Drug resistance significantly limits the long-term effectiveness of targeted therapeutics for cancer patients. Recent experimental studies have demonstrated that cancer cell heterogeneity and microenvironment adaptations to targeted therapy play important roles in promoting the rapid acquisition of drug resistance and in increasing cancer metastasis. The systematic development of effective therapeutics to overcome drug resistance mechanisms poses a major challenge. In this study, we used a modeling approach to connect cellular mechanisms underlying cancer drug resistance to population-level patient survival. To predict progression-free survival in cancer patients with metastatic melanoma, we developed a set of stochastic differential equations to describe the dynamics of heterogeneous cell populations while taking into account micro-environment adaptations. Clinical data on survival and circulating tumor cell DNA (ctDNA) concentrations were used to confirm the effectiveness of our model. Moreover, our model predicted distinct patterns of dose-dependent synergy when evaluating a combination of BRAF and MEK inhibitors versus a combination of BRAF and PI3K inhibitors. These predictions were consistent with the findings in previously reported studies. The impact of the drug metabolism rate on patient survival was also discussed. The proposed model might facilitate the quantitative evaluation and optimization of combination therapeutics and cancer clinical trial design.

We consider the Monge-Ampère equation det(D 2 u) = f where f is a positive function in R n and f = 1 + O(|x| −β ) for some β > 2 at infinity. If the equation is globally defined on R n we classify the asymptotic behavior of solutions at infinity. If the equation is defined outside a convex bounded set we solve the corresponding exterior Dirichlet problem. Finally we prove for n ≥ 3 the existence of global solutions with prescribed asymptotic behavior at infinity. The assumption β > 2 is sharp for all the results in this article.

We establish upper bounds on the blow up rate of the gradients of solutions of the Lamé system with partially infinite coefficients in dimension two as the distance between the surfaces of discontinuity of the coefficients of the system tends to zero.

Cancer is widely recognized as a genetic disease in which somatic mutations are sequentially accumulated to drive tumor progression. Although genomic landscape studies are informative for individual cancer types, a comprehensive comparative study of tumorigenic mutations across cancer types based on integrative data sources is still a pressing need. We systematically analyzed ~106 non-synonymous mutations extracted from COSMIC, involving ~8000 genome-wide screened samples across 23 major human cancers at both the amino acid and gene levels. Our analysis identified cancer-specific heterogeneity that traditional nucleotide variation analysis alone usually overlooked. Particularly, the amino acid arginine (R) turns out to be the most favorable target of amino acid alteration in most cancer types studied (P < 10−9, binomial test), reflecting its important role in cellular physiology. The tumor suppressor gene TP53 is mutated exclusively with the HYDIN, KRAS, and PTEN genes in large intestine, lung, and endometrial cancers respectively, indicating that TP53 takes part in different signaling pathways in different cancers. While some of our analyses corroborated previous observations, others indicated relevant candidates with high priority for further experimental validation. Our findings have many ramifications in understanding the etiology of cancer and the underlying molecular mechanisms in particular cancers.

Osteogenetic microenvironment is a complex constitution in which extracellular matrix (ECM) molecules, stem cells and growth factors each interact to direct the coordinate regulation of bone tissue development. Importantly, angiogenesis improvement and revascularization are critical for osteogenesis during bone tissue regeneration processes. In this study, we developed a three-dimensional (3D) multi-scale system model to study cell response to growth factors released from a 3D biodegradable porous calcium phosphate (CaP) scaffold. Our model reconstructed the 3D bone regeneration system and examined the effects of pore size and porosity on bone formation and angiogenesis. The results suggested that scaffold porosity played a more dominant role in affecting bone formation and angiogenesis compared with pore size, while the pore size could be controlled to tailor the growth factor release rate and release fraction. Furthermore, a combination of gradient VEGF with BMP2 and Wnt released from the multi-layer scaffold promoted angiogenesis and bone formation more readily than single growth factors. These results demonstrated that the developed model can be potentially applied to predict vascularized bone regeneration with specific scaffold and growth factors.

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BackgroundThe epidermal growth factor receptor (EGFR) signaling pathway and angiogenesis in brain cancer act as an engine for tumor initiation, expansion and response to therapy. Since the existing literature does not have any models that investigate the impact of both angiogenesis and molecular signaling pathways on treatment, we propose a novel multi-scale, agent-based computational model that includes both angiogenesis and EGFR modules to study the response of brain cancer under tyrosine kinase inhibitors (TKIs) treatment.ResultsThe novel angiogenesis module integrated into the agent-based tumor model is based on a set of reaction–diffusion equations that describe the spatio-temporal evolution of the distributions of micro-environmental factors such as glucose, oxygen, TGFα, VEGF and fibronectin. These molecular species regulate tumor growth during angiogenesis. Each tumor cell is equipped with an EGFR signaling pathway linked to a cell-cycle pathway to determine its phenotype. EGFR TKIs are delivered through the blood vessels of tumor microvasculature and the response to treatment is studied.ConclusionsOur simulations demonstrated that entire tumor growth profile is a collective behaviour of cells regulated by the EGFR signaling pathway and the cell cycle. We also found that angiogenesis has a dual effect under TKI treatment: on one hand, through neo-vasculature TKIs are delivered to decrease tumor invasion; on the other hand, the neo-vasculature can transport glucose and oxygen to tumor cells to maintain their metabolism, which results in an increase of cell survival rate in the late simulation stages.

We are concerned with properties of (convex) solutions to the Hessian quotient equation S n,k ( D 2 u ) = φ, 1 ≤ k < n . As our first main result we prove some regularity of strong solutions, while the second states that for φ ≡ 1 an entire convex solution with a quadratic growth bound must be a quadratic polynomial.