During its first solar encounter, the Parker Solar Probe (PSP) acquired unprecedented up-close imaging of a small Coronal Mass Ejection (CME) propagating in the forming slow solar wind. The CME originated as a cavity imaged in extreme ultraviolet that moved very slowly ($<50$ km/s) to the 3-5 solar radii (R$_odot$) where it then accelerated to supersonic speeds. We present a new model of an erupting Flux Rope (FR) that computes the forces acting on its expansion with a computation of its internal magnetic field in three dimensions. The latter is accomplished by solving the Grad-Shafranov equation inside two-dimensional cross sections of the FR. We use this model to interpret the kinematic evolution and morphology of the CME imaged by PSP. We investigate the relative role of toroidal forces, momentum coupling, and buoyancy for different assumptions on the initial properties of the CME. The best agreement between the dynamic evolution of the observed and simulated FR is obtained by modeling the two-phase eruption process as the result of two episodes of poloidal flux injection. Each episode, possibly induced by magnetic reconnection, boosted the toroidal forces accelerating the FR out of the corona. We also find that the drag induced by the accelerating solar wind could account for about half of the acceleration experienced by the FR. We use the model to interpret the presence of a small dark cavity, clearly imaged by PSP deep inside the CME, as a low-density region dominated by its strong axial magnetic fields.
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