Solid tumors are poroelastic solids with a chemo-mechanical feedback on growth

A mechanobiological model of bone metastasis reveals that mechanical stimulation inhibits the pro-osteolytic effects of breast cancer cells

Bone is highly susceptible to cancer metastasis, and both tumor and bone cells enable tumor invasion through a "vicious cycle" of biochemical signaling. Tumor metastasis into bone also alters biophysical cues to both tumor and bone cells, which are highly sensitive to their mechanical environment. However, the mechanobiological feedback between these cells that perpetuate this cycle has not been studied. Here, we develop highly advanced in vitro and computational models to provide an advanced understanding of how tumor growth is regulated by the synergistic influence of tumor-bone cell signaling and mechanobiological cues. In particular, we develop a multicellular healthy and metastatic bone model that can account for physiological mechanical signals within a custom bioreactor. These models successfully recapitulated mineralization, mechanobiological responses, osteolysis, and metastatic activity. Ultimately, we demonstrate that mechanical stimulus provided protective effects against tumor-induced osteolysis, confirming the importance of mechanobiological factors in bone metastasis development.

Quantifying fat zonation in liver lobules: an integrated multiscale in silico model combining disturbed microperfusion and fat metabolism via a continuum biomechanical bi-scale, tri-phasic approach

Metabolic zonation refers to the spatial separation of metabolic functions along the sinusoidal axes of the liver. This phenomenon forms the foundation for adjusting hepatic metabolism to physiological requirements in health and disease (e.g., metabolic dysfunction-associated steatotic liver disease/MASLD). Zonated metabolic functions are influenced by zonal morphological abnormalities in the liver, such as periportal fibrosis and pericentral steatosis. We aim to analyze the interplay between microperfusion, oxygen gradient, fat metabolism and resulting zonated fat accumulation in a liver lobule. Therefore we developed a continuum biomechanical, tri-phasic, bi-scale, and multicomponent in silico model, which allows to numerically simulate coupled perfusion-function-growth interactions two-dimensionally in liver lobules. The developed homogenized model has the following specifications: (i) thermodynamically consistent, (ii) tri-phase model (tissue, fat, blood), (iii) penta-substances (glycogen, glucose, lactate, FFA, and oxygen), and (iv) bi-scale approach (lobule, cell). Our presented in silico model accounts for the mutual coupling between spatial and time-dependent liver perfusion, metabolic pathways and fat accumulation. The model thus allows the prediction of fat development in the liver lobule, depending on perfusion, oxygen and plasma concentration of free fatty acids (FFA), oxidative processes, the synthesis and the secretion of triglycerides (TGs). The use of a bi-scale approach allows in addition to focus on scale bridging processes. Thus, we will investigate how changes at the cellular scale affect perfusion at the lobular scale and vice versa. This allows to predict the zonation of fat distribution (periportal or pericentral) depending on initial conditions, as well as external and internal boundary value conditions.

Predicting stress and interstitial fluid pressure in tumors based on biphasic theory

The uncontrolled proliferation of cancer cells causes the growth of the tumor mass. Consequently, the normal surrounding tissue exerts a compressive force on the tumor mass to oppose its expansion. These stresses directly promote tumor metastasis and invasion and affect drug delivery. In the past, the mechanical behavior of solid tumors has been extensively studied using linear elastic and nonlinear hyperelastic constitutive models. In this study, we develop a two-dimensional biomechanical model based on the biphasic assumption of the solid matrix and fluid phase of the tissues. Heterogeneous vasculature and nonuniform blood perfusion are also investigated by incorporating in the model a necrotic core and a well-vascularized zone. The findings of our study demonstrate a significant difference between the linear and nonlinear tissue responses to stress, while the interstitial fluid pressure (IFP) distribution is found to be independent of the constitutive model. The proposed biphasic model may be useful for elasticity imaging techniques aiming at predicting stress and IFP in tumors.

Ultrasound stiffness and perfusion markers correlate with tumor volume responses to immunotherapy

Immunotherapy has revolutionized the treatment of dozens of cancers and became a standard of care for some tumor types. However, the majority of patients do not benefit from current immunotherapeutics and many develop severe toxicities. Therefore, the identification of biomarkers to classify patients as likely responders or non-responders to immunotherapy is a timely task. Here, we test ultrasound imaging markers of tumor stiffness and perfusion. Ultrasound imaging is non-invasive and clinically available and can be used both for stiffness and perfusion evaluation. In this study, we employed syngeneic orthotopic models of two breast cancers, a fibrosarcoma and melanoma, to demonstrate that ultrasound-derived measures of tumor stiffness and perfusion (i.e., blood volume) correlate with the efficacy of immune checkpoint inhibition (ICI) in terms of changes in primary tumor volume. To modulate tumor stiffness and perfusion and thus, get a range of therapeutic outcomes, we employed the mechanotherapeutic tranilast. Mechanotherapeutics combined with ICI are advancing through clinical trials, but biomarkers of response have not been tested until now. We found the existence of linear correlations between tumor stiffness and perfusion imaging biomarkers as well as strong linear correlations between the stiffness and perfusion markers with ICI efficacy on primary tumor growth rates. Our findings set the basis for ultrasound imaging biomarkers predictive of ICI therapy in combination with mechanotherapeutics. STATEMENT OF SIGNIFICANCE: Hypothesis: Monitoring Tumor Microenvironment (TME) mechanical abnormalities can predict the efficacy of immune checkpoint inhibition (ICI) and provide biomarkers predictive of response. Tumor stiffening and solid stress elevation are hallmarks of tumor patho-physiology in desmoplastic tumors. They induce hypo-perfusion and hypoxia by compressing tumor vessels, posing major barriers to immunotherapy. Mechanotherapeutics is a new class of drugs that target the TME to reduce stiffness and improve perfusion and oxygenation. In this study, we show that measures of stiffness and perfusion derived from ultrasound shear wave elastography and contrast enhanced ultrasound can provide biomarkers of tumor response.

A mechanobiological model for tumor spheroid evolution with application to glioblastoma: A continuum multiphysics approach

BACKGROUND

Spheroids are in vitro quasi-spherical structures of cell aggregates, eventually cultured within a hydrogel matrix, that are used, among other applications, as a technological platform to investigate tumor formation and evolution. Several interesting features can be replicated using this methodology, such as cell communication mechanisms, the effect of gradients of nutrients, or the creation of realistic 3D biological structures. The main objective of this work is to link the spheroid evolution with the mechanical activity of cells, coupled with nutrient consumption and the subsequent cell dynamics.

METHOD

We propose a continuum mechanobiological model which accounts for the most relevant phenomena that take place in tumor spheroid evolution under in vitro suspension, namely, nutrient diffusion in the spheroid, kinetics of cellular growth and death, and mechanical interactions among the cells. The model is qualitatively validated, after calibration of the model parameters, versus in vitro experiments of spheroids of different glioblastoma cell lines.

RESULTS

Our model is able to explain in a novel way quite different setups, such as spheroid growth (up to six times the initial configuration for U-87 MG cell line) or shrinking (almost half of the initial configuration for U-251 MG cell line); as the result of the mechanical interplay of cells driven by cellular evolution.

CONCLUSIONS

Glioblastoma tumor spheroid evolution is driven by mechanical interactions of the cell aggregate and the dynamical evolution of the cell population. All this information can be used to further investigate mechanistic effects in the evolution of tumors and their role in cancer disease.

Mechanotransduction in tumor dynamics modeling

Mechanotherapy is a groundbreaking approach to impact carcinogenesis. Cells sense and respond to mechanical stimuli, translating them into biochemical signals in a process known as mechanotransduction. The impact of stress on tumor growth has been studied in the last three decades, and many papers highlight the role of mechanics as a critical self-inducer of tumor fate at the in vitro and in vivo biological levels. Meanwhile, mathematical models attempt to determine laws to reproduce tumor dynamics. This review discusses biological mechanotransduction mechanisms and mathematical-biomechanical models together. The aim is to provide a common framework for the different approaches that have emerged in the literature from the perspective of tumor avascularity and to provide insight into emerging mechanotherapies that have attracted interest in recent years.

Two-phase model of compressive stress induced on a surrounding hyperelastic medium by an expanding tumour

In vitro experiments in which tumour cells are seeded in a gelatinous medium, or hydrogel, show how mechanical interactions between tumour cells and the tissue in which they are embedded, together with local levels of an externally-supplied, diffusible nutrient (e.g., oxygen), affect the tumour's growth dynamics. In this article, we present a mathematical model that describes these in vitro experiments. We use the model to understand how tumour growth generates mechanical deformations in the hydrogel and how these deformations in turn influence the tumour's growth. The hydrogel is viewed as a nonlinear hyperelastic material and the tumour is modelled as a two-phase mixture, comprising a viscous tumour cell phase and an isotropic, inviscid interstitial fluid phase. Using a combination of numerical and analytical techniques, we show how the tumour's growth dynamics change as the mechanical properties of the hydrogel vary. When the hydrogel is soft, nutrient availability dominates the dynamics: the tumour evolves to a large equilibrium configuration where the proliferation rate of nutrient-rich cells on the tumour boundary balances the death rate of nutrient-starved cells in the central, necrotic core. As the hydrogel stiffness increases, mechanical resistance to growth increases and the tumour's equilibrium size decreases. Indeed, for small tumours embedded in stiff hydrogels, the inhibitory force experienced by the tumour cells may be so large that the tumour is eliminated. Analysis of the model identifies parameter regimes in which the presence of the hydrogel drives tumour elimination.

How dynamic prestress governs the shape of living systems, from the subcellular to tissue scale

Cells and tissues change shape both to carry out their function and during pathology. In most cases, these deformations are driven from within the systems themselves. This is permitted by a range of molecular actors, such as active crosslinkers and ion pumps, whose activity is biologically controlled in space and time. The resulting stresses are propagated within complex and dynamical architectures like networks or cell aggregates. From a mechanical point of view, these effects can be seen as the generation of prestress or prestrain, resulting from either a contractile or growth activity. In this review, we present this concept of prestress and the theoretical tools available to conceptualize the statics and dynamics of living systems. We then describe a range of phenomena where prestress controls shape changes in biopolymer networks (especially the actomyosin cytoskeleton and fibrous tissues) and cellularized tissues. Despite the diversity of scale and organization, we demonstrate that these phenomena stem from a limited number of spatial distributions of prestress, which can be categorized as heterogeneous, anisotropic or differential. We suggest that in addition to growth and contraction, a third type of prestress-topological prestress-can result from active processes altering the microstructure of tissue.

Energetic convenience of cell division in biological tissues

A typical feature of living tissues is proliferation by division: it is a fundamental aspect of many biological processes, including embryonic development, morphogenesis, and cancer growth. Here, we study the energetics of cell division occurring in epithelia, highlighting the interplay of the key parameters ruling mitosis. We find the existence of a region, in the parameter space, which is independent of the cell elasticity and weakly dependent on the ratio between mother and daughter cells areas. In this region, cell division is energetically favorable. Our results may lead to an exact characterization of cells having anomalous proliferation.

Coupling solid and fluid stresses with brain tumour growth and white matter tract deformations in a neuroimaging-informed model

Brain tumours are among the deadliest types of cancer, since they display a strong ability to invade the surrounding tissues and an extensive resistance to common therapeutic treatments. It is therefore important to reproduce the heterogeneity of brain microstructure through mathematical and computational models, that can provide powerful instruments to investigate cancer progression. However, only a few models include a proper mechanical and constitutive description of brain tissue, which instead may be relevant to predict the progression of the pathology and to analyse the reorganization of healthy tissues occurring during tumour growth and, possibly, after surgical resection. Motivated by the need to enrich the description of brain cancer growth through mechanics, in this paper we present a mathematical multiphase model that explicitly includes brain hyperelasticity. We find that our mechanical description allows to evaluate the impact of the growing tumour mass on the surrounding healthy tissue, quantifying the displacements, deformations, and stresses induced by its proliferation. At the same time, the knowledge of the mechanical variables may be used to model the stress-induced inhibition of growth, as well as to properly modify the preferential directions of white matter tracts as a consequence of deformations caused by the tumour. Finally, the simulations of our model are implemented in a personalized framework, which allows to incorporate the realistic brain geometry, the patient-specific diffusion and permeability tensors reconstructed from imaging data and to modify them as a consequence of the mechanical deformation due to cancer growth.

Stress generation, relaxation and size control in confined tumor growth

Experiments on tumor spheroids have shown that compressive stress from their environment can reversibly decrease tumor expansion rates and final sizes. Stress release experiments show that nonuniform anisotropic elastic stresses can be distributed throughout. The elastic stresses are maintained by structural proteins and adhesive molecules, and can be actively relaxed by a variety of biophysical processes. In this paper, we present a new continuum model to investigate how the growth-induced elastic stresses and active stress relaxation, in conjunction with cell size control feedback machinery, regulate the cell density and stress distributions within growing tumors as well as the tumor sizes in the presence of external physical confinement and gradients of growth-promoting chemical fields. We introduce an adaptive reference map that relates the current position with the reference position but adapts to the current position in the Eulerian frame (lab coordinates) via relaxation. This type of stress relaxation is similar to but simpler than the classical Maxwell model of viscoelasticity in its formulation. By fitting the model to experimental data from two independent studies of tumor spheroid growth and their cell density distributions, treating the tumors as incompressible, neo-Hookean elastic materials, we find that the rates of stress relaxation of tumor tissues can be comparable to volumetric growth rates. Our study provides insight on how the biophysical properties of the tumor and host microenvironment, mechanical feedback control and diffusion-limited differential growth act in concert to regulate spatial patterns of stress and growth. When the tumor is stiffer than the host, our model predicts tumors are more able to change their size and mechanical state autonomously, which may help to explain why increased tumor stiffness is an established hallmark of malignant tumors.

Impact of axisymmetric deformation on MR elastography of a nonlinear tissue-mimicking material and implications in peri-tumour stiffness quantification

Solid tumour growth is often associated with the accumulation of mechanical stresses acting on the surrounding host tissue. Due to tissue nonlinearity, the shear modulus of the peri-tumoural region inherits a signature from the tumour expansion which depends on multiple factors, including the soft tissue constitutive behaviour and its stress/strain state. Shear waves used in MR-elastography (MRE) sense the apparent change in shear modulus along their propagation direction, thereby probing the anisotropic stiffness field around the tumour. We developed an analytical framework for a heterogeneous shear modulus distribution using a thick-shelled sphere approximation of the tumour and soft tissue ensemble. A hyperelastic material (plastisol) was identified to validate the proposed theory in a phantom setting. A balloon-catheter connected to a pressure sensor was used to replicate the stress generated from tumour pressure and growth while MRE data were acquired. The shear modulus anisotropy retrieved from the reconstructed elastography data confirmed the analytically predicted patterns at various levels of inflation. An alternative measure, combining the generated deformation and the local wave direction and independent of the reconstruction strategy, was also proposed to correlate the analytical findings with the stretch probed by the waves. Overall, this work demonstrates that MRE in combination with non-linear mechanics, is able to identify the apparent shear modulus variation arising from the strain generated by a growth within tissue, such as an idealised model of tumour. Investigation in real tissue represents the next step to further investigate the implications of endogenous forces in tissue characterisation through MRE.

Physical principles of morphogenesis in mushrooms

Mushroom species display distinctive morphogenetic features. For example, Amanita muscaria and Mycena chlorophos grow in a similar manner, their caps expanding outward quickly and then turning upward. However, only the latter finally develops a central depression in the cap. Here we use a mathematical approach unraveling the interplay between physics and biology driving the emergence of these two different morphologies. The proposed growth elastic model is solved analytically, mapping their shape evolution over time. Even if biological processes in both species make their caps grow turning upward, different physical factors result in different shapes. In fact, we show how for the relatively tall and big A. muscaria a central depression may be incompatible with the physical need to maintain stability against the wind. In contrast, the relatively short and small M. chlorophos is elastically stable with respect to environmental perturbations; thus, it may physically select a central depression to maximize the cap volume and the spore exposure. This work gives fully explicit analytic solutions highlighting the effect of the growth parameters on the morphological evolution, providing useful insights for novel bio-inspired material design.

Fractional Mathematical Oncology: On the potential of non-integer order calculus applied to interdisciplinary models

Mathematical Oncology investigates cancer-related phenomena through mathematical models as comprehensive as possible. Accordingly, an interdisciplinary approach involving concepts from biology to materials science can provide a deeper understanding of biological systems pertaining the disease. In this context, fractional calculus (also referred to as non-integer order) is a branch in mathematical analysis whose tools can describe complex phenomena comprising different time and space scales. Fractional-order models may allow a better description and understanding of oncological particularities, potentially contributing to decision-making in areas of interest such as tumor evolution, early diagnosis techniques and personalized treatment therapies. By following a phenomenological (i.e. mechanistic) approach, the present study surveys and explores different aspects of Fractional Mathematical Oncology, reviewing and discussing recent developments in view of their prospective applications.

Gap junctions amplify spatial variations in cell volume in proliferating tumor spheroids

Sustained proliferation is a significant driver of cancer progression. Cell-cycle advancement is coupled with cell size, but it remains unclear how multiple cells interact to control their volume in 3D clusters. In this study, we propose a mechano-osmotic model to investigate the evolution of volume dynamics within multicellular systems. Volume control depends on an interplay between multiple cellular constituents, including gap junctions, mechanosensitive ion channels, energy-consuming ion pumps, and the actomyosin cortex, that coordinate to manipulate cellular osmolarity. In connected cells, we show that mechanical loading leads to the emergence of osmotic pressure gradients between cells with consequent increases in cellular ion concentrations driving swelling. We identify how gap junctions can amplify spatial variations in cell volume within multicellular spheroids and, further, describe how the process depends on proliferation-induced solid stress. Our model may provide new insight into the role of gap junctions in breast cancer progression.

Biomechanical modelling of spinal tumour anisotropic growth

Biomechanical abnormalities of solid tumours involve stiffening of the tissue and accumulation of mechanical stresses. Both abnormalities affect cancer cell proliferation and invasiveness and thus, play a crucial role in tumour morphology and metastasis. Even though, it has been known for more than two decades that high mechanical stresses reduce cancer cell proliferation rates driving growth towards low-stress regions, most biomechanical models of tumour growth account for isotropic growth. This cannot be valid, however, in tumours that grow within multiple host tissues of different mechanical properties, such as the spine. In these cases, structural heterogeneity would result in anisotropic growth of tumours. To this end, we present a biomechanical, biphasic model for anisotropic growth of spinal tumours. The model that accounts for both the fluid and the solid phase of the tumour was used to predict the evolution of solid stress and interstitial fluid pressure in intramedullary spinal tumours and highlight the differences between isotropic and anisotropic growth. Varying the degree of anisotropy, we found considerable differences in the shape of the tumours, leading to tumours of more realistic ellipsoidal shapes.

Investigating the Physical Effects in Bacterial Therapies for Avascular Tumors

Tumor-targeting bacteria elicit anticancer effects by infiltrating hypoxic regions, releasing toxic agents and inducing immune responses. Although current research has largely focused on the influence of chemical and immunological aspects on the mechanisms of bacterial therapy, the impact of physical effects is still elusive. Here, we propose a mathematical model for the anti-tumor activity of bacteria in avascular tumors that takes into account the relevant chemo-mechanical effects. We consider a time-dependent administration of bacteria and analyze the impact of bacterial chemotaxis and killing rate. We show that active bacterial migration toward tumor hypoxic regions provides optimal infiltration and that high killing rates combined with high chemotactic values provide the smallest tumor volumes at the end of the treatment. We highlight the emergence of steady states in which a small population of bacteria is able to constrain tumor growth. Finally, we show that bacteria treatment works best in the case of tumors with high cellular proliferation and low oxygen consumption.

Computational models of melanoma

Genes, proteins, or cells influence each other and consequently create patterns, which can be increasingly better observed by experimental biology and medicine. Thereby, descriptive methods of statistics and bioinformatics sharpen and structure our perception. However, additionally considering the interconnectivity between biological elements promises a deeper and more coherent understanding of melanoma. For instance, integrative network-based tools and well-grounded inductive in silico research reveal disease mechanisms, stratify patients, and support treatment individualization. This review gives an overview of different modeling techniques beyond statistics, shows how different strategies align with the respective medical biology, and identifies possible areas of new computational melanoma research.

Growth and remodelling of living tissues: perspectives, challenges and opportunities

One of the most remarkable differences between classical engineering materials and living matter is the ability of the latter to grow and remodel in response to diverse stimuli. The mechanical behaviour of living matter is governed not only by an elastic or viscoelastic response to loading on short time scales up to several minutes, but also by often crucial growth and remodelling responses on time scales from hours to months. Phenomena of growth and remodelling play important roles, for example during morphogenesis in early life as well as in homeostasis and pathogenesis in adult tissues, which often adapt to changes in their chemo-mechanical environment as a result of ageing, diseases, injury or surgical intervention. Mechano-regulated growth and remodelling are observed in various soft tissues, ranging from tendons and arteries to the eye and brain, but also in bone, lower organisms and plants. Understanding and predicting growth and remodelling of living systems is one of the most important challenges in biomechanics and mechanobiology. This article reviews the current state of growth and remodelling as it applies primarily to soft tissues, and provides a perspective on critical challenges and future directions.

Experimental pancreatic cancer develops in soft pancreas: novel leads for an individualized diagnosis by ultrafast elasticity imaging

Rapid, easy and early pancreatic cancer diagnosis and therapeutic follow up continue to necessitate an increasing attention towards the development of effective treatment strategies for this lethal disease. The non invasive quantitative assessment of pancreatic heterogeneity is limited. Here, we report the development of a preclinical imaging protocol using ultrasonography and shear wave technology in an experimental in situ pancreatic cancer model to measure the evolution of pancreatic rigidity.

Methods: Intrapancreatic tumors were genetically induced by mutated Kras and p53 in KPC mice. We evaluated the feasiblity of a live imaging protocol by assessing pancreas evolution with Aixplorer technology accross 36 weeks. Lethality induced by in situ pancreatic cancer was heterogeneous in time.

Results: The developed method successfully detected tumor mass from 26 weeks onwards at minimal 0.029 cm³ size. Elastography measurements using shear wave methodology had a wide detection range from 4.7kPa to 166.1kPa. Protumorigenic mutations induced a significant decrease of the rigidity of pancreatic tissue before tumors developed in correlation with the detection of senescent marker p16-positive cells. An intratumoral increased rigidity was quantified and found surprisingly heterogeneous. Tumors also presented a huge inter-individual heterogeneity in their rigidity parameters; tumors with low and high rigidity at detection evolve very heterogeneously in their rigidity parameters, as well as in their volume. Increase in rigidity in tumors detected by ultrafast elasticity imaging coincided with detection of tumors by echography and with the detection of the inflammatory protumoral systemic condition by non invasive follow-up and of collagen fibers by post-processing tumoral IHC analysis.

Conclusion: Our promising results indicate the potential of the shear wave elastography to support individualization of diagnosis in this most aggressive disease.

On the Impact of Chemo-Mechanically Induced Phenotypic Transitions in Gliomas

Tumor microenvironment is a critical player in glioma progression, and novel therapies for its targeting have been recently proposed. In particular, stress-alleviation strategies act on the tumor by reducing its stiffness, decreasing solid stresses and improving blood perfusion. However, these microenvironmental changes trigger chemo-mechanically induced cellular phenotypic transitions whose impact on therapy outcomes is not completely understood. In this work we analyze the effects of mechanical compression on migration and proliferation of glioma cells. We derive a mathematical model of glioma progression focusing on cellular phenotypic plasticity. Our results reveal a trade-off between tumor infiltration and cellular content as a consequence of stress-alleviation approaches. We discuss how these novel findings increase the current understanding of glioma/microenvironment interactions and can contribute to new strategies for improved therapeutic outcomes.

On the existence of elastic minimizers for initially stressed materials

A soft solid is said to be initially stressed if it is subjected to a state of internal stress in its unloaded reference configuration. In physical terms, its stored elastic energy may not vanish in the absence of an elastic deformation, being also dependent on the spatial distribution of the underlying material inhomogeneities. Developing a sound mathematical framework to model initially stressed solids in nonlinear elasticity is key for many applications in engineering and biology. This work investigates the links between the existence of elastic minimizers and the constitutive restrictions for initially stressed materials subjected to finite deformations. In particular, we consider a subclass of constitutive responses in which the strain energy density is taken as a scalar-valued function of both the deformation gradient and the initial stress tensor. The main advantage of this approach is that the initial stress tensor belongs to the group of divergence-free symmetric tensors satisfying the boundary conditions in any given reference configuration. However, it is still unclear which physical restrictions must be imposed for the well-posedness of this elastic problem. Assuming that the constitutive response depends on the choice of the reference configuration only through the initial stress tensor, under given conditions we prove the local existence of a relaxed state given by an implicit tensor function of the initial stress distribution. This tensor function is generally not unique, and can be transformed according to the symmetry group of the material at fixed initial stresses. These results allow one to extend Ball's existence theorem of elastic minimizers for the proposed constitutive choice of initially stressed materials. This article is part of the theme issue 'Rivlin's legacy in continuum mechanics and applied mathematics'.

An adjoint-based method for a linear mechanically-coupled tumor model: Application to estimate the spatial variation of murine glioma growth based on diffusion weighted magnetic resonance imaging

We present an efficient numerical method to quantify the spatial variation of glioma growth based on subject-specific medical images using a mechanically-coupled tumor model. The method is illustrated in a murine model of glioma in which we consider the tumor as a growing elastic mass that continuously deforms the surrounding healthy-appearing brain tissue. As an inverse parameter identification problem, we quantify the volumetric growth of glioma and the growth component of deformation by fitting the model predicted cell density to the cell density estimated using the diffusion-weighted magnetic resonance imaging (DW-MRI) data. Numerically, we developed an adjoint-based approach to solve the optimization problem. Results on a set of experimentally measured, in vivo rat glioma data indicate good agreement between the fitted and measured tumor area and suggest a wide variation of in-plane glioma growth with the growth-induced Jacobian ranging from 1.0 to 6.0.

High-Frequency Mechanical Properties of Tumors Measured by Brillouin Light Scattering

The structure of tumors can be recapitulated as an elastic frame formed by the connected cytoskeletons of the cells invaded by interstitial and intracellular fluids. The low-frequency mechanics of this poroelastic system, dictated by the elastic skeleton only, control tumor growth, penetration of therapeutic agents, and invasiveness. The high-frequency mechanical properties containing the additional contribution of the internal fluids have also been posited to participate in tumor progression and drug resistance, but they remain largely unexplored. Here we use Brillouin light scattering to produce label-free images of tumor microtissues based on the high-frequency viscoelastic modulus as a contrast mechanism. In this regime, we demonstrate that the modulus discriminates between tissues with altered tumorigenic properties. Our micrometric maps also reveal that the modulus is heterogeneously altered across the tissue by drug therapy, revealing a lag of efficacy in the core of the tumor. Exploiting high-frequency poromechanics should advance present theories based on viscoelasticity and lead to integrated descriptions of tumor response to drugs.

Computational Modeling in Liver Surgery

The need for extended liver resection is increasing due to the growing incidence of liver tumors in aging societies. Individualized surgical planning is the key for identifying the optimal resection strategy and to minimize the risk of postoperative liver failure and tumor recurrence. Current computational tools provide virtual planning of liver resection by taking into account the spatial relationship between the tumor and the hepatic vascular trees, as well as the size of the future liver remnant. However, size and function of the liver are not necessarily equivalent. Hence, determining the future liver volume might misestimate the future liver function, especially in cases of hepatic comorbidities such as hepatic steatosis. A systems medicine approach could be applied, including biological, medical, and surgical aspects, by integrating all available anatomical and functional information of the individual patient. Such an approach holds promise for better prediction of postoperative liver function and hence improved risk assessment. This review provides an overview of mathematical models related to the liver and its function and explores their potential relevance for computational liver surgery. We first summarize key facts of hepatic anatomy, physiology, and pathology relevant for hepatic surgery, followed by a description of the computational tools currently used in liver surgical planning. Then we present selected state-of-the-art computational liver models potentially useful to support liver surgery. Finally, we discuss the main challenges that will need to be addressed when developing advanced computational planning tools in the context of liver surgery.

A nonlinear poroelastic theory of solid tumors with glycosaminoglycan swelling

Mechanics plays a crucial role in the growth, development, and therapeutics of tumors. In this paper, a nonlinear poroelastic theory is established to describe the mechanical behaviors of solid tumors. The free-swollen state of a tumor is chosen as the reference state, which enables us to avoid pursuing a dry and stress-free state that is hard to achieve for living tissues. Our results reveal that the compression resistance of a tumor is primarily attributed to glycosaminoglycan (GAG) swelling, and the compactness of cell aggregates is found to affect tumor consolidation. Over-expressed GAGs and dense cell aggregates can stiffen the tumor, a remodeling mechanism that makes the tumor with higher elastic modulus than its surrounding host tissues. Glycosaminoglycan chains also influence the transient mechanical response of the tumor by modulating the tissue permeability. The theoretical results show good agreement with relevant experimental observations. This study may not only deepen our understanding of tumorigenesis but also provide cues for developing novel anticancer strategies.