Classical mathematical models for description and prediction of experimental tumor growth.

Despite internal complexity, tumor growth kinetics follow relatively simple laws that can be expressed as mathematical models. To explore this further, quantitative analysis of the most classical of these were performed. The models were assessed against data from two in vivo experimental systems: an ectopic syngeneic tumor (Lewis lung carcinoma) and an orthotopically xenografted human breast carcinoma. The goals were threefold: 1) to determine a statistical model for description of the measurement error, 2) to establish the descriptive power of each model, using several goodness-of-fit metrics and a study of parametric identifiability, and 3) to assess the models' ability to forecast future tumor growth. The models included in the study comprised the exponential, exponential-linear, power law, Gompertz, logistic, generalized logistic, von Bertalanffy and a model with dynamic carrying capacity. For the breast data, the dynamics were best captured by the Gompertz and exponential-linear models. The latter also exhibited the highest predictive power, with excellent prediction scores (≥80%) extending out as far as 12 days in the future. For the lung data, the Gompertz and power law models provided the most parsimonious and parametrically identifiable description. However, not one of the models was able to achieve a substantial prediction rate (≥70%) beyond the next day data point. In this context, adjunction of a priori information on the parameter distribution led to considerable improvement. For instance, forecast success rates went from 14.9% to 62.7% when using the power law model to predict the full future tumor growth curves, using just three data points. These results not only have important implications for biological theories of tumor growth and the use of mathematical modeling in preclinical anti-cancer drug investigations, but also may assist in defining how mathematical models could serve as potential prognostic tools in the clinic.

Molecular signaling network complexity is correlated with cancer patient survivability.

The 5-y survival for cancer patients after diagnosis and treatment is strongly dependent on tumor type. Prostate cancer patients have a >99% chance of survival past 5 y after diagnosis, and pancreatic patients have <6% chance of survival past 5 y. Because each cancer type has its own molecular signaling network, we asked if there are "signatures" embedded in these networks that inform us as to the 5-y survival. In other words, are there statistical metrics of the network that correlate with survival? Furthermore, if there are, can such signatures provide clues to selecting new therapeutic targets? From the Kyoto Encyclopedia of Genes and Genomes Cancer Pathway database we computed several conventional and some less conventional network statistics. In particular we found a correlation (R(2) = 0.7) between degree-entropy and 5-y survival based on the Surveillance Epidemiology and End Results database. This correlation suggests that cancers that have a more complex molecular pathway are more refractory than those with less complex molecular pathway. We also found potential new molecular targets for drugs by computing the betweenness--a statistical metric of the centrality of a node--for the molecular networks.

Quantitative modeling of chronic myeloid leukemia: insights from radiobiology.

Mathematical models of chronic myeloid leukemia (CML) cell population dynamics are being developed to improve CML understanding and treatment. We review such models in light of relevant findings from radiobiology, emphasizing 3 points. First, the CML models almost all assert that the latency time, from CML initiation to diagnosis, is at most ∼10 years. Meanwhile, current radiobiologic estimates, based on Japanese atomic bomb survivor data, indicate a substantially higher maximum, suggesting longer-term relapses and extra resistance mutations. Second, different CML models assume different numbers, between 400 and 10(6), of normal HSCs. Radiobiologic estimates favor values>10(6) for the number of normal cells (often assumed to be the HSCs) that are at risk for a CML-initiating BCR-ABL translocation. Moreover, there is some evidence for an HSC dead-band hypothesis, consistent with HSC numbers being very different across different healthy adults. Third, radiobiologists have found that sporadic (background, age-driven) chromosome translocation incidence increases with age during adulthood. BCR-ABL translocation incidence increasing with age would provide a hitherto underanalyzed contribution to observed background adult-onset CML incidence acceleration with age, and would cast some doubt on stage-number inferences from multistage carcinogenesis models in general.

Angiogenic impairment of the vascular endothelium: a novel mechanism and potential therapeutic target in muscular dystrophy.

OBJECTIVE: Dystrophin, the missing or defective protein in Duchenne muscular dystrophy, is expressed not only in muscle cells but also in vascular endothelial cells (ECs). In this study, we assessed the effects of dystrophin deficiency on the angiogenic capacities of ECs. APPROACH AND RESULTS: We isolated vascular ECs from mdx mice, the murine equivalent of Duchenne muscular dystrophy in humans, and wild-type controls, and we found that mdx-derived ECs have impaired angiogenic properties, in terms of migration, proliferation, and tube formation. They also undergo increased apoptosis in vitro compared with wild-type cells and have increased senescence-associated ß-galactosidase activity. Mdx-derived ECs also display reduced ability to support myoblast proliferation when cocultured with satellite cell-derived primary myoblasts. These endothelial defects are mirrored by systemic impairment of angiogenesis in vivo, both on induction of ischemia, stimulation with growth factors in the corneal model and matrigel plug assays, and tumor growth. We also found that dystrophin forms a complex with endothelial NO synthase and caveolin-1 in ECs, and that NO production and cGMP formation are compromised in ECs isolated from mdx mice. Interestingly, treatment with aspirin enhances production of both cGMP and NO in dystrophic ECs, whereas low-dose aspirin improves the dystrophic phenotype of mdx mice in vivo, in terms of resistance to physical exercise, muscle fiber permeability, and capillary density. CONCLUSIONS: These findings demonstrate that impaired angiogenesis is a novel player and potential therapeutic target in Duchenne muscular dystrophy.

A multicompartment mathematical model of cancer stem cell-driven tumor growth dynamics.

Tumors are appreciated to be an intrinsically heterogeneous population of cells with varying proliferation capacities and tumorigenic potentials. As a central tenet of the so-called cancer stem cell hypothesis, most cancer cells have only a limited lifespan, and thus cannot initiate or reinitiate tumors. Longevity and clonogenicity are properties unique to the subpopulation of cancer stem cells. To understand the implications of the population structure suggested by this hypothesis--a hierarchy consisting of cancer stem cells and progeny non-stem cancer cells which experience a reduction in their remaining proliferation capacity per division--we set out to develop a mathematical model for the development of the aggregate population. We show that overall tumor progression rate during the exponential growth phase is identical to the growth rate of the cancer stem cell compartment. Tumors with identical stem cell proportions, however, can have different growth rates, dependent on the proliferation kinetics of all participating cell populations. Analysis of the model revealed that the proliferation potential of non-stem cancer cells is likely to be small to reproduce biologic observations. Furthermore, a single compartment of non-stem cancer cell population may adequately represent population growth dynamics only when the compartment proliferation rate is scaled with the generational hierarchy depth.

Sex differences in the incidence of chronic myeloid leukemia.

The incidence of chronic myeloid leukemia (CML), which is caused by BCR/ABL chimeric oncogene formation in a pluripotent hematopoietic stem cell (HSC), increases with age and exposure to ionizing radiation. CML is a comparatively well-characterized neoplasm, important for its own sake and useful for insights into other neoplasms. Here, Surveillance, Epidemiology and End Results (SEER) CML data are analyzed after considering possible misclassification of chronic myelo-monocytic leukemia as CML. For people older than 25 years, plots of male and female CML log incidences versus age at diagnosis are approximately parallel straight lines with males either above or to the left of females. This is consistent with males having a higher risk of developing CML or a shorter latency from initiation to diagnosis of CML. These distinct mechanisms cannot be distinguished using SEER data alone. Therefore, CML risks among male and female Japanese A-bomb survivors are also analyzed. The present analyses suggest that sex differences in CML incidence more likely result from differences in risk than in latency. The simplest but not the sole interpretation of this is that males have more target cells at risk to develop CML. Comprehensive mathematical models of CML could lead to a better understanding of the role of HSCs in CML and other preleukemias that can progress to acute leukemia.

An integrated multidisciplinary model describing initiation of cancer and the Warburg hypothesis.

BACKGROUND: In this paper we propose a chemical physics mechanism for the initiation of the glycolytic switch commonly known as the Warburg hypothesis, whereby glycolytic activity terminating in lactate continues even in well-oxygenated cells. We show that this may result in cancer via mitotic failure, recasting the current conception of the Warburg effect as a metabolic dysregulation consequent to cancer, to a biophysical defect that may contribute to cancer initiation. MODEL: Our model is based on analogs of thermodynamic concepts that tie non-equilibrium fluid dynamics ultimately to metabolic imbalance, disrupted microtubule dynamics, and finally, genomic instability, from which cancers can arise. Specifically, we discuss how an analog of non-equilibrium Rayleigh-Benard convection can result in glycolytic oscillations and cause a cell to become locked into a higher-entropy state characteristic of cancer. CONCLUSIONS: A quantitative model is presented that attributes the well-known Warburg effect to a biophysical mechanism driven by a convective disturbance in the cell. Contrary to current understanding, this effect may precipitate cancer development, rather than follow from it, providing new insights into carcinogenesis, cancer treatment, and prevention.

Combined therapy with sonic hedgehog gene transfer and bone marrow-derived endothelial progenitor cells enhances angiogenesis and myogenesis in the ischemic skeletal muscle.

We have previously demonstrated that sonic hedgehog (Shh) gene transfer improves angiogenesis in the setting of ischemia by upregulating the expression of multiple growth factors and enhancing the incorporation of endogenous bone marrow (BM)-derived endothelial progenitor cells (EPCs). In this study, we hypothesized that combined therapy with Shh gene transfer and BM-derived EPCs is more effective than Shh gene therapy alone in an experimental model of peripheral limb ischemia. We used old mice, which have a significantly reduced angiogenic response to ischemia, and compared the ability of Shh gene transfer, exogenous EPCs, or both to improve regeneration after ischemia. We found a significantly higher capillary density in the Shh + EPC-treated muscles compared to the other experimental groups. We also found that Shh gene transfer increases the incorporation and survival of transplanted EPCs. Finally, we found a significantly higher number of regenerating myofibers in the ischemic muscles of mice receiving combined treatment with Shh and BM-derived EPCs. In summary, the combination of Shh gene transfer and BM-derived EPCs more effectively promotes angiogenesis and muscle regeneration than each treatment individually and merits further investigation for its potential beneficial effects in ischemic diseases.

Immunoediting: evidence of the multifaceted role of the immune system in self-metastatic tumor growth.

BACKGROUND: The role of the immune system in tumor progression has been a subject for discussion for many decades. Numerous studies suggest that a low immune response might be beneficial, if not necessary, for tumor growth, and only a strong immune response can counter tumor growth and thus inhibit progression. METHODS: We implement a cellular automaton model previously described that captures the dynamical interactions between the cancer stem and non-stem cell populations of a tumor through a process of self-metastasis. By overlaying on this model the diffusion of immune reactants into the tumor from a peripheral source to target cells, we simulate the process of immune-system-induced cell kill on tumor progression. RESULTS: A low cytotoxic immune reaction continuously kills cancer cells and, although at a low rate, thereby causes the liberation of space-constrained cancer stem cells to drive self-metastatic progression and continued tumor growth. With increasing immune system strength, however, tumor growth peaks, and then eventually falls below the intrinsic tumor sizes observed without an immune response. With this increasing immune response the number and proportion of cancer stem cells monotonically increases, implicating an additional unexpected consequence, that of cancer stem cell selection, to the immune response. CONCLUSIONS: Cancer stem cells and immune cytotoxicity alone are sufficient to explain the three-step "immunoediting" concept - the modulation of tumor growth through inhibition, selection and promotion.

Non-stem cancer cell kinetics modulate solid tumor progression.

BACKGROUND: Solid tumors are heterogeneous in composition. Cancer stem cells (CSCs) are believed to drive tumor progression, but the relative frequencies of CSCs versus non-stem cancer cells span wide ranges even within tumors arising from the same tissue type. Tumor growth kinetics and composition can be studied through an agent-based cellular automaton model using minimal sets of biological assumptions and parameters. Herein we describe a pivotal role for the generational life span of non-stem cancer cells in modulating solid tumor progression in silico. RESULTS: We demonstrate that although CSCs are necessary for progression, their expansion and consequently tumor growth kinetics are surprisingly modulated by the dynamics of the non-stem cancer cells. Simulations reveal that slight variations in non-stem cancer cell proliferative capacity can result in tumors with distinctly different growth kinetics. Longer generational life spans yield self-inhibited tumors, as the emerging population of non-stem cancer cells spatially impedes expansion of the CSC compartment. Conversely, shorter generational life spans yield persistence-limited tumors, with symmetric division frequency of CSCs determining tumor growth rate. We show that the CSC fraction of a tumor population can vary by multiple orders of magnitude as a function of the generational life span of the non-stem cancer cells. CONCLUSIONS: Our study suggests that variability in the growth rate and CSC content of solid tumors may be, in part, attributable to the proliferative capacity of the non-stem cancer cell population that arises during asymmetric division of CSCs. In our model, intermediate proliferative capacities give rise to the fastest-growing tumors, resulting in self-metastatic expansion driven by a balance between symmetric CSC division and expansion of the non-stem cancer population. Our results highlight the importance of non-stem cancer cell dynamics in the CSC hypothesis, and may offer a novel explanation for the large variations in CSC fractions reported in vivo.

A rapid-mutation approximation for cell population dynamics.

Carcinogenesis and cancer progression are often modeled using population dynamics equations for a diverse somatic cell population undergoing mutations or other alterations that alter the fitness of a cell and its progeny. Usually it is then assumed, paralleling standard mathematical approaches to evolution, that such alterations are slow compared to selection, i.e., compared to subpopulation frequency changes induced by unequal subpopulation proliferation rates. However, the alterations can be rapid in some cases. For example, results in our lab on in vitro analogues of transformation and progression in carcinogenesis suggest there could be periods where rapid alterations triggered by horizontal intercellular transfer of genetic material occur and quickly result in marked changes of cell population structure.We here initiate a mathematical study of situations where alterations are rapid compared to selection. A classic selection-mutation formalism is generalized to obtain a "proliferation-alteration" system of ordinary differential equations, which we analyze using a rapid-alteration approximation. A system-theoretical estimate of the total-population net growth rate emerges. This rate characterizes the diverse, interacting cell population acting as a single system; it is a weighted average of subpopulation rates, the weights being components of the Perron-Frobenius eigenvector for an ergodic Markov-process matrix that describes alterations by themselves. We give a detailed numerical example to illustrate the rapid-alteration approximation, suggest a possible interpretation of the fact that average aneuploidy during cancer progression often appears to be comparatively stable in time, and briefly discuss possible generalizations as well as weaknesses of our approach.

Stochastic population dynamic effects for lung cancer progression.

The multistage paradigm is widely used in quantitative analyses of radiation-influenced carcinogenesis. Steps such as initiation, promotion and transformation have been investigated in detail. However, progression, a later step during which malignant cells produced in the earlier steps can develop into clinical cancer, has received less attention in computational radiobiology; it has often been approximated deterministically as a fixed, comparatively short, lag time. This approach overlooks important mechanisms in progression, including stochastic extinction, possible radiation effects on tumor growth, immune suppression and angiogenic bottlenecks. Here we analyze tumor progression in background and in radiation-induced lung cancers, emphasizing tumor latent times and the stochastic extinction of malignant lesions. A Monte Carlo cell population dynamics formalism is developed by supplementing the standard two-stage clonal expansion (TSCE) model with a stochastic birth-death model for proliferation of malignant cells. Simulation results for small cell lung cancers and lung adenocarcinomas show that the effects of stochastic malignant cell extinction broaden progression time distributions drastically. We suggest that fully stochastic cancer progression models incorporating malignant cell kinetics, dormancy (a phase in which tumors remain asymptomatic), escape from dormancy, and invasiveness, with radiation able to act directly on each phase, need to be considered for a better assessment of radiation-induced lung cancer risks.

Chromosomes are predominantly located randomly with respect to each other in interphase human cells.

To test quantitatively whether there are systematic chromosome-chromosome associations within human interphase nuclei, interchanges between all possible heterologous pairs of chromosomes were measured with 24-color whole-chromosome painting (multiplex FISH), after damage to interphase lymphocytes by sparsely ionizing radiation in vitro. An excess of interchanges for a specific chromosome pair would indicate spatial proximity between the chromosomes comprising that pair. The experimental design was such that quite small deviations from randomness (extra pairwise interchanges within a group of chromosomes) would be detectable. The only statistically significant chromosome cluster was a group of five chromosomes previously observed to be preferentially located near the center of the nucleus. However, quantitatively, the overall deviation from randomness within the whole genome was small. Thus, whereas some chromosome-chromosome associations are clearly present, at the whole-chromosomal level, the predominant overall pattern appears to be spatially random.

A new view of radiation-induced cancer: integrating short- and long-term processes. Part II: second cancer risk estimation.

As the number of cancer survivors grows, prediction of radiotherapy-induced second cancer risks becomes increasingly important. Because the latency period for solid tumors is long, the risks of recently introduced radiotherapy protocols are not yet directly measurable. In the accompanying article, we presented a new biologically based mathematical model, which, in principle, can estimate second cancer risks for any protocol. The novelty of the model is that it integrates, into a single formalism, mechanistic analyses of pre-malignant cell dynamics on two different time scales: short-term during radiotherapy and recovery; long-term during the entire life span. Here, we apply the model to nine solid cancer types (stomach, lung, colon, rectal, pancreatic, bladder, breast, central nervous system, and thyroid) using data on radiotherapy-induced second malignancies, on Japanese atomic bomb survivors, and on background US cancer incidence. Potentially, the model can be incorporated into radiotherapy treatment planning algorithms, adding second cancer risk as an optimization criterion.

A new view of radiation-induced cancer: integrating short- and long-term processes. Part I: approach.

Mathematical models of radiation carcinogenesis are important for understanding mechanisms and for interpreting or extrapolating risk. There are two classes of such models: (1) long-term formalisms that track pre-malignant cell numbers throughout an entire lifetime but treat initial radiation dose-response simplistically and (2) short-term formalisms that provide a detailed initial dose-response even for complicated radiation protocols, but address its modulation during the subsequent cancer latency period only indirectly. We argue that integrating short- and long-term models is needed. As an example of this novel approach, we integrate a stochastic short-term initiation/inactivation/repopulation model with a deterministic two-stage long-term model. Within this new formalism, the following assumptions are implemented: radiation initiates, promotes, or kills pre-malignant cells; a pre-malignant cell generates a clone, which, if it survives, quickly reaches a size limitation; the clone subsequently grows more slowly and can eventually generate a malignant cell; the carcinogenic potential of pre-malignant cells decreases with age.

PPARgamma ligands inhibit primary tumor growth and metastasis by inhibiting angiogenesis.

Several drugs approved for a variety of indications have been shown to exhibit antiangiogenic effects. Our study focuses on the PPARgamma ligand rosiglitazone, a compound widely used in the treatment of type 2 diabetes. We demonstrate, for the first time to our knowledge, that PPARgamma is highly expressed in tumor endothelium and is activated by rosiglitazone in cultured endothelial cells. Furthermore, we show that rosiglitazone suppresses primary tumor growth and metastasis by both direct and indirect antiangiogenic effects. Rosiglitazone inhibits bovine capillary endothelial cell but not tumor cell proliferation at low doses in vitro and decreases VEGF production by tumor cells. In our in vivo studies, rosiglitazone suppresses angiogenesis in the chick chorioallantoic membrane, in the avascular cornea, and in a variety of primary tumors. These results suggest that PPARgamma ligands may be useful in treating angiogenic diseases such as cancer by inhibiting angiogenesis.

Interpreting chromosome aberration spectra.

Ionizing radiation can damage cells by breaking both strands of DNA in multiple locations, essentially cutting chromosomes into pieces. The cell has enzymatic mechanisms to repair such breaks; however, these mechanisms are imperfect and, in an exchange process, may produce a large-scale rearrangement of the genome, called a chromosome aberration. Chromosome aberrations are important in killing cells, during carcinogenesis, in characterizing repair/misrepair pathways, in retrospective radiation biodosimetry, and in a number of other ways. DNA staining techniques such as mFISH (multicolor fluorescent in situ hybridization) provide a means for analyzing aberration spectra by examining observed final patterns. Unfortunately, an mFISH observed final pattern often does not uniquely determine the underlying exchange process. Further, resolution limitations in the painting protocol sometimes lead to apparently incomplete final patterns. We here describe an algorithm for systematically finding exchange processes consistent with any observed final pattern. This algorithm uses aberration multigraphs, a mathematical formalism that links the various aspects of aberration formation. By applying a measure to the space of consistent multigraphs, we will show how to generate model-specific distributions of aberration processes from mFISH experimental data. The approach is implemented by software freely available over the internet. As a sample application, we apply these algorithms to an aberration data set, obtaining a distribution of exchange cycle sizes, which serves to measure aberration complexity. Estimating complexity, in turn, helps indicate how damaging the aberrations are and may facilitate identification of radiation type in retrospective biodosimetry.

Prolonged dormancy of human liposarcoma is associated with impaired tumor angiogenesis.

The disease state of cancer appears late in tumor development. Before being diagnosed, a tumor can remain for prolonged periods of time in a dormant state. Dormant human cancer is commonly defined as a microscopic tumor that does not expand in size and remains asymptomatic. Dormant tumors represent an early stage in tumor development and may therefore be a potential target for nontoxic, antiangiogenic therapy that could prevent tumor recurrence. Here, we characterize an experimental model that recapitulates the clinical dormancy of human tumors in mice. We demonstrate that these microscopic dormant cancers switch to the angiogenic phenotype at a predictable time. We further show that while angiogenic liposarcomas expand rapidly after inoculation of tumor cells in mice, nonangiogenic dormant liposarcomas remain microscopic up to one-third of the normal severe combined immune deficiency (SCID) mouse life span, although they contain proliferating tumor cells. Nonangiogenic dormant tumors follow a similar growth pattern in subcutaneous (s.c.) and orthotopic environments. Throughout the dormancy period, development of intratumoral vessels is impaired. In nonangogenic dormant tumors, small clusters of endothelial cells without lumens are observed early after tumor cell inoculation, but the nonangiogenic tumor cannot sustain these vessels, and they disappear within weeks. There is a concomitant decrease in microvessel density, and the nonangiogenic dormant tumor remains harmless to the host. In contrast, microvessel density in tumors increases rapidly after the angiogenic switch and correlates with rapid expansion of tumor mass. Both tumor types cultured in vitro contain fully transformed cells, but only cells from the nonangiogenic human liposarcoma secrete relatively high levels of the angiogenesis inhibitors thrombospondin-1 and TIMP-1. This model suggests that as improved blood or urine molecular biomarkers are developed, the microscopic, nonangiogenic, dormant phase of human cancer may be vulnerable to antiangiogenic therapy years before symptoms, or before anatomical location of a tumor can be detected, by conventional methods.

Mathematical Modeling of Tumor-Tumor Distant Interactions Supports a Systemic Control of Tumor Growth.

Interactions between different tumors within the same organism have major clinical implications, especially in the context of surgery and metastatic disease. Three main explanatory theories (competition, angiogenesis inhibition, and proliferation inhibition) have been proposed, but precise determinants of the phenomenon remain poorly understood. Here, we formalized these theories into mathematical models and performed biological experiments to test them with empirical data. In syngeneic mice bearing two simultaneously implanted tumors, growth of only one of the tumors was significantly suppressed (61% size reduction at day 15, P < 0.05). The competition model had to be rejected, whereas the angiogenesis inhibition and proliferation inhibition models were able to describe the data. Additional models including a theory based on distant cytotoxic log-kill effects were unable to fit the data. The proliferation inhibition model was identifiable and minimal (four parameters), and its descriptive power was validated against the data, including consistency in predictions of single tumor growth when no secondary tumor was present. This theory may also shed new light on single cancer growth insofar as it offers a biologically translatable picture of how local and global action may combine to control local tumor growth and, in particular, the role of tumor-tumor inhibition. This model offers a depiction of concomitant resistance that provides an improved theoretical basis for tumor growth control and may also find utility in therapeutic planning to avoid postsurgery metastatic acceleration. Cancer Res; 77(18); 5183-93. ©2017 AACR.