Raimondo Penta

Multiscale homogenization for fluid and drug transport in vascularized malignant tissues

A system of differential equations for coupled fluid and drug transport in vascularized (malignant) tissues is derived by a multiscale expansion. We start from mass and momentum balance equations, stated in the physical domain, geometrically characterized by the intercapillary distance (the microscale). The Kedem–Katchalsky equations are used to account for blood and drug exchange across the capillary walls. The multiscale technique (homogenization) is used to formulate continuum equations describing the coupling of fluid and drug transport on the tumor length scale (the macroscale), under the assumption of local periodicity; macroscale variations of the microstructure account for spatial heterogeneities of the angiogenic capillary network. A double porous medium model for the fluid dynamics in the tumor is obtained, where the drug dynamics is represented by a double advection–diffusion–reaction model. The homogenized equations are straightforward to approximate, as the role of the vascular geometry is recovered at an average level by solving standard cell differential problems. Fluid and drug fluxes now read as effective mass sources in the macroscale model, which upscale the interplay between blood and drug dynamics on the tissue scale. We aim to provide a theoretical setting for a better understanding of the design of effective anti-cancer therapies.

The role of the microvascular tortuosity in tumor transport phenomena

The role of the microvascular network geometry in transport phenomena in solid tumors and its interplay with the leakage and pressure drop across the vessels is qualitatively and quantitatively discussed. Our starting point is a multiscale homogenization, suggested by the sharp length scale separation that exists between the characteristic vessels and the tumor tissue spatial scales, referred to as the microscale and the macroscale, respectively. The coupling between interstitial and capillary compartment is described by a double Darcy model on the macroscale, whereas the geometric information on the microvascular structure is encoded in the effective hydraulic conductivities, which are numerically computed by solving classical differential problems on the microscale representative cell. Then, microscale information is injected into the macroscopic model, which is analytically solved in a prototypical geometry and compared with previous experimentally validated, phenomenological models. In this way, we are able to capture the role of the standard blood flow determinants in the tumor, such as tumor radius, tissue hydraulic conductivity and vessels permeability, as well as influence of the vascular tortuosity on fluid convection. The results quantitatively confirm that transport of blood (and, as a consequence, of any advected anti-cancer drug) can be dramatically impaired by increasing the geometrical complexity of the microvasculature. Hence, our quantitative analysis supports the argument that geometric regularization of the capillary network improves blood transport and drug delivery in the tumor mass.

Investigation of the potential of asymptotic homogenization for elastic composites via a three-dimensional computational study

Asymptotic homogenization is employed assuming a sharp length scale separation between the periodic structure (fine scale) and the whole composite (coarse scale). A classical approach yields the linear elastic-type coarse scale model, where the effective elastic coefficients are computed solving fine scale periodic cell problems. We generalize the existing results by considering an arbitrary number of subphases and general periodic cell shapes. We focus on the stress jump conditions arising in the cell problems and explicitly compute the corresponding interface loads. The latter represent a key driving force to obtain nontrivial cell problems solutions whenever discontinuities of the coefficients between the host medium (matrix) and the subphases occur. The numerical simulations illustrate the geometrically induced anisotropy and foster the comparison between asymptotic homogenization and well established Eshelby based techniques. We show that the method can be routinely implemented in three dimensions and should be applied to hierarchical hard tissues whenever the precise shape and arrangement of the subphases cannot be ignored. Our numerical results are benchmarked exploiting the semi-analytical solution which holds for cylindrical aligned fibers.

The role of the microvascular network structure on diffusion and consumption of anticancer drugs

We investigate the impact of microvascular geometry on the transport of drugs in solid tumors, focusing on the diffusion and consumption phenomena. We embrace recent advances in the asymptotic homogenization literature starting from a double Darcy-double advection-diffusion-reaction system of partial differential equations that is obtained exploiting the sharp length separation between the intercapillary distance and the average tumor size. The geometric information on the microvascular network is encoded into effective hydraulic conductivities and diffusivities, which are numerically computed by solving periodic cell problems on appropriate microscale representative cells. The coefficients are then injected into the macroscale equations, and these are solved for an isolated, vascularized spherical tumor. We consider the effect of vascular tortuosity on the transport of anticancer molecules, focusing on Vinblastine and Doxorubicin dynamics, which are considered as a tracer and as a highly interacting molecule, respectively. The computational model is able to quantify the treatment performance through the analysis of the interstitial drug concentration and the quantity of drug metabolized in the tumor. Our results show that both drug advection and diffusion are dramatically impaired by increasing geometrical complexity of the microvasculature, leading to nonoptimal absorption and delivery of therapeutic agents. However, this effect apparently has a minor role whenever the dynamics are mostly driven by metabolic reactions in the tumor interstitium, eg, for highly interacting molecules. In the latter case, anticancer therapies that aim at regularizing the microvasculature might not play a major role, and different strategies are to be developed.

Can a continuous mineral foam explain the stiffening of aged bone tissue? A micromechanical approach to mineral fusion in musculoskeletal tissues

UNLABELLED Recent experimental data revealed a stiffening of aged cortical bone tissue, which could not be explained by common multiscale elastic material models. We explain this data by incorporating the role of mineral fusion via a new hierarchical modeling approach exploiting the asymptotic (periodic) homogenization (AH) technique for three-dimensional linear elastic composites. We quantify for the first time the stiffening that is obtained by considering a fused mineral structure in a softer matrix in comparison with a composite having non-fused cubic mineral inclusions. We integrate the AH approach in the Eshelby-based hierarchical mineralized turkey leg tendon model (Tiburtius et al 2014 Biomech. MODEL Mechanobiol. 13 1003-23), which can be considered as a base for musculoskeletal mineralized tissue modeling. We model the finest scale compartments, i.e. the extrafibrillar space and the mineralized collagen fibril, by replacing the self-consistent scheme with our AH approach. This way, we perform a parametric analysis at increasing mineral volume fraction, by varying the amount of mineral that is fusing in the axial and transverse tissue directions in both compartments. Our effective stiffness results are in good agreement with those reported for aged human radius and support the argument that the axial stiffening in aged bone tissue is caused by the formation of a continuous mineral foam. Moreover, the proposed theoretical and computational approach supports the design of biomimetic materials which require an overall composite stiffening without increasing the amount of the reinforcing material.

The role of malignant tissue on the thermal distribution of cancerous breast

The present work focuses on the integration of analytical and numerical strategies to investigate the thermal distribution of cancerous breasts. Coupled stationary bioheat transfer equations are considered for the glandular and heterogeneous tumor regions, which are characterized by different thermophysical properties. The cross-section of the cancerous breast is identified by a homogeneous glandular tissue that surrounds the heterogeneous tumor tissue, which is assumed to be a two-phase periodic composite with non-overlapping circular inclusions and a square lattice distribution, wherein the constituents exhibit isotropic thermal conductivity behavior. Asymptotic periodic homogenization method is used to find the effective properties in the heterogeneous region. The tissue effective thermal conductivities are computed analytically and then used in the homogenized model, which is solved numerically. Results are compared with appropriate experimental data reported in the literature. In particular, the tissue scale temperature profile agrees with experimental observations. Moreover, as a novelty result we find that the tumor volume fraction in the heterogeneous zone influences the breast surface temperature.

An Introduction to Asymptotic Homogenization

We present an introduction to the asymptotic homogenization technique as a follow up of the four hours course held at the International Workshop on Multiscale Models in Mechano and Tumor Biology: Modeling, Homogenization, and Applications (M3TB2015) at TU Darmstadt (Germany) on 28 September 2015. The content is well-known, although revisited to provide a first insight to scientists and (especially) students who are unaware of the topic. We present the technique via three simple and instructive examples (one-dimensional and multi-dimensional diffusion and the Stokes’s problem for porous media) remarking on the role of regularity assumptions (periodicity vs. local boundedness) and non-dimensionalization, which are sometimes not sufficiently clarified in the existing literature.

MATHEMATICAL MODELING OF THE INTERPLAY BETWEEN STRESS AND ANISOTROPIC GROWTH OF AVASCULAR TUMORS

In this work, we propose a new mathematical framework for the study of the mutual interplay between anisotropic growth and stresses of an avascular tumor surrounded by an external medium. The mechanical response of the tumor is dictated by anisotropic growth, and reduces to that of an elastic, isotropic, and incompressible material when the latter is not taking place. Both proliferation and death of tumor cells are in turn assumed to depend on the stresses. We perform a parametric analysis in terms of key parameters representing growth anisotropy and the influence of stresses on tumor growth in order to determine how these effects affect tumor progression. We observe that tumor progression is enhanced when anisotropic growth is considered, and that mechanical stresses play a major role in limiting tumor growth.