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Purpose To introduce a novel imaging and parameter estimation framework for accurate multi-shot diffusion MRI. Theory and Methods We propose a new framework called ADEPT (Accurate Diffusion Echo-Planar imaging with multi-contrast shoTs) that enables fast diffusion MRI by allowing diffusion contrast settings to change between shots in a multi-shot EPI acquisition (i.e., intra-scan modulation). The framework estimates diffusion parameter maps directly from the acquired intra-scan modulated k-space data, while simultaneously accounting for shot-to-shot phase inconsistencies. The performance of the estimation framework is evaluated using Monte Carlo simulation studies and in-vivo experiments and compared to that of reference methods that rely on parallel imaging for shot-to-shot phase correction. Results Simulation and real-data experiments show that ADEPT yields more accurate and more precise estimates of the diffusion metrics in multi-shot EPI data in comparison with the reference methods. Conclusion ADEPT allows fast multi-shot EPI diffusion MRI without significantly degrading the accuracy and precision of the estimated diffusion maps.