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Intravascular ultrasound (IVUS) elastography visualizes local radial strain of arteries in so-called elastograms to detect rupture-prone plaques. However, due to the unknown arterial stress distribution these elastograms cannot be directly interpreted as a morphology and material composition image. To overcome this limitation we have developed a method that reconstructs a Young's modulus image from an elastogram. This method is especially suited for thin-cap fibroatheromas (TCFAs), i.e., plaques with a media region containing a lipid pool covered by a cap. Reconstruction is done by a minimization algorithm that matches the strain image output, calculated with a parametric finite element model (PFEM) representation of a TCFA, to an elastogram by iteratively updating the PFEM geometry and material parameters. These geometry parameters delineate the TCFA media, lipid pool and cap regions by circles. The material parameter for each region is a Young's modulus, EM, EL, and EC, respectively. The method was successfully tested on computer-simulated TCFAs (n=2), one defined by circles, the other by tracing TCFA histology, and additionally on a physical phantom (n=1) having a stiff wall (measured EM=16.8 kPa) with an eccentric soft region (measured EL=4.2 kPa). Finally, it was applied on human coronary plaques in vitro (n=1) and in vivo (n=1). The corresponding simulated and measured elastograms of these plaques showed radial strain values from 0% up to 2% at a pressure differential of 20, 20, 1, 20, and 1 mmHg respectively. The used/reconstructed Young's moduli [kPa] were for the circular plaque EL=50/66, EM=1500/1484, EC=2000/2047, for the traced plaque EL=25/1, EM=1000/1148, EC=1500/1491, for the phantom EL=4.2/4 kPa, EM=16.8/16, for the in vitro plaque EL=n.a./29, EM=n.a./647, EC=n.a./1784 kPa and for the in vivo plaque EL=n.a./2, EM=n.a./188, EC=n.a./188 kPa.