The interplay of blood flow and temperature in regional hyperthermia: a mathematical approach — TEI-tagged view

Jesús J. BosquepersName, Gabriel F. CalvopersName, Víctor M. Pérez-GarcíapersName, María Cruz NavarropersName (2021date). “The interplay of blood flow and temperature in regional hyperthermia: a mathematical approach.” Royal Society Open Sciencetitle·j.

Object of study — a lumped three-compartment model of temperatureterm·dom and perfusionterm·key in brain tumoursterm·key under regional hyperthermiaterm·key

Research question — how temperatureterm·dom and perfusionterm·key in a brain tumourterm·key evolve during and after regional hyperthermiaterm·key

Contribution — model evidence that well-perfused brain tumoursterm·key reach therapeutic temperaturesterm·dom only if their vasculatureterm·key is highly disrupted

Full text — tagged

The interplay of blood flowterm·dom and temperatureterm·dom in regional hyperthermiaterm·key: a mathematical approach Abstract In recent decades, hyperthermiaterm·key has been used to raise oxygenationterm·key levels in tumours undergoing other therapeutic modalities, of which radiotherapyterm·key is the most prominent one. It has been hypothesized that oxygenationterm·key increases would come from improved blood flowterm·dom associated with vasodilation. However, no test has determined whether this is a relevant assumption or other mechanisms might be acting. Additionally, since hyperthermiaterm·key and radiotherapyterm·key are not usually co-administered, the crucial question arises as to how temperatureterm·dom and perfusionterm·key in tumours will change during and after hyperthermiaterm·key. Overall, it would seem necessary to find a research framework that clarifies the current knowledge, delimits the scope of the different effects and guides future research. Here, we propose a simple mathematical modelterm·key to account for temperatureterm·dom and perfusionterm·key dynamics in brain tumoursterm·key subjected to regional hyperthermiaterm·key. Our results indicate that tumours in well-perfused organs like the brain might only reach therapeutic temperaturesterm·dom if their vasculatureterm·key is highly disrupted. Furthermore, the characteristic times of return to normal temperatureterm·dom levels are markedly shorter than those required to deliver adjuvant radiotherapyterm·key. According to this, a mechanistic coupling of perfusionterm·key and temperatureterm·dom would not explain any major oxygenationterm·key boost in brain tumoursterm·key immediately after hyperthermiaterm·key.

Keywords: hyperthermiaterm·key, perfusionterm·key, blood flowterm·dom, mathematical modelterm·key Introduction Hyperthermiaterm·key treatment (HT) in cancer has been regarded as an adjuvant treatment to improve the efficacy of other traditional therapies, most often radiation therapy [ 1 , 2 ]. A number of preclinical studies have been carried out showing the benefits of using radiotherapyterm·key on target tissues that also receive thermal doses [ 3 ]. In addition, various randomized clinical trials on different tumours have shown partial/total remissions [ 4 , 5 ]. This radio-sensitization is attributed to several biological mechanisms and interactions [ 6 ], such as direct cell killing by heat [ 7 ], DNA repair inhibition [ 8 ], short-term [ 9 ] and long-term improved oxygenationterm·key [ 10 , 11 ], protein denaturation [ 12 ], preferential vascular damage [ 13 ] or other effects like immune system activation [ 14 ]. Despite the long history of this therapeutic modality, currently there is no conclusive data making it possible to determine which of these HT-induced mechanisms has the most substantial impact on treatment response.

Blood flowterm·dom is widely recognized as being influenced by heat and plays a twofold role in hyperthermiaterm·key. Firstly, it is a key factor in heat dissipation within a tissue [ 15 ]. Secondly, it has the biological function of delivering nutrients and oxygen, the latter being of great importance when hyperthermiaterm·key is used in combination with radiotherapyterm·key, as oxygenated tissues display a better response to radiotherapyterm·key [ 16 ]. Blood flowterm·dom acts as a convective sink for heat in tissues and thus it may greatly affect the temperatureterm·dom of tumour regions during HT. In addition, it has been frequently reported that a local higher temperatureterm·dom leads to an increase in the blood flowterm·dom irrigating the tissue [ 17 ]. Thus, temperatureterm·dom and blood flowterm·dom are two mutually interacting quantities that have to be considered in a unified way. Hence, increases in perfusionterm·key, prompted by higher temperaturesterm·dom, are expected to improve oxygenationterm·key and, consequently, the response to radiation therapy. However, in the clinic, hyperthermiaterm·key and radiotherapyterm·key are not administered simultaneously [ 18 ]; depending on the specific institution they are sequentially applied in a different order and with varying time intervals. Therefore, the important and practical problem emerges of determining how temperatureterm·dom and perfusionterm·key will evolve both during and after treatment, according to their mutual mechanistic association, and also how an elevation in perfusionterm·key could impact on a sustained enhanced oxygenationterm·key after hyperthermiaterm·key is delivered.

Here, we address the use of regional hyperthermiaterm·key in brain tumoursterm·key. Malignancies of the central nervous system are rare, representing about 1.5 % of new cancer diagnoses in 2020, but they make up over 3 % of all cancer-related deaths [ 19 ], and are a significant cause of morbidity and mortality in children and young adults [ 20 ]. Due to the challenges that exist in treating these malignancies, particularly in high-grade gliomas, the possibility of taking advantage of elevated temperaturesterm·dom to improve the outcome of standard therapies is frequently used in the clinic. For instance, previous clinical trials with 79 randomized glioblastoma patients showed a significant benefit, both in time to progression and overall survival, for the group receiving hyperthermiaterm·key plus radiotherapyterm·key when compared with the group that only received radiotherapyterm·key [ 21 ]. In that study, heat was administered via an invasive intracranial device consisting of interstitial antennas, both before and after the application of brachytherapy. Currently, a number of different and less invasive procedures are employed in the clinic [ 22 ], but there is still great interest in how heat can improve the outcomes of brain tumourterm·key patients.

Mathematical modelsterm·key have been proposed as a way to gain insight into the hallmarks of cancer and the therapeutic approaches [ 23 , 24 ]. Specifically, in hyperthermiaterm·key, the process of heating tissues and the interaction with other therapies has received attention from the mathematical standpoint [ 25 , 26 ]. Mechanistic models, with varying levels of complexity, have been developed aimed at calculating temperaturesterm·dom [ 27 ] and, less frequently, to quantify the relevance of the biological effects intervening during treatment [ 28 ]. In this study, we put forward a lumped parameter mathematical modelterm·key based on energy conservation laws that captures the essential features of the interplay between temperatureterm·dom and blood perfusionterm·key in heated tumour tissue. We focus our study on the coupling between both variables, and evaluate their dynamics during and after treatment in brain tumoursterm·key with the goal of identifying limitations in current procedures as well as pinpointing possible room for improvement in the clinical setting.

Methods Model formulation To model the temperatureterm·dom evolution in a tumour subjected to regional hyperthermiaterm·key and the variations in the blood flowterm·dom in response to the administration of heat, we use the law of conservation of energy, accounting for three separate macroscopic compartments which exchange heat through conduction and convection ( figure 1 a , c ): tumour tissue (no subscript for variables), healthy tissue surrounding the tumour (subscript s ) and blood feeding both types of tissues (subscript b ). Each compartment will be assigned to a single representative average value of the temperatureterm·dom ( T , T s , T b [°C]) and the compartments for the tumour and healthy tissues will be characterized by a single value for the baseline blood perfusionterm·key within each tissue ( w , w b [kg s −1 m −3 ], though usually reported as ml of blood per 100 g of tissue per minute).

Graphical representation of the mathematical modelterm·key for temperatureterm·dom and perfusionterm·key evolution during hyperthermiaterm·key treatment. ( a ) The human silhouette shows the distinct body regions related to each compartment. The tumour is considered to be located within the brain (for instance, a glioma), and corresponds to the first compartment, while the healthy brain parenchyma (surrounding tissue), is the second. The blood feeding both the brain and the tumour, which circulates through the rest of the body, exchanging heat with it and with the environment, constitutes the third compartment. ( b ) The response of the compartments to heat will be highly dependent on the functionality of their vasculatureterm·key. Malignant tumours are characterized by aberrant blood vessels and different tumours will respond disparately to heat according to the functionality of their vasculatureterm·key, which translates into starkly dissimilar behaviour over time of the associated perfusionterm·key w . ( c ) The mathematical modelterm·key given by equations (2.1) incorporates the energy relations represented and aims at providing the temperatureterm·dom and blood flowterm·dom variation in the three compartments during heat administration and subsequently. ( d ) Linear dependence between blood flowterm·dom and temperatureterm·dom for different tissues considered in our mathematical modelterm·key. The colour code is consistent with the subsequent figures. The blue line represents the healthy tissue whereas the pink and purple lines represent two tumours with distinct vasculatureterm·key functionality status. For impaired vasculatureterm·key, the uncertainty associated with the tumour perfusionterm·key increases, while its heat response capacity (the slope of the line) is lower.

The first compartment (tumour) has a characteristic density ρ [kg m −3 ] and a heat capacity c [J °C −1 kg −1 ]. It receives energy from the external source, which induces an internal effective power density P [W m −3 ], and by its own metabolic heat production Q [W m −3 ]. The tumour exchanges energy with the blood entering through the arteries with a temperatureterm·dom T b [°C] at a rate w [kg s −1 m −3 ]. This represents a contribution of wc b T b [W m −3 ], where c b [J °C −1 kg −1 ] is the heat capacity of the blood. We follow Pennes’ assumption that blood leaves tissue through the veins at the tissue temperatureterm·dom T [ 29 ], explained by the large heat exchange surface between tumour and blood in the vessel network. Even though Pennes’ framework has been criticized (e.g. [ 30 ]), it is important to note that our assumptions do not hold locally at every point of the tissue, as his equation does, but rather it comprises all possible aggregate macroscopic contributions for the tissue as an energy rate balance for a control volume.

The second compartment, corresponding to the healthy tissue surrounding the tumour (mass density ρ s , heat capacity c s ), shares the same modelling scheme as the previous one, being heated by an internal metabolic rate of Q s [W m −3 ] and by an external radio-frequency source that produces an effective internal power density P s [W m −3 ] that could be different from that of the tumour tissue. Healthy tissue exchanges heat with blood, yielding a term w s c b ( T b − T s ) [W m −3 ] for the gross energy gain. The first and second compartments would also exchange heat, by thermal conduction, both with each other and with the environment, and diffusive terms could be included to account for this process. However, those contributions are neglected here as previous works have shown that perfusionterm·key is the main process in heat transfer within the brain and conduction is only important in a limited region of only a few millimetres [ 31 , 32 ]. Furthermore, since the healthy tissue is part of an internal organ, the brain, all the losses to the environment are accounted for via the blood that carries the heat to the skin. The mathematical description of other organs with a larger interface with the environment, as would be the case of the breast, may require those contributions to be accounted for.

The third compartment (blood) receives the heat that the other two have exchanged according to the amount of flow reaching the first and second compartments proportionally to their respective volumes, V and V s . The total mass affected by the heat exchange in this compartment is the total body mass m T [kg] minus the mass of the first and second compartments already considered, ρV and ρ s V s , respectively. The heat capacity in this case is the effective one for the body c T [J °C −1 kg −1 ]. This compartment is also fed by the metabolic body heat MET, excluding what has already been attributed to the other compartments, QV and Q s V s [W]. Lastly, this compartment accounts for the overall losses of thermal energy to the environment, which, according to Newton’s Law, depends on the difference between its average temperatureterm·dom and that of the environment T E [°C], the surface of the body A [m 2 ] and an effective heat transfer coefficient by the skin h [W m −2 °C −1 ].

The aforementioned relationships among the three different compartments in a brain tumourterm·key are illustrated in figure 1 c . These considerations are expressed by the following system of three ordinary differential equationsterm·mth (ODEsterm·mth) for the temperaturesterm·dom in each compartment System (2.1) represents a lumped parameter model which provides spatially averaged temperatureterm·dom dynamics in the three main compartments considered.

Blood perfusionterm·key The perfusionterm·key terms w , w s play an essential role in equations (2.1). They are assumed to display a moderate dependence on temperatureterm·dom, thus making 2.1 a nonlinear ODEterm·mth system. Moriyama studied the changes in cerebral blood flowterm·dom (CBF) in monkeys exposed to interstitial microwave hyperthermiaterm·key [ 33 ]. The author found that in those animals the CBF increased with temperatureterm·dom at a rate of 10% for each degree increase in temperatureterm·dom. Normal flow levels were restored after concluding heat administration. Rosomoff studied the variation in CBF during hypothermia in dogs and found a proportional decrease in temperatureterm·dom variation of around 7% per degree [ 34 ]. Overall, these studies suggest that the variation in CBF is proportional to the temperatureterm·dom change and that the functional dependence of that change is somewhat similar whether the variation is upward or downward. Based on those studies, we propose a change in perfusionterm·key rates proportional to the temperatureterm·dom with a linear variation of the characteristic basal perfusionterm·key w 0 (in units [kg s −1 m −3 ]) of the form and where T eq and T seq refer to the homeostatic equilibrium values of the temperatureterm·dom in the tumour and its surrounding tissue, respectively, whereas the parameters γ and χ (in units [°C −1 ]) are the proportionality constants of these linear dependencies. Furthermore, an additional parameter ζterm·mth has been introduced to reflect the vascular impairment status within the tumour region, which is expected to display, on average, a different net blood flowterm·dom in comparison to the surrounding healthy tissue. The functional relation of blood flowterm·dom values to temperatureterm·dom is graphically illustrated in figure 1 d for three tissues: the healthy brain and two abnormal tissues with disparate degrees of impairment and fluctuations. Such fluctuations incorporate temporal stochastic components into the blood flowterm·dom of tumours, which are often characterized by a tortuous vasculatureterm·key that cannot continuously satisfy the requirements imposed by the cells [ 35 ].

Vascular network model In order to provide biological insight into the role of the phenomenological parameter ζterm·mth in our model, we will consider two scenarios ( figure 2 ), each involving a simple vascular network: one functional and the other dysfunctional crossing a tumour region. Both networks will be assumed to be tree-like, sufficiently large and of similar size. Consequently, an equal reference pressure difference Δ P , corresponding to the difference between a generic feeding artery and a draining vein, can be established so that the comparison in performance of the two networks is meaningful. Moreover, each parent arterial vessel will bifurcate into two daughter branches down to the capillary levels which will subsequently merge to form venous vessels. The number of arterial and venous levels, when either a branching or a merging takes place, will be N . Although the two networks considered will be relatively symmetric, we allow for the possibility of changes both in the radius r k ( j ) and length l k ( j ) of each vessel at level k , with k = 0, 1, …, N − 1 for arterial vessels, and k = N , N + 1, …, 2 N − 1 for venous vessels. At each level k , the index j will run over the ranges j = 1, 2, …, 2 k and j = 1, 2, …, 2 2 N −1− k for arterial and venous vessels, respectively. The blood flowterm·dom through each network will be approximated as an ideal laminar flow obeying the Hagen–Poiseuille Law [ 36 ], where the flow rate Q k ( j ) at each vessel, labelled by integers ( k , j ), satisfies Q k ( j ) = Δ P k ( j ) / Z k ( j ) , where Δ P k ( j ) is the corresponding pressure drop and Z k ( j ) the resistance, given by Z k ( j ) = 8 μ l k ( j ) / π ( r k ( j ) ) 4 , where μ is the dynamic viscosity of the fluid.

( a ) Tree-like vascular network in which parent arterial vessels bifurcate into two daughter branches down to the capillary levels which then merge to form venous vessels. The networks are constructed starting from the capillary levels with characteristic radii and lengths of r N − 1 ( c ) = 5 μ m and l N − 1 ( c ) = 75 μ m and recursively applying well-known scaling laws for the previous levels. At each level, stochastic changes are allowed for both the radii r k ( j ) and lengths l k ( j ) of each vessel ( k , j ), the variability being higher for dysfunctional (tumour) vasculaturesterm·key. ( b ) Histograms for the total resistances (see the text) Z tot ( nor ) and Z tot ( tum ) in functional (normal) and a dysfunctional (tumour) vasculaturesterm·key with N = 6 arterial levels (same number for venous levels). The pressure difference Δ P between some reference input arterial vessel and an output venous vessel in both networks is the same. The mean value of Z tot ( nor ) is 2.547 × 10 14 Pa s m −3 , with a standard deviation of 0.091 × 10 14 Pa s m −3 , while the mean value for Z tot ( tum ) is 3.744 × 10 14 Pa s m −3 , with a standard deviation of 0.862 × 10 14 Pa s m −3 .

If in the networks described we count over all possible paths along the 2 N vessel levels, each path being associated with a total pressure drop Δ P , we can write where the first and second terms come from adding up the ongoing (arterial) and the return (venous) trees, respectively. Next, we impose conservation of blood flowterm·dom at each junction point, i.e. Q k ( j ) = Q k + 1 ( 2 j − 1 ) + Q k + 1 ( 2 j ) and Q k ( 2 j − 1 ) + Q k ( 2 j ) = Q k + 1 ( j ) , at arterial and venous sections, respectively. If we further assume that at each bifurcation of the network, either at the arterial or venous sections, Q k + 1 ( 2 j − 1 ) = Q k + 1 ( 2 j ) holds, then it follows that Q k ( j ) = Q 0 ( 1 ) / 2 k , for k = 0, 1, …, N − 1 and Q k ( j ) = Q 2 N − 1 ( 1 ) / 2 2 N − 1 − k , for k = N …, 2 N − 1, respectively. Here, Q 0 ( 1 ) and Q 0 ( 2 N − 1 ) denote the input and output flow rates in the network, and since the network is lossless, they are both equal, say, to Q 0 . Hence, expression ( 2.3 ) can be approximated by where Z tot is the total resistance of the network.

Given the morphometric data of a specific vascular network, Z tot can be evaluated using the expression between square brackets in ( 2.4 ). However, for sufficiently large networks, the number of levels arising implies that the factors 2 k and 2 2 N −1− k will be much greater than one. Hence, we may approximate the summation terms, involving the index j , by integrals and thus write where f k ( r , l ) and f 2 N −1− k ( r , l ) denote the joint distribution functions of radii and lengths for arterial and venous vessels at levels k and 2 N − 1 − k , respectively, whereas r k ( min ) , r k ( max ) , l k ( min ) and l k ( max ) represent the minimum and maximum radii and lengths of the vascular network at vessel level k . These distribution functions account for the fact that, in general, the radii and lengths of the vessels can be regarded as random variables fluctuating around average geometric values obeying intraspecific scaling laws of the vasculatureterm·key [ 36 ].

The characteristic hierarchy found in normal vasculatureterm·key, which is evenly spaced and displays well-differentiated arteries, arterioles, capillaries, venules and veins, becomes highly disorganized in tumour vessel networks [ 37 ]. In general, it is observed that the relation between radii and lengths in successive levels can be considered to strongly fluctuate with respect to the average geometric values encountered in the normal case. This implies that the corresponding joint distribution functions in ( 2.4 ) will show relatively larger deviations in the aberrant than in the normal vasculatureterm·key. Thus, the total resistances Z tot ( nor ) and Z tot ( tum ) in normal and tumour vasculaturesterm·key will be significantly different, with Z tot ( tum ) expected to be greater than Z tot ( nor ) . Therefore, if Q 0 ( nor ) and Q 0 ( tum ) are the flow rates in normal and tumour vascular networks, and since by assumption both networks have an equal pressure difference Δ P , the following relation will hold for the impairment parameter ζterm·mth In figure 2 b , we show an example of two histograms for Z tot ( nor ) and Z tot ( tum ) calculated for two stochastic networks of similar size and equal pressure difference Δ P . It is apparent that ζ < 1 in most of the resulting outcomes.

Heat loss In order to quantify heat loss to the environment, we assume that its variation is a function of blood temperatureterm·dom, which in the present model will act as a surrogate for the body’s internal temperatureterm·dom. We consider two ways of losing energy to the environment: one by evaporative cooling (via sweating) and the second by convection on the skin [ 38 ]. Variation of sweat rate is taken to be as in [ 39 ] where we have neglected the contribution of temperatureterm·dom variation in the skin.

Finally, we model the variation of heat loss by convection on the skin. When the body’s internal temperatureterm·dom increases, it reacts by diverting blood flowterm·dom to the skin, which leads to vessel dilation through hypothalamic signalling. We account for this effect by varying the heat transfer coefficient h according to the blood temperatureterm·dom T b , which can be thought of as a surrogate for core temperatureterm·dom. By resorting to the fit for the relative blood flowterm·dom in the skin, given in [ 38 ], which used data from [ 40 ], we assume that the heat transfer coefficient varies depending on the blood flowterm·dom temperatureterm·dom, i.e.

where κ and ν are constants given in table 1 .

Variables and parameters of the mathematical modelterm·key.

Implementation and numerical solution Our model equations were implemented using the scientific software environment Matlab (R2018b, MathWorks) and run on a 16 GB memory 3.2 GHz iMac machine. The code is available for public access on the websites specified in the Data accessibility section. We developed the wrapper HTforBrainTumour to frame all the different conditions presented in this paper, keep values of the parameters, and enclose the data flow. The parameters are grouped in different data structs related to different categories (geometrical, heat transfer, applied power, etc.) which in the default setting contain the values from table 1 . These data structs are passed from the higher-level function HTforBrainTumour to EqSolver whenever the solution of equations (2.1) and (2.2) is needed. EqSolver performs the calculations and returns the results back to the main function, which in turn calls specific graphical routines. In order to provide a more realistic scenario, stochastic fluctuations of the blood flowterm·dom that would happen in a real tumour [ 50 ] were added. These fluctuations are generated through a multiplicative factor of the blood flowterm·dom that consists of a purely harmonic part with a 30 min period, and uniformly distributed random fluctuations centred around zero that vary every 1.5 min. The amplitude of the latter depends on the parameter ζterm·mth , being larger for increasingly impaired vasculaturesterm·key (i.e. for lower ζ ), and the evolution is taken to be linear between consecutive changes. To solve the model equations describing the evolution of temperaturesterm·dom and blood flowterm·dom, EqSolver resorts to the ode23tb built-in Matlab function, which deals with stiff differential equations using a combination of the trapezoidal rule and a backward differentiation scheme. The function is based on the algorithm TR-BDF2 (trapezoidal rule-second order backward differentiation formula) developed by Randolph E. Bank and co-workers for the simulation of circuits in semiconductor devices [ 51 , 52 ]. The actual expression of the equations that EqSolver uses is coded in Matlab language in the function ThermalModel.

Results Parametrization Solving the model equation (2.1), allows us to evaluate the temperatureterm·dom changes and blood flowterm·dom during and after HT. We consider the values of tissue density and heat capacity of the tumour tissue and its surrounding tissue to be equal, with values of 1050 kg m −3 for densities ρ s and ρ [ 41 ], and 3.86 MJ °C −1 m −3 for the heat capacities c s and c [ 42 ]. Additionally, we use a value of 3.82 MJ °C −1 m −3 for the heat capacity of the blood c b , and 4.12 MJ °C −1 m −3 as an average for the heat capacity of the entire body c T [ 42 ]. The average human body mass is taken to be m T = 70 kg [ 42 ] and its surface area A = 1.91 m 2 [ 43 ]. We consider a tumour having a volume V of 80 cm 3 located in a brain of 1280 cm 3 [ 44 ]. The surrounding tissue is taken to be the whole brain since the heating device is considered non-specific, as it happens with some capacitive heating devices, which are the most widespread methods of applying loco-regional hyperthermiaterm·key [ 53 ] due to its ease of use and simplicity [ 54 ]. Therefore, the heated surrounding tissue V s is 1200 cm 3 . The metabolic heat produced by the body at rest (e.g. during hyperthermiaterm·key administration) is approximately 85 W [ 42 , 45 ], which is the value taken for MET, while the brain has a metabolic heat generation of 11 W kg −1 ( Q s ), corresponding to 21 mol ATP g −1 min −1 [ 46 ] and 30.5 kJ (mol ATP) −1 [ 47 ]. For simplicity, the amount of heat ( Q ) produced in the tumour is assumed to be equal to that of the healthy brain ( Q s ).

The proportionality χ is taken as 0.1 °C −1 following [ 33 ]. For the blood flowterm·dom we use a baseline value of 49.2 ml min − 1 per 100 ml of tissue [ 49 ]. As mentioned above, we use the parameter γ for the proportionality between blood flowterm·dom and temperatureterm·dom in tumours, which is expected to be less sensitive to temperatureterm·dom changes than χ . Furthermore, the baseline value for the blood flowterm·dom in the tumour will be different from that for the healthy tissue [ 35 ]. This is accounted for by parameter ζterm·mth which weighs the baseline perfusionterm·key value ( w 0 ) with the one under pathological conditions. The parameter ζterm·mth is expected to be less than 1, which corresponds to a tumour having an impaired blood flowterm·dom with respect to the healthy tissue, although we do not exclude the possibility that, in some cases, it may temporarily be greater than 1, for tumours characterized by an elevated blood flowterm·dom [ 55 ].

Typical power values applied to tissue in HTs range between 20 and 40 W kg −1 [ 45 ]. Since the brain is a very well-perfused organ, the elevation of temperatureterm·dom will be smaller than in other organs and therefore we choose 40 W kg −1 for P , but will also explore the values of 25 and 50 W kg −1 . Henceforth, we consider a treatment with no focusing of power on the tumour, where the surrounding brain will receive the same power as the tumour, setting P s equal to P . Regarding the heat transfer to the environment through sweating and convection, which is modelled by equation ( 2.7 ), the sweating rate h e is taken to be 80.25 W m −2 °C −1 [ 39 ].

We performed a sensitivity analysis to evaluate the relative influence that variations of each parameter around its prescribed nominal value (as referred to in table 1 ) has on the output of our model. We used an iterative approximation based on adaptive increment directional derivatives of the output [ 56 ] as programmed by [ 57 ]. The system was analysed using the nominal values of every parameter, and the variation rate of the output temperaturesterm·dom of the model, due to perturbations of the parameters, was assessed. The relative percentage error in the output δT , due to the input uncertainty in each parameter p i , denoted by Δ p i , was calculated as To make a meaningful inter-parameter comparison among all the Δ p i , we set an equal relative value r = 10 % for all of them. The resulting influence of every parameter on each of the three output temperaturesterm·dom of our model are shown in table 2 . It can be readily seen that the uncertainties of most of the parameters have a limited influence and that the most relevant ones are T seq that acts as a reference level, the baseline blood flowterm·dom w 0 that is transporting heat, and the power levelterm·dom introduced in the system P . Note that the three resulting δT s, which are expressed as relative errors, did not exceed r = 10 % . This implies that the model is well defined and does not possess critical parameters in the explored parameter space.

Sensitivity analysis of the model parameters to determine their impact on the three temperaturesterm·dom (tumour, healthy tissue and blood compartments). The figures are relative percentage errors in each output for a 10% uncertainty in the considered parameter.

In addition to that, we performed the same kind of analysis for the parameters defining the vascular state of the tumour γ and ζ . Since the range of variation of those is much broader (due to the limited information), we analysed all their range of variation. The results are shown in figure 3 . The uncertainties in both of them have very little influence on the healthy tissue and the incoming blood temperaturesterm·dom, although it does have some impact on the temperatureterm·dom of the tumour. Variations around the chosen nominal values of ζ become specially important as it decreases, while variations of γ are more relevant as this parameter increases.

Results of the sensitivity analysis performed on the vascular parameters γ and ζ , related to the variation of tumour blood flowterm·dom due to the elevation of temperatureterm·dom, and the impairment in the tumour blood flowterm·dom with respect to that of the healthy tissue, respectively. The percentage errors δT , δT s and δT b in the corresponding output variables, due to the input uncertainty in each of the parameters, is depicted. The variation in any of these parameters shows a limited influence on the healthy tissue and blood temperaturesterm·dom (centre and right columns), while it is more prominent when considering the temperatureterm·dom of the tumour (left column). The uncertainty in the value of ζ becomes more important at lower values of the parameter, while for γ the uncertainty gains relevance at high levels.

Equilibrium point and initial conditions When no external power is applied ( P = 0), the temperaturesterm·dom vary—due to random fluctuations in blood flowterm·dom—around the following equilibrium points of the system (2.1) Equations (3.2) represent the thermal homeostatic conditions to which the unaltered system tends, i.e. the steady-state temperaturesterm·dom when there are no external energy sources. Typical brain temperaturesterm·dom T seq are around 36.9°C [ 48 ]. From the known parameter values and by using equation ( 3.2c ), we can solve for the temperatureterm·dom T beq that the incoming blood must have to meet the physiological equilibrium conditions of the model. The same goes for the initial temperatureterm·dom of the tumour T eq which can be calculated from equation ( 3.2b ). The temperatureterm·dom of the environment ( T E ) is set to 22°C and the value of the heat exchange parameter h 0 is chosen to equilibrate the metabolic heat for a given external temperatureterm·dom.

Computational simulation with a functional vasculatureterm·key First, we consider a tumour with a fairly intact vasculatureterm·key that responds to temperatureterm·dom in a similar way to the healthy tissue and that has a similar level of baseline perfusionterm·key, but lower than the healthy tissue. According to this, we take ζ = 0.85 ± 0.06 and γ = 0.08 ± 0.008°C −1 . The confidence intervals here represent the intervals within which the parameters lie in different simulations. We simulate a treatment lasting 55 min that starts at time t = 5 min and finishes at t = 60 min, with the first 10 min being allotted to progressively warming up the tissue (the power increases linearly from zero to the nominal value).

The results for the evolution of temperaturesterm·dom are shown in figure 4 a , where the bands account for the envelopes of the temperaturesterm·dom obtained through multiple simulations using parameters within the indicated ranges, while the solid lines show the results with the nominal value of each parameter. The onset temperaturesterm·dom for the simulation are the pseudo-equilibrium values from equation (3.2) which are only perturbed by the variations of blood flowterm·dom within the tumours. When the heating is started, there is an increase in temperatureterm·dom that follows the elevation of power. Shortly after the power has reached its full value, the tumour temperatureterm·dom attains a transient maximum. Neglecting the small fluctuations, a new equilibrium exists when the power is on that corresponds to the situation where the heat entering the system is the same as that lost by exchange with the environment through the skin. Therefore, there is a maximum temperatureterm·dom that can be reached, which is imposed by the conditions of the problem, and this maximum temperatureterm·dom is achieved early during heat administration. In this case, for a well-perfused tumour, the increase in temperatureterm·dom is 1.42°C and the observed plateau remains at values of mild hyperthermiaterm·key, well below those usually reported of 42°C or higher. The temperatureterm·dom of the surrounding healthy tissue experiences a similar rise, but always lower than that of the tumour tissue due to its better ability to remove heat through its vasculatureterm·key. A rise in the temperatureterm·dom of the entering blood is observed, albeit it is almost imperceptible. The maintenance of this temperatureterm·dom profile would be consistent with observed fact for the application of local hyperthermiaterm·key, where the core body temperatureterm·dom of the patient remains at a fairly constant temperatureterm·dom.

Left column: Evolution of temperaturesterm·dom ( a ) and perfusionterm·key ( b ) of an in silico brain tumourterm·key subjected to hyperthermiaterm·key with an effective internal power density P of 40 W kg −1 . The parameters correspond to a tumour with a fairly intact vasculatureterm·key (i. v.) characterized by an impairment parameter ζterm·mth = 0.85 ± 0.06 and a response parameter γ = 0.08 ± 0.008°C −1 . Treatment starts at minute 5 and power is increased until full power is reached at minute 15. The dark-pink curve corresponds to the tumour tissue while the dark-blue curve shows the values for the healthy surrounding tissue, which is also heated. The light-blue line represents the temperatureterm·dom of the blood entering into the tumour region and circulating around the body. The panels on the right display the same concepts—temperaturesterm·dom in ( c ), and blood flowterm·dom in ( d )—for a tumour with a dysfunctional vasculatureterm·key (d. v., ζ = 0.44 ± 0.13, γ = 0.06 ± 0.006°C −1 ) whose evolution is shown by the purple line. Multiple runs are computed with different values of γ , ζ within their range of variation and the resulting envelope is represented by light colour bands. The result obtained with the nominal values of the parameters is plotted in solid line.

The results for the evolution of perfusionterm·key in the tumour and in its surrounding tissue are shown in figure 4 b . The increase in temperatureterm·dom produces a corresponding rise in blood flowterm·dom, both in the tumour and in the healthy tissue. Since the vasculatureterm·key of the healthy tissue is assumed to be more functional and responsive than that of the tumour, the values of perfusionterm·key are steadily higher. Leaving aside the fluctuations inherent in the tumour blood flowterm·dom, the values in both tissues reach stability shortly after the onset of heating, matching the stabilization of temperatureterm·dom. After the cessation of the external source of heat, both the temperatureterm·dom and the blood flowterm·dom return to their baseline figures at a high rate, doing so in less than 15 min.

We study an in silico tumour with similar characteristics defined in this case by the nominal values of the parameters and subjected to two more levels of maximum power, namely 25 and 50 W kg −1 , applied following the same administration mode. The results are shown in figure 5 . The overall behaviour of temperatureterm·dom and perfusionterm·key exhibits the same progression, but it is apparent that the maximum temperatureterm·dom is highly affected by the power levelterm·dom, which is echoed by the levels of perfusionterm·key. However, even with the highest power the maximum achievable temperatureterm·dom is rather low by the standards of therapy, as is the time that it takes for the temperatureterm·dom and perfusionterm·key to return to basal levels.

Different evolution of temperatureterm·dom and perfusionterm·key as a function of applied power. All the panels correspond to the case of a tumour with functional vascular status ( ζ = 0.85, γ = 0.08°C −1 ). Panels ( a ) and ( b ) correspond to a heating with a maximum effective internal power density P of 25 W kg −1 administered from t = 15 min to t = 60 min, with a linear ramp from zero to the maximum power between t = 5 min and t = 15 min. The right-hand panels show the same concept for temperatureterm·dom ( c ), and perfusionterm·key ( d ) in a treatment with power density P = 50 W kg −1 .

Computational simulation with impaired vasculatureterm·key In the next scenario, we examine the heating process of a tumour with a dysfunctional perfusionterm·key due to a higher disruption of its vasculatureterm·key. We use the parameter ζterm·mth = 0.44 ± 0.13 to quantify the impairment of tumour blood flowterm·dom with respect to that of the normal tissue and the parameter γ = 0.06 ± 0.006°C −1 to describe the variation of blood flowterm·dom for each degree change in temperatureterm·dom. The application of heat follows the same pattern as in the previous case (10 min warm-up with linearly increasing power and 45 more minutes at nominal power) and the results of temperatureterm·dom elevation and changes in perfusionterm·key for this case are shown in figure 4 c , d , respectively. With impaired blood flowterm·dom, the ability of the tissue to remove heat is lower and thus a higher temperatureterm·dom is achieved. It is apparent that the temperatureterm·dom of the tumour is much higher than that of the healthy tissue in this case. Nevertheless, the possibility of improving the perfusionterm·key in the tumour is not very high, since its response is lower than that of the healthy tissue. With a baseline perfusionterm·key of 3.79 ml kg s −1 m −3 , the corresponding maximum blood flowterm·dom obtained during the heating is 4.38 ml kg s −1 m −3 . Moreover, the decrease after the cessation of the external power source is still very steep and both the blood flowterm·dom and the temperatureterm·dom return to normal values in around 20 min (slightly longer than in the functional scenario).

To test the influence of the power density in this scenario, we ran the model with nominal values of the parameters ( ζ = 0.44, γ = 0.06°C −1 ) for two values of power density, 25 and 50 W. The results are shown in figure 6 where it can be observed that power has a large influence on the temperaturesterm·dom that can be attained, but it does not significantly change the perfusionterm·key values. The attainable temperaturesterm·dom for this case are undoubtedly higher than for a tumour with a functional vasculatureterm·key, but are still far from the temperaturesterm·dom typically sought of around 43°C, which underlines how difficult it is to achieve therapeutically significant temperaturesterm·dom for such a well-perfused organ as the brain.

Difference in heating and blood perfusionterm·key for two different levels of applied powers on a tumour with impaired vasculatureterm·key ( ζ = 0.44, γ = 0.06°C −1 ). The left-hand column shows the temperatureterm·dom evolution in ( a ) and blood flowterm·dom evolution in ( b ) for a tumour that undergoes hyperthermiaterm·key treatment with an effective internal power P of 25 W kg −1 applied between t = 15 min and t = 60 min, and a linearly increasing power from t = 5 min until t = 15 min. ( c , d ) The same concept when the heating power is larger, P = 50 W kg −1 .

Map of variation of vascular parameters Since the parameters that define the impairment of the flow ( ζ , γ ) strongly affect the outcome of the treatment, we analyse their influence on the maximum temperatureterm·dom reached in the tumour during the treatment ( figure 7 ). The parameter that defines the proportional variation of perfusionterm·key for each degree Celsius change in temperatureterm·dom ( γ ) is varied from 0, for a tissue that does not respond to temperatureterm·dom, to 0.1, a value observed in healthy brains [ 33 ]. The impairment in blood flowterm·dom at the homeostatic temperatureterm·dom ζ ranges from 0.2, meaning that the tumour has a perfusionterm·key of about one fifth that of healthy tissue, to 1.2, for tumours that have an increased blood flowterm·dom with respect to healthy tissue. The figure shows that the blood flowterm·dom has a great impact on the temperaturesterm·dom that can be reached during the treatment. The parameter that shows the greatest influence is the initial impairment of the blood flowterm·dom. Normal values similar to healthy tissue give maximum temperaturesterm·dom below 39°C, far from what is considered therapeutically relevant. To achieve temperaturesterm·dom greater than 41°C, the blood flowterm·dom must be lower than about one-half of that of the surrounding healthy tissue, which corresponds to extremely impaired perfusionterm·key values; when ζ < 0.3 this can lead to temperaturesterm·dom greater than 42°C. The effect of γ on temperatureterm·dom is lower than that of ζ , particularly when the latter takes normal values. However, lower values of γ imply higher temperatureterm·dom values, since the blood flowterm·dom is not able to respond to the rise in temperatureterm·dom as efficiently as healthy tissue.

Map of temperaturesterm·dom in a tumour as a function of the vascular impairment parameters ζ and γ entering in equation ( 2.2a ). The parameter ζterm·mth accounts for the impairment in the baseline perfusionterm·key value ( w 0 ), while γ gives the proportional variation of the blood flowterm·dom with each degree Celsius change in temperatureterm·dom, with the value ( χ ) for the healthy tissue estimated to be 0.1°C −1 . These parameters greatly influence the maximum temperaturesterm·dom reached.

Variation of basal perfusionterm·key The previous results suggest that the typical values of blood flowterm·dom in the brain make it very difficult to attain therapeutically relevant temperaturesterm·dom unless the vasculatureterm·key of the tumour treated is highly impaired. Another immediate implication is that the time window during which the tumour region remains well above normal temperaturesterm·dom after local hyperthermiaterm·key is very small (of the order of minutes). To analyse the response of the system to other values of the basal perfusionterm·key w 0 , we ran simulations for a broad range of values of this parameter. Simulations were carried out using four different values of the vascular impairment parameter and for each pair of values ( w 0 , ζ ) we performed 14 different trials to account for the variability of the random effects of blood flowterm·dom considered in our model. In these simulations, we quantified the maximum temperatureterm·dom reached over the heating time and the standard deviation ( figure 8 a ). Temperaturesterm·dom higher than the usual target of 43°C were observed when both the basal perfusionterm·key and the vascular impairment parameter were lower. Additionally, in those simulations, we determined the time during which the tumour region remained hot after the cessation of the treatment. Results in figure 8 b indicate that the time required to return to normal temperaturesterm·dom is very short whenever the values of perfusionterm·key are high, as already anticipated. As the basal perfusionterm·key of the organ decreases, the heated tumour remains at higher temperaturesterm·dom for longer times. However, none of the scenarios explored with our modelling approach support that the times of return to normal temperatureterm·dom might be higher than 1 h, thus setting an upper bound on the therapeutic time window.

Maximum achievable temperatureterm·dom ( a ) and time that the tumour takes to return to normal temperatureterm·dom after the cessation of heating ( b ) as a function of basal perfusionterm·key w 0 for different values of the impairment parameter ζterm·mth . Solid lines indicate the mean of 14 simulations for each ( w 0 , ζ ) pair and bands represent the standard deviation of each. Both maximum temperatureterm·dom and time to return to basal temperatureterm·dom are greatly improved when basal perfusionterm·key decreases.

Discussion In this study, we put forward a mathematical modelterm·key based on energy conservation laws to analyse the evolution of temperatureterm·dom and blood perfusionterm·key during regional hyperthermiaterm·key of brain tumoursterm·key (e.g. gliomas), motivated by the potential synergy with radiotherapyterm·key when treating this type of tumour, which is characterized by a low response and development of resistance [ 58 ]. Our model predicts a quasi-homeostatic equilibrium condition for the temperatureterm·dom of the brain and the blood vasculatureterm·key feeding it, as given by equations ( 3.2a ) and ( 3.2c ), where the temperatureterm·dom of the brain is higher than the incoming blood by a number of 0.37°C, consistent with other models [ 32 ] and confirmed by experimental results [ 59 ]. In [ 31 ], the authors rationalize the shielding effect of blood flowterm·dom in the brain which protects it from temperatureterm·dom variations induced by external cooling or heating, an effect that has been previously reported [ 60 ]. This effect, therefore, supports one of the assumptions of our model in which we attribute the variation of temperatureterm·dom in the heated tissue only to blood perfusionterm·key, and shift the dissipation of energy by conduction to the rest of the body. However, more detailed analyses of the heat losses might be needed to apply our model to different organs.

The elevation of temperatureterm·dom caused by HT is assumed to produce a subsequent rise in blood flowterm·dom. Moriyama evaluated the CBF response to a rise in temperatureterm·dom in monkeys, finding a linear relation with the CBF increasing 10% for each degree of temperatureterm·dom rise [ 33 ]. Furthermore, in [ 34 ], the authors studied the decrease of CBF in dogs when exposed to cold temperaturesterm·dom and found a similar linear behaviour in which there was an approximate decrease of 7% in CBF for each degree of decrease, at least at temperaturesterm·dom not far from the homeostatic equilibrium. Taking these previous studies as a starting point, we have modelled the dependence of blood flowterm·dom as a linear relation with temperatureterm·dom and assumed that the behaviour of tumour tissue would be similar but less reactive. While some authors have also used linear relations [ 61 , 62 ], others have resorted to more sophisticated functional forms connecting blood flowterm·dom and temperatureterm·dom, such as in [ 63 ], although a linear relationship was finally employed. Alternatively, some authors have used other more complex empirical formulae [ 64 , 65 ] which ultimately derive from the experimental information contained in [ 66 ]. Nevertheless, what those models try to reproduce is mostly an extrapolation of the observed effects, since the actual data covers only a few points and does not seem to justify fits beyond those based on linear regression.

When hyperthermiaterm·key applied to cancer started receiving attention at the end of the 1970s, the target temperaturesterm·dom were very high, usually higher than 43°C, with the objective of killing cells directly by the effect of heat [ 7 ]. Later, the interest turned to milder temperaturesterm·dom, below 41°C [ 5 ], which were easier to achieve and had the purpose of improving the synergy with chemo and radiotherapyterm·key [ 67 ]. The results of our model show that, due to the high blood flowterm·dom that feeds the brain, it will be very difficult to achieve temperaturesterm·dom in the therapeutic range for brain tumoursterm·key at the radiofrequency standard powers, unless the vasculatureterm·key of the tumour is highly compromised. A good knowledge of the pre-treatment blood flowterm·dom is also essential. Our mathematical modelterm·key keeps track of the temperatureterm·dom of the blood cooling the tissue, which is also circulating all over the body and losing heat to the environment. After solving the equations, it is observed that its temperatureterm·dom remains almost unaffected by the treatment and thus it could have been considered constant with no great difference in the results. As to the heated tissues, there is a pseudo-equilibrium around which the temperatureterm·dom oscillates, and that temperatureterm·dom is reached soon after starting the treatment. The pseudo-equilibrium appears due to the thermoregulation of the body, which is able to lose heat to the environment through the skin to compensate the power applied. This has direct relevance to the clinical setting, since it implies that, regarding the functionality of the vasculatureterm·key, the temperatureterm·dom is bounded and there is control over the heating of the tissue.

The overall duration at which the vasculatureterm·key is maintained at an elevated temperatureterm·dom may be important. In [ 68 ], the authors measured the evolution of CBF in adult male Wistar rats subjected to 8 MHz radio-frequency-induced localized cerebral hyperthermiaterm·key. The results show a rise in CBF when the temperatureterm·dom is increased to temperaturesterm·dom lower than 43°C. After treatment ceases, CBF remains raised for some minutes, even though the temperatureterm·dom seems to return to baseline figures much faster. Also, in [ 69 ], the authors show that CBF might follow dynamics other than temperatureterm·dom when heating the cortex of eight-week-old male Sprague–Dawley rats. Therefore, we might encounter inertial effects not taken into account in the present form of our mathematical modelterm·key. These mechanisms would be easy to add, provided that they are confirmed in humans and the correct parameters can be estimated. It must be taken into account that the sustained effect in CBF after hyperthermiaterm·key might be only a reflection of increased heart rate, which stresses the importance of measuring heart rate in the patients receiving the treatment. Therefore, more experimental research on human patients is needed to confirm or reject the functional perfusion-temperature dependency that has been chosen here according to the current literature.

Another heating-mediated effect is the reduction in blood flowterm·dom after exposure to heat [ 66 , 70 , 71 ]. This occurs when the temperaturesterm·dom reached are high enough or when the exposure time is long, pointing either to a destruction of the blood vessels caused by heat shock or to a vasoconstriction effect at high temperaturesterm·dom. Vujaskovic et al. suggested that a decrease in oxygenationterm·key was seen after hyperthermiaterm·key in murine models at lower temperaturesterm·dom than in spontaneous tumours [ 72 ]. Again, this indicates a likely greater brittleness of rodent vessels (specially in xenograft models), which would attribute the decrease in oxygenationterm·key to the destruction of the vessels. The implications of this destruction would be enormous, since a reduction in blood flowterm·dom implies a greater increase in temperatureterm·dom which feedbacks the process. Nonetheless, interest over the past few years has been directed at mild temperatureterm·dom hyperthermiaterm·key rather than high temperaturesterm·dom (e.g. thermal ablation). Thus it seems more difficult that the potential damage to blood vessels is a main factor in the overall outcome of the treatment. Due to the low temperaturesterm·dom obtained with regular values of the parameters, we have not considered this possible destruction in our model. If it were to be included, a vasculatureterm·key injury function of temperatureterm·dom and time should be derived based on experimental results that, for the moment, are not sufficiently systematized. This damage would act as a nonlinearity on the temperatureterm·dom, tending to increase its value when it is already sufficiently high.

To the best of our knowledge, no previous studies have clearly addressed the effects of heat on blood flowterm·dom in humans. Due to the current availability of imaging techniques for non-invasively measuring blood flowterm·dom [ 73 ], it would be advisable to test the differential response in perfusionterm·key after HT both in the tumour and in the healthy tissue. In [ 74 ], the authors presented one single case evaluating the perfusionterm·key in a head and neck cancer patient for whom there was a clear increase in blood flowterm·dom immediately after heating the thorax using water-filtered IR-A radiation, and also five days after the treatment. This is a remarkable result that, if reproduced in other patients and further studied, would help to clarify many aspects of hyperthermiaterm·key. Furthermore, it has been possible for many years to make use of magnetic resonance imaging techniques to evaluate the evolution of temperaturesterm·dom in tissues subjected to external heating non-invasively [ 75 ]. Despite the problems that may arise due to the obscuring influence of a changing perfusionterm·key [ 76 , 77 ], the evolution obtained from imaging, in combination with mathematical modelsterm·key that set the relations between the variables, may be very useful to clarify the open questions related to the operating mode of the treatment. In the same way, the utilization of two-point blood perfusionterm·key images [ 78 ], although not challenge-free [ 79 ], could elucidate current questions related to the evolution of blood flowterm·dom under thermal treatment that have been addressed in this paper.

Traditionally, HT in oncology has been deemed to promote the increase in blood flowterm·dom, which would lead to better oxygenationterm·key in the tissue [ 6 ], thereby improving the outcome of radiotherapyterm·key, which, as is well known, has improved efficiency in the presence of oxygen, in opposition to hypoxic microenvironments. Hyperthermiaterm·key is said to be a very good way to overcome hypoxia, improving the outcome of radiotherapyterm·key [ 2 ]. However, the actual role of improved perfusionterm·key is controversial [ 3 , 79 ]. Our computational modelterm·mth takes into account only mechanistic effects of heat on perfusionterm·key where a variation in temperatureterm·dom has immediate and direct effect on the modification of blood flowterm·dom. Assuming this hypothesis, the results indicate that the decrease in temperatureterm·dom is very fast after the cessation of the external power, which would be accompanied by a parallel drop in blood flowterm·dom. Usually, the application of radiotherapyterm·key after hyperthermiaterm·key in the clinic is not immediate and takes place around an hour later [ 80 ]. There is currently controversy on whether a fast application of radiotherapyterm·key after hyperthermiaterm·key does have an effect on the outcome [ 81 , 82 ]. These late works show the current lack of knowledge on which is the best scheduling for joint radio-hyperthermia, even though the work by Overgaard [ 83 , 84 ] has been very often cited as an answer to this regard. According to our results, considering the aforementioned clinical time gaps and the fast drop in temperatureterm·dom, it seems unlikely that improved oxygenationterm·key caused by an elevated perfusionterm·key is behind the benefits of HT, and other effects might have much higher relevance. A validation of the evolution of perfusionterm·key in humans patients who have undergone HT would very much clarify some of these current issues.

Our approach spatially averages the contribution in each of the compartments. The heterogeneous microenvironment typical of tumours ensures variability in the vascular conditions at different points that will lead to local differences in temperatureterm·dom and possibly overheated areas. At points where the vasculatureterm·key is highly impaired or very leaky the flow might at times go to zero, creating hot-spots. These zones, however, would not benefit from an immediate improvement in oxygenationterm·key caused by an elevated temperatureterm·dom as their blood vessels cannot provide the required flow. Other local effects, like the induction of cell death followed by subsequent reoxygenation, or the turn to a glycolitic metabolism, might arise, but in these cases the time scales needed to achieve observable gains from a subsequent administration of radiotherapyterm·key would be much longer.

In conclusion, we have developed a mathematical modelterm·key for the temperatureterm·dom and perfusionterm·key evolution of a brain tumourterm·key subjected to external heating. Our model emphasizes the importance of the status of tumour vasculatureterm·key in the response to regional hyperthermiaterm·key and points to the necessity of relating perfusionterm·key maps to patient outcome. Our results suggest that it may be difficult to significantly increase the temperatureterm·dom of such a well-perfused organ and show the existence of a quasi-steady-state maximum temperatureterm·dom that is reached fairly soon after the initiation of heat administration. Furthermore, the fast decay to baseline perfusionterm·key and temperatureterm·dom values after heating cessation apparently eliminates a raise in blood perfusionterm·key as a main factor for the favourable outcome of patients subjected to co-adjuvant hyperthermiaterm·key with radiotherapyterm·key. Therefore, a mechanistic augmented perfusionterm·key due to heat does not appear to be the reason behind a better oxygenationterm·key, and the time scale of the latter would not correspond to the former. According to these results, other contributions seem more likely to be responsible and their time scales will be dominant for the combined treatment with radiotherapyterm·key.

Supplementary Material Acknowledgements The authors thank Juan Jiménez-Sánchez and Carmen Ortega-Sabater for their fruitful comments on the biology of cancer and the potential implications of thermal therapy.

Data accessibility Data and relevant code for this research work are stored on the group’s website: http://matematicas.uclm.es/molab/ThermalModelForHT.zip and have been archived within the Zenodo repository: https://doi.org/10.5281/zenodo.4059158 Authors' contributions J.J.B. and G.F.C. performed the analyses and drafted the manuscript. J.J.B. and M.C.N wrote the numerical code. G.F.C., V.M.P.-G. and M.C.N conceived and supervised the study and critically revised the manuscript. All authors gave final approval for publication.

Competing interests We declare we have no competing interests.

Funding This research has been supported by the James S. McDonnell Foundation Twenty-first Century Science Initiative in Mathematical and Complex Systems Approaches for Brain Cancer (Collaborative awards 220020560 and 220020450), Ministerio de Economía y Competitividad/FEDER, Spain (grant nos. MTM2015-71200-R), Junta de Comunidades de Castilla-La Mancha (grant nos. SBPLY/17/180501/000154 and SBPLY/19/180501/000211) and the Research grant no. PID2019-109652GB-I00 from the MICINN of the Spanish Government, which includes RDEF funds. J.J.B. is supported by funding from the University of Castilla-La Mancha (grant reference 2018-CPUCLM-7798).

References Associated Data Supplementary Materials Data Availability Statement Data and relevant code for this research work are stored on the group’s website: http://matematicas.uclm.es/molab/ThermalModelForHT.zip and have been archived within the Zenodo repository: https://doi.org/10.5281/zenodo.4059158