Objective: In this paper, we propose an algorithm for the generation of a patient-specific cardiac vascular network starting from segmented epicardial vessels down to the arterioles. Method: We extend a tree generation method based on satisfaction of functional principles, named Constrained Constructive Optimization (CCO), to account for multiple, competing vascular trees. The algorithm simulates angiogenesis under vascular volume minimization with flow-related and geometrical constraints adapting the simultaneous tree growths to patient priors. The generated trees fill the entire left ventricle myocardium up to the arterioles. Results: From actual vascular tree models segmented from CT images, we generated networks with 6000 terminal segments for 6 patients. These networks contain between 33 and 62 synthetic trees. All vascular models match morphometry properties previously described. Conclusion and significance: Image-based models derived from CT angiography are being used clinically to simulate blood flow in the coronary arteries of individual patients to aid in the diagnosis of disease and planning treatments. However, image resolution limits vessel segmentation to larger epicardial arteries. The generated model can be used to simulate blood flow and derived quantities from the aorta into the myocardium. This is an important step for diagnosis and treatment planning of coronary artery disease.