Global constructive myocardial work predicts reduction of ejection fraction in patients with heart failure with preserved ejection fraction

Background

Despite advances in treatment of heart failure with preserved ejection fraction (HFpEF) its management remains challenging. SGLT2 inhibitors benefits across the full range of ejection fraction, and sacubitril/valsartan benefits up to the lower end of preserved EF <57% implies that in some patients with HFpEF some pathophysiological mechanisms of HFrEF might co-exist, and some subset of HFpEF patients might benefit from proven treatment of HFrEF, particularly those with EF deterioration over time. We aimed to found out predictors of EF deterioration in HFpEF patients assuming that we can start treating them earlier with therapies of HFrEF, preventing further deterioration.

Methods

We studied 215 patients (63% women) 73±8 years with HFpEF. All patients had records of comorbidity Charlson index (CI), glomerular filtration rate (GFR). Echocardiography (EchoCG) was performed with offline analysis, including calculations of myocardial work (MW), global longitudinal (LS), radial (RS), circumferential (SS) and area strain (AS) by one experienced specialist. GW index was obtained from pressure-strain loops derived from speckle tracking analysis multiplied by brachial systolic blood pressure. Global constructive work (GCW) as the sum of positive work due to myocardial shortening during systole and negative work due to lengthening during isovolumic relaxation, global wasted work as energy loss by myocardial lengthening in systole and shortening in isovolumic relaxation, and GW efficiency as the percentage ratio of constructive work to the sum of constructive work and wasted work were obtained. RS, SS and AS were calculated in 3D by dedicated software. Patients followed up for 3 years.

Results

5 patients developed myocardial infarction and were excluded from the study. Baseline EF was higher in women (61,2±3,1 vs 56,4±2,7; P<0.002), in patients >70 years (62,4±2,1 vs 57,1±2,3; p<0.005), and with end-diastolic volume index <60 ml/m2 (56,1±3,2 vs 63,4±2,3; p<0.001). Overall decline in EF compare to baseline was −7.3±1.6%, p<0.01. Reduction in EF was more prominent in patients >70 years (−6,9±1,8 vs −5,7±1,7; P<0,002), and in patients with coronary artery disease (CAD) (−7,2±1,9 vs −5,8±1,6; P<0,001) and did not relate to sex, LV size, CI, and GFR. During follow up 58 (27%) patients had EF <50%. We observed significant worsening in AS (−27.9±8.5% vs −24.7±5.3%, p<0.003), LS (−19.7±2.4% vs −17.1±1.6%, p<0.005), and GCW (GCW 2378±117 vs 2107±102 mmHg%, p<0.002). Patients with EF <50% at the end of the study had significantly less AS and GCW baseline values compared with patients with EF>50% (22.4±7.2% vs −27.6±8.1%, p<0.002; 2081±92 vs 2489±127 mmHg%, p<0.001). GCW was the predictor of EF deterioration (area under curve 0,875).

Conclusion

GCW predicts reduction of EF in patients with HFpEF which may help earlier identify the subset of HFpEF patients who may benefit from proven therapies for HFrEF and prevent upcoming deterioration.

Funding Acknowledgement

Type of funding sources: None.

Summation of a Schlömilch type series

The ability to accurately and efficiently characterize multiple scattering of waves of different nature attracts substantial interest in physics. The advent of photonic crystals has created additional impetus in this direction. An efficient approach in the study of multiple scattering originates from the Rayleigh method, which often requires the summation of conditionally converging series. Here summation formulae have been derived for conditionally convergent Schlömilch type series ∑ s = − ∞ ∞ Z n ( | s D − x | ) × e − i n arg ⁡ ( s D − x )   e i s D sin ⁡ θ 0 , where Z n ( z ) stands for any of the following cylindrical functions of integer order: Bessel functions J n ( z ), Neumann functions Y n ( z ) or Hankel functions of the first kind H n ( 1 ) ( z ) = J n ( z ) + i Y n ( z ) . These series arise in two-dimensional scattering problems on diffraction gratings with multiple inclusions per unit cell. It is shown that the Schlömilch series involving Hankel functions or Neumann functions can be expressed as an absolutely converging series of elementary functions and a finite sum of Lerch transcendent functions, while the Schlömilch series of Bessel functions can be transformed into a finite sum of elementary functions. The closed-form expressions for the Coates's integrals of integer order have also been found. The derived equations have been verified numerically and their accuracy and efficiency has been demonstrated.

Gap-edge Asymptotics of defect modes in 2D Photonic Crystals

We consider defect modes created in complete gaps of 2D photonic crystals by perturbing the dielectric constant in some region. We study their evolution from a band edge with increasing perturbation using an asymptotic method that approximates the Green function by its dominant component which is associated with the bulk mode at the band edge. From this, we derive a simple exponential law which links the frequency difference between the defect mode and the band edge to the relative change in the electric energy. We present numerical results which demonstrate the accuracy of the exponential law, for TE and TM polarizations, hexagonal and square arrays, and in each of the first and second band gaps.

Evidence of a mobility edge for photons in two dimensions

A scaling analysis of conductance for photons in two dimensions is carried out and, contrary to widely held belief, we find strong evidence of a mobility edge. Such behavior is compatible with the existence of an Anderson transition for electronic systems under symplectic symmetry, and indeed we show that the transfer matrix in the photonic system we have modelled has such a symmetry. We verify single parameter scaling of the conductance and demonstrate the transition from the metallic phase to localization. Key parameters, including the critical disorder, the conductance, and the critical exponent of the localization length are calculated, and it is shown that the value of the critical exponent is similar to that for electronic systems with symplectic symmetry.

Conductance of photons in disordered photonic crystals

The conductance of photons in two-dimensional disordered photonic crystals is calculated using an exact multipole-plane wave method that includes all multiple scattering processes. Conductance fluctuations, the universal nature of which has been established for electrons in the diffusive regime, are studied for photons, in both principal polarizations and for varying disorder. Our simulations show that universal conductance fluctuations can be observed in H(||) (TE) polarization for weak and intermediate disorder while, for E(||) (TM) polarization, we show that the conductance variance is essentially independent of sample size but strongly dependent on disorder. The probability distribution of the conductance is also calculated in the diffusive and localized regimes, and also at their transition, for which the distributions for both polarizations are seen to be very similar.

Bloch mode scattering matrix methods for modeling extended photonic crystal structures. II. Applications

The Bloch mode scattering matrix method is applied to several photonic crystal waveguide structures and devices, including waveguide dislocations, a Fabry-Pérot resonator, a folded directional coupler, and a Y-junction design. The method is an efficient tool for calculating the properties of extended photonic crystal (PC) devices, in particular when the device consists of a small number of distinct photonic crystal structures, or for long propagation lengths through uniform PC waveguides. The physical insight provided by the method is used to derive simple, semianalytic models that allow fast and efficient calculations of complex photonic crystal structures. We discuss the situations in which such simplifications can be made and provide examples.

Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. Theory

We present a rigorous Bloch mode scattering matrix method for modeling two-dimensional photonic crystal structures and discuss the formal properties of the formulation. Reciprocity and energy conservation considerations lead to modal orthogonality relations and normalization, both of which are required for mode calculations in inhomogeneous media. Relations are derived for studying the propagation of Bloch modes through photonic crystal structures, and for the reflection and transmission of these modes at interfaces with other photonic crystal structures.

Modes of coupled photonic crystal waveguides

We consider the modes of coupled photonic crystal waveguides. We find that the fundamental modes of these structures can be either even or odd, in contrast with the behavior in coupled conventional waveguides, in which the fundamental mode is always even. We explain this finding using an asymptotic model that is valid for long wavelengths.

Density of states functions for photonic crystals

We discuss density of states functions for photonic crystals, in the context of the two-dimensional problem for arrays of cylinders of arbitrary cross section. We introduce the mutual density of states (MDOS), and show that this function can be used to calculate both the local density of states (LDOS), which gives position information for emission of radiation from photonic crystals, and the spectral density of states (SDOS), which gives angular information. We establish the connection between MDOS, LDOS, SDOS and the conventional density of states, which depends only on frequency. We relate all four functions to the band structure and propagating states within the crystal, and give numerical examples of the relation between band structure and density of states functions.

Effects of disorder in two-dimensional photonic crystal waveguides

The effects of randomness on the guiding properties of waveguides embedded in disordered two-dimensional photonic crystals composed of a finite cluster of circular cylinders of infinite length are investigated for TM-polarized radiation. Different degrees of disorder in the radius, filling fraction, refractive index, and position are considered for both straight and 90 degrees bent guides. The crystals exhibit similar sensitivity to refractive index and radius disorder, with a degree of disorder from 15%-20% yielding little substantial change in the guiding properties. A smaller range of position disorder is also considered. For strong disorder in radius and refractive index, the guide effectively closes. These results were obtained by a Monte Carlo simulation method, and the performance of this method is analyzed. The method requires at least ten realizations in some cases for convergence to commence; substantially more realizations are required for moderate and strong disorder to achieve accurate results.

Semianalytic treatment for propagation in finite photonic crystal waveguides

We present a semianalytic theory for the properties of two-dimensional photonic crystal waveguides of finite length. For single-mode guides, the transmission spectrum and field intensity can be accurately described by a simple two-parameter model. Analogies are drawn with Fabry-Perot interferometers, and generalized Fresnel coefficients for the interfaces are calculated.

Three-dimensional local density of states in a finite two-dimensional photonic crystal composed of cylinders

The three-dimensional local density of states (LDOS), which determines the radiation dynamics of a point source, is presented here for a finite two-dimensional photonic crystal as a function of space and frequency. The LDOS is obtained from the dyadic Green's function, which is calculated exactly using the multipole method. Maximum suppression in the LDOS occurs at the high frequency edge of the complete two-dimensional band gap and varies smoothly about this frequency. Macroporous silicon is shown to suppress the LDOS by one order of magnitude at the center of its air pores.

Diffusion and anomalous diffusion of light in two-dimensional photonic crystals

The transport properties of electromagnetic waves in disordered, finite, two-dimensional photonic crystals composed of circular cylinders are considered. Transport parameters such as the transport and scattering mean free paths and the transport velocity are calculated, for the case where the electromagnetic radiation has its electric field along the cylinder axes. The range of the parameters in which the diffusion process can take place is specified. It is shown that the transport velocity upsilon(E) can be as much as 10(8) times less than its free space value, while just outside the cluster upsilon(E) can be 0.3c. The effects of weak and strong disorders on the transport velocity are investigated. Different regimes of the wave transport-ordered propagation, diffusion, and anomalous diffusion-are demonstrated, and it is inferred that Anderson localization is incipient in the latter regime. Exact numerical calculations from the Helmholtz equation are shown to be in good agreement with the diffusion approximation.

Photonic band structure calculations using scattering matrices

We consider band structure calculations of two-dimensional photonic crystals treated as stacks of one-dimensional gratings. The gratings are characterized by their plane wave scattering matrices, the calculation of which is well established. These matrices are then used in combination with Bloch's theorem to determine the band structure of a photonic crystal from the solution of an eigenvalue problem. Computationally beneficial simplifications of the eigenproblem for symmetric lattices are derived, the structure of eigenvalue spectrum is classified, and, at long wavelengths, simple expressions for the positions of the band gaps are deduced. Closed form expressions for the reflection and transmission scattering matrices of finite stacks of gratings are established. A new, fundamental quantity, the reflection scattering matrix, in the limit in which the stack fills a half space, is derived and is used to deduce the effective dielectric constant of the crystal in the long wavelength limit.

Two-dimensional Green's function and local density of states in photonic crystals consisting of a finite number of cylinders of infinite length

Using the exact theory of multipole expansions, we construct the two-dimensional Green's function for photonic crystals, consisting of a finite number of circular cylinders of infinite length. From this Green's function, we compute the local density of states (LDOS), showing how the photonic crystal affects the radiation properties of an infinite fluorescent line source embedded in it. For frequencies within the photonic band gap of the infinite crystal, the LDOS decreases exponentially inside the crystal; within the bands, we find "hot" and "cold" spots. Our method can be extended to three dimensions as well as to treating disorder and represents an important and efficient tool for the design of photonic crystal devices.

Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part I. Method

We present a formulation for wave propagation and scattering through stacked gratings comprising metallic and dielectric cylinders. By modeling a photonic crystal as a grating stack of this type, we thus formulate an efficient and accurate method for photonic crystal calculations that allows us to calculate reflection and transmission matrices. The stack may contain an arbitrary number of gratings, provided that each has a common period. The formulation uses a Green's function approach based on lattice sums to obtain the scattering matrices of each layer, and it couples these layers through recurrence relations. In a companion paper [J. Opt Soc. Am. A 17, 2177 (2000)] we discuss the numerical implementation of the method and give a comprehensive treatment of its conservation properties.

Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part II. Properties and implementation

A numerical implementation and generalized conservation properties of a formulation for calculating wave propagation through stacked gratings comprising metallic and dielectric cylinders are presented. The basic formulation of the method was given in a companion paper [J. Opt. Soc. Am. A. 17, 2165 (2000)]. Here, details of the numerical implementation of the method are discussed and are illustrated for the ensemble average of a strongly scattering structure with refractive index and radius disorder. Also presented are a comprehensive treatment of energy conservation and generalized phase relations, as well as reciprocity.

Effects of geometric and refractive index disorder on wave propagation in two-dimensional photonic crystals

The effects of disorder in the geometry and refractive index on the transmittance of two-dimensional photonic crystals composed of dielectric circular cylinders are considered, including randomness of radii, positions of the cylinder centers, and thickness of each layer of the photonic crystal. The effects of combinations of different types of strong disorder are also considered. The localization and homogenization properties of disordered photonic crystals are investigated. Analytical expressions for the two-dimensional localization length in the form of integrals are presented for both polarizations. It is shown numerically that the slope of the exponential divergence of the localization length in two dimensions is proportional to the inverse of the square of randomness for strong disorder and proportional to the inverse of the randomness for weak disorder. The effective dielectric constants for both polarizations in the case of strong disorder are also found. The transition from localization to homogenization is discussed and the terms responsible for this transition are identified and investigated.

Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders

A method is developed to calculate electromagnetic properties of arrays of metallic and dielectric cylinders. It incorporates and exploits cylindrical boundary conditions and Rayleigh identities for efficient, high-accuracy calculation of scattering off individual layers that are stacked into arrays using scattering matrices. The method enables absorption, dispersion, and randomness to be incorporated efficiently, and reproduces known results with vastly improved speed and accuracy. It is used to demonstrate existence of states introduced into photonic band gaps of a dielectric array by disorder, and anomalous absorption behavior in arrays of aluminum cylinders.

Effects of disorder on wave propagation in two-dimensional photonic crystals

The electromagnetic transmittance of disordered two-dimensional photonic crystals composed of circular cylinders is investigated as a function of wavelength and polarization. At short wavelengths, the transmittance shows a band structure similar to that found in the optical absorption spectrum of amorphous semiconductors, with impurity states increasingly appearing on the long wavelength side of the band gaps as the degree of disorder is increased. In the long-wavelength limit, Anderson localization of waves is found, provided that the wavelength is not so large that the random photonic crystal can be viewed as homogeneous. The localization properties in this regime are studied and an analytic expression for the dependence of the localization length on wavelength is derived. In the limit of extremely long wavelengths, the system homogenizes and can be replaced by an equivalent one with uniform effective refractive index, whose form is derived for both polarizations. Analysis of the crossover between localization and homogenization is also presented.

Ordered and Disordered Photonic Band Gap Materials

We discuss a formulation and computer implementation of a new method that can be used to determine the electromagnetic properties of ordered and disordered dielectric and metallic cylinders, using periodic boundary conditions in one direction. We show results which exhibit strong parallels with the behaviour of electrons in disordered semiconductors, but also illustrate some characteristics which clearly differentiate between photonic and electronic behaviour. Among these are strong polarisation sensitivity and effects due to metallic absorption.