Dynamic changes during the treatment of pancreatic cancer

This manuscript follows a single patient with pancreatic adenocarcinoma for a five year period, detailing the clinical record, pathology, the dynamic evolution of molecular and cellular alterations as well as the responses to treatments with chemotherapies, targeted therapies and immunotherapies. DNA and RNA samples from biopsies and blood identified a dynamic set of changes in allelic imbalances and copy number variations in response to therapies. Organoid cultures established from biopsies over time were employed for extensive drug testing to determine if this approach was feasible for treatments. When an unusual drug response was detected, an extensive RNA sequencing analysis was employed to establish novel mechanisms of action of this drug. Organoid cell cultures were employed to identify possible antigens associated with the tumor and the patient\u27s T-cells were expanded against one of these antigens. Similar and identical T-cell receptor sequences were observed in the initial biopsy and the expanded T-cell population. Immunotherapy treatment failed to shrink the tumor, which had undergone an epithelial to mesenchymal transition prior to therapy. A warm autopsy of the metastatic lung tumor permitted an extensive analysis of tumor heterogeneity over five years of treatment and surgery. This detailed analysis of the clinical descriptions, imaging, pathology, molecular and cellular evolution of the tumors, treatments, and responses to chemotherapy, targeted therapies, and immunotherapies, as well as attempts at the development of personalized medical treatments for a single patient should provide a valuable guide to future directions in cancer treatment

S-Lemma with Equality and Its Applications

Let $f(x)=x^TAx+2a^Tx+c$ and $h(x)=x^TBx+2b^Tx+d$ be two quadratic functions having symmetric matrices $A$ and $B$. The S-lemma with equality asks when the unsolvability of the system $f(x)<0, h(x)=0$ implies the existence of a real number $\mu$ such that $f(x) + \mu h(x)\ge0, ~\forall x\in \mathbb{R}^n$. The problem is much harder than the inequality version which asserts that, under Slater condition, $f(x)<0, h(x)\le0$ is unsolvable if and only if $f(x) + \mu h(x)\ge0, ~\forall x\in \mathbb{R}^n$ for some $\mu\ge0$. In this paper, we show that the S-lemma with equality does not hold only when the matrix $A$ has exactly one negative eigenvalue and $h(x)$ is a non-constant linear function ($B=0, b\not=0$). As an application, we can globally solve $\inf\{f(x)\vert h(x)=0\}$ as well as the two-sided generalized trust region subproblem $\inf\{f(x)\vert l\le h(x)\le u\}$ without any condition. Moreover, the convexity of the joint numerical range $\{(f(x), h_1(x),\ldots, h_p(x)):~x\in\Bbb R^n\}$ where $f$ is a (possibly non-convex) quadratic function and $h_1(x),\ldots,h_p(x)$ are affine functions can be characterized using the newly developed S-lemma with equality.Comment: 34 page