Using Fractal Geometry and Universal Growth Curves as Diagnostics for Comparing Tumor Vasculature and Metabolic Rate With Healthy Tissue and for Predicting Responses to Drug Therapies

Efficient vasculature investment in tissues can be determined without global information

Cells are the fundamental building blocks of organs and tissues. Information and mass flow through cellular contacts in these structures is vital for the orchestration of organ function. Constraints imposed by packing and cell immobility limit intercellular communication, particularly as organs and organisms scale up to greater sizes. In order to transcend transport limitations, delivery systems including vascular and respiratory systems evolved to facilitate the movement of matter and information. The construction of these delivery systems has an associated cost, as vascular elements do not perform the metabolic functions of the organs they are part of. This study investigates a fundamental trade-off in vascularization in multicellular tissues: the reduction of path lengths for communication versus the cost associated with producing vasculature. Biologically realistic generative models, using multicellular templates of different dimensionalities, revealed a limited advantage to the vascularization of two-dimensional tissues. Strikingly, scale-free improvements in transport efficiency can be achieved even in the absence of global knowledge of tissue organization. A point of diminishing returns in the investment of additional vascular tissue to the increased reduction of path length in 2.5- and three-dimensional tissues was identified. Applying this theory to experimentally determined biological tissue structures, we show the possibility of a co-dependency between the method used to limit path length and the organization of cells it acts upon. These results provide insight as to why tissues are or are not vascularized in nature, the robustness of developmental generative mechanisms and the extent to which vasculature is advantageous in the support of organ function.

Nanoparticle Optimization for Enhanced Targeted Anticancer Drug Delivery

Nanoparticle (NP)-based drug delivery is a promising method to increase the therapeutic index of anticancer agents with low median toxic dose. The delivery efficiency, corresponding to the fraction of the injected NPs that adhere to the tumor site, depends on NP size a and aspect ratio AR. Values for these variables are currently chosen empirically, which may not result in optimal targeted drug delivery. This study applies rigorous optimization to the design of NPs. A preliminary investigation revealed that delivery efficiency increases monotonically with a and AR. However, maximizing a and AR results in nonuniform drug distribution, which impairs tumor regression. Therefore, a multiobjective optimization (MO) problem is formulated to quantify the trade-off between NPs accumulation and distribution. The MO is solved using the derivative-free mesh adaptive direct search algorithm. Theoretically, the Pareto-optimal set consists of an infinite number of mathematically equivalent solutions to the MO problem. However, interesting design solutions can be identified subjectively, e.g., the ellipsoid with a major axis of 720 nm and an aspect ratio of 7.45, as the solution closest to the utopia point. The MO problem formulation is then extended to optimize NP biochemical properties: ligand-receptor binding affinity and ligand density. Optimizing physical and chemical properties simultaneously results in optimal designs with reduced NP sizes and thus enhanced cellular uptake. The presented study provides an insight into NP structures that have potential for producing desirable drug delivery.

Shapes and distributions of soft tissue scatterers

What causes scattering of ultrasound from normal soft tissues such as the liver, thyroid, and prostate? Commonly, the answer is formulated around the properties of spherical scatterers, related to cellular shapes and sizes. However, an alternative view is that the closely packed cells forming the tissue parenchyma create the reference media, and the long cylindrical-shaped fluid vessels serve as the scattering sites. Under a weak scattering or Born approximation for the extracellular fluid in the vessels, and assuming an isotropic distribution of cylindrical channels across a wide range of diameters, consistent with a fractal branching pattern, some simple predictions can be made about the nature of backscatter as a function of frequency in soft tissues. Specifically, a number of plausible shapes would predict that backscatter increases as a power law of frequency, where the power law is determined by the function governing the number density of the vessels versus diameter. These results are compared with some historical models developed over the last 100 years in scattering theory and point to the need for higher spatial resolution and higher bandwidths to obtain more precise measures of the key parameters in normal tissues, and to better identify the dominant structures responsible for backscatter in everyday clinical imaging.

Energy oversupply to tissues: a single mechanism possibly underlying multiple cancer risk factors

BACKGROUND AND OBJECTIVES

Several major risk factors for cancer involve vascular oversupply of energy to affected tissues. These include obesity, diabetes and chronic inflammation. Here, we propose a potential mechanistic explanation for the association between energy oversupply and cancer risk, which we call the metabolic cancer suppression hypothesis: We hypothesize that oncogenesis is normally suppressed by organismal physiology that regulates and strictly limits normal energy supply to somatic cells, and that this protection is removed by abnormal oversupply of energy.

METHODOLOGY

We evaluate this hypothesis using a computational model of somatic cell evolution to simulate experimental manipulation of the vascular energy supply to a tissue. The model simulates the evolutionary dynamics of somatic cells during oncogenesis.

RESULTS

In our simulation experiment, we found that under plausible biological assumptions, elevated energy supply to a tissue led to the evolution of elevated energy uptake by somatic cells, leading to the rapid evolution of both defining traits of cancer cells: hyperproliferation, and tissue invasion.

CONCLUSIONS AND IMPLICATIONS

Our results support the hypothesis of metabolic cancer suppression, suggesting that vascular oversupply of energetic resources to somatic cells removes normal energetic limitations on cell proliferation, and that this accelerates cellular evolution toward cancer. Various predictions of this hypothesis are amenable to empirical testing, and have promising implications for translational research toward clinical cancer prevention.

Clinical biomarkers for cancer recognition and prevention: A novel approach with optical measurements

Cancer is the most important cause of death worldwide, and early cancer detection is the most fundamental factor for efficacy of treatment, prognosis, and increasing survival rate. Over the years great effort has been devoted to discovering and testing new biomarkers that can improve its diagnosis, especially at an early stage. Here we report the potential usefulness of new, easily applicable, non-invasive and relatively low-cost clinical biomarkers, based on abnormalities of oral mucosa spectral reflectance and fractal geometry of the vascular networks in several different tissues, for identification of hereditary non-polyposis colorectal cancer carriers as well for detection of other tumors, even at an early stage. In the near future the methodology/technology of these procedures should be improved, thus making possible their applicability worldwide as screening tools for early recognition and prevention of cancer.

Modeling of Interactions between Cancer Stem Cells and their Microenvironment: Predicting Clinical Response

Mathematical models of cancer stem cells are useful in translational cancer research for facilitating the understanding of tumor growth dynamics and for predicting treatment response and resistance to combined targeted therapies. In this chapter, we describe appealing aspects of different methods used in mathematical oncology and discuss compelling questions in oncology that can be addressed with these modeling techniques. We describe a simplified version of a model of the breast cancer stem cell niche, illustrate the visualization of the model, and apply stochastic simulation to generate full distributions and average trajectories of cell type populations over time. We further discuss the advent of single-cell data in studying cancer stem cell heterogeneity and how these data can be integrated with modeling to advance understanding of the dynamics of invasive and proliferative populations during cancer progression and response to therapy.

Fractal dimension and universality in avascular tumor growth

For years, the comprehension of the tumor growth process has been intriguing scientists. New research has been constantly required to better understand the complexity of this phenomenon. In this paper, we propose a mathematical model that describes the properties, already known empirically, of avascular tumor growth. We present, from an individual-level (microscopic) framework, an explanation of some phenomenological (macroscopic) aspects of tumors, such as their spatial form and the way they develop. Our approach is based on competitive interaction between the cells. This simple rule makes the model able to reproduce evidence observed in real tumors, such as exponential growth in their early stage followed by power-law growth. The model also reproduces (i) the fractal-space distribution of tumor cells and (ii) the universal growth behavior observed in both animals and tumors. Our analyses suggest that the universal similarity between tumor and animal growth comes from the fact that both can be described by the same dynamic equation-the Bertalanffy-Richards model-even if they do not necessarily share the same biological properties.

Asymmetries arising from the space-filling nature of vascular networks

Cardiovascular networks span the body by branching across many generations of vessels. The resulting structure delivers blood over long distances to supply all cells with oxygen via the relatively short-range process of diffusion at the capillary level. The structural features of the network that accomplish this density and ubiquity of capillaries are often called space-filling. There are multiple strategies to fill a space, but some strategies do not lead to biologically adaptive structures by requiring too much construction material or space, delivering resources too slowly, or using too much power to move blood through the system. We empirically measure the structure of real networks (18 humans and 1 mouse) and compare these observations with predictions of model networks that are space-filling and constrained by a few guiding biological principles. We devise a numerical method that enables the investigation of space-filling strategies and determination of which biological principles influence network structure. Optimization for only a single principle creates unrealistic networks that represent an extreme limit of the possible structures that could be observed in nature. We first study these extreme limits for two competing principles, minimal total material and minimal path lengths. We combine these two principles and enforce various thresholds for balance in the network hierarchy, which provides a novel approach that highlights the tradeoffs faced by biological networks and yields predictions that better match our empirical data.

Fractal dimension and lacunarity of tumor microscopic images as prognostic indicators of clinical outcome in early breast cancer

AIM

Research in the field of breast cancer outcome prognosis has been focused on molecular biomarkers, while neglecting the discovery of novel tumor histology structural clues. We thus aimed to improve breast cancer prognosis by fractal analysis of tumor histomorphology.

PATIENTS & METHODS

This retrospective study included 92 breast cancer patients without systemic treatment.

RESULTS

Fractal dimension and lacunarity of the breast tumor microscopic histology possess prognostic value comparable to the major clinicopathological prognostic parameters.

CONCLUSION

Fractal analysis was performed for the first time on routinely produced archived pan-tissue stained primary breast tumor sections, indicating its potential for clinical use as a simple and cost-effective prognostic indicator of distant metastasis risk to complement the molecular approaches for cancer risk prognosis.

Models for Allee effect based on physical principles

We propose some models of single species with Allee effect based on physical principles. A method is used to obtain the expression for the per capita growth rate (a macroscopic information) starting from the characteristics of interactions between the individuals (a microscopic information). We assume that the agents in a model of a single species interact according to the distance between them. Moreover these agents must (i) cooperate with their nearest neighbors, (ii) compete with neighbors at an intermediate distance, and (iii) being indifferent to those who are far away. Using these assumptions and based on fundamental physical principles, we find what appears to be a new way of establishing models of single species with Allee effect.

Testing Foundations of Biological Scaling Theory Using Automated Measurements of Vascular Networks

Scientists have long sought to understand how vascular networks supply blood and oxygen to cells throughout the body. Recent work focuses on principles that constrain how vessel size changes through branching generations from the aorta to capillaries and uses scaling exponents to quantify these changes. Prominent scaling theories predict that combinations of these exponents explain how metabolic, growth, and other biological rates vary with body size. Nevertheless, direct measurements of individual vessel segments have been limited because existing techniques for measuring vasculature are invasive, time consuming, and technically difficult. We developed software that extracts the length, radius, and connectivity of in vivo vessels from contrast-enhanced 3D Magnetic Resonance Angiography. Using data from 20 human subjects, we calculated scaling exponents by four methods—two derived from local properties of branching junctions and two from whole-network properties. Although these methods are often used interchangeably in the literature, we do not find general agreement between these methods, particularly for vessel lengths. Measurements for length of vessels also diverge from theoretical values, but those for radius show stronger agreement. Our results demonstrate that vascular network models cannot ignore certain complexities of real vascular systems and indicate the need to discover new principles regarding vessel lengths.

Vascular networks distribute resources and constrain metabolic rate. Founded on a few key principles, biological scaling theories predict characteristic patterns for vascular networks as they branch from large to small vessels. These theories also predict seemingly unrelated phenomena, such as size limits on mammals. However, vascular networks are difficult to measure because there are billions of vessels that range in size from meters to micrometers. To test the foundations of biological scaling theories, we developed software that quickly measures thousands of in vivo vessels based on MRI. Data for vessel radii match predicted patterns but lengths do not. Our work suggests the need for new theoretical principles and should facilitate comparisons across organisms, spatial scales, and healthy and diseased tissue.