2.1.1 Terminology and definitions

The thermal conductivity, k, of a substance is defined as the quantity of heat, Q, transmitted due to a unit temperature gradient, in a direction normal to a surface of unit area in unit time under steady–state conditions and where heat transfer is dependent only on the temperature gradient.

(2.1)

where ∂T/∂x is the temperature gradient in the direction of heat flow, and A the cross–sectional area. The negative sign indicates that the heat flux is in the direction of decreasing temperature. The SI unit for thermal conductivity is watt per metre kelvin (W m−1 K−1). Other units, and conversion factors are given in Table 2.1.

Where steady–state conditions are not relevant, the quantity thermal diffusivity, α = k/ρC, is used, where C is the specific heat capacity and ρ is the density, α is related to the spatial and temporal variation of temperature, T, in the medium by

(2.2)

The numeric value of α determines the relative time rate of temperature change, and is thus a measure of the ability of a thermally perturbed system to relax back to steady–state conditions. The SI unit for α is metre2 per second (m2 s−1). Other units, and conversion factors are given in Table 2.1.

Living tissue is perfused, and the passage of blood modifies the heat transfer process. Furthermore, metabolic processes generate heat within the tissue. A formulation taking these factors into account has been given by Pennes (1948) and Perl (1962) as follows:

(2.3)

where mb, Cb and Tb are the mass flow rate, specific heat and temperature of the perfusing blood respectively, and Qm is the rate of metabolic heat production. Equation 2.3 is the so–called bio–heat equation. The measurement of thermal conductivity in–vivo using thermal clearance methods has been widely used as a technique to estimate tissue blood flow both of the skin and of deeper tissue. A practical quantity, the effective thermal conductivity, keff, is used under these circumstances to give an equation with a parameter which is flow dependent.

(2.4)

keff is found to vary proportionally with the square root of the perfusion rate (Jain et al 1979).

An equivalent effective thermal diffusivity, αeff = keff/ρC, can also be defined.

Another thermal quantity, that of thermal inertia, was introduced by Buettner (1951) in considering the thermal response of skin during exposure to radiation. The temperature rise, ΔT, at the surface of an opaque semi–infinite solid following irradiation during time t is:

(2.5)

where H is the irradiating intensity and A the surface absorptance. The product kρC is termed the thermal inertia and has SI units of watt2 second per metre4 kelvin2 (W2 s m−4 K−2). Other units and conversion factors are given in Table 2.1.

2.1.2 Measurement of thermal conductivity

An extensive review of methods which may be used to measure the thermal properties of materials is given by Touloukian (1964), and a useful summary of methods relevant for biological materials given in a general survey by Bowman et al (1975). The techniques used may be categorised as invasive or non–invasive, and in each case may enable steady–state or non–steady–state measurements to be made. Historically, many measurements of k were made using a guarded hot plate (e.g. Poppendiek et al, 1966). The experimental geometry is intended to ensure that heat flow between two conductive plates is entirely in the axial direction of the specimen. Equation 2.1 is applied directly to calculate k. The method has been applied exclusively to in–vitro samples of tissue. More recently, invasive probes have been developed which involve the implantation within the specimen of heat sources (or sinks) which may also serve as temperature sensors. Invasive thermal diffusion methods use a heated thermocouple (e.g. Grayson et al, 1971) or a thermistor probe (Chen et al, 1981; Balasubramaniam and Bowman, 1977; Valvano et al, 1984). The thermistor probe, typically 0.5 mm diameter, is inserted into the tissue through a hypodermic needle, which is then removed to minimise the heat conduction along the metal. The thermistor is first used passively to measure the ambient temperature, and then heated. The electrical power required to maintain the thermistor at a steady temperature is related to the effective thermal conductivity of the surrounding tissue (Valvano et al, 1984). Alternatively, the time change of temperature following a transient pulse of energy to the probe is analysed (Chen et al, 1981). In either case the method depends upon adequate modelling of the thermal characteristics of the probe and surrounding tissue. In spite of the possible criticism that some tissue damage may occur during probe insertion and cause some modification in thermal behaviour, reliable values have been obtained using these probes in both perfused and non–perfused tissues.

A set of ‘semi–invasive’ techniques has been investigated in which temperatures have been measured using cutaneous and subcutaneous thermocouples with surface heat fluxes provided by various non–invasive sources. Henriques (1947) and Hensel and Bender (1956) describe the use of contact heating. Other techniqes have used electromagnetic irradiation: for instance infrared (Dersken et al, 1957), microwave (Cook, 1952) or optical radiation (Kraning, 1973).

Completely non–invasive methods depend upon the fact that when two homogeneous solids initially at different temperatures are brought into contact the interface temperature takes an intermediate value dependent upon the thermal inertias of the two contacting bodies. Tanasawa and Katsuda (1972) have used a technique based upon that described by Vendrik and Vos (1957). Totally non–contact methods use external radiation to heat tissue, and observe the subsequent time–course of skin temperature with a radiometer. Results of measurements of kρC using this method of measurement have been reported by Lipkin and Hardy (1954), Dersken et al (1957) and Kraning (1973) and have been used to give estimates of k using the assumption that ρC for tissue has the same value as that for water.

2.1.3 Historical background

Earliest measurements of the conduction of heat through skin were reported by Klug in 1874. Bordier (1898) showed that beef muscle conducted heat almost twice as well as fat. A value of 5 × 10−4 cal cm−1 s−1 °C−1 was given by Lefevre (1901) for the average conductivity of peripheral tissues in man, based on the assumption that the thermal gradient extended 20 mm inwards from the skin surface. Subsequent work has been tabulated by Chato (1969) and with...