Locally induced hypothermia for treatment of acute ischaemic stroke: a physical feasibility study

Abstract

During the treatment of stroke by local intra-arterial thrombolysis (LIT) it is frequently possible to pass the blood clot with a micro-catheter, allowing perfusion of brain tissue distally to the occlusion. This possibility allows for new early treatments of ischaemic brain tissue, even before the blood clot has been removed. One potential new approach to preserve brain tissue at risk may be locally induced endovascular hypothermia. Physical parameters such as the required micro-catheter input pressure, output velocity and flow rates, and a heat exchange model, applicable in the case of a micro-catheter placed within a guiding catheter, are presented. Also, a simple cerebral temperature model is derived that models the temperature response of the brain to the perfusion with coolant fluids. Based on this model, an expression has been derived for the time needed to reach a certain cerebral target temperature. Experimental in vitro measurements are presented that confirm the usability of standard commercially available micro-catheters to induce local hypothermia of the brain. If applied in vivo, the model predicts a local cooling rate of ischaemic brain tissue of 300 g of approximately 1°C in 1 min, which is up to a factor 30-times faster than the time-consuming systemic hypothermia via the skin. Systemic body temperature is only minimally affected by application of local hypothermia, thus avoiding many limitations and complications known in systemic hypothermia.

Appendices

Appendix 1

Radiation

The body radiates electromagnetic waves. This radiated power by the human body, denoted by P_radiation, can be calculated by:

$$ P_{{\text{radiation}}} = \varepsilon \sigma S\left( {T_{{\text{body}}}^4 - T_{{\text{ambient}}}^4 } \right) $$

(5)

where ɛ is the emissivity of the human skin having a value of 0.97, σ is Boltzmann’s constant (5.67×10⁻⁸ W m⁻² K⁻⁴), S is the surface of the human body (assumed to be roughly 2.0 m² for an adult), T_body is the skin temperature, and T_ambient is the ambient temperature. For a skin temperature of 34.0°C and an ambient temperature of 23°C, P_radiation is 132.9 W.

Conduction

The heat transfer rate of the human body due to conduction can be calculated in the following way:

$$ P_{{\text{conduction}}} = \frac{{k_{{\text{air}}} S\left( {T_{{\text{body}}} - T_{{\text{ambient}}} } \right)}} {d} $$

(6)

in which k_air=24×10⁻³ W m⁻¹ K⁻¹. The distance parameter d accounts for the distance from the skin surface where the air temperature drops from T_body to T_ambient. Assuming d=0.045 m and the same ambient and skin temperature as in the radiation example, P_conduction=11.7 W.

Convection

Convection involves the transport of energy by means of the motion of the heat transfer medium. In the case of the human body this is the surrounding air. The heat losses due to convection are more difficult to calculate. Convection can be approximated using (Eq. 6) by introducing an effective heated air layer d_eff. The stronger the air motion, the smaller the heated air layer, and the larger the energy loss due to convection. Under surgical conditions, this term is normally negligible.

Perspiration

As soon as the skin temperature reaches 37°C, the skin will begin to sweat. Perspiration will increase rapidly with increasing skin temperature. A healthy person has a perspiration volume rate $\dot M$ of approximately 600 g/day, but the perspiration volume rate can be up to 1.5 l/h under extreme conditions [31]. The body is cooled by the vaporization of water. The vaporization heat C_vap of water at body temperature is 2.436×10³ kJ kg⁻¹. The perspiration heat transfer rate P_perspiration, expressed in watts, of a healthy person can thus be calculated as:

$$ P_{{\text{perspiration}}} = C_{{\text{vap}}} \dot M = 16.9\,{\text{W}} $$

(7)

If the skin temperature drops below 37°C, sweating will stop. A direct consequence of the physical facts presented above is that, under surgical conditions, the heat loss due to radiation alone (132.9 W) is larger than the basal metabolic rate (80–100 W), which means that the patients have to be warmed actively in order to prevent unwanted systemic hypothermia.

Appendix 2

Required micro-catheter input pressure

Since the inner radii of micro-catheters are very small, the input pressures needed to pump coolant fluid through them are very high. The relationship between the liquid volume per time unit ϕ_V (flow rate) that passes through a circular tube and the required pressure difference ΔP over the tube is given by the Poiseuille–Hagen formula:

$$ \phi _{\text{V}} = \Delta P\frac{{8\eta L}} {{\pi R^4 }} $$

(8)

in which η is the viscosity of the fluid transported through the micro-catheter, L is the length of the micro-catheter, and R is the inner radius of the micro-catheter.

Coolant fluid outflow velocity

The output speed, v_out, with which the coolant leaves the micro-catheter, measured in metres per second, depends on the radius R of the micro-catheter as well as on the coolant flow rate ϕ_V and is given by:

$$ v_{{\text{out}}} = \frac{{\phi _{\text{V}} }} {{\pi R^2 }} $$

(9)

In order to prevent damage to the blood vessel wall distal to the occlusion site, the maximum outflow velocity should not exceed a certain maximum value. Realistic outflow velocity values for micro-catheters are in the order of 1–5 m s⁻¹. The maximum tolerable outflow velocity will depend on the ratio between the micro-catheter’s inner radius and the arterial inner radius. The smaller this ratio, the higher the maximum tolerable micro-catheter output velocity, as the coolant fluid jet at the micro-catheter tip will be slowed down very rapidly by the eddies that are produced in the non-moving blood behind the occlusion. To the best of the authors’ knowledge, there exists no safety guideline for the maximum tolerable micro-catheter output velocity. When the technique is used clinically, before the therapeutic injection of cool saline solution is started, the position of the tip of the catheter relative to the vessel wall, as well as the size of each, has to be visualized by subtle contrast injection as shown in Figs. 2b and 3c.

Appendix 3

The bio-heat equation

For the time-dependent and spatially dependent temperature T in any point of living tissue to be calculated, the bio-heat equation needs to be solved and is given by:

$$ \rho _{{\text{IB}}} c_{{\text{IB}}} \frac{{\partial T}} {{\partial t}} = \nabla \left( {k_{{\text{IB}}} \nabla T} \right) - c_{{\text{Blood}}} W_{{\text{Blood}}} \left( {T - T_{{\text{Blood}}} } \right) + P $$

(10)

in which ρ _IB is the density of the infarcted brain tissue (kg m⁻³), c_IB the specific heat (J kg⁻¹ K⁻¹), k_IB the thermal conductivity (W K⁻¹ m⁻¹), c_Blood the specific heat (J kg⁻¹ K⁻¹) of blood, and W_Blood the volumetric perfusion rate (kg m⁻³ s⁻¹). The first term on the right-hand side accounts for thermal conductivity, the second term for heat exchange of the tissue with the arterial blood and P is the brain tissue’s metabolic heat production. Finding solutions to the partial differential equation is far from trivial. Here we present an approximate solution by making the following assumptions. Since the blood vessel is totally occluded the volumetric perfusion rate of the blood is zero. The ischaemic brain tissue that was perfused by the artery before it became occluded is assumed to have total mass M_IB. This brain tissue is during hypothermia treatment solely artificially perfused by the micro-catheter having an output flow rate ϕ_mc,out (ml min⁻¹). Since the tissue is in an ischaemic state we assume the metabolic rate P to be negligible. We also assume conductive heat transport to be negligible and further assume that the heat is uniformly in space transferred via the arterioles and capillaries. Under these assumptions, Eq. 10 can be written as

$$ M_{{\text{IB}}} c_{{\text{IB}}} \frac{{{\text{d}}T_{{\text{IB}}} }} {{{\text{d}}t}} = \phi _{{\text{mc,out}}} c\left( {T_{{\text{IB}}} - T_{{\text{Artery}}} } \right) $$

(11)

This equation can be rewritten as:

$$ \frac{{{\text{d}}T_{{\text{IB}}} }} {{{\text{d}}t}} = \frac{{\phi _{{\text{mc,out}}} }} {{M_{{\text{IB}}} }}\left( {T_{{\text{IB}}} - T_{{\text{mc,out}}} } \right) = \kappa \left( {T_{{\text{IB}}} - T_{{\text{mc,out}}} } \right) $$

(12)

with c≈ c_Blood. With T_IB (0)=T_{IB, 0} and T_mc,out not time dependent, the following solution is found:

$$ T_{{\text{IB}}} (t) = T_{{\text{IB,}}0} + \left( {{\text{exp}}\,( - \kappa \,t) - 1} \right)\left( {T_{{\text{IB,}}0} - T_{{\text{mc,out}}} } \right) $$

(13)

Time to reach target hypothermia temperature

Of more practical use than Eq. 13 is an expression that relates to the time necessary to reach a specific hypothermia temperature T_hypothermia, while given the mass of the tissue jeopardized by ischaemia, the coolant temperature, and the coolant flow rate. It can be proven that the time t needed to reach a target temperature of T_hypothermia (with the additional condition that T_mc,out < T_hypothermia < T_IB) is given by:

$$ t\left( {T_{{\text{hypothermia}}} } \right) = - \frac{{M_{{\text{IB}}} }} {{\phi _{{\text{mc,out}}} }}\ln \left( {\frac{{T_{{\text{hypothermia}}} - T_{{\text{IB}}} (0)}} {{T_{{\text{IB}}} (0) - T_{{\text{mc,out}}} }} + 1} \right) $$

(14)