Home > Mathematica, Mathematics > Thermodilution

## Thermodilution

In this post I seek to explore the equations governing thermodilution cardiac output assessment.

Essentially, this is a modified Stewart-Hamilton equation used in indicator dilution methods.  Temperature is used as the indicator. However, in contrast to an indicator where amount and concentration are well defined, conservation of heat energy allow us to derive the form of the equations (with a constant for energy losses).

Let:

• $\sigma_{\text{injectate}}$ be the specific heat of the injectate (for water 4200 $\text{J}\cdot\text{kg}^{-1}\cdot\text{K}^{-1}$)
• $\rho_{\text{injectate}}$ be the density of the injectate (for 0.9% saline 1004.6 $\text{kg}\cdot\text{m}^{-3}$)
• $V_{\text{injectate}}$ be the volume of injectate
• $T_{\text{injectate}}$ temperature of injectate
• $\sigma_{\text{blood}}$ be the specific heat of blood ( 3780 $\text{J}\cdot\text{kg}\cdot\text{K}^{-1}$)
• $\rho_{\text{blood}}$ be the density of blood (1025 $\text{kg}\cdot\text{m}^{-3}$ )
• $V_{\text{blood}}$ be the blood volume
• $T_{\text{blood}}$ blood temperature
• $T_{\text{mix}}(t)$ the temperature effect of mixing of injectate bolus

The technique assumes that cardiac output $\text{Q}=\frac{dV_{\text{blood}}}{dt} =\text{constant}$

From conservation of energy:

$\sigma_{\text{injectate}}\rho_{\text{injectate}}\frac{dV_{\text{injectate}}}{dt}(T_{\text{blood}}-T_{\text{injectate}})= \sigma_{\text{blood}}\rho_{\text{blood}} \text{Q}(T_{\text{blood}}-T_{\text{mix}}(t))$

Let $U(t) =T_{\text{blood}}-T_{\text{mix}}(t)$ and note that the injectate is administered as a bolus (i.e. $\frac{dV_{\text{injectate}}}{dt}=0$ after finite time and further, $\int_0^\infty\frac{dV_{\text{injectate}}}{dt}\,dt =V_{\text{injectate}}$

Solving for $\text{Q}$:

$\text{Q} =\frac{\sigma_{\text{injectate}}\rho_{\text{injectate}}V_{\text{injectate}}(T_{\text{blood}}-T_{\text{injectate}})} {\sigma_{\text{blood}}\rho_{\text{blood}}\int_0^\infty U(t)\,dt}$

The integral in the denominator is the area under the curve (temperature drop versus time)  recorded with monitoring equipment  during  the assessment using the dedicated pulmonary artery floatation (Swan Ganz) catheter.   The injectate is injected in the right atrium and thermistor detects temperature  change.

Below is a simulation of temperature change time curves and cardiac outputs.

References for Stewart-Hamilton equations:

HAMILTON, W. F., MOORE, J. W., KINSMAN, J. M. & SPURLING, R. G.(1932). Studies on the circulation. IV. Further analysis of the injection method, and of changes in hemodynamics under physiological and pathological conditions. American Journal of Physiology 99, 534-551.

STEWART, G. N. (1897). Researches on the circulation time and on the influences which affect it. Journal of Physiology 22, 159-183.