Dynamics of McMullen maps

In this article, we develop the Yoccoz puzzle technique to study a family of rational maps termed McMullen maps. We show that the boundary of the immediate basin of infinity is always a Jordan curve if it is connected. This gives a positive answer to a question of Devaney. Higher regularity of this boundary is obtained in almost all cases. We show that the boundary is a quasi-circle if it contains neither a parabolic point nor a recurrent critical point. For the whole Julia set, we show that the McMullen maps have locally connected Julia sets except in some special cases.Comment: Complex dynamics, 51 pages, 13 figure

Quasisymmetric geometry of the Cantor circles as the Julia sets of rational maps

We give three families of parabolic rational maps and show that every Cantor set of circles as the Julia set of a non-hyperbolic rational map must be quasisymmetrically equivalent to the Julia set of one map in these families for suitable parameters. Combining a result obtained before, we give a complete classification of the Cantor circles Julia sets in the sense of quasisymmetric equivalence. Moreover, we study the regularity of the components of the Cantor circles Julia sets and establish a sufficient and necessary condition when a component of a Cantor circles Julia set is a quasicircle.Comment: 39 pages, 10 figures and 1 table, to appear in Discrete and Continous Dynamical Systems-

Real-time Temperature Imaging Using Ultrasonic Change in Backscattered Energy

Thermal therapy from low-temperature cryosurgery to high-temperature ablation of tumors and unwanted electrical pathways has gained increased attention. Temperature imaging (TI) from magnetic resonance studies is the de facto standard for volumetric estimation of temperature. Ultrasound has the advantages of being cheap, portable, non-invasive and non-ionizing. Our group showed in predictions for single scatterers, simulations of scatterer populations and measurements in 1D, 2D and 3D, that CBE changed monotonically with temperature with 1oC accuracy. An obstacle to clinical application of CBE TI is estimation of temperature in real time, which is limited by time for motion compensation (MC). To achieve real-time TI, we implemented a two-computer architecture. Our Terason 3000 ultrasonic imaging system collected and sent raw images over a jtcp connection to a TI computer with a GeForce GTX 770 GPU card. The TI computer performed motion compensation and extracted temperature images. Turkey specimens were imaged during heating with hot water (75oC) in 1 cm tube. Total heating time was 1200 sec, with a 30 sec interval between image acquisitions; tissue temperature was monitored with thermocouples. Over six experiments at 3 thermocouple sites, the accuracy of CBE TI was 0.8±0.7oC. Using its CPU, the TI computer updated temperature images using rigid MC in 4 sec, and using more nearly accurate nonrigid MC in 7 sec. Nonrigid MC time was reduced to 0.2 sec using the GPU processor along with optimization of the MC algorithm. Calculation of CBE in MC images and conversion of CBE to TI takes less than an additional 0.1 sec. With TI time reduced to \u3c 0.3 sec, the limit to real-time CBE TI now lies with the Terason 3000 system. It takes about 5 sec to transform an ultrasonic image in its native format to Matlab before sending it to the TI computer. Therefore, we believe CBE TI can be done at a 1 Hz frame rate with \u3c 1oC error if conversion to Matlab in the Terason 3000 can be reduced to less than 0.7 sec

GDF1 AS A REGULATOR OF CERAMIDE METABOLISM AND HEMATOPOIESIS IN ACUTE MYELOID LEUKEMIA

Acute myeloid leukemia (AML) is cancer of the myeloid lineage of blood cells. In AML, hematopoietic precursors acquire mutations or chromosomal changes that cause differentiation arrest. This results in an uncontrolled proliferation of these malignant cells in the bone marrow that can interfere with normal blood cell production. AML progresses rapidly and can be fatal within weeks if left untreated. The only current curative treatment is bone marrow transplant, which has potentially life-threatening side effects. Thus, a better understanding of the underlying biology of AML is needed to enable the development of better therapeutic modalities.Ceramides are a family of wax-like lipids in the broader category of sphingolipids. Ceramide is comprised of a sphingoid base linked to a fatty acid through an amide group. Recently, ceramides have been appreciated for their bioactive functions including as regulators of apoptosis. This is significant because ceramide metabolic routes persist in AML that may be exploited for therapeutic development. Recently, it was shown that nanoliposomal C6-ceramide (Lip-C6) exerts unique therapeutic efficacy towards AML with myelodysplastic syndrome-related changes (AML-MRC). In contrast, other forms of AML were resistant to Lip-C6 due to enhanced ceramide metabolism. Interestingly, in these Lip-C6-resistant AMLs the gene encoding for growth/differentiation factor 1 (GDF1) is downregulated. GDF1 is encoded from a rare bicistronic gene that also encodes for ceramide synthase 1. GDF1 is presently only appreciated to have roles in embryonic and cardiac development. However, its genetic link to a ceramide biosynthesizing enzyme as well as its inverse relation to Lip-C6-resistant AML suggest that it may also exert a role in ceramide metabolism. This dissertation research first explored a hypothesis that GDF1 regulates ceramide detoxification in AML. GDF1 expression was variably expressed across AML subtypes but was mostly downregulated in Lip-C6-resistant AML. Moreover, there was an inverse relationship between GDF1 expression and genes encoding for ceramide neutralizing enzymes. This inverse relationship was validated as treatment of AML cell lines with recombinant GDF1 downregulated the expression of these same genes. Interestingly, recombinant GDF1 was also able to uniquely promote SMAD2/3 phosphorylation while concurrently downregulating STAT2 tyrosine phosphorylation in a transforming growth factor beta receptor 1 (TGFR1)-dependent manner. Next, this dissertation research evaluated the ability of GDF1 to regulate hematopoiesis and exert an anti-AML therapeutic effect. Recombinant GDF1 restored hematopoiesis and promote erythropoiesis (red blood cell development) both in vitro and in vivo. GDF1 was unable to impact hematopoiesis in normal bone marrow, suggesting that its hematopoietic-regulatory therapeutic effect was specific to abnormal and malignant situations. Furthermore, GDF1 treatment exerted combinatorial anti-AML efficacy with cytarabine and extended the overall survival of mice engrafted with a highly aggressive AML. The final aspect of this dissertation studied a novel transgenic obese AML mouse model to evaluate links between obesity and AML. Obesity provokes profound changes in lipid homeostasis including by upregulating sphingolipid biosynthesis. Transgenic obese AML mice developed a robust leukemia burden compared with non-obese counterparts. More so, genes responsible for regulation of the ceramide-mediated NADPH oxidase 2 were upregulated in these transgenic obese AML mice. This demonstrated a further sphingolipid-mediated dysfunction that can contribute to the development and progression of AML. Overall, this dissertation research has uncovered important sphingolipid metabolic biology that underlies the development and progression of AML. This has revealed pathways associated with resistance to ceramide-elevating therapeutics such as Lip-C6. Most noteworthy, this research has identified for the first time an ability for GDF1 to regulate ceramide metabolism, hematopoiesis, and erythropoiesis. It has also shown that GDF1 can exert effects through a unique TGFR1-dependent mechanism regulating both SMAD2/3 and STAT3 signaling. Collectively, this has revealed an anti-AML therapeutic ability for GDF1 linked to the regulation of ceramide metabolism