Comparing Powers of Edge Ideals

Given a nontrivial homogeneous ideal $I\subseteq k[x_1,x_2,\ldots,x_d]$, a problem of great recent interest has been the comparison of the $r$th ordinary power of $I$ and the $m$th symbolic power $I^{(m)}$. This comparison has been undertaken directly via an exploration of which exponents $m$ and $r$ guarantee the subset containment $I^{(m)}\subseteq I^r$ and asymptotically via a computation of the resurgence $\rho(I)$, a number for which any $m/r > \rho(I)$ guarantees $I^{(m)}\subseteq I^r$. Recently, a third quantity, the symbolic defect, was introduced; as $I^t\subseteq I^{(t)}$, the symbolic defect is the minimal number of generators required to add to $I^t$ in order to get $I^{(t)}$. We consider these various means of comparison when $I$ is the edge ideal of certain graphs by describing an ideal $J$ for which $I^{(t)} = I^t + J$. When $I$ is the edge ideal of an odd cycle, our description of the structure of $I^{(t)}$ yields solutions to both the direct and asymptotic containment questions, as well as a partial computation of the sequence of symbolic defects.Comment: Version 2: Revised based on referee suggestions. Lemma 5.12 was added to clarify the proof of Theorem 5.13. To appear in the Journal of Algebra and its Applications. Version 1: 20 pages. This project was supported by Dordt College's undergraduate research program in summer 201

The Effect of Anesthetic Drugs During Craniotomy on Patient Outcomes

Glioblastoma remains an aggressive, highly malignant brain cancer with poor prognosis despite treatment options including surgery, radiotherapy, and chemotherapy. The objective of this retrospective chart review study was to evaluate if there is a correlation between the type and dose of anesthetic drugs administered during craniotomy surgery for glioblastoma and the time to first postoperative tumor progression and mortality. Based on preliminary data in mice, it was hypothesized that the use of propofol during surgery is associated with slower tumor progression and lower mortality, when compared with other anesthetic drugs. In the observed time frame, 133 patients were diagnosed with glioblastoma and underwent a total of 161 craniotomy surgeries (23 awake, 138 asleep) at Henry Ford Hospital. Propofol was utilized in 97.5% of these surgeries. Other anesthetics such as isoflurane (53.4%), dexmedetomidine (37.7%), sevoflurane (31.0%), desflurane (14.3%), and nitrous oxide (4.9%) were used less frequently. Data analysis regarding time to progression and mortality after use of each anesthetic remains in progress. This study will guide the administration of anesthetic drugs during surgery to treat glioblastoma, with potential to improve the poor prognosis for these patients

Tumor Vasculature Changes Before or During Treatment to Predict Response to Systemic Therapy

A diagnosis of non-small cell lung cancer (NSCLC) carries a grim prognosis, with 5-year survival rates of 25%. 25-30% of NSCLC patients have brain metastases at initial presentation, which carries an even worse prognosis. New systemic therapies such as targeted-therapies and immuno-therapies have potential to provide better outcomes, but are not without challenges. First, efficacy is limited to a subset of patients. Second, the blood-brain barrier limits penetration, which varies among patients. Third, toxicities can be considerable. Current practice involves waiting 3-6 months to follow-up and assess tumor response; however, by then, it is later than ideal to try other therapies, and too late to limit toxicity. Establishing a non-invasive early predictor of response will accelerate the use of new promising agents and could improve tumor response and outcomes. Pre-clinical studies demonstrate changes in tumor vasculature hours after treatment are predictive of long-term treatment response. The aim of this study is to use dynamic-contrast enhanced magnetic resonance imaging (DCE-MRI) to evaluate both pre-treatment and post-treatment vascular measures (vascular volume, vascular permeability, interstitial tumor pressure) as predictors of long-term response. This exploratory clinical study will enroll 20 patients to complete 3 DCE-MRI studies. The response variables will be modeled against the vascular measures at three timepoints (pre-treatment, immediate post-treatment, and standard follow-up interval (6-8 weeks)) to assess the predictive ability of tumor vascular characteristics on survival, tumor progression, and imaging response. The data acquired in this study will be used in planning larger and more comprehensive trials in the future