Some implications of Scale Relativity theory in avascular stages of growth of solid tumors in the presence of an immune system response

We present a traveling-wave analysis of a reduced mathematical model describing the growth of a solid tumor in the presence of an immune system response in the framework of Scale Relativity theory. Attention is focused upon the attack of tumor cells by tumor-infiltrating cytotoxic lymphocytes (TICLs), in a small multicellular tumor, without necrosis and at some stage prior to (tumor-induced) angiogenesis. For a particular choice of parameters, the underlying system of partial differential equations is able to simulate the well-documented phenomenon of cancer dormancy and propagation of a perturbation in the tumor cell concentration by cnoidal modes, by depicting spatially heterogeneous tumor cell distributions that are characterized by a relatively small total number of tumor cells. This behavior is consistent with several immunomorphological investigations. Moreover, the alteration of certain parameters of the model is enough to induce soliton like modes and soliton packets into the system, which in turn result in tumor invasion in the form of a standard traveling wave. In the same framework of Scale Relativity theory, a very important feature of malignant tumors also results, that even in avascular stages they might propagate and invade healthy tissues, by means of a diffusion on a Newtonian fluid.     

Computational Modeling of 3D Tumor Growth and Angiogenesis for Chemotherapy Evaluation

Solid tumors develop abnormally at spatial and temporal scales, giving rise to biophysical barriers that impact anti-tumor chemotherapy. This may increase the expenditure and time for conventional drug pharmacokinetic and pharmacodynamic studies. In order to facilitate drug discovery, we propose a mathematical model that couples three-dimensional tumor growth and angiogenesis to simulate tumor progression for chemotherapy evaluation. This application-oriented model incorporates complex dynamical processes including cell- and vascular-mediated interstitial pressure, mass transport, angiogenesis, cell proliferation, and vessel maturation to model tumor progression through multiple stages including tumor initiation, avascular growth, and transition from avascular to vascular growth. Compared to pure mechanistic models, the proposed empirical methods are not only easy to conduct but can provide realistic predictions and calculations. A series of computational simulations were conducted to demonstrate the advantages of the proposed comprehensive model. The computational simulation results suggest that solid tumor geometry is related to the interstitial pressure, such that tumors with high interstitial pressure are more likely to develop dendritic structures than those with low interstitial pressure.     

The role of stress in the growth of a multicell spheroid

Rather recent experimental results demonstrate the non–negligible role of mechanical stress in the growth of a multicell spheroid. In this paper we discuss a theoretical framework for volumetric growth suitable for modeling the growth of soft tissues exhibiting the properties of a solid. After a proper kinematic decomposition, balance equations for mass, momentum and energy are discussed together with constitutive relationships. The mathematical model is then applied to avascular tumor growth. We show by numerical simulation that, under assumption of spherical symmetry, the mathematical model is able to reproduce the experimental data with a satisfying qualitative agreement.     

Evolution of osmotic pressure in solid tumors

The mechanical microenvironment of solid tumors includes both fluid and solid stresses. These stresses play a crucial role in cancer progression and treatment and have been analyzed rigorously both mathematically and experimentally. The magnitude and spatial distribution of osmotic pressures in tumors, however, cannot be measured experimentally and to our knowledge there is no mathematical model to calculate osmotic pressures in the tumor interstitial space. In this study, we developed a triphasic biomechanical model of tumor growth taking into account not only the solid and fluid phase of a tumor, but also the transport of cations and anions, as well as the fixed charges at the surface of the glycosaminoglycan chains. Our model predicts that the osmotic pressure is negligible compared to the interstitial fluid pressure for values of glycosaminoglycans (GAGs) taken from the literature for sarcomas, melanomas and adenocarcinomas. Furthermore, our results suggest that an increase in the hydraulic conductivity of the tumor, increases considerably the intratumoral concentration of free ions and thus, the osmotic pressure but it does not reach the levels of the interstitial fluid pressure.     

Elastic Free Energy Drives the Shape of Prevascular Solid Tumors

It is well established that the mechanical environment influences cell functions in health and disease. Here, we address how the mechanical environment influences tumor growth, in particular, the shape of solid tumors. In an in vitro tumor model, which isolates mechanical interactions between cancer tumor cells and a hydrogel, we find that tumors grow as ellipsoids, resembling the same, oft-reported observation of in vivo tumors. Specifically, an oblate ellipsoidal tumor shape robustly occurs when the tumors grow in hydrogels that are stiffer than the tumors, but when they grow in more compliant hydrogels they remain closer to spherical in shape. Using large scale, nonlinear elasticity computations we show that the oblate ellipsoidal shape minimizes the elastic free energy of the tumor-hydrogel system. Having eliminated a number of other candidate explanations, we hypothesize that minimization of the elastic free energy is the reason for predominance of the experimentally observed ellipsoidal shape. This result may hold significance for explaining the shape progression of early solid tumors in vivo and is an important step in understanding the processes underlying solid tumor growth.     

Does tumor growth follow a "universal law"?

A general model for the ontogenetic growth of living organisms has been recently proposed. Here we investigate the extension of this model to the growth of solid malignant tumors. A variety of in vitro and in vivo data are analysed and compared with the prediction of a "universal" law, relating properly rescaled tumor masses and tumor growth times. The results support the notion that tumor growth follows such a universal law. Several important implications of this finding are discussed, including its relevance for tumor metastasis and recurrence, cell turnover rates, angiogenesis and invasion.     

The evolving concept of cancer and metastasis stem cells

The cancer stem cell (CSC) concept, which arose more than a decade ago, proposed that tumor growth is sustained by a subpopulation of highly malignant cancerous cells. These cells, termed CSCs, comprise the top of the tumor cell hierarchy and have been isolated from many leukemias and solid tumors. Recent work has discovered that this hierarchy is embedded within a genetically heterogeneous tumor, in which various related but distinct subclones compete within the tumor mass. Thus, genetically distinct CSCs exist on top of each subclone, revealing a highly complex cellular composition of tumors. The CSC concept has therefore evolved to better model the complex and highly dynamic processes of tumorigenesis, tumor relapse, and metastasis.     

Evidence for the isolation,growth, and characterization of malignant cells in primary cultures of human tumors

Isolation and growth of malignant cells from solid tumors have often met with disappointing results. Consequently, we have developed a cell culture methodology based on ex vivo explantation of tumor tissue, with subsequent monolayer cell outgrowth. In an attempt to assess methods for detection of malignant cells in these cultures, we analyzed and compared the results of cytopathology, growth in soft agar, and detection of telomerase activity with those of standard immunohistochemistry (IHC) techniques for the detection of cytokeratins, tumor marker p53, and proliferation marker Ki-67. The sensitivity of detection of malignant cells was 85% (22/26) for cytopathological examination, 30% (3/10) for soft agar growth, and 100% (12/12) for detection of telomerase activity. From these data, we concluded that both cytopathological examination and assessment of telomerase activity contribute to the detection of malignant cells in primary cultures of human solid tumors, whereas growth in soft agar was not a good indicator of malignant cells. Although not specific for malignant cells per se, IHC detection for epithelial cell cytokeratins showed a high degree of sensitivity (100%, 23/23), whereas the sensitivity for detection of tumor marker p53 and proliferation marker Ki-67 was 30% (7/23) and 70% (16/23), respectively. These data also provide proof that malignant tumor cells, derived from a diverse number of human solid tumors, can be isolated and grown in primary cell culture.     

The strain energy function in axial plant growth

A model for axial plant growth is formulated based on conservation of energy. The model derivation assumes that a strain energy function exists to describe the dissipation of potential energy associated with water uptake, mechanical deformation, and biosynthesis during growth. The derivation does not, however, make any further assumption on the mathematical form of this constitutive relation. The model is employed to investigate possible forms of the strain energy function as applied to steady root growth. Solutions of the nonlinear partial differential equations governing growth are given for cases when the third derivative of the strain energy function is >, <, or =0. These three cases encompass a multitude of mathematical forms of the strain energy function. The resulting solutions are compared with the realization of steady axial root growth. The results of this analysis indicate that a quadratic form of the strain energy function best described steady growth. This conclusion is consistent with previous assumptions on the form of constitutive relations for growth, and allows further interpretation on the water relations, mechanical, and biosynthetic energies associated with plant growth.Research support provided by state and federal funds appropriated to the OSU/OARDC. Journal article no. 12–88     

Optimal and suboptimal protocols for a class of mathematical models of tumor anti-angiogenesis

Tumor anti-angiogenesis is a cancer treatment approach that aims at preventing the primary tumor from developing its own vascular network needed for further growth. In this paper the problem of how to schedule an a priori given amount of angiogenic inhibitors in order to minimize the tumor volume is considered for three related mathematical formulations of a biologically validated model developed by Hahnfeldt et al. [1999. Tumor development under angiogenic signalling: a dynamical theory of tumor growth, treatment response, and postvascular dormancy. Cancer Res. 59, 4770-4775]. Easily implementable piecewise constant protocols are compared with the mathematically optimal solutions. It is shown that a constant dosage protocol with rate given by the averaged optimal control is an excellent suboptimal protocol for the original model that achieves tumor values that lie within 1% of the theoretically optimal values. It is also observed that the averaged optimal dose is decreasing as a function of the initial tumor volume.     

Incorporating observability thresholds of tumors into the two-stage carcinogenesis model

This paper discusses a general way of incorporating the growth kinetics of malignant tumors with the two-stage carcinogenesis model. The model is presented using time-homogeneous rate parameters. In that case, the differential equations comprising the model are straightforward to solve using standard numerical techniques and software. An extension of the method to time-dependent rate parameters is included in Appendix A. Allowing the rate parameters to be time-dependent does incur computational cost. An expression is given for the expected time without visible tumor, a generalization of the expected time to an observable tumor that includes the possibility of tumor regression. The model is illustrated using incidental liver tumor data in control rats from NTP rodent carcinogenicity studies, using linear birth-death kinetics of tumors combined with a non-absorbing detection limit. The approach is also shown to be potentially useful with tumor observability thresholds having more complicated features.     

Computational challenges of tumor spheroid modeling

The speed and the versatility of today's computers open up new opportunities to simulate complex biological systems. Here we review a computational approach recently proposed by us to model large tumor cell populations and spheroids, and we put forward general considerations that apply to any fine-grained numerical model of tumors. We discuss ways to bypass computational limitations and discuss our incremental approach, where each step is validated by experimental observations on a quantitative basis. We present a few results on the growth of tumor cells in closed and open environments and of tumor spheroids. This study suggests new ways to explore the initial growth phase of solid tumors and to optimize antitumor treatments.     

CD9 expression in solid non-neuroepithelial tumors and infiltrative astrocytic tumors.

The tetraspan membrane protein CD9 is normally expressed in the mature myelin sheath and is believed to suppress the metastatic potential of certain human tumors. In this study we identified CD9 in a variety of brain tumors by immunohistochemical (IHC) and immunoblotting analyses. We examined 96 tumor samples and three glioma cell lines in addition to a murine brain tumor model of transplanted glioma cells in CD9-deficient mice and control mice. CD9 was expressed not only in solid non-neuroepithelial tumors but also in infiltrative malignant neuroepithelial tumors. Among the neuroepithelial tumors, high-grade astrocytic tumors, including glioblastomas and anaplastic astrocytomas, showed higher immunoreactivity than low-grade cerebral astrocytomas. Thus, CD9 expression in astrocytic tumors correlated with their malignancy. In the murine brain tumor model, transplanted glioma cells were shown to grow and spread through myelinated areas irrespective of the presence or absence of CD9 expression in the recipient's brain. These results indicate that the CD9 expression of astrocytic tumors plays a significant role in the malignancy independent of CD9 expression in the surrounding tissue. This might be explained by the observation that the CD9 molecule is associated with a mitogenic factor, membrane-anchored heparin-binding epidermal growth factor, which is known to be upregulated in malignant gliomas.     

Blocking tumor cell eicosanoid synthesis by GP x 4 impedes tumor growth and malignancy

Using tumor cell-restricted overexpression of glutathione peroxidase 4 (GP x 4), we investigated the contribution of tumor cell eicosanoids to solid tumor growth and malignant progression in two tumor models differing in tumorigenic potential. By lowering cellular lipid hydroperoxide levels, GP x 4 inhibits cyclooxygenase (COX) and lipoxygenase (LOX) activities. GP x 4 overexpression drastically impeded solid tumor growth of weakly tumorigenic L929 fibrosarcoma cells, whereas B16BL6 melanoma solid tumor growth was unaffected. Yet, GP x 4 overexpression did markedly increase the sensitivity of B16BL6 tumors to angio-destructive TNF-alpha therapy and abolished the metastatic lung colonizing capacity of B16BL6 cells. Furthermore, the GP x 4-mediated suppression of tumor cell prostaglandin E(2) (PGE(2)) production impeded the induction of COX-2 expression by the tumor stress conditions hypoxia and inflammation. Thus, our results reflect a PGE(2)-driven positive feedback loop for COX-2 expression in tumor cells. This was further supported by the restoration of COX-2 induction capacity of GP x 4-overexpressing L929 tumor cells when cultured in the presence of exogenous PGE(2). Thus, although COX-2 expression and eicosanoid production may be enabled by PGE(2) from the tumor microenvironment, our results demonstrate the predominant tumor cell origin of protumoral eicosanoids, promoting solid tumor growth of weakly tumorigenic tumors and malignant progression of strongly tumorigenic tumors.     

Physical aspects of cancer invasion

Invasiveness, one of the hallmarks of tumor progression, represents the tumor's ability to expand into the host tissue by means of several complex biochemical and biomechanical processes. Since certain aspects of the problem present a striking resemblance with well-known physical mechanisms, such as the mechanical insertion of a solid inclusion in an elastic material specimen (G Eaves 1973 The invasive growth of malignant tumours as a purely mechanical process J. Pathol. 109 233; C Guiot, N Pugno and P P Delsanto 2006 Elastomechanical model of tumor invasion Appl. Phys. Lett. 89 233901) or a water drop impinging on a surface (C Guiot, P P Delsanto and T S Deisboeck 2007 Morphological instability and cancer invasion: a 'splashing water drop' analogy Theor. Biol. Med. Model 4 4), we propose here an analogy between these physical processes and a cancer system's invasive branching into the surrounding tissue. Accounting for its solid and viscous properties, we then arrive, as a unifying model, to an analogy with a granular solid. While our model has been explicitly formulated for multicellular tumor spheroids in vitro, it should also contribute to a better understanding of tumor invasion in vivo.