fluid pressure (IFP) is high and relatively uniform within solid tumour masses and suddenly drops at the tumour periphery [1,2]. This implies that any given bloodborne macromolecular drug has difficulties in reaching the centre of a tumour and this negatively affects the therapeutic efficacy of the drug. Possible strategies to overcome this limitation include transient modulation of the vascular pressure within the tumour vasculature [3–5] (table 1).

Table 1.

Nomenclature.

Roman letters
cp	heat capacity, J kg−1 K−1
E, e	strain tensor and volumetric strain
I	identity matrix
k	thermal conductivity, W m−1 K−1
K	hydraulic conductivity, m2 Pa−1 s−1
Lp	hydraulic permeability, m Pa−1 s−1
pv, p	vascular and hydraulic pressure, Pa
Q	heat generation, W m−3
r	radial coordinate, m
S/V	specific surface area, 1 m−1
S	stress tensor, Pa
t	time, s
T	temperature, K
u, u	displacement tensor, vector and scalar, m
v, v	velocity vector and scalar, m s−1
x	vascular pressure step
Greek letters
α	thermal expansion coefficient, 1 K−1
αdiff	thermal diffusivity, m2 s−1
ϕ	porosity
λ, μ	Lamé parameters, Pa
Ω	Starling source term, 1 s−1
ρ	density, kg m−3
ω	dimensionless pulsation
subscripts
eff	effective
hyp	hyperthermia
gen	generation
met	metabolic
ref	reference
rel	relative
dimensionless numbers
$\beta =R\sqrt{(L_p/K)(S/V)}$	interstitial to transcapillar resistance
$\gamma =\text{Fo}/t^{\ast }={\alpha }_{\mathrm{diff}}/K(2\mu +\lambda )$	thermal to poroelastic time number
$\mathrm{\Delta }{T}^{\ast }=\mathrm{\Delta }T/T_{\mathrm{ref}}$	temperature
$\text{Fo}={\alpha }_{\mathrm{diff}}t/R^2$	heating time (Fourier number)
$\text{Gl}=(3\lambda +2\mu )\alpha T_{\mathrm{rif}}/(2\mu \hspace{0.17em}+\hspace{0.17em}\lambda )$	thermal to mechanical stresses (Gay-Lussac number)
$p^{\ast }=p/(2\mu +\lambda )$	pressure
$p_v^{\ast }=p_v/(2\mu +\lambda )$	vascular pressure
$\text{P}{\text{e}}_{\mathrm{rel}}=\mathrm{R}{v}_{\mathrm{rel}}/{\alpha }_{\mathrm{diff}}$	heat convection number (Peclet number)
$\text{P}{\text{o}}_{\mathrm{hyp}}=Q_{\mathrm{hyp}}(r^{\ast })R^2/T_{\mathrm{ref}}{k}_{\mathrm{eff}}$	hyperthermia heat generation (Pomerantsev number)
$\text{P}{\text{o}}_{\mathrm{gen}}=Q_{\mathrm{gen}}{R}^2/T_{\mathrm{reif}}{k}_{\mathrm{eff}}$	total heat generation (Pomerantsev number)
$r^{\ast }=r/R$	radius
$t^{\ast }=t/[R^2/K(2\mu +\lambda )]$	poroelastic time
$u^{\ast }=u/R$	displacement
$v^{\ast }=vR/[K(2\mu +\lambda )]$	velocity

In order to describe fluid transport in a tumour, various models have been considered through the years. One approach consists of modelling the whole tumour as a deformable medium in which two phases can be identified, i.e. a deformable porous medium [5–7]. Both fluid flow and mechanical deformation are mutually related, and if a steady-state equilibrium situation is considered then this equilibrium can be perturbed by various disturbances like for example interruption of tumour blood flow (TBF) or an increase/decrease of the vascular pressure [5]. Besides the modulation of vascular pressure, temperature modulation might also play a role since it can induce transient tissue deformations. The theory behind the study of fluid flow, thermo-mechanical deformations and heat transfer problem is the thermoporoelasticity theory [8], which couples everything by considering also matrix thermal deformation. In human tissues and by considering various applications, thermal effects can strongly influence various phenomena. In microwave thermal ablation, Keangin et al. [9] showed that the accuracy of predictions increases if thermal deformations are included in the model. For laser angioplasty, temperature enhancement due to hyperthermia conditions can enhance low-density lipoprotein accumulation in the arterial wall due to both thermal deformations [10] and Soret effect [10,11]. Finally, for interstitial fluid flow transport in a thermoporoelasticity model under tumour blood flow arrest conditions, Andreozzi et al. [12] showed that interstitial flow and pressure profiles are influenced by applied heating because of matrix thermal dilation.

Modulating-heat procedures have been proposed for various fields in bioengineering applications. Generally speaking, they are useful to applications in which temperature peaks need to be avoided or for other purposes. Such procedures have been used by Sluijter [13,14] for spinal pain treatment, showing that neural damage can be avoided because of the lower temperature achieved during the process. Cohen & Foester [15] achieved complete pain relief in three patients with groin pain and orchialgia by employing a pulsating heat procedure. For tumour ablation, Goldberg et al. [16] performed experiments with RF ablation pulsating procedures in animals both ex vivo and in vivo. They concluded that if a variable peak current procedure is used then larger tumours can be treated. Fukushima et al. [17] employed either a controlled-temperature procedure or an impedance-controlled procedure to treat hepatocellular carcinoma, showing that controlled-impedance procedures provide bigger ablation zones with smaller ablation times. For microwave ablation, Bedoya et al. [18] performed ex vivo and in vivo experiments on both bovine and porcine livers, respectively. They concluded that it is possible to achieve larger ablation zones at lower average power with a pulsed protocol.

When the tumour heat flux is modulated in time, matrix thermal dilation and all other related variables are changing with time. The objective of this paper is to analyse if a modulating heat procedure in a tumour tissue can have a positive impact on interstitial fluid velocities and pressure via thermal dilation. A transient effect is included in terms of increase/decrease of vascular pressure. After setting the coupled equations under thermoporoelasticity theory and solving the equations with a numerical scheme, different cases are analysed in order to appreciate if and for what conditions a pulsating heat procedure customizes velocity and pressure profiles for applications like macromolecules diffusion or drug delivery.

2. Mathematical model

2.1. Governing equations

The whole tumour is here modelled as a spherical porous medium that consists of two phases [5,12]. The fluid phase is the interstitial fluid that moves together with its solutes, while the solid phase is all the rest that completes the tissue here considered, i.e. the cellular volume with capillaries and so on [5,12]. It is assumed that tissue variations with time are negligible because that timescales considered are much smaller than those. For the solid phase, the intercapillary distance is much smaller than the characteristic length of the investigated domain, thus capillaries are considered to be point mass sources and no mutual interactions are present. For the fluid phase, the interstitium is a fluid-saturated homogeneous poroelastic medium. With this approach, effects of microscale variables will be considered as closure terms for governing equations in order to appreciate features like geometry and tortuosity of capillaries [19].

Governing equations here employed are basically similar to Andreozzi et al. [12], where details can be found. With reference to a representative elementary volume (REV) of the porous medium, one can write mass equations for fluid and solid phases, under the assumption of negligible lymphatic drainage [5,12]

$$\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }t}=-\mathrm{\nabla }\cdot (\varphi \mathbf{v})$$ 2.1

and

$$\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }t}=\mathrm{\nabla }\cdot \left[(1-\varphi )\frac{\mathrm{\partial }\mathbf{u}}{\mathrm{\partial }t}\right]-\mathit{\Omega }(r,t).$$ 2.2

By making equations (2.1) and (2.2) equal, and by noting that the fluid source term is a function of the average transmural pressure $\mathit{\Omega }(r,t)=L_p(S/V)(\hspace{0.17em}{p}_v-p)$ also known as the Starling or transcapillary term, the mass equation for both phases is derived as follows

$$\mathrm{\nabla }\cdot \left[\varphi \mathbf{v}+(1-\varphi )\frac{\mathrm{\partial }\mathbf{u}}{\mathrm{\partial }t}\right]=L_p\frac{S}{V}(\hspace{0.17em}{p}_v-p).$$ 2.3

The momentum equation for the fluid phase in the deformable porous medium is Darcy's law

$$\varphi \left(\mathbf{v}-\frac{\mathrm{\partial }\mathbf{u}}{\mathrm{\partial }t}\right)=-K\mathrm{\nabla }p.$$ 2.4

For the solid phase, the momentum equation is written by assuming negligible inertial forces, which is reasonable for the problem investigated here [12,20–22]

$$\mathrm{\nabla }\cdot \mathit{S}=\text{0, }$$ 2.5

where the stress tensor, which respectively considers deformation, thermal and porous contributions, is equal to [12]

$$\mathit{S}=(2\mu \mathbf{E}+\lambda e\mathbf{I})-\alpha \mathrm{\Delta }T(2\mu +3\lambda )\mathbf{I}-p\mathbf{I}.$$ 2.6

It is reminded that in this paper a Terzaghi poroelastic model that considers also thermal effects is here used, with Biot modulus and coefficient respectively equal to infinite and one. In other words, it is assumed that both the phases are intrinsically incompressible. Recently, based on multiscale and micromechanical studies, it has been shown that compressibility effects might arise [23,24] because of lower Poisson ratios, which can vary from 0.35 to 0.49 for biological tissues. Besides, it is widely known that a lower Poisson ratio produces a smaller Biot coefficient [19]. However, for rat brain μ = 684 mm Hg and λ = 15.2 mm Hg [25], obtaining a Poisson coefficient of 0.49, for which incompressibility assumption is reliable.

For the infinitesimal strain theory we know that $e=tr(\mathbf{E})=\mathrm{\nabla }\cdot \mathbf{u}$ and $\mathbf{E}=1/2\hspace{0.17em}(\mathrm{\nabla }\mathbf{u}\mathbf{+}\mathrm{\nabla }{\mathbf{u}}^T),$ and after some manipulations and under the assumption that radial normal stress vanishes on the boundary of the sphere [12], the final form can be derived in order to obtain the following set of differential equations in 1D spherical coordinates (r is the radial coordinate) for volumetric strain e, pressure p, temperature difference ΔT and displacement u

$$\begin{array}{rl}& \frac{\mathrm{\partial }e}{\mathrm{\partial }t}-K(2\mu +\lambda )\frac{1}{r^2}\frac{\mathrm{\partial }}{\mathrm{\partial }r}\left(r^2\frac{\mathrm{\partial }e}{\mathrm{\partial }r}\right)+L_p\frac{S}{V}(2\mu +\lambda )\\ e& =L_p\frac{S}{V}[\hspace{0.17em}{p}_v+(3\lambda +2\mu )\alpha \mathrm{\Delta }T]+-K(3\lambda +2\mu )\alpha \frac{1}{r^2}\frac{\mathrm{\partial }}{\mathrm{\partial }r}\left(r^2\frac{\mathrm{\partial }\mathrm{\Delta }T}{\mathrm{\partial }r}\right),\end{array}$$ 2.7

$$p=(2\mu +\lambda )e-(3\lambda +2\mu )\alpha \mathrm{\Delta }T$$ 2.8

$$\text{and}e=\frac{\mathrm{\partial }u}{\mathrm{\partial }r}+2\frac{u}{r}.$$ 2.9

Energy equation needs to be solved in order to quantify thermal dilation and so on. Under the assumption of local thermal equilibrium (LTE) between the two phases, the energy equation is derived from an energy balance in spherical coordinates

$$\begin{array}{rl}& {(\rho c_p)}_{\mathrm{eff}}\left[\frac{\mathrm{\partial }T}{\mathrm{\partial }t}+\left(v-\frac{\mathrm{\partial }u}{\mathrm{\partial }t}\right)\frac{\mathrm{\partial }T}{\mathrm{\partial }r}\right]=\frac{1}{r^2}\frac{\mathrm{\partial }}{\mathrm{\partial }r}\left(k_{\mathrm{eff}}{r}^2\frac{\mathrm{\partial }T}{\mathrm{\partial }r}\right)\\ & +Q_{\mathrm{met}}(1-\varphi )+Q_{\mathrm{hyp}}(r,t)-{(\rho c_p)}_{\mathrm{eff}}{L}_p\frac{S}{V}(\hspace{0.17em}{p}_v-p)(T-T_{\mathrm{ref}}).\end{array}$$ 2.10

In this equation, the last three terms on the right are respectively metabolic heat generation (that depends on porosity ϕ), hyperthermia heating generation term and the heat transfer due to the Starling mass source term previously introduced. By following [12], the heat generation term might be induced by an antenna placed in the centre of the tumour region, as for cancer ablation. If radiofrequencies are employed, a DC (direct current) approach is used, and it is possible to assume that $Q_{\mathrm{hyp}}(r,t)\propto r^{-4}$ [12], where details about functions employed will be explained later, together with heat generation term variation due to pulsating heat effect. Finally, it is noticed that the last term on the right side of equation (2.10) has negligible effects on temperature fields.

Equations (2.2), (2.7)–(2.10) here are scaled as in Andreozzi et al. [12], in order to obtain the final dimensionless form

$$\varphi \left(v^{\ast }-\frac{\mathrm{\partial }{u}^{\ast }}{\mathrm{\partial }{t}^{\ast }}\right)=-\frac{\mathrm{\partial }{p}^{\ast }}{\mathrm{\partial }{r}^{\ast }},$$ 2.11

$$\frac{\mathrm{\partial }e}{\mathrm{\partial }{t}^{\ast }}-\frac{1}{r^{\ast 2}}\frac{\mathrm{\partial }}{\mathrm{\partial }{r}^{\ast }}\left(r^{\ast 2}\frac{\mathrm{\partial }e}{\mathrm{\partial }{r}^{\ast }}\right)={\beta }^2(\hspace{0.17em}{p}_v^{\ast }+\text{Gl}\mathrm{\Delta }{T}^{\ast }-e)-\frac{1}{r^{\ast 2}}\frac{\mathrm{\partial }}{\mathrm{\partial }{r}^{\ast }}\left[r^{\ast 2}\frac{\mathrm{\partial }(\text{Gl}\mathrm{\Delta }{T}^{\ast })}{\mathrm{\partial }{r}^{\ast }}\right],$$ 2.12

$$p^{\ast }=e-\text{Gl}\mathrm{\Delta }{T}^{\ast },$$ 2.13

$$\frac{\mathrm{\partial }{u}^{\ast }}{\mathrm{\partial }{r}^{\ast }}=e-2\frac{u^{\ast }}{r^{\ast }}$$ 2.14

and

$$\frac{\mathrm{\partial }\mathrm{\Delta }{T}^{\ast }}{\mathrm{\partial }{t}^{\ast }}\left[\frac{1}{\gamma }\right]+\text{P}{\text{e}}_{\mathrm{rel}}\frac{\mathrm{\partial }\mathrm{\Delta }{T}^{\ast }}{\mathrm{\partial }{r}^{\ast }}=\frac{1}{r^{\ast 2}}\frac{\mathrm{\partial }}{\mathrm{\partial }{r}^{\ast }}\left[r^{\ast 2}\frac{\mathrm{\partial }\mathrm{\Delta }{T}^{\ast }}{\mathrm{\partial }{r}^{\ast }}\right]+\text{P}{\text{o}}_{\mathrm{gen}}(r^{\ast },t^{\ast })+\frac{{\beta }^2}{\gamma }(\hspace{0.17em}{p}_v^{\ast }-e+\text{Gl}\mathrm{\Delta }{T}^{\ast })\mathrm{\Delta }{T}^{\ast }$$ 2.15

in which the vascular pressure $p_v^{\ast }$ is multiplied by a constant x with the following definition

$$x=\frac{\hspace{0.17em}{p}_v^{\ast }(t^{\ast }=1)}{\hspace{0.17em}{p}_v^{\ast }(t^{\ast }=0)},$$ 2.16

where $p_v^{\ast }(t^{\ast }=1)$ refers to the vascular pressure in the new final steady condition, while $p_v^{\ast }(t^{\ast }=0)$ is the vascular pressure at the beginning of the transient here analysed. The constant x takes into account the step increase/decrease of vascular pressure for the transient.

Governing equations (2.11)–(2.15) can be solved in the coupled unknowns $v^{\ast },\hspace{0.17em}e,\hspace{0.17em}{p}^{\ast },\hspace{0.17em}{u}^{\ast },\hspace{0.17em}\mathrm{\Delta }{T}^{\ast };$ in particular, $e=f\hspace{0.17em}(t^{\ast },$ $r^{\ast },\hspace{0.17em}\beta ,p_v^{\ast },\hspace{0.17em}\text{Gl}\mathrm{\Delta }{T}^{\ast })$ and $\mathrm{\Delta }{T}^{\ast }=f\hspace{0.17em}(t^{\ast },\hspace{0.17em}\gamma ,\hspace{0.17em}\text{P}{\text{e}}_{\mathrm{rel}},\hspace{0.17em}{r}^{\ast },\hspace{0.17em}\text{P}{\text{o}}_{\mathrm{gen}}(r^{\ast },\hspace{0.17em}{t}^{\ast })).$ The porosity is a function of volumetric strain in terms of ϕ = (ϕ₀ + e)/(1 + e), with ϕ₀ = 0.20 [26]. Finally, it is reasonable to assume that metabolic heat generation is negligible with respect to induced hyperthermia [12], so $\text{P}{\text{o}}_{\mathrm{gen}}(r^{\ast },\hspace{0.17em}{t}^{\ast })\approx \text{P}{\text{o}}_{\mathrm{hyp}}(r^{\ast },\hspace{0.17em}{t}^{\ast })$ and $\mathrm{\Delta }{T}^{\ast }=f\hspace{0.17em}(t^{\ast },\hspace{0.17em}\gamma ,\hspace{0.17em}\text{P}{\text{e}}_{\mathrm{rel}},\hspace{0.17em}{r}^{\ast },$ $\text{P}{\text{o}}_{\mathrm{hyp}}(r^{\ast },t^{\ast })).$

2.2. Modulating-heat procedure modelling

Space and time dependence of the heat source $\text{P}{\text{o}}_{\mathrm{hyp}}(r^{\ast },t^{\ast })$ is described as follows. In particular, the final expression becomes

$$\text{P}{\text{o}}_{\mathrm{hyp}}(r^{\ast },t^{\ast })=\text{P}{\text{o}}_{\mathrm{hyp}}(r^{\ast }=0)f(r^{\ast })f(t^{\ast }),$$ 2.17

where the value Po_hyp, which refers to the heat generation source value at the centre of the sphere $(r^{\ast }=0),$ is multiplied by a function of dimensionless radius and time. With reference to the radial dependence $f(r^{\ast }),$ as in [12] this function needs to be equal to 1 for $r^{\ast }=0$ and equal to 0 for $r^{\ast }=1,$ since it is assumed that the maximum heat generation is at the centre, while the heat generation approaches zero at the boundary of the domain. The fourth-grade hyperbolic function $f(r^{\ast }),$ that has to guarantee $f\hspace{0.17em}(r^{\ast }=0)=1$ and $f\hspace{0.17em}(r^{\ast }=1)=0$ is

$$f(r^{\ast })=\frac{1}{{(r-r_0)}^4}+f_0,$$ 2.18

where r₀ = −0.9845 and f₀ = −0.0645 [12]. Since modulating heat effects are analysed, the hyperthermia heat generation varies also through the time by following the function $f(t^{\ast }).$ It is assumed that heat is modulated by means of a pulsating function. By assuming that a harmonic function with zero-phase is used, the function employed to describe the variation with time is:

$$f(t^{\ast })=\frac{1}{2}+\frac{1}{2}\cos (\omega t^{\ast })$$ 2.19

with ω dimensionless pulsation. The harmonic function $f(t^{\ast })$ employed is presented in figure 1, for different pulsations and radial position (figure 1b). It is shown that the higher the pulsation, the lower the period, then heat is delivered in a shorter time. Besides, the function is always equal to 1 for $t^{\ast }=0.$

Figure 1.

(a) Pulsating functions for different values of ω and (b) dimensionless hyperthermia Po_hyp for different $r^{\ast }$ and $t^{\ast }.$

Pulsating functions employed here are chosen with a certain criterion. It is important to choose pulsations/periods that are of the same order of magnitude as the phenomena investigated. In this paper, values of ω = 0.1, 500, 2500 and 1×10⁴ are investigated. With properties taken from Netti et al. [5], these values roughly correspond to periods of 2123717 s, 425 s, 85 s and 21 s, respectively. It is important to guarantee that the heat pulsation ω has the same order of magnitude as the characteristic evolution times of the physical phenomena involved here. By employing data from Netti et al. [5] and Andreozzi et al. [12], vascular characteristic time, percolation characteristic time and thermal characteristic time are of the order of magnitude of 20 s, 3500 s and 70 s. This means that vascular time is comparable with the ω = 1×10⁴ case analysed here, as they have the same order of magnitude. Thus, for the vascular pressure enhance/decrease, the time needed to reach steady state equilibrium condition for the transport problem is comparable to the characteristic times for pulsating heating. In figure 2, pressure versus time referred to the analytical solution of Netti et al. [5] is compared with pulsatile function employed here. It is clear that the heat delivered has a pulsation of the same order of magnitude scale required to reach the new steady state pressure.

Figure 2.

Pressure and pulsating-heat functions for different $t^{\ast }.$

2.3. Boundary conditions and numerical modelling

Details on boundary conditions herein employed are in Andreozzi et al. [12] and these are briefly summarized in the following. An axial symmetry condition is invoked on the symmetry axis. On the boundaries, it is assumed that $\mathrm{\Delta }{T}^{\ast }=0$ for $r^{\ast }=1.$ Reliability of this boundary condition has been already discussed in Andreozzi et al. [12]. For the strains, a stress-free boundary condition is generally used, thus the whole tissue is insulated from the surrounding tissue [5,26]. Noting that $u^{\ast }$ approaches to zero at the boundary, with μ ≪ λ [27,28], one can conclude that for $r^{\ast }=1$

$$e=\text{Gl}\mathrm{\Delta }{T}^{\ast }$$ 2.20

that of course becomes zero because of the assumption made on temperature at the boundary.

The initial condition is derived by solving the steady-state problem. For this case, no heat generation is assumed, thus all the tissue is assumed to be at uniform 37°C temperature [12]. After obtaining the steady state condition, a perturbation is applied to the system by following equation (2.16). This means that x ≠ 1 only for $t^{\ast }>0$ in order to simulate the transient effect of vascular pressure step increase/decrease. In particular, for x < 1 there is a sudden vascular pressure drop, while for x > 1 vascular pressure increases.

Finally, scaled governing equations (2.11)–(2.15) are solved with a finite-element scheme. High order (quintic) Lagrange polynomials have been employed in the finite-element scheme. A free variable time stepping between 10⁻⁵ and 0.1 was used. Time step convergence was checked, while RMS absolute tolerance referred to each time step was set to 10⁻⁴. A 300 element grid that follows a symmetric arithmetic sequence with element ratio of 200 was used in order to appreciate gradients.

3. Comparisons with literature data

The solution derived here was validated in a previous study for a TBF arrest case [12], and it is here compared with a blood vascular pressure increase/decrease case. As per the authors' knowledge, no data for the non-isothermal case are available, thus comparisons have been performed separately for the isothermal increase/decrease of vascular pressure case, and for well-established solutions for temperature fields. In particular, the isothermal solution is an analytical solution from Netti et al. [5]. This analytical solution has been already experimentally validated in Netti et al. [5] with ex vivo experiments on human colon adenocarcinoma LS174T transplanted in athymic mice for vascular pressure step change, which is the case analysed here, instantaneous arrest of tumour blood flow and abrupt cessation of tumour blood flow. With reference to the vascular pressure step change, the authors found that their analytical solution matched the evolution of experimental infusion pressure with time [5]. For the temperature field, the well-established analytical solution from Jaeger & Carslaw [29] is employed for comparisons since it has been found to be the closest validating case available in the literature. This solution refers to a conductive sphere with a uniform heat generation. Comparisons with the latter case have been done since a low Peclet number makes heat conduction dominant on advection (see Andreozzi et al. [12]).

Comparisons with Netti et al. [5] and Jaeger & Carslaw [29] are shown in figure 3 for pressure, velocity and temperature profiles. In all the cases, very good agreement has been found.

Figure 3.

Comparisons between numerical solution and analytical solution from literature.

4. Results and discussion

The objective of this work was to analyse the effects of modulated tissue heating on interstitial tumour fluid redistribution. Sensitivity analyses are done for different dimensionless numbers in order to appreciate which effects are relevant. Temperature distribution for various Fourier numbers are presented in figure 4. These are presented for different values of pulsation ω, with γ = 40 and $\text{P}{\text{o}}_{\mathrm{hyp}}(r^{\ast }=0)=5.$ Case with ω = 0.1 (figure 4a) refers to a quasi-steady state regime. In particular, it is shown that the higher the Fo, the higher the temperatures because the steady state is reached at about Fo = 0.4. This Fo value corresponds to $t^{\ast }=0.05,$ if γ = 40, since $t^{\ast }=\text{Fo}/\gamma .$ By comparing different values of ω (figure 4b–d), it is shown that the temperature trend versus the Fo number is not monotone because of the pulsating heat effect. With reference to temperature peaks reached, these are avoided especially for ω = 2500 and ω = 1×10⁴ (figure 4c,d). Indeed, there is a difference on the overall maximum temperature between the case with ω = 0.1 and ω = 1×10⁴ of about 50%.

Figure 4.

Dimensionless temperature difference for different hyperthermia conditions and Fo = 0, 0.02, 0.4, 2, 4 and 10 with $\text{P}{\text{o}}_{\mathrm{hyp}}(r^{\ast }=0)=5$ and γ = 40: (a) ω = 0.1, (b) ω = 500, (c) ω = 2500 and (d) ω = 1×10⁴.

Pulsation ω effects on thermal dilation are presented in figure 5 with γ = 40, $\text{P}{\text{o}}_{\mathrm{hyp}}(r^{\ast }=0)=3,$ Gl = 0.05 and β = 50. Two different pulsations, ω = 0.1 (quasi-state case) and ω = 1×10⁴ (time-phase case), are analysed under the condition of vascular pressure increase (x = 2) and decrease (x = 0.5). Times $t^{\ast }$ are referred to 0, 0.0002, 0.0005, 0.001, 0.01 and 0.05. With reference to the vascular pressure increase cases (figure 5a,b), it is shown that strain increases with different times $t^{\ast }.$ In particular, the strain increase in the proximity of the sphere centre is due to the fact that there is heat generation which contributes to the matrix dilation. The situation becomes slightly different for the vascular pressure decrease cases presented in figure 5c,d. First of all, the trend with time $t^{\ast }$ is not monotonic at all. For the first part of the transients, i.e. from $t^{\ast }=0$ and $t^{\ast }=0.001,$ there is a slight decrease in volumetric strain. However, strain decrease in the proximity of $r^{\ast }=0$ is balanced by matrix thermal dilation, which has the effect of increasing volumetric strain. In other words, competition between matrix contraction due to vascular pressure decrease and expansion due to applied heat is found. For $t^{\ast }=0.01$ and 0.05, volumetric strain increases in both cases because the thermal dilation effect overcomes the vascular pressure reduction. This effect is more remarkable for ω = 0.1 (figure 5c) than for ω = 1×10⁴ (figure 5d). Finally, it has to be noticed that in all cases strain is the same for $t^{\ast }=0.01$ and 0.05 because both transvascular and thermal problems reach a new equilibrium condition.

Figure 5.

Volumetric strain with Gl = 0.05, γ = 40, β = 50 and $\text{P}{\text{o}}_{\mathrm{hyp}}(r^{\ast }=0)=3,$ for $t^{\ast }=0$ (solid-black), 0.0002 (dotted-black), 0.0005 (solid-cyan), 0.001 (dotted-cyan), 0.01 (solid-red) and 0.05 (dotted-red): (a) ω = 0.1 and x = 2, (b) ω = 1×10⁴ and x = 2, (c) ω = 0.1 and x = 0.5 and (d) ω = 1×10⁴ and x = 0.5.

Fluid pressure and velocity as a function of dimensionless radius $r^{\ast }$ is presented in figure 6, for β = 50, Gl = 0.05, x = 0.5 and $\text{P}{\text{o}}_{\mathrm{hyp}}(r^{\ast }=0)=3.$ In particular, the effects of γ, that is the ratio between percolation and timescales [12], are highlighted together with effects of ω. With reference to the cases with low values of γ (figure 6a,c), it is shown that different ω does not provide differences in terms of pressure and velocity profile. This because if the constant γ is very low, then thermal effects on the pressure profile are not remarkable since heat diffusion is too small compared to transcapillary exchange typical time, making heat transfer effects almost negligible with timescales typical of pressure variations. Indeed, in energy equation (2.15), γ has the same meaning as volumetric heat capacity, thus for very small γ the heat capacity is very big and the transient too long to be compared with typical timescales of the present work. However, it is expected that thermal effects are relevant after a certain $t^{\ast },$ thus the results obtained would only be similar to higher γ cases but with higher times. This aspect will be analysed later. By analysing pressure and velocity changes for these cases, it is shown that pressure decreases because of the step function x, reaching a new equilibrium condition. On the other hand, velocity increases from the steady state value at $t^{\ast }=0.0001,$ obtaining positive values after a certain time, which means a change in flow direction. The situation becomes different for higher values of γ (say, 150), in which thermal effects have a role. With reference to the case with γ = 150, in the quasi-initial solution $(t^{\ast }=0.0001),$ it is possible to see that pressure drops faster in closer proximity to the centre $(r^{\ast }=0)$ for both values of ω. This because heat propagation starts, and timescales are not as small as for γ = 0.1. The matrix thermal dilation helps pressure to drop faster. At $t^{\ast }=0.005,$ pressure value equals vascular pressure value, as for lower values of γ. This means that heat has the role of changing transient evolution between $p_v^{\ast }$ and $x{p}_v^{\ast }.$ Similar conclusions can be done for velocity profiles. With reference to pulsating heat effects, in figure 6b it is shown that pulsating heat changes pressure drop with time. For $t^{\ast }=0.001,$ pressure for ω = 1×10⁴ is higher than for ω = 0.1; on the other hand, the steady condition referred to $t^{\ast }=0.005$ is not still reached for ω = 1×10⁴, probably because the periodic pulsed heat changes times required to reach the steady condition of $p^{\ast }=x{p}_v^{\ast }.$ For velocity profiles (figure 6d), it is shown that velocities for ω = 1×10⁴ at $t^{\ast }=0.005$ become higher than for ω = 0.1 because of the aforementioned effect of pulsing heat.

Figure 6.

Dimensionless pressure and velocity with Gl = 0.05, β = 50, x = 0.5 and $\text{P}{\text{o}}_{\mathrm{hyp}}(r^{\ast }=0)=3$: (a,c) γ = 0.1, (b,d) γ = 150.

In figure 7, the effects of the variable β on pressure and velocity are shown. Values of γ = 40, $\text{P}{\text{o}}_{\mathrm{hyp}}(r^{\ast }=0)=3,$ x = 2.0 and Gl = 0.05 are set. The variable β is the ratio between interstitial and transcapillary resistance, and proportional to the ratio between interstitial and transcapillary time constants [5]. With reference to the figures, it is shown that higher values of β always present higher pressures and velocities, (figure 7b,d). On the other hand, lower values make pressure (relative) and velocities negative (figure 7a,c). This because there is less flow percolation inside the interstitium in the tumour. In other words, for higher values of β transcapillary resistance is smaller so the vascular (Starling) term dominates transport. Negative values of pressure might be achieved due to heat application that dominates over poroelastic deformation, and this effect is emphasized with the time. Similar results have been found in Andreozzi et al. [12] for the tumour blood flow arrest case. With reference to applied heat, it is shown that small differences can be achieved for high β cases in figure 7b,d. First of all, pressure and velocity increase and decrease with time in all the cases, respectively. However, for modulated heat (ω = 1×10⁴), pressure might be almost equal for $t^{\ast }=0.001$ and slightly higher for $t^{\ast }=0.005$ compared to the heating stationary case at ω = 0.1. This pulsating effect is not as emphasized as for lower β (figure 7a,c). Indeed, in such cases there is more difference if different ω are compared. This because transcapillary forces are smaller, thus thermal deformation have a more important role which changes pressure and velocities. In particular it is clear that pulsating heat makes pressure higher and velocity higher, i.e. closer to zero in all cases. Finally, it has to be noticed that scale lengths for very small β are typically longer since β is also the ratio between interstitial and transvascular timescales [5], thus a new steady state condition would be achieved for higher values of $t^{\ast }.$ However, this aspect will be discussed later.

Figure 7.

Dimensionless pressure and velocity with Gl = 0.05, γ = 40, x = 2.0 and $\text{P}{\text{o}}_{\mathrm{hyp}}(r^{\ast }=0)=3$: (a,c) β = 0.0001, (b,d) β = 50.

In order to appreciate transient evolution, strain and pressure versus time for $r^{\ast }=0$ are shown in figure 8 for different values of γ and β, for $\text{P}{\text{o}}_{\mathrm{hyp}}(r^{\ast }=0)=3,$ x = 2.0 and Gl = 0.05, and various pulsations, say ω = 0.1, 10 and 50. It is reminded that γ and β represent timescales ratios [5,12]. With reference to volumetric strains, it is possible to see that for low γ the volumetric strain (figure 8a) slowly reaches an asymptotic value that is slightly higher than about two times initial strain over long durations, since the thermal timescale becomes very long due to the small value of γ. This slight difference with two times the initial strain is caused by the matrix dilation. For low values of β (figure 8b), maximum volumetric strain is reached in shorter times because γ is higher. With reference to pulsation effects, one can remark that strong strain oscillations can be found for ω = 10, while shorter amplitudes are evident for higher ω = 50, which might reach an asymptotic condition for very high ω. This asymptotic effect might be evident everywhere, especially for low γ because of the higher thermal effect due to timescales. If one assumes very high ω values, it seems that there is no time for the physical system to absorb and release heat due to the very small heating periods. In all cases, the higher strain values are achieved for smaller ω, which roughly refer to steady-state cases. Pressure versus time is presented in figure 8c,d for the same γ and β values. For small γ (figure 8c), it is shown that a steady condition is immediately reached, with a pressure roughly equal to two times the initial pressure because heating effects are very small compared to the transcapillar effect, obtaining a profile that is roughly equal to an isothermal case for all pulsations investigated. With reference to pressure evaluation with small β (figure 8d), it seems that all the pressure curves have an asymptotic value for ω = 0.1 (figure 8d), which is the quasi-steady value. For a fixed $t^{\ast },$ maximum and minimum pressure values are achieved for ω = 50, because the heating damping effect is not the same as for the quasi-steady case. Besides, one can remark that pressure rapidly drops even if x = 2 (figure 7), then it rises up because of the combined effect of vascular pressure increase and heating effects. In particular, the latter is more evident with higher ω because of the heat source variation with time.

Figure 8.

Volumetric strain and dimensionless pressure versus time with Gl = 0.05, x = 2.0 and $\text{P}{\text{o}}_{\mathrm{hyp}}(r^{\ast }=0)=3$: (a,c) β = 50 and γ = 0.1, (b,d) β = 0.0001 and γ = 150.

Effects of heat generation $\text{P}{\text{o}}_{\mathrm{hyp}}(r^{\ast }=0)$ are presented in figure 9 to underline the importance of applied heat. Figures are presented for γ = 40, β = 50 and Gl = 0.05. It is shown that for low values of $\text{P}{\text{o}}_{\mathrm{hyp}}(r^{\ast }=0),$ results do not change that much between ω = 0.1 and ω = 1×10⁴ because heat does not have an effect on flow fields. On the other hand, for higher values of $\text{P}{\text{o}}_{\mathrm{hyp}}(r^{\ast }=0),$ the differences are remarkable. First of all, for ω = 0.1, one can notice a small increase of $p^{\ast }$ along $r^{\ast }$ axis until $r^{\ast }$ about 0.9 for growing values of $t^{\ast }.$ This because the matrix thermal dilation effect tends to reduce pressure to maintain force balance. With reference to ω = 1×10⁴, modulating heat seems to change profiles; in particular, for $t^{\ast }=0.001,$ pressure is almost the same, while for $t^{\ast }=0.005$ pressure becomes slightly higher for ω = 1×10⁴. For velocity profiles presented in figure 9d, it is shown that velocities are smaller for $t^{\ast }=0.005$ with ω = 1×10⁴. This means that modulating heat makes velocity to become zero in a faster way.

Figure 9.

Dimensionless pressure and velocity with Gl = 0.05, γ = 40, x = 2.0 and β = 50: (a,c) $\text{P}{\text{o}}_{\mathrm{hyp}}(r^{\ast }=0)=1,$ (b,d) $\text{P}{\text{o}}_{\mathrm{hyp}}(r^{\ast }=0)=5.$

Finally, a comparison between various pulsations in terms of pressure and strain evolution versus time is presented for x = 2.0 in figure 10, with Gl = 0.05, $\text{P}{\text{o}}_{\mathrm{hyp}}(r^{\ast }=0)=3,$ γ = 40 and β = 50. In particular, volumetric strain is presented in figure 10a,b, while pressure is shown in figure 10c,d; figure 10b and d are respectively an enlargement of figure 10a and c. Volumetric strains always increase with time because of thermal deformation. Maximum values are achieved for ω = 0.1 because of the quasi-steady condition. When pulsating heat is introduced, volumetric strain is generally smaller because of the lower temperature achieved, and it generally follows the pulsating wave. Volumetric strain oscillations are due to the fact that both pressure and heating are deforming the tissue. In particular, this reduction is more evident for ω = 500 because of the longer temperature decay period. Pressure trends are presented in figure 10c,d. The pulsating effect here is less evident than for volumetric strain, and pressures achieved are more or less the same. However, higher pulsations cause more oscillations in terms of pressure, even if they are small (figure 10d).

Figure 10.

(a,b) Volumetric strain, (c,d) dimensionless pressure versus time with Gl = 0.05, x = 2.0, γ = 40, β = 50 and $\text{P}{\text{o}}_{\mathrm{hyp}}(r^{\ast }=0)=3.$

Volumetric strain and pressure as a function of time, for different pulsations ω, with Gl = 0.05, $\text{P}{\text{o}}_{\mathrm{hyp}}(r^{\ast }=0)=3,$ γ = 40 and β = 50 and x = 0.5, are represented in figure 11. Again, figure 11b and d are respectively enlarged pictures of figure 11a and c. Because of vascular pressure instantaneous decrease, strain and pressure tend to reduce. In particular, for an isothermal case they should reach half of both volumetric strain and pressure at $t^{\ast }=0.$ However, volumetric strains are higher than half of the volumetric strain at $t^{\ast }=0$ because of the heating effect, which makes the tissue dilated and causes strain (figure 11a,b). With reference to the pressure, heating makes pressure just a little bit higher than this half. In other words, heat has a damping effect on strain/pressure decrease. Effects of pulsation are also analysed. With reference to the strain (figure 11a,b), it is shown that ω = 0.1 provides the highest values of strain compared to the other pulsations, because temperatures are higher. Pulsations cause more fluctuations in the strain, and for very high values of ω strains are almost monotonically increasing after about $t^{\ast }=0.001,$ because the frequency is as high as to make temperature not so high. With reference to pressure evolution (figure 11c,d), the pulsating effect is not as marked as for volumetric strain. However, modulating heat still causes pressure oscillations that follows the heat wave.

Figure 11.

(a,b) Volumetric strain, (c,d) dimensionless pressure versus time with Gl = 0.05, x = 0.5, γ = 40, β = 50 and $\text{P}{\text{o}}_{\mathrm{hyp}}(r^{\ast }=0)=3.$

Finally, in order to appreciate the importance of modulating heat in terms of physical units, an example by employing variables generally taken after Andreozzi et al. [12] is shown in the following. Results are shown in terms of temperature, pressure and velocity profiles for different times in figure 12, with Gl = 0.05, x = 0.5, β = 50, $\text{P}{\text{o}}_{\mathrm{hyp}}(r^{\ast }=0)=3$ and γ respectively equal to 0.1 and 150. Values of Gl, x, β and $\text{P}{\text{o}}_{\mathrm{hyp}}(r^{\ast }=0)$ are taken from variables after Andreozzi et al. [12]. For the variable γ, these values have been chosen in order to underline the effects of thermal characteristic times, with no relevant variation remarkable after γ = 40. From the figures, it is shown that maximum temperature peaks of 60–70°C can be achieved. With reference to the various criteria employed for thermal damage, it is widely known that necrosis is a combination between temperature achieved and application times, thus pulsating heat could be useful to avoid necrosis even if large temperature gradients are induced. By comparing cases with γ = 0.1 and γ = 150, one can observe that the former refers to an isothermal case for the times observed since heating propagation times are much higher than percolation time. Indeed, for the cases with γ = 0.1 (figure 12a–c), no relevant differences can be noticed. The situation changes for γ = 150, thus for the case in which heating time becomes more relevant, as shown in figure 12d. It is also noticed in this figure that modulated heat makes it possible to avoid temperature peaks, thus for the times observed here temperature peaks drop from about 65°C to about 55°C. With reference to pressure, one can notice that for t = 3.39 s pressure at the core of the tumour changes from 9 mm Hg (figure 12b) to 8 mm Hg (figure 12e) due to heating effects. The same occurs also for other times, especially for t = 33.89 s. When t = 169.46, results between figure 12b and e become similar because a new equilibrium state is reached for the pressure. By comparing steady heating with modulated heating, one can notice differences for t = 33.89 s, where pulsating heat makes pressure higher at equal time. Finally, pressure variations also have an effect on velocity, as evident in figure 12c,f. For t = 3.39 s, there is a slight change in the proximity of the tumour centre, making the velocity modulus slightly lower for the isothermal case in figure 12c. Differences can be also observed for t = 33.89 s, where the velocity modulus is generally higher for the isothermal case, while if the steady heating case (ω = 0.1) is compared with the modulated heating case (ω = 1×10⁴) as in figure 12f, one can observe that modulated heating makes the velocity modulus higher at equal time.

Figure 12.

Temperature, pressure and velocity along tumour radius with Gl = 0.05, β = 50, x = 0.5 and $\text{P}{\text{o}}_{\mathrm{hyp}}(r^{\ast }=0)=3$: (a–c) γ = 0.1, (d–f) γ = 150.

5. Conclusion

Modulating heat effects on induced hyperthermia applied on tumour tissue have been analysed here. The aim was to ascertain if modulating tissue heating can influence hydrodynamic features of interstitial fluid (pressure and velocity), which play a crucial role in macromolecule transport and drug delivery. The problem is modelled by assuming a spherical tumour made up of a fluid phase (interstitial fluid) and a solid phase (tissue) with the vascular network assumed to be distributed fluid-point sources. Coupled thermoporoelasticity scaled governing equations are written to take into account fluid flow, deformation and heat transfer, which all happen at the same time. Modulated heat is modulated by employing a time-pulsatile function, in which the period is of the same order of magnitude as the typical time-evolution of physics involved. A transient condition of sudden increase/decrease in vascular pressure is simulated.

Results have been presented in terms of temperature fields, deformations, flow and velocity profiles, for different dimensionless constants defined through the paper. It has been shown that temperature peaks are avoided if heat is modulated with high pulsations. When vascular pressure is suddenly increased, volumetric strain deformations are higher with reference to an isothermal case; on the other hand, their reduction during a sudden reduction in vascular pressure are dumped by hyperthermia, especially for lower pulsations. With reference to the pressure and velocity profiles achieved, it has been shown that hyperthermia generally does not affect profiles at the beginning and at the end of the transient, i.e. for $t^{\ast }=0$ and very high $t^{\ast }.$ However, it can change transient evolution between these times by making velocity and pressure higher with respect to the radial coordinate in the tumour. With reference to pulsatile heating, this could enhance this transient-changing effect. It has been also showed that very low values of dimensionless parameter β could make pressure (relative) negative, and velocities much smaller than for higher β values. With reference to transient effects, it has been shown that timescales are strongly affected by dimensionless parameters, and that pulsating heat can help pressure modulation through time. Different pulsations have been compared in order to emphasize that modulating heat could vary pressure evolution versus time by following heating wave propagation. Finally, an example of variables presented in dimensional units is shown in order to quantify differences caused by modulated heating. The insights of this paper could be useful to optimize a modulated-heat procedure to optimize advantages due to hyperthermia for velocity and pressure fields referred to interstitial fluid for macromolecules transport and drug delivery.

Data accessibility

Programs and codes used in this research are the commercial software Comsol Multiphysics and Matlab. All data needed are available upon request from the authors.

Authors' contributions

A.A. helped with simulations and general conversations about the research. M.I. conducted the simulations, designed and produced the figures. P.A.N. supervised the research and took part in general conversations about it. All authors conceived the study and wrote the manuscript, and they gave final approval for publication.

Competing interests

We declare we have no competing interests.

Funding

We received no funding for this study.