In mathematics, the Menger curvature of three points in n-dimensional Euclidean space is the reciprocal of the radius of the circle that passes through the three points. Intuitively, curvature is the amount by which a geometric object deviates from being:

• a flat plane in case of the surface,
• a straight line in case of the curve.

In this project, the curvature measurement is applied to human left ventricle traces, in order to determine the occurrence of the left ventricular basal septal hypertrophy from 2 dimensional echocardiography (2D echo). The described methods can be applied to any traces of the left ventricle.

Localized basal septal hypertrophy (BSH) is a known marker of a more advanced impact of afterload on cardiac function in hypertension. There is variability in criteria used for defining BSH, mainly based on ratios of multiple septal wall thickness measurements with high inter-observer variability. The aim is to investigate septal curvature as a novel, semiautomated method for better recognition of patients with BSH.

Examples of differences in septal curvature among 3 patients: healthy, hypertensive and hypertensive with septal bulge:

Call

Curvature(line=[(x1, y1), (x2, y2), (x3, y3) ... (xn, yn)] 

Input

Input argument line is a list of tuples. Each tuple contains 2 values, i.e. X and Y position on the 2D plane.

Output

Numpy array. Menger's curvature value for each tuple in the input list, except first and the last one.

Methods

Calculates the curvature of the line. It is defined as the reciprocal of the radius of a circle intersecting three points in 2D space. Optional parameter gap sets the number of points away from the processed point, based on which the curvature is calculated:

• if gap = 0, three consecutive points are used,
• if gap = 1, for point number 2, points 0 and 4 are used, for point number 3, points 1 and 5 are used and so on,
• if gap = 2, for point number 3, points 0 and 6 are used, for point number 4, points 1 and 7 are used and so on.

It has been included as a smoothing option, with the trade-off on information loss.

Plots the curvature values as a line plot.

Example (quadratic parabola)

import numpy as np
from curvature import Curvature

x = np.linspace(-5, 5, 1001)
y = (x ** 2)
xy = list(zip(x, y))  # list of points in 2D space

curv = Curvature(line=xy)
curv.calculate_curvature(gap=0)

print('Curvature values (first 10 points): {}'.format(curv.curvature[:10]))
print('Curvature values (10 middle points): {}'.format(curv.curvature[int(len(x)/2-5):int(len(x)/2+5)]))
print('Maximum curvature: {}'.format(max(curv.curvature)))
print('Minimum curvature: {}'.format(min(curv.curvature)))

curv.plot_curvature()

Results:

Curvature values (first 10 points): [0.00198212 0.00199397 0.00200591 0.00201794 0.00203007 0.0020423
 0.00205463 0.00206705 0.00207958 0.00209221]
 
Curvature values (10 middle points): [1.98075815 1.98905163 1.99501104 1.99860098 1.99980002 1.99860098
 1.99501104 1.98905163 1.98075815 1.97017947]
 
Maximum curvature: 1.9998000199980006
Minimum curvature: 0.0019821241706415283

The aim of this class is to calculate the curvature of the trace of left ventricle and find indices that are useful for classification of the basal septal hypertrophy in hypertensive patients. The indices include maximum and minimum curvature, the changes in curvature over the cycle and interactions between them. The derived indices are useful for the statistical analysis, and show potential to unveil the basal septal hypertrophy setting in a robust, unbiased way.

Call

class Cohort(source_path='path_to_data', view='4C', output_path='path_to_store_results', 
output='name_of_output_file.csv')

Input

source_path: path to the .csv files containing the myocardial trace obtained with speckle tracing in EchoPAC.

view: the view in which the image was taken; 4-chamber ('4C'), 3-chamber ('3C'), or 2-chamber ('2C')

output_path: path to a folder where the results of computation and plots will be stored

Output

1 Tables:

• File names in the input directory with corresponding IDs of cases,

• Curvature values of the trace changing in the cardiac cycle,

• All derived indices calculated in available views,

• Simple statistical values of the derived indices,

• Lists of most prevalent cases, in terms of the derived indices.

2 Plots:

• Inidividual plots of the trace and the curvature throughout the cardiac cycle,

• Distributions of the derived indices in the population.

Methods

Creates (or prints) the table with names of the files and corresponding IDs. Useful when the clinician provides tables with different IDs, unrelated to one another.

to_file controls whether the table is saved to file, or is printed in the console.

views list is used to choose the relevant views to print out. The function prints the names and IDs for all views by default.

Example:

Cohort.print_names_and_ids(views=['4C'])

Saves the curvature of individual trace over 1 cycle. Rows denote the frames and columns are the separate points of the trace.

Example:

Cohort.save_curvatures() 

In ~/output_curvature/curvatures/ABCDE0123.csv:

Saves the derived indices of the current view to a .csv file. The meaning of the indices is listed in the table below.

Example:

In ~/output_curvature/_all_cases.csv:

Builds a table with means and standard deviations of the derived indices for different labelled cohorts. It is useful for quick hypothesis testing.

Example:

Creates a table with IDs of cases with most prevalent indices and interactions. It is useful for the analyst to decide on which indices are relevant for the classification.

n is the number of cases to print for each index.

Example:

Plots the with traces in a given view and the curvature of each points in the trace in each frame. The traces can be coloured according to the value of the curvature, or the frame number. This function also creates heatmaps showing the curvature of the trace in the given view changing in time. End-diastolic and end-systolic frames are found by calculating the maximum and minimum area covered by the trace.

coloring_scheme - if set to 'curvautre', the plot shows the positive curvature as red and negative curvature as blue. Otherwise, by default, the colors correspond to the frame number. This feature is not perfect, as it is not possible to relate the frame number to cycle makers, such as aortic valve closure, mitral valve opening, etc.

Example:

Cohort.plot_curvatures(coloring_scheme='curvature')

Plots curvature, where for each point in the the trace a mean of the curvature over the full cycle is calculated.

Example:

Cohort.plot_mean_curvature()

Plots the distributions of the derived indices, for univariate and bivariate exploratory data analysis.

plot_data - if set to true, plots the univariate plots of available indices from a single view. With *table_name set to 'master_table.csv', it will plot the bivariate interactions between the indices (provided that the 'master_table exists)

plot_master - if set to true, also plots the univariate plots, but from all the available views. With *table_name set to 'master_table.csv',the function will plot bivariate interactions between the indices (provided that the 'master_table exists and different views indices are avaiable).

Examples:

Cohort.plot_distributions(plot_data=True, table_name=all_cases_with_labels.csv)

Cohort.plot_distributions(plot_data=True, table_name=all_cases_with_labels.csv)

Full example

import os
from bsh import Cohort

source = os.path.join('~/Python/data/curvature')
target_path = os.path.join('~/Python/data/curvature/')

cohort = Cohort(source_path=source, view=_view, output_path=target_path)

cohort.get_extemes(32)
cohort.plot_curvatures('asf')
cohort.plot_curvatures(coloring_scheme='curvature')
cohort.save_curvatures()
cohort.plot_distributions(plot_data=True, table_name='_all_cases_with_labels.csv')
cohort.print_names_and_ids(to_file=True)

Abstract