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Despite the great progress that has been made in understanding cancer biology and the potential molecular targets for its treatment, the majority of drugs fail in the clinical trials. This may be attributed (at least in part) to the complexity of interstitial drug transport in the patient’s body, which is hard to test experimentally. Similarly, recent advances in molecular imaging have led to the development of targeted biomarkers that can predict pharmacological responses to therapeutic interventions. However, both the drug and biomarker molecules need to access the tumor tissue and be taken up into individual cells in concentrations sufficient to exert the desired effect. To investigate the process of drug penetration at the mesoscopic level we developed a computational model of interstitial transport that incorporates the biophysical properties of the tumor tissue, including its architecture and interstitial fluid flow, as well as the properties of the agents. This model is based on the method of regularized Stokeslets to describe the fluid flow coupled with discrete diffusion-advection-reaction equations to model the dynamics of the drugs. Our results show that the tissue cellular porosity and density influence the depth of penetration in a non-linear way, with sparsely packed tissues being traveled through more slowly than the denser tissues. We demonstrate that irregularities in the cell spatial configurations result in the formation of interstitial corridors that are followed by agents leading to the emergence of tissue zones with less exposure to the drugs. We describe how the model can be integrated with in vivo experiments to test the extravasation and penetration of the targeted biomarkers through the tumor tissue. A better understanding of tissue- or compound-specific factors that limit the penetration through the tumors is important for non-invasive diagnoses, chemotherapy, the monitoring of treatment responses and the detection of tumor recurrence.