Charged particles accelerated by interplanetary shocks can escape from the shock without returning to it. However, the simplest version of the model of Diffusive Shock Acceleration (DSA) does not include an energy-dependent escape from the foreshock region. We present a model for interplanetary shock acceleration that includes such escape and expands upon DSA. Building off our past research, we analytically solve a one-dimensional transport equation that includes an escape-time dependent on both particle position and momentum. In addition to previous work, we consider the case where a shock encounters a population of preexisting charged particles with a power law energy distribution. We find that at lower energies our solution is concave, whereas at higher energies it asymptotically approaches a power law whose slope depends on the original energy spectrum’s power law index and shock parameters. We fitted the solution obtained from this transport equation to ACE/EPAM shock data measured from multiple shock events. We also compared the best fit parameters to the predicted parameter values, with the latter being derived from measured shock properties. We find that for the shock events considered, our model’s best fit parameters match very well with the predicted values. From this model, we can better understand the mechanism of interplanetary shock acceleration and how this phenomenon energizes charged particles near other objects such as blazars and supernova remnants. This work is supported by the NSF-REU solar physics program at SAO, grant number AGS-1850750. This work is also partially supported by the NSF under grant 1850774.