Perfusion and thermal field during hyperthermia. Experimental measurements and modelling in recurrent breast cancer.

Recurrences of malignant tumours in the chest wall are proposed as a valuable model of tissue mainly perfused by small size vessels (the so-called 'phase III' vessels). Invasive thermal measurements have been performed on two patients affected by cutaneous metastasis of malignant tumours during hyperthermic sessions. Thermal probes were inserted into catheters implanted into the tissue at different depths. In one of the catheters a probe connected with laser-Doppler equipment was inserted to assess blood perfusion in the tumour periphery. The perfusion was monitored throughout the sessions, and a noticeable temporal variability was observed. The effect of the perfusion on the thermal map in the tissue was evaluated locally and the 'effective conductivity' of the perfused tissue was estimated by means of the numerical integration of the 'bio-heat' equation. The tumour temperature, at the site where the perfusion probe is located, can be predicted by the numerical model provided two free parameters, alpha and beta, are evaluated with a fitting procedure. Alpha is related to the effective conductivity and beta to the SAR term of the bio-heat equation. The model aimed at estimating the 'effective conductivity' K(eff) of the perfused tissue, and average values of K(eff) of 0.27 +/- 0.03 W m(-1) degrees C(-1) in Patient 1 and of 0.665 +/- 0.005 W m(-1) degrees C(-1) in Patient 2 were obtained throughout the treatment. However, when the average temperature in a larger tumour volume is to be predicted but only a single, 'local' measurement of the perfusion is available and is assumed to be representative for the whole region, the model results are far less satisfactory. This is probably due to the fact that changes of blood perfusion throughout hyperthermic sessions occur to different extents within the tumour volume, and the differences in perfusion cannot be ignored. The above result suggests that, in addition to the 'temperature map', also a 'perfusion map' within the heated volume should be monitored routinely throughout hyperthermic sessions.

Study of a regional circulatory system based on the "equivalent tube" concept: the feto-placental circulation.

The fluid-dynamical description of a regional circulatory system, based on the numerical solution of the dynamical equations applied to an equivalent tube is proposed. The feto-placental circulation is explicitly studied with this technique as an example, and results are discussed and compared with experimental data. The original features of this work are the quantitative description of the terminal load of the vascular system on the basis of morphological data and the attention paid to the formulation of physiologically sensible boundary conditions for the dynamical equations.

Modelling the feto-placental circulation: 1. A distributed network predicting umbilical haemodynamics throughout pregnancy.

The modifications of the Doppler flow velocity parameters occurring in the feto-placental circulation throughout pregnancy have been reproduced on the basis of a mathematical model. Some simple assumptions were made, such as the progressive development of a dichotomous villous vessel network and the increase of the perfusion pressure and of the umbilical arteries dimensions throughout pregnancy. Moreover, both the viscous and capacitive characteristics of the vascular bed were taken into consideration in order to predict the mean values of blood volume, flow and velocity and the pulsatility index. Their value is shown to depend on few parameters, and mainly on the cross-sectional area ratio between the vessels belonging to two succeeding generations.

Modelling the feto-placental circulation: 2. A continuous approach to explain normal and abnormal flow velocity waveforms in the umbilical arteries.

A mathematical model of the feto-placental circulation which describes the development of the placental vasculature throughout pregnancy on the basis of simple assumptions is used to simulate some abnormalities of the villous vascular tree which may affect the values of Doppler indices. In normal cases, the model shows a decreasing trend of the pulsatility index (PI) throughout pregnancy which is comparable to that observed in vivo. When a pathological interruption of the villous growth is simulated, the PI does not decrease any further, unless the input pressure keeps increasing. When various degrees of obliteration of the villous tree are simulated, either through an occlusive process or a reduction of the lumen of the vessels, the PI values increase and the volume flow decreases to a greater extent. The data predicted by the model are compared to those obtained in clinical studies and in experimental animal models.

A mathematical model of the flow of blood through the left coronary artery.

Several models of the coronary circulation have been proposed, relating arterial flow waveform and cardiac contraction. Some of them have produced new concepts, such as back pressure, and zero flow pressure, founded on an over-simplified description of vascular collapse. Experimental studies have shown that the actual dynamics of a collapsible tube cannot be adequately described using a lumped parameter model. A distributed parameter model is therefore proposed and compared with the previous models.

Prediction of coronary blood flow with a numerical model based on collapsible tube dynamics.

We present a theoretical, hydrodynamic model of the vascular system feeding the left ventricle from which the inflow and outflow waveforms can be predicted given the waveforms of aortic and left ventricular pressure. The main feature of the model is that the central portion of the tubes representing intramyocardial vessels is subjected to an external pressure equal to left ventricular pressure, and they therefore collapse (and empty) when that pressure exceeds the internal pressure. The model is a one-dimensional model, so that the propagation of the collapse waves into the vessels can be properly described; this process takes a finite time, and volume change is not in phase with transmural pressure change. Parameters of the model are assessed from independent physiological data. The predicted inflow waveform is compared with experimental data, and the model is shown to reproduce all the main features, in particular the second minimum of flow rate in late systole as well as the first minimum in early systole. The corresponding lumped-parameter model, which cannot take account of wave propagation, is shown not to agree with experiments and in particular to predict unphysiological spikes in the inflow waveform.