A detailed understanding of the drug transport barriers within granulomas can illuminate causes underlying the necessary prolonged treatment for TB, and suggest approaches to alleviate these physiological abnormalities of the GME that hinder transport. Thus, analogous modeling approaches from cancer research are applied here, based on the structural and morphological similarities between TB granulomas and tumors 14 – shared hallmarks in these dense tissue masses despite the obvious differences between the two disease etiologies. The approximate analytical solution obtained previously 13, for the case of diffusion-limited reaction, was able to predict the size of hypoxic and necrotic regions in good agreement with our experimental results from the rabbit TB model. Here, we extended our mathematical framework to include the effect of convective transport within the GME as a result of its abnormal vasculature, and to determine its impact on oxygen, nutrient, and drug delivery.

*Model Features, Findings, and Limitations*

Transport phenomena including pressure rise, velocity profile, and interstitial diffusion, are dependent on the structural properties of the GME. We assume that the TB granuloma structure is similar to that of a tumor 17, which is comprised of three regions 13, 14: 1) the vascular region, limited to the periphery and devoid of functional lymphatic vessels; 2) the cellular region containing the cells; and 3) the interstitial (extra-cellular) space that contains an extracellular matrix network that imparts mechanical rigidity, bathed in interstitial fluid. We have shown previously that granuloma vessels are structurally and functionally abnormal, leading to inhomogeneous transport 14. In tumors, this aberrant vasculature compromises transvascular and interstitial convection due to high vessel permeability and an increase in IFP from plasma leakage and a lack of functional lymphatics. Together, these effects result in an outward convective interstitial velocity that opposes inward transport. By assuming that the GME recapitulates these transvascular and interstitial transport limitations seen in tumors based on our experimental evidence, the objective of our modeling effort was to predict these parameters and to indicate fruitful directions for future experimental investigations.

We provide here a comprehensive theoretical model of oxygen transport and reaction within a granuloma (or a tumor) that accounts for non-uniform vasculature, transcapillary exchange, plasma dilution, and interstitial convection and diffusion. Three limiting models of vasculature distribution were considered simultaneously for the first time: a model of uniform MVD distribution, a shell-core model, and a non-uniform MVD distribution model. Based on our analyses for IFP and IFV, we can conclude that the simpler case of uniform MVD distribution is adequate for predicting species transport and reaction. In contrast to earlier tumor IFP/IFV models 2, 3, our model accounts for plasma dilution (which is often excluded as negligible). Indeed, our model of interstitial transport was found to be in accord with tumor experiments. Predictions of convective zone thickness were found to be in agreement with experimental data from granulomas. The model was also utilized to theoretically investigate the effect of enhanced tissue hydraulic conductivity for overcoming transport barriers. Finally, our model accurately predicted the delivery of two anti-TB agents – CFZ and RIF – with surprisingly good agreement given the simplicity of the mathematics describing the underlying transport phenomena.

In short, this model sheds light on the limitations of oxygen, nutrient, and drug transport within granulomas and tumors, and how such barriers might be overcome. For example, it predicts an IFP rise towards the granuloma core as a result of the known abnormal vasculature. This in turn results in a reduced or even stagnated IFV, compromising interstitial convective transport and thus depriving the granuloma core – where the bacilli hide – of oxygen, nutrients, and drugs. Modulating the interstitial hydraulic conductivity, e.g., with an agent targeting the interstitial components (such as losartan, a widely-prescribed, safe and inexpensive anti-hypertensive drug 9), could prove to be an effective host-directed therapeutic strategy to modulate the GME and improve drug delivery and efficacy.

There are some limitations to the model. As described in detail in the Supplementary Materials, the non-uniform and/or shell-core models of perfusion better represent the physiological reality of blood vessel distribution in granulomas than the uniform perfusion model. However, the simpler limiting case of uniform perfusion advantageously allows for analytical solutions that 1) do not require artificially defining regions where convection or diffusion dominate, 2) are in good agreement when fitted with the non-uniform perfusion numerical solutions, and, most importantly, 3) accurately predict IFP/IFV experimental data. We also ignore cell membrane transport as a factor in the mass balance equation; however, we account for the cellular reaction of the species (i.e., oxygen consumption). Because we do not have experimental parameter measurements for TB granulomas (e.g., membrane and interstitial hydraulic conductivities) we apply tumor parameter values to the model given the similarities between granulomas and tumors. We also do not consider variable oxygen/nutrient/drug uptake rates by different types of immune cells, e.g., macrophages vs. T cells, which we have recently demonstrated experimentally can contribute to heterogeneous drug distribution within the GME 4. Finally, because we do not consider the transport of large molecules here, we ignore any retardation factor. Indeed, because anti-TB agents often bind to proteins such as albumin 1, it may become necessary to model the unbound vs. bound drug fractions in future considerations.

*Future Directions*

To the best of our knowledge, we provide here the first consideration of convective transport in the GME, and initial predictions for IFP and IFV profiles and their associated consequences in TB granulomas. To further support these findings, it should be confirmed if – like tumors – TB granulomas lack functional lymphatics. It should be noted that the first predictions of tumor IFP/IFV 3 preceded, yet accurately predicted, the first experimental measurements of these parameters by a number of years 5-8, 20, 22. We posit that future IFP/IFV measurements in TB granulomas will similarly confirm our predictions here. Furthermore, our findings support future testing of host-directed therapies that can modulate the GME to overcome transport barriers and improve treatment outcomes for this virulent and deadly disease. Indeed, following recent efforts by others to computationally optimize the implementation of multiple antibiotic treatments 11, future modeling efforts should provide rational basis for multi-drug dosing and scheduling.