Here we present a mathematical model of movement in an abstract space representing states of cellular differentiation. We motivate this work with recent examples that demonstrate a continuum of ...cellular differentiation using single-cell RNA-sequencing data to characterize cellular states in a high-dimensional space, which is then mapped into
or
with dimension reduction techniques. We represent trajectories in the differentiation space as a graph, and model directed and random movement on the graph with partial differential equations. We hypothesize that flow in this space can be used to model normal and abnormal differentiation processes. We present a mathematical model of haematopoiesis parameterized with publicly available single-cell RNA-Seq data and use it to simulate the pathogenesis of acute myeloid leukaemia (AML). The model predicts the emergence of cells in novel intermediate states of differentiation consistent with immunophenotypic characterizations of a mouse model of AML.
Gliomas are notoriously aggressive, malignant brain tumors that have variable response to treatment. These patients often have poor prognosis, informed primarily by histopathology. Mathematical ...neuro-oncology (MNO) is a young and burgeoning field that leverages mathematical models to predict and quantify response to therapies. These mathematical models can form the basis of modern "precision medicine" approaches to tailor therapy in a patient-specific manner. Patient-specific models (PSMs) can be used to overcome imaging limitations, improve prognostic predictions, stratify patients, and assess treatment response in silico. The information gleaned from such models can aid in the construction and efficacy of clinical trials and treatment protocols, accelerating the pace of clinical research in the war on cancer. This review focuses on the growing translation of PSM to clinical neuro-oncology. It will also provide a forward-looking view on a new era of patient-specific MNO.
Glioblastomas are aggressive primary brain tumors known for their inter- and intratumor heterogeneity. This disease is uniformly fatal, with intratumor heterogeneity the major reason for treatment ...failure and recurrence. Just like the nature vs nurture debate, heterogeneity can arise from intrinsic or environmental influences. Whilst it is impossible to clinically separate observed behavior of cells from their environmental context, using a mathematical framework combined with multiscale data gives us insight into the relative roles of variation from different sources. To better understand the implications of intratumor heterogeneity on therapeutic outcomes, we created a hybrid agent-based mathematical model that captures both the overall tumor kinetics and the individual cellular behavior. We track single cells as agents, cell density on a coarser scale, and growth factor diffusion and dynamics on a finer scale over time and space. Our model parameters were fit utilizing serial MRI imaging and cell tracking data from ex vivo tissue slices acquired from a growth-factor driven glioblastoma murine model. When fitting our model to serial imaging only, there was a spectrum of equally-good parameter fits corresponding to a wide range of phenotypic behaviors. When fitting our model using imaging and cell scale data, we determined that environmental heterogeneity alone is insufficient to match the single cell data, and intrinsic heterogeneity is required to fully capture the migration behavior. The wide spectrum of in silico tumors also had a wide variety of responses to an application of an anti-proliferative treatment. Recurrent tumors were generally less proliferative than pre-treatment tumors as measured via the model simulations and validated from human GBM patient histology. Further, we found that all tumors continued to grow with an anti-migratory treatment alone, but the anti-proliferative/anti-migratory combination generally showed improvement over an anti-proliferative treatment alone. Together our results emphasize the need to better understand the underlying phenotypes and tumor heterogeneity present in a tumor when designing therapeutic regimens.
Glioblastoma multiforme (GBM) is the most common malignant primary brain tumor associated with a poor median survival of 15–18 months, yet there is wide heterogeneity across and within patients. This ...heterogeneity has been the source of significant clinical challenges facing patients with GBM and has hampered the drive toward more precision or personalized medicine approaches to treating these challenging tumors. Over the last two decades, the field of Mathematical Neuro-oncology has grown out of desire to use (often patient-specific) mathematical modeling to better treat GBMs. Here, we will focus on a series of clinically relevant results using patient-specific mathematical modeling. The core model at the center of these results incorporates two hallmark features of GBM, proliferation
(
ρ
)
and invasion (
D
), as key parameters. Based on routinely obtained magnetic resonance images, each patient’s tumor can be characterized using these two parameters. The Proliferation-Invasion (PI) model uses
ρ
and
D
to create patient-specific growth predictions. The PI model, its predictions, and parameters have been used in a number of ways to derive biological insight. Beyond predicting growth, the PI model has been utilized to identify patients who benefit from different surgery strategies, to prognosticate response to radiation therapy, to develop a treatment response metric, and to connect clinical imaging features and genetic information. Demonstration of the PI model’s clinical relevance supports the growing role for it and other mathematical models in routine clinical practice.
Gliomas are uniformly fatal forms of primary brain neoplasms that vary from low- to high-grade (glioblastoma). Whereas low-grade gliomas are weakly angiogenic, glioblastomas are among the most ...angiogenic tumors. Thus, interactions between glioma cells and their tissue microenvironment may play an important role in aggressive tumor formation and progression. To quantitatively explore how tumor cells interact with their tissue microenvironment, we incorporated the interactions of normoxic glioma cells, hypoxic glioma cells, vascular endothelial cells, diffusible angiogenic factors, and necrosis formation into a first-generation, biologically based mathematical model for glioma growth and invasion. Model simulations quantitatively described the spectrum of in vivo dynamics of gliomas visualized with medical imaging. Furthermore, we investigated how proliferation and dispersal of glioma cells combine to induce increasing degrees of cellularity, mitoses, hypoxia-induced neoangiogenesis and necrosis, features that characterize increasing degrees of "malignancy," and we found that changes in the net rates of proliferation (ρ) and invasion (D) are not always necessary for malignant progression. Thus, although other factors, including the accumulation of genetic mutations, can change cellular phenotype (e.g., proliferation and invasion rates), this study suggests that these are not required for malignant progression. Simulated results are placed in the context of the current clinical World Health Organization grading scheme for studying specific patient examples. This study suggests that through the application of the proposed model for tumor-microenvironment interactions, predictable patterns of dynamic changes in glioma histology distinct from changes in cellular phenotype (e.g., proliferation and invasion rates) may be identified, thus providing a powerful clinical tool.
Sediment cores were taken in 2002 in Lakes Michigan and Huron at six locations. A total of 75 samples were characterized, dated using 210Pb, and analyzed for 10 congeners of polybromodiphenyl ether ...(PBDE) including BDE209, as well as 39 congeners of polychlorinated biphenyls (PCBs). The concentrations of nine tri- through hepta-BDE congeners (Sigma9PBDE) in the surficial sediments range from 1.7 to 4 ng g(-1) for Lake Michigan and from 1.0 to 1.9 ng g(-1) for Lake Huron, on the basis of the dry sediment weight. The Sigma9PBDEs fluxes to the sediment around the year 2002 are from 36 to 109 pg cm(-2) yr(-1) in Lake Michigan and from 30 to 73 pg cm(-2) yr(-1) in Lake Huron, with spatial variations in both lakes. The flux of BDE209 ranges from 0.64 to 2.04 ng cm(-2) yr(-1) and from 0.67 to 1.41 ng cm(-2) yr(-1) in Lake Michigan and Lake Huron, respectively. Dramatic increases in PBDE concentrations and fluxes upward toward the sediment surface and the present time are evident at all locations. The inventory of PBDEs in both lakes appears to be dependent upon latitude and the proximity to populated areas, implying that north-bound air plumes from urban areas are the major sources of PBDEs found in the lake sediments at locations away from the shores. Heavier congeners are more abundant in the sediments than in air and fish samples in the region. BDE209 is about 96% and 91% of the total PBDEs on a mass basis in Lake Michigan and Lake Huron, respectively; both are higher than the 89% found in Lake Superior, although a t test shows that the value for Lake Huron is not statistically different from that for Lake Superior at the 95% confidence level.
Accurate clinical assessment of a patient's response to treatment for glioblastoma multiforme (GBM), the most malignant type of primary brain tumor, is undermined by the wide patient-to-patient ...variability in GBM dynamics and responsiveness to therapy. Using computational models that account for the unique geometry and kinetics of individual patients' tumors, we developed a method for assessing treatment response that discriminates progression-free and overall survival following therapy for GBM. Applying these models as untreated virtual controls, we generate a patient-specific "Days Gained" response metric that estimates the number of days a therapy delayed imageable tumor progression. We assessed treatment response in terms of Days Gained scores for 33 patients at the time of their first MRI scan following first-line radiation therapy. Based on Kaplan-Meier analyses, patients with Days Gained scores of 100 or more had improved progression-free survival, and patients with scores of 117 or more had improved overall survival. Our results demonstrate that the Days Gained response metric calculated at the routinely acquired first post-radiation treatment time point provides prognostic information regarding progression and survival outcomes. Applied prospectively, our model-based approach has the potential to improve GBM treatment by accounting for patient-to-patient heterogeneity in GBM dynamics and responses to therapy.
Sediment cores were taken in 2001 and 2002 in Lake Superior at six locations away from lakeshores and segmented at 0.5-5 cm intervals. The year of sediment deposition was estimated for each segment ...of four cores using the 210Pb dating technique. Samples were Soxhlet-extracted and cleaned up by silica gel fractionation, and the concentrations of 10 polybrominated diphenyl ethers (PBDEs) and 19 polychlorinated biphenyls (PCBs) were measured by GC-MS in SIM mode. In contrast to recent declining or level-off trends in PCB fluxes, the sedimentary records of PBDEs generally show a significant increase in recent years. The load of total PBDEs to Lake Superior was estimated to be 2-6 metric tons, and the current loading rate was about 80-160 kg yr(-1). With the exclusion of decabromodiphenyl ether (BDE209), the surficial concentration of sumPBDE (sum of 9 congeners) ranged from 0.5 to 3 ng g(-1), and the current sumPBDEs flux was 8-31 pg cm(-2) yr(-1). The concentrations of BDE209 were about an order of magnitude higher than the sum of other congeners, comprising 83-94% of the total PBDE inventory in the sediments. Among the other nine PBDEs detected, congeners 47 and 99 were the most abundant, and congeners 100, 153, 154, and 183 were also detected in all the cores. Congener analysis demonstrated that the pattern of PBDEs in Lake Superior sediments differs from those in air and fish.
Sediment cores were taken in 2002 in Lakes Ontario and Erie at four locations. A total of 48 sediment samples were characterized, dated using 210Pb, and analyzed for 10 congeners of polybrominated ...diphenyl ethers (PBDEs) including BDE209 as well as 39 congeners of polychlorinated biphenyls (PCBs). The surficial concentrations of nine tri- through hepta-BDE congeners (sigma9PBDE) are 4.85 and 6.33 ng g(-1), at sampling sites ON40 and ON30 in Lake Ontario, and 1.83 and 1.95 ng g(-1) at ER37 and ER09 in Lake Erie, respectively, based on dry sediment weight. The surficial BDE209 concentrations are 242 and 211 ng g(-1) at ON40 and ON30 and 50 and 55 ng g(-1) at ER37 and ER09. The sigma(9-) PBDEs fluxes to the sediment around 2002 are 147 and 195 pg cm(-2) year(-1) at ON40 and ON30 and 136 and 314 pg cm(-2) year(-1) at ER37 and ER09, respectively. The fluxes of BDE209 are 6.5 and 7.3 ng cm(-2) year(-1) at ON30 and ON40 and 3.7 and 8.9 ng cm(-2) year(-1) at ER37 and ER09, respectively. Dramatic increases in PBDE concentrations and fluxes upward toward the sediment surface and the present time are evident at both locations in Lake Ontario, while PCBs concentrations peak in the middle of sediment cores around the dated time of 1970s and 1960s. For both locations of Lake Erie, the increasing trends of both PBDEs and PCBs from the bottom to the surficial segments were distorted by sediment mixing. BDE209 is the most abundant congener among PBDEs in the sediments, constituting about 96 and 91% of the total PBDEs on mass basis in Lakes Ontario and Erie, respectively.
The 2019 mathematical oncology roadmap Rockne, Russell C; Hawkins-Daarud, Andrea; Swanson, Kristin R ...
Physical biology,
06/2019, Letnik:
16, Številka:
4
Journal Article
Recenzirano
Odprti dostop
Whether the nom de guerre is Mathematical Oncology, Computational or Systems Biology, Theoretical Biology, Evolutionary Oncology, Bioinformatics, or simply Basic Science, there is no denying that ...mathematics continues to play an increasingly prominent role in cancer research. Mathematical Oncology-defined here simply as the use of mathematics in cancer research-complements and overlaps with a number of other fields that rely on mathematics as a core methodology. As a result, Mathematical Oncology has a broad scope, ranging from theoretical studies to clinical trials designed with mathematical models. This Roadmap differentiates Mathematical Oncology from related fields and demonstrates specific areas of focus within this unique field of research. The dominant theme of this Roadmap is the personalization of medicine through mathematics, modelling, and simulation. This is achieved through the use of patient-specific clinical data to: develop individualized screening strategies to detect cancer earlier; make predictions of response to therapy; design adaptive, patient-specific treatment plans to overcome therapy resistance; and establish domain-specific standards to share model predictions and to make models and simulations reproducible. The cover art for this Roadmap was chosen as an apt metaphor for the beautiful, strange, and evolving relationship between mathematics and cancer.
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