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Register a new userArrhythmogenicity of cardiac fibrosis: fractal measures and Betti numbers
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Added by
Mahesh Kumar Mulimani
Publication date
2021
fields
Physics
and research's language is
English
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Infarction- or ischaemia-induced cardiac fibrosis can be arrythmogenic. We use mathematcal models for diffuse fibrosis ($mathcal{DF}$), interstitial fibrosis ($mathcal{IF}$), patchy fibrosis ($mathcal{PF}$), and compact fibrosis ($mathcal{CF}$) to study patterns of fibrotic cardiac tissue that have been generated by new mathematical algorithms. We show that the fractal dimension $mathbb{D}$, the lacunarity $mathcal{L}$, and the Betti numbers $beta_0$ and $beta_1$ of such patterns are textit{fibrotic-tissue markers} that can be used to characterise the arrhythmogenicity of different types of cardiac fibrosis. We hypothesize, and then demonstrate by extensive textit{in silico} studies of detailed mathematical models for cardiac tissue, that the arrhytmogenicity of fibrotic tissue is high when $beta_0$ is large and the lacunarity parameter $b$ is small.
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We present the relation between the genus in cosmology and the Betti numbers for excursion sets of three- and two-dimensional smooth Gaussian random fields, and numerically investigate the Betti numbers as a function of threshold level. Betti numbers are topological invariants of figures that can be used to distinguish topological spaces. In the case of the excursion sets of a three-dimensional field there are three possibly non-zero Betti numbers; $beta_0$ is the number of connected regions, $beta_1$ is the number of circular holes, and $beta_2$ is the number of three-dimensional voids. Their sum with alternating signs is the genus of the surface of excursion regions. It is found that each Betti number has a dominant contribution to the genus in a specific threshold range. $beta_0$ dominates the high-threshold part of the genus curve measuring the abundance of high density regions (clusters). $beta_1$ dominates the genus near the median thresholds which measures the topology of negatively curved iso-density surfaces, and $beta_2$ corresponds to the low-threshold part measuring the void abundance. We average the Betti number curves (the Betti numbers as a function of the threshold level) over many realizations of Gaussian fields and find that both the amplitude and shape of the Betti number curves depend on the slope of the power spectrum $n$ in such a way that their shape becomes broader and their amplitude drops less steeply than the genus as $n$ decreases. This behaviour contrasts with the fact that the shape of the genus curve is fixed for all Gaussian fields regardless of the power spectrum. Even though the Gaussian Betti number curves should be calculated for each given power spectrum, we propose to use the Betti numbers for better specification of the topology of large scale structures in the universe.
We study the expected behavior of the Betti numbers of arrangements of the zeros of random (distributed according to the Kostlan distribution) polynomials in $mathbb{R}mathrm{P}^n$. Using a random spectral sequence, we prove an asymptotically exact estimate on the expected number of connected components in the complement of $s$ such hypersurfaces in $mathbb{R}mathrm{P}^n$. We also investigate the same problem in the case where the hypersurfaces are defined by random quadratic polynomials. In this case, we establish a connection between the Betti numbers of such arrangements with the expected behavior of a certain model of a randomly defined geometric graph. While our general result implies that the average zeroth Betti number of the union of random hypersurface arrangements is bounded from above by a function that grows linearly in the number of polynomials in the arrangement, using the connection with random graphs, we show an upper bound on the expected zeroth Betti number of random quadrics arrangements that is sublinear in the number of polynomials in the arrangement. This bound is a consequence of a general result on the expected number of connected components in our random graph model which could be of independent interest.
We introduce a new class of monomial ideals which we call symmetric shifted ideals. Symmetric shifted ideals are fixed by the natural action of the symmetric group and, within the class of monomial ideals fixed by this action, they can be considered as an analogue of stable monomial ideals within the class of monomial ideals. We show that a symmetric shifted ideal has linear quotients and compute its (equivariant) graded Betti numbers. As an application of this result, we obtain several consequences for graded Betti numbers of symbolic powers of defining ideals of star configurations.
Let $(A, m, k)$ be a Gorenstein local ring of dimension $ dgeq 1.$ Let $I$ be an ideal of $A$ with $htt(I) geq d-1.$ We prove that the numerical function [ n mapsto ell(ext_A^i(k, A/I^{n+1}))] is given by a polynomial of degree $d-1 $ in the case when $ i geq d+1 $ and $curv(I^n) > 1$ for all $n geq 1.$ We prove a similar result for the numerical function [ n mapsto ell(Tor_i^A(k, A/I^{n+1}))] under the assumption that $A$ is a CM ~ local ring. oindent We note that there are many examples of ideals satisfying the condition $curv(I^n) > 1,$ for all $ n geq 1.$ We also consider more general functions $n mapsto ell(Tor_i^A(M, A/I_n)$ for a filtration ${I_n }$ of ideals in $A.$ We prove similar results in the case when $M$ is a maximal CM ~ $A$-module and ${I_n=overline{I^n} }$ is the integral closure filtration, $I$ an $m$-primary ideal in $A.$
An electrocardiogram (EKG) is a common, non-invasive test that measures the electrical activity of a patients heart. EKGs contain useful diagnostic information about patient health that may be absent from other electronic health record (EHR) data. As multi-dimensional waveforms, they could be modeled using generic machine learning tools, such as a linear factor model or a variational autoencoder. We take a different approach:~we specify a model that directly represents the underlying electrophysiology of the heart and the EKG measurement process. We apply our model to two datasets, including a sample of emergency department EKG reports with missing data. We show that our model can more accurately reconstruct missing data (measured by test reconstruction error) than a standard baseline when there is significant missing data. More broadly, this physiological representation of heart function may be useful in a variety of settings, including prediction, causal analysis, and discovery.
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