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This paper investigates changes in the surface morphology, internal structure, and the molecular mass distribution of amorphous poly-DL-lactides during their hydrolytic degradation in the presence of extra-germinal mesenchymal stem cells (MSCs) (Wharton’s jelly of umbilical cord) of the rat and their metabolic products. It is shown that the degradation of initially monolithic polymer samples in culture and conditioned media is almost identical. However, in a culture medium containing MSCs, this process is much more intense. This effect can be interpreted in terms of the influence of enzymes secreted by living cells, which diffuse from the surface deep into the polymer sample and accelerate its hydrolysis, participating in a catalytic reaction with the ester bonds of polylactide molecules. We developed and verified a mathematical model that takes into account both noncatalytic and catalytic channels of hydrolysis, changes in the porosity of the polymer sample, and diffusion of short-length oligomers; this model is used to adequately interpret the experimental results.