On Random Planar Curves and Their Scaling Limits

Planar curves arise naturally as interfaces between two regions of the plane. An important part of statistical physics is the study of lattice models. This thesis is about the interfaces of 2D lattice models. The scaling limit is an infinite system limit which is taken by letting the lattice mesh decrease to zero. At criticality, the scaling limit of an interface is one of the SLE curves (Schramm-Loewner evolution), introduced by Oded Schramm. This family of random curves is parametrized by a real variable, which determines the universality class of the model. The first and the second paper of this thesis study properties of SLEs. They contain two different methods to study the whole SLE curve, which is, in fact, the most interesting object from the statistical physics point of view. These methods are applied to study two symmetries of SLE: reversibility and duality. The first paper uses an algebraic method and a representation of the Virasoro algebra to find common martingales to different processes, and that way, to confirm the symmetries for polynomial expected values of natural SLE data. In the second paper, a recursion is obtained for the same kind of expected values. The recursion is based on stationarity of the law of the whole SLE curve under a SLE induced flow. The third paper deals with one of the most central questions of the field and provides a framework of estimates for describing 2D scaling limits by SLE curves. In particular, it is shown that a weak estimate on the probability of an annulus crossing implies that a random curve arising from a statistical physics model will have scaling limits and those will be well-described by Loewner evolutions with random driving forces.

Clinical Studies on Epidural and Spinal Postoperative Analgesia : With Special Reference to Continuous Techniques of Administration and Adjuvant Drugs

Continuous epidural analgesia (CEA) and continuous spinal postoperative analgesia (CSPA) provided by a mixture of local anaesthetic and opioid are widely used for postoperative pain relief. E.g., with the introduction of so-called microcatheters, CSPA found its way particularly in orthopaedic surgery. These techniques, however, may be associated with dose-dependent side-effects as hypotension, weakness in the legs, and nausea and vomiting. At times, they may fail to offer sufficient analgesia, e.g., because of a misplaced catheter. The correct position of an epidural catheter might be confirmed by the supposedly easy and reliable epidural stimulation test (EST). The aims of this thesis were to determine a) whether the efficacy, tolerability, and reliability of CEA might be improved by adding the α2-adrenergic agonists adrenaline and clonidine to CEA, and by the repeated use of EST during CEA; and, b) the feasibility of CSPA given through a microcatheter after vascular surgery. Studies I IV were double-blinded, randomized, and controlled trials; Study V was of a diagnostic, prospective nature. Patients underwent arterial bypass surgery of the legs (I, n=50; IV, n=46), total knee arthroplasty (II, n=70; III, n=72), and abdominal surgery or thoracotomy (V, n=30). Postoperative lumbar CEA consisted of regular mixtures of ropivacaine and fentanyl either without or with adrenaline (2 µg/ml (I) and 4 µg/ml (II)) and clonidine (2 µg/ml (III)). CSPA (IV) was given through a microcatheter (28G) and contained either ropivacaine (max. 2 mg/h) or a mixture of ropivacaine (max. 1 mg/h) and morphine (max. 8 µg/h). Epidural catheter tip position (V) was evaluated both by EST at the moment of catheter placement and several times during CEA, and by epidurography as reference diagnostic test. CEA and CSPA were administered for 24 or 48 h. Study parameters included pain scores assessed with a visual analogue scale, requirements of rescue pain medication, vital signs, and side-effects. Adrenaline (I and II) had no beneficial influence as regards the efficacy or tolerability of CEA. The total amounts of epidurally-infused drugs were even increased in the adrenaline group in Study II (p=0.02, RM ANOVA). Clonidine (III) augmented pain relief with lowered amounts of epidurally infused drugs (p=0.01, RM ANOVA) and reduced need for rescue oxycodone given i.m. (p=0.027, MW-U; median difference 3 mg (95% CI 0 7 mg)). Clonidine did not contribute to sedation and its influence on haemodynamics was minimal. CSPA (IV) provided satisfactory pain relief with only limited blockade of the legs (no inter-group differences). EST (V) was often related to technical problems and difficulties of interpretation, e.g., it failed to identify the four patients whose catheters were outside the spinal canal already at the time of catheter placement. As adjuvants to lumbar CEA, clonidine only slightly improved pain relief, while adrenaline did not provide any benefit. The role of EST applied at the time of epidural catheter placement or repeatedly during CEA remains open. The microcatheter CSPA technique appeared effective and reliable, but needs to be compared to routine CEA after peripheral arterial bypass surgery.

Brain and Spinal Cavernomas : Helsinki Experience

Cavernomas are rare neurovascular lesions, encountered in up to 10% of patients harboring vascular abnormalities of the CNS. Cavernomas consist of dilated thin-walled sinusoids or caverns covered by a single layer of endothelium. Due to advancements in neuroradiology, the number of cavernoma patients coming to be evaluated in neurosurgical practice is increasing. In the present work, we summarized our results on the treatment of cavernomas. Particular attention was paid to uncommon locations or insufficiently investigated cavernomas, including 1. Intraventricular cavernomas; 2. Multiple cavernomas; 3. Spinal cavernomas; and 4. Temporal lobe cavernomas. After analyzing the patient series with these lesions, we concluded that: 1. IVCs are characterized by a high tendency to cause repetitive hemorrhages in a short period of time after the first event. In most patients, hemorrhages were not life-threatening. Surgery is indicated when re-bleedings are frequent and the mass-effect causes progressive neurological deterioration. Modern microsurgical techniques allow safe removal of the IVC, but surgery on fourth ventricle cavernomas carries increased risk of postoperative cranial nerve deficits. 2. In MC cases, when the cavernoma bleeds or generates drug-resistant epilepsy, microsurgical removal of the symptomatic lesion is beneficial to patients. In our series, surgical removal of the most active cavernoma usually the biggest lesion with signs of recent hemorrhage - was safe and prevented further bleedings. Epilepsy outcome showed the effectiveness of active treatment of MCs. However, due to the remaining cavernomas, epileptogenic activity can persist postoperatively, frequently necessitating long-term use of antiepileptic drugs. 3. Spinal cavernomas can cause severe neurological deterioration due to low tolerance of the spinal cord to mass-effect with progressive myelopathy. When aggravated by extralesional massive hemorrhage, neurological decline is usually acute and requires immediate treatment. Microsurgical removal of a cavernoma is effective and safe, improving neurological deficits. Sensorimotor deficits and pain improved postoperatively at a high rate, whereas bladder dysfunction remained essentially unchanged, causing social discomfort to patients. 4. Microsurgical removal of temporal lobe cavernomas is beneficial for patents suffering from drug-resistant epilepsy. In our series, 69% of patients with this condition became seizure-free postoperatively. Duration of epilepsy did not correlate with seizure prognosis. The most frequent disabling symptom at follow-up was memory disorder, considered to be the result of a complex interplay between chronic epilepsy and possible damage to the temporal lobe during surgery.