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A new algorithm for the construction of artificial blood vessel networks is presented. The algorithm produces three-dimensional (3-D) geometrical representations of both arterial and venous networks. The key ingredient of the algorithm is a 3-D potential function defined in the tissue volume. This potential function controls the paths by which points are connected to existing vessels, thereby producing new vessel segments. The potential function has no physiological interpretation, but, by adjustment of parameters governing the potential, it is possible to produce networks that have physiologically meaningful geometrical properties. If desired, the veins can be generated counter current to the arteries. Furthermore, the potential function allows fashioning of the networks to the presence of bone or air cavities. The resulting networks can be used for thermal simulations of hyperthermia treatment.