Incorporating spatial dependence into a multicellular tumor spheroid growth model

Recent models for organism and tumor growth yield simple scaling laws based on conservation of energy. Here, we extend such a model to include spatial dependence to model necrotic core formation. We adopt the allometric equation for tumor volume with a reaction-diffusion equation for nutrient concentration. In addition, we assume that the total metabolic energy and average cellular metabolic rate depend on nutrient concentration in a Michaelis-Menten-like manner. From experimental results, we relate the necrotic volume to nutrient consumption and estimate both the time and nutrient concentration at necrotic core formation. Based on experimental results, we demand that the necrotic core radius varies linearly with tumor radius after core formation and extend the equations for tumor volume and nutrient concentration to the postnecrotic core regime. In particular, we obtain excellent agreement with experimental data and the final steady-state viable rim thickness.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/87333/2/124701_1.pd

Haematopoietic stem cells do not asymmetrically segregate chromosomes or retain BrdU

Stem cells are proposed to segregate chromosomes asymmetrically during self-renewing divisions so that older ('immortal') DNA strands are retained in daughter stem cells whereas newly synthesized strands segregate to differentiating cells(1-6). Stem cells are also proposed to retain DNA labels, such as 5-bromo-2-deoxyuridine (BrdU), either because they segregate chromosomes asymmetrically or because they divide slowly(5,7-9). However, the purity of stem cells among BrdU-label-retaining cells has not been documented in any tissue, and the 'immortal strand hypothesis' has not been tested in a system with definitive stem cell markers. Here we tested these hypotheses in haematopoietic stem cells (HSCs), which can be highly purified using well characterized markers. We administered BrdU to newborn mice, mice treated with cyclophosphamide and granulocyte colony-stimulating factor, and normal adult mice for 4 to 10 days, followed by 70 days without BrdU. In each case, less than 6% of HSCs retained BrdU and less than 0.5% of all BrdU-retaining haematopoietic cells were HSCs, revealing that BrdU has poor specificity and poor sensitivity as an HSC marker. Sequential administration of 5-chloro-2-deoxyuridine and 5-iodo-2-deoxyuridine indicated that all HSCs segregate their chromosomes randomly. Division of individual HSCs in culture revealed no asymmetric segregation of the label. Thus, HSCs cannot be identified on the basis of BrdU-label retention and do not retain older DNA strands during division, indicating that these are not general properties of stem cells.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/62821/1/nature06115.pd

Topography of Extracellular Matrix Mediates Vascular Morphogenesis and Migration Speeds in Angiogenesis

The extracellular matrix plays a critical role in orchestrating the events necessary for wound healing, muscle repair, morphogenesis, new blood vessel growth, and cancer invasion. In this study, we investigate the influence of extracellular matrix topography on the coordination of multi-cellular interactions in the context of angiogenesis. To do this, we validate our spatio-temporal mathematical model of angiogenesis against empirical data, and within this framework, we vary the density of the matrix fibers to simulate different tissue environments and to explore the possibility of manipulating the extracellular matrix to achieve pro- and anti-angiogenic effects. The model predicts specific ranges of matrix fiber densities that maximize sprout extension speed, induce branching, or interrupt normal angiogenesis, which are independently confirmed by experiment. We then explore matrix fiber alignment as a key factor contributing to peak sprout velocities and in mediating cell shape and orientation. We also quantify the effects of proteolytic matrix degradation by the tip cell on sprout velocity and demonstrate that degradation promotes sprout growth at high matrix densities, but has an inhibitory effect at lower densities. Our results are discussed in the context of ECM targeted pro- and anti-angiogenic therapies that can be tested empirically

Vascular tumor growth and treatment: Consequences of polyclonality, competition and dynamic vascular support

 A mathematical model is presented to describe the evolution of a vascular tumor in response to traditional chemotherapeutic treatment. Particular attention is paid to the effects of a dynamic vascular support system in a tumor comprised of competing cell populations that differ in proliferation rates and drug susceptibility. The model consists of a system of partial differential equations governing intratumoral drug concentration, cancer cell density, and blood vessel density. The balance between cell proliferation and death along with vessel production and destruction within the tumor generates a velocity field which drives the expansion or regression of the neoplasm. Radially symmetric solutions are obtained for the case when only one cell type is present and when the proportion of the tumor occupied by blood vessels remains constant. The stability of these solutions to asymmetric perturbations and to a small semi-drug resistant cell population is then investigated. The analysis shows that drug concentrations which are sufficient to insure eradication of a spherical tumor may be inadequate for the successful treatment of non-spherical tumors. When the drug is continuously infused, linear analysis predicts that whether or not a cure is possible is crucially dependent on the proliferation rate of the semi-resistant cells and on the competitive effect of the sensitive cells on the resistant population. When the blood vessel density is allowed to change dynamically, the model predicts a dramatic increase in the tumors growth and decrease in its response to therapy.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/42106/1/285-44-3-201_20440201.pd

Mechanisms that enhance sustainability of p53 pulses.

The tumor suppressor p53 protein shows various dynamic responses depending on the types and extent of cellular stresses. In particular, in response to DNA damage induced by γ-irradiation, cells generate a series of p53 pulses. Recent research has shown the importance of sustaining repeated p53 pulses for recovery from DNA damage. However, far too little attention has been paid to understanding how cells can sustain p53 pulses given the complexities of genetic heterogeneity and intrinsic noise. Here, we explore potential molecular mechanisms that enhance the sustainability of p53 pulses by developing a new mathematical model of the p53 regulatory system. This model can reproduce many experimental results that describe the dynamics of p53 pulses. By simulating the model both deterministically and stochastically, we found three potential mechanisms that improve the sustainability of p53 pulses: 1) the recently identified positive feedback loop between p53 and Rorα allows cells to sustain p53 pulses with high amplitude over a wide range of conditions, 2) intrinsic noise can often prevent the dampening of p53 pulses even after mutations, and 3) coupling of p53 pulses in neighboring cells via cytochrome-c significantly reduces the chance of failure in sustaining p53 pulses in the presence of heterogeneity among cells. Finally, in light of these results, we propose testable experiments that can reveal important mechanisms underlying p53 dynamics

A mathematical model of cancer stem cell driven tumor initiation: implications of niche size and loss of homeostatic regulatory mechanisms.

Hierarchical organized tissue structures, with stem cell driven cell differentiation, are critical to the homeostatic maintenance of most tissues, and this underlying cellular architecture is potentially a critical player in the development of a many cancers. Here, we develop a mathematical model of mutation acquisition to investigate how deregulation of the mechanisms preserving stem cell homeostasis contributes to tumor initiation. A novel feature of the model is the inclusion of both extrinsic and intrinsic chemical signaling and interaction with the niche to control stem cell self-renewal. We use the model to simulate the effects of a variety of types and sequences of mutations and then compare and contrast all mutation pathways in order to determine which ones generate cancer cells fastest. The model predicts that the sequence in which mutations occur significantly affects the pace of tumorigenesis. In addition, tumor composition varies for different mutation pathways, so that some sequences generate tumors that are dominated by cancerous cells with all possible mutations, while others are primarily comprised of cells that more closely resemble normal cells with only one or two mutations. We are also able to show that, under certain circumstances, healthy stem cells diminish due to the displacement by mutated cells that have a competitive advantage in the niche. Finally, in the event that all homeostatic regulation is lost, exponential growth of the cancer population occurs in addition to the depletion of normal cells. This model helps to advance our understanding of how mutation acquisition affects mechanisms that influence cell-fate decisions and leads to the initiation of cancers

Pathways to Tumorigenesis—Modeling Mutation Acquisition in Stem Cells and Their Progeny1

Most adult tissues consist of stem cells, progenitors, and mature cells, and this hierarchical architecture may play an important role in the multistep process of carcinogenesis. Here, we develop and discuss the important predictions of a simple mathematical model of cancer initiation and early progression within a hierarchically structured tissue. This work presents a model that incorporates both the sequential acquisition of phenotype altering mutations and tissue hierarchy. The model simulates the progressive effect of accumulating mutations that lead to an increase in fitness or the induction of genetic instability. A novel aspect of the model is that symmetric self-renewal, asymmetric division, and differentiation are all incorporated, and this enables the quantitative study of the effect of mutations that deregulate the normal, homeostatic stem cell division pattern. The model is also capable of predicting changes in both tissue composition and in the progression of cells along their lineage at any given time and for various sequences of mutations. Simulations predict that the specific order in which mutations are acquired is crucial for determining the pace of cancer development. Interestingly, we find that the importance of genetic stability differs significantly depending on the physiological expression of mutations related to symmetric self-renewal and differentiation of stem and progenitor cells. In particular, mutations that lead to the alteration of the stem cell division pattern or the acquisition of some degree of immortality in committed progenitors lead to an early onset of cancer and diminish the impact of genetic instability

Mutation acquisition in stem cells and the formation of abnormal progeny.

<p>Stem cells acquire mutations with small probability during each division and pass on mutations to their progeny. Terminally differentiated cells are fully mature, and therefore, do not divide and acquire additional mutations.</p

Increased symmetric self-renewal speeds cancer onset more than increased proliferation rate.

<p>The time to first cancer stem cell is faster for increased symmetric self-renewal when all mutations are not advantageous (Case 2B) even when compared to the case where all mutations are advantageous with increased proliferation rate (Case 1A).</p