Thermal effect model

Cumulative Equivalent Minutes at 43°C

To estimate quantitatively thermal damages to tissues, an empirical measure has been developed for clinical use (especially for the use of Magnetic Resonance Imaging, or for hyperthermic treatment of tumors) [1] [2] . Based on experimentations, and on the fact that observed thermal damage follow an exponential function with a breaking point at 43°C, the following formula was shown to describe quite accurately the deleterious effects of hyperthermia on human tissues in vivo:

\[\begin{aligned} \text{CEM43°C} \space \text{dose} = \sum_{(\Delta t_{i},T_{i})}\Delta t_{i} \times R_{i}^{43-T_{i}} \space with \begin{cases}R_{i} = 0.25 \space if & T_{i} < 43°C\\R_{i} = 0.50 \space if & T_{i} > 43°C\end{cases}\end{aligned}\]

According to Van Rhoon, Gerard C. et al. [2], the highest CEM43°C value below which no thermal damage was not observe to any tissue in large animals and humans is \[\text{dose}_{max} = 10 \space \text{CEM43°C} \] We will use this value as a limit for the exposition of healthy tissues nearby to the tumor.

Naive model of uniform temperature

Let us begin with a very simple calculation in which we consider that heating the area where our bacteria have colonized will lead to a uniform temperature elevation that will affect nearby healthy tissues. This places us into the worst situation possible, and we will be very happy if the thermal doses are bearable for the healthy tissues. Considering our experimental data, we need to apply the induction temperature during 3 hours. With this naive model, the thermal trigger would lead to the following CEM43°C doses for healthy tissues:

As we can see, in the case of this simple model, it is conceivable to apply 41°C during 3h (although we get borderline to the acceptable limit dose), but totally impossible to apply the heat induction during 3 hours at 45°C, under penalty of causing irreparable thermal damages. From this first simple model, we can also conclude that the maximum temperature experienced by the healthy tissues should not exceed 41°C during 3 hours.

Thermal diffusion model

A more realistic and precise description of the thermal profile in and around the tumor when we heat the area colonized area can be achieved implementing a thermic diffusion model. With this more sophisticated model, we can verify whether we exceed the critical temperature of 41°C deduced from the first part.

Assumptions about thermal coefficients

In vivo thermal diffusion models can be rather complex, depending on whether they take into account vascularisation on a more (locally) or less (globally) precise way. Experimentally, thermal diffusivity coefficients have been measured in dead [3] [4] or living tissues [4], with the main difference between both cases being a perfusion effect (thermal exchanges with flowing blood) being respectively present or not. It is however considered that, unless the perfusion flow rate is particularly high (>10 kg/m³/s [4]), it won't have a significant effect on thermal diffusion in vivo. This perfusion effect would all the more play in our favor if we took it into account, as it would participate in the dissipation of thermic energy coming from the focused ultrasounds.

For these reasons, we will take as a parameter in our model a typical thermal conductivity coefficient of a human organ, which is 0.5 W/m/°C [5]. We also take as typical thermal capacity 3.6 kJ/kg/°C and 1.1 kg/L as typical density [5].

Geometry of the system

We take as a geometry for our problem the same one that we already defined from the literature for our Tumor Sensing module:

Radius of tumor Tumor radius

20 mm

Radius of colony shell (concentric to tumor)

10 mm

Width of colony shell

0.5 mm

For the simulation, we will impose the temperature of induction of 45°C to the colonization area (green on the figure) during 3 hours while imposing a surrounding temperature of 37°C very far from the tumor (applied to the internal face of a sphere of 1 m diameter in the model).

Result of the simulation

Thanks to this simulation, we can conclude that if we selectively heat to 45°C the area colonized by bacteria, something we can do thanks to the magnetic resonance imaging capability of our bacteria and the use of a MRI-guided focused ultrasound machine, neighboring helathy tissues experience temperatures which do not exceed 40.5°C, below the maximum temperature of 41°C above which thermally induced damages appear when applied for 3 hours.