On the applicability of mathematics : philosophical and historical perspectives

The present thesis is a contribution to the debate on the applicability of mathematics; it examines the interplay between mathematics and the world, using historical case studies. The first part of the thesis consists of four small case studies. In chapter 1, I criticize "ante rem structuralism", proposed by Stewart Shapiro, by showing that his so-called "finite cardinal structures" are in conflict with mathematical practice. In chapter 2, I discuss Leonhard Euler's solution to the Königsberg bridges problem. I propose interpreting Euler's solution both as an explanation within mathematics and as a scientific explanation. I put the insights from the historical case to work against recent philosophical accounts of the Königsberg case. In chapter 3, I analyze the predator-prey model, proposed by Lotka and Volterra. I extract some interesting philosophical lessons from Volterra's original account of the model, such as: Volterra's remarks on mathematical methodology; the relation between mathematics and idealization in the construction of the model; some relevant details in the derivation of the Third Law, and; notions of intervention that are motivated by one of Volterra's main mathematical tools, phase spaces. In chapter 4, I discuss scientific and mathematical attempts to explain the structure of the bee's honeycomb. In the first part, I discuss a candidate explanation, based on the mathematical Honeycomb Conjecture, presented in Lyon and Colyvan (2008). I argue that this explanation is not scientifically adequate. In the second part, I discuss other mathematical, physical and biological studies that could contribute to an explanation of the bee's honeycomb. The upshot is that most of the relevant mathematics is not yet sufficiently understood, and there is also an ongoing debate as to the biological details of the construction of the bee's honeycomb. The second part of the thesis is a bigger case study from physics: the genesis of GR. Chapter 5 is a short introduction to the history, physics and mathematics that is relevant to the genesis of general relativity (GR). Chapter 6 discusses the historical question as to what Marcel Grossmann contributed to the genesis of GR. I will examine the so-called "Entwurf" paper, an important joint publication by Einstein and Grossmann, containing the first tensorial formulation of GR. By comparing Grossmann's part with the mathematical theories he used, we can gain a better understanding of what is involved in the first steps of assimilating a mathematical theory to a physical question. In chapter 7, I introduce, and discuss, a recent account of the applicability of mathematics to the world, the Inferential Conception (IC), proposed by Bueno and Colyvan (2011). I give a short exposition of the IC, offer some critical remarks on the account, discuss potential philosophical objections, and I propose some extensions of the IC. In chapter 8, I put the Inferential Conception (IC) to work in the historical case study: the genesis of GR. I analyze three historical episodes, using the conceptual apparatus provided by the IC. In episode one, I investigate how the starting point of the application process, the "assumed structure", is chosen. Then I analyze two small application cycles that led to revisions of the initial assumed structure. In episode two, I examine how the application of "new" mathematics - the application of the Absolute Differential Calculus (ADC) to gravitational theory - meshes with the IC. In episode three, I take a closer look at two of Einstein's failed attempts to find a suitable differential operator for the field equations, and apply the conceptual tools provided by the IC so as to better understand why he erroneously rejected both the Ricci tensor and the November tensor in the Zurich Notebook.

Early white matter alterations in men predisposed to FXTAS revealed by MT imaging.

Introduction: The Fragile X - associated Tremor Ataxia Syndrome (FXTAS) is a recently described, and under-diagnosed, late onset (≈ 60y) neurodegenerative disorder affecting male carriers of a premutation in the Fragile X Mental Retardation 1 (FMR1) gene. The premutation is an CGG (Cytosine-Guanine-Guanine) expansion (55 to 200 CGG repeats) in the proximal region of the FMR1 gene. Patients with FXTAS primarily present with cerebellar ataxia and intention tremor. Neuroradiological features of FXTAS include prominent white matter disease in the periventricular, subcortical, middle cerebellar peduncles and deep white matter of the cerebellum on T2-weighted or FLAIR MR imaging (Jacquemmont 2007, Loesch 2007, Brunberg 2002, Cohen 2006). We hypothesize that a significant white matter alteration is present in younger individuals many years prior to clinical symptoms and/or the presence of visible lesions on conventional MR sequences and might be detectable by magnetization transfer (MT) imaging. Methods: Eleven asymptomatic premutation carriers (mean age = 55 years) and seven intra-familial controls participated to the study. A standardized neurological examination was performed on all participants and a neuropsychological evaluation was carried out before MR scanning performed on a 3T Siemens Trio. The protocol included a sagittal T1-weighted 3D gradient-echo sequence (MPRAGE, 160 slices, 1 mm^3 isotropic voxels) and a gradient-echo MTI (FA 30, TE 15, matrix size 256*256, pixel size 1*1 mm, 36 slices (thickness 2mm), MT pulse duration 7.68 ms, FA 500, frequency offset 1.5 kHz). MTI was performed by acquiring consecutively two set of images; first with and then without the MT saturation pulse. MT images were coregistered to the T1 acquisition. The MTR for every intracranial voxel was calculated as follows: MTR = (M0 - MS)/M0*100%, creating a MTR map for each subject. As first analysis, the whole white matter (WM) was used to mask the MTR image in order to create an histogram of the MTR distribution in the whole tissue class over the two groups examined. Then, for each subject, we performed a segmentation and parcellation of the brain by means of Freesurfer software, starting from the high resolution T1-weighted anatomical acquisition. Cortical parcellations was used to assign a label to the underlying white matter by the construction of a Voronoi diagram in the WM voxels of the MR volume based on distance to the nearest cortical parcellation label. This procedure allowed us to subdivide the cerebral WM in 78 ROIs according to the cortical parcellation (see example in Fig 1). The cerebellum, by the same procedure, was subdivided in 5 ROIs (2 per each hemisphere and one corresponding to the brainstem). For each subject, we calculated the mean value of MTR within each ROI and averaged over controls and patients. Significant differences between the two groups were tested using a two sample T-test (p<0.01). Results: Neurological examination showed that no patient met the clinical criteria of Fragile X Tremor and Ataxia Syndrome yet. Nonetheless, premutation carriers showed some subtle neurological signs of the disorder. In fact, premutation carriers showed a significant increase of tremor (CRST, T-test p=0.007) and increase of ataxia (ICARS, p=0.004) when compared to controls. The neuropsychological evaluation was normal in both groups. To obtain general characterizations of myelination for each subject and premutation carriers, we first computed the distribution of MTR values across the total white matter volume and averaged for each group. We tested the equality of the two distributions with the non parametric Kolmogorov-Smirnov test and we rejected the null-hypothesis at a p=0.03 (fig. 2). As expected, when comparing the asymptomatic permutation carriers with control subjects, the peak value and peak position of the MTR values within the whole WM were decreased and the width of the distribution curve was increased (p<0.01). These three changes point to an alteration of the global myelin status of the premutation carriers. Subsequently, to analyze the regional myelination and white matter integrity of the same group, we performed a ROI analysis of MTR data. The ROI-based analysis showed a decrease of mean MTR value in premutation carriers compared to controls in bilateral orbito-frontal and inferior frontal WM, entorhinal and cingulum regions and cerebellum (Fig 3). The detection of these differences in these regions failed with other conventional MR techniques. Conclusions: These preliminary data confirm that in premutation carriers, there are indeed alterations in "normal appearing white matter" (NAWM) and these alterations are visible with the MT technique. These results indicate that MT imaging may be a relevant approach to detect both global and local alterations within NAWM in "asymptomatic" carriers of premutations in the Fragile X Mental Retardation 1 (FMR1) gene. The sensitivity of MT in the detection of these alterations might point towards a specific physiopathological mechanism linked to an underlying myelin disorder. ROI-based analyses show that the frontal, parahippocampal and cerebellar regions are already significantly affected before the onset of symptoms. A larger sample will allow us to determine the minimum CGG expansion and age associated with these subclinical white matter alterations.

Segmentation and grid generation for numerical simulations of vad connections with patient-specific data

Objectives: We are interested in the numerical simulation of the anastomotic region comprised between outflow canula of LVAD and the aorta. Segmenta¬tion, geometry reconstruction and grid generation from patient-specific data remain an issue because of the variable quality of DICOM images, in particular CT-scan (e.g. metallic noise of the device, non-aortic contrast phase). We pro¬pose a general framework to overcome this problem and create suitable grids for numerical simulations.Methods: Preliminary treatment of images is performed by reducing the level window and enhancing the contrast of the greyscale image using contrast-limited adaptive histogram equalization. A gradient anisotropic diffusion filter is applied to reduce the noise. Then, watershed segmentation algorithms and mathematical morphology filters allow reconstructing the patient geometry. This is done using the InsightToolKit library (www.itk.org). Finally the Vascular Model¬ing ToolKit (www.vmtk.org) and gmsh (www.geuz.org/gmsh) are used to create the meshes for the fluid (blood) and structure (arterial wall, outflow canula) and to a priori identify the boundary layers. The method is tested on five different patients with left ventricular assistance and who underwent a CT-scan exam.Results: This method produced good results in four patients. The anastomosis area is recovered and the generated grids are suitable for numerical simulations. In one patient the method failed to produce a good segmentation because of the small dimension of the aortic arch with respect to the image resolution.Conclusions: The described framework allows the use of data that could not be otherwise segmented by standard automatic segmentation tools. In particular the computational grids that have been generated are suitable for simulations that take into account fluid-structure interactions. Finally the presented method features a good reproducibility and fast application.

Stoneley wave modeling in heterogeneous porous media with viscous pore fluids

We implemented Biot-type porous wave equations in a pseudo-spectral numerical modeling algorithm for the simulation of Stoneley waves in porous media. Fourier and Chebyshev methods are used to compute the spatial derivatives along the horizontal and vertical directions, respectively. To prevent from overly short time steps due to the small grid spacing at the top and bottom of the model as a consequence of the Chebyshev operator, the mesh is stretched in the vertical direction. As a large benefit, the Chebyshev operator allows for an explicit treatment of interfaces. Boundary conditions can be implemented with a characteristics approach. The characteristic variables are evaluated at zero viscosity. We use this approach to model seismic wave propagation at the interface between a fluid and a porous medium. Each medium is represented by a different mesh and the two meshes are connected through the above described characteristics domain-decomposition method. We show an experiment for sealed pore boundary conditions, where we first compare the numerical solution to an analytical solution. We then show the influence of heterogeneity and viscosity of the pore fluid on the propagation of the Stoneley wave and surface waves in general.

Laparoscopic inguinal hernia repair with extraperitoneal double-mesh technique.

Totally extraperitoneal laparoscopic hernia repair is an efficient but technically demanding procedure. As mechanisms of hernia recurrence may be related to these technical difficulties, we have modified a previously described double-mesh technique in an effort to simplify the procedure. Extraperitoneal laparoscopic hernia repairs were performed in 82 male and 17 female patients having inguinal, femoral, and recurrent bilateral hernias. A standard propylene mesh measuring 15 x 15 cm was cut into two pieces of 4 x 15 cm and 11 x 15 cm. The smaller mesh was placed over both inguinal rings without splitting. The larger mesh was then inserted over the first mesh and stapled to low-risk zones, reinforcing the large-vessel area and the nerve transition zone. The mean procedure duration was 60 minutes for unilateral and 100 minutes for bilateral hernia repair. Patients were discharged from the hospital within 48 hours. The mean postoperative follow-up was 22 months, with no recurrences, neuralgia, or bleeding complications. Over a 2-year period, this technique was found to be satisfactory without recurrences or significant complications. In our hands, this technique was easier to perform: it allows for a less than perfect positioning of the meshes and avoids most of the stapling to crucial zones.

A collagen-poly(lactic acid-co-ɛ-caprolactone) hybrid scaffold for bladder tissue regeneration.

Scaffold materials should favor cell attachment and proliferation, and provide designable 3D structures with appropriate mechanical strength. Collagen matrices have proven to be beneficial scaffolds for tissue regeneration. However, apart from small intestinal submucosa, they offer a limited mechanical strength even if crosslinking can enhance their mechanical properties. A more cell-friendly way to increase material strength is to combine synthetic polymer meshes with plastic compressed collagen gels. This work describes the potential of plastic compressed collagen-poly(lactic acid-co-ɛ-caprolactone) (PLAC) hybrids as scaffolds for bladder tissue regeneration. Human bladder smooth muscle and urothelial cells were cultured on and inside collagen-PLAC hybrids in vitro. Scaffolds were analyzed by electron microscopy, histology, immunohistochemistry, and AlamarBlue assay. Both cell types proliferated in and on the hybrid, forming dense cell layers on top after two weeks. Furthermore, hybrids were implanted subcutaneously in the backs of nude mice. Host cell infiltration, scaffold degradation, and the presence of the seeded bladder cells were analyzed. Hybrids showed a lower inflammatory reaction in vivo than PLAC meshes alone, and first signs of polymer degradation were visible at six months. Collagen-PLAC hybrids have potential for bladder tissue regeneration, as they show efficient cell seeding, proliferation, and good mechanical properties.

A novel technique for the reconstruction of infected full-thickness chest wall defects.

BACKGROUND: Chest wall resection and reconstruction can be performed with minimal mortality and excellent functional and cosmetic results using synthetic meshes, methylmethacrylate, or other substitutes. However, these techniques are less easily applicable if chest wall resections have to be performed for infections. METHODS: We report a novel technique for this purpose using a modified latissimus dorsi flap harvested in continuity with the thoracolumbar fascia. The vascularized fascia was sutured into the chest wall defect, providing a stable base for the muscular component of the flap. Three patients requiring large full-thickness resections of the anterolateral chest wall for chronic infections were treated accordingly, two presenting with chronic radionecrosis and osteomyelitis and one with chest wall invasion by pulmonary aspergillosis. RESULTS: There were no intraoperative or postoperative complications and immediate extubation was possible in all 3 patients without the need for postoperative ventilation or tracheotomy. Healing of the infected chest wall was observed in all 3 patients. Postoperative cinemagnetic resonance imaging revealed concordant movements of the replaced segments without evidence of paradoxical motion during inspiration and expiration. CONCLUSIONS: This technique is easy and safe. It allows a stable and satisfactory reconstruction after large anterolateral full-thickness chest wall resections of infected, previously irradiated tissues, using only well-vascularized autologous tissue.