Kidney cortex shear wave motion simulations based on segmented biopsy histology

Abstract

Background and Objective:

Biopsy stands as the gold standard for kidney transplant assessment, yet its invasive nature restricts frequent use. Shear wave elastography (SWE) is emerging as a promising alternative for kidney transplant monitoring. A parametric study involving 12 biopsy data sets categorized by standard biopsy scores (3 with normal histology, 3 with interstitial inflammation (i), 3 with interstitial fibrosis (ci), and 3 with tubular atrophy (ct)), was conducted to evaluate the interdependence between microstructural variations triggered by chronic allograft rejection and corresponding alterations in SWE measurements.

Methods:

Heterogeneous shear wave motion simulations from segmented kidney cortex sections were performed employing the staggered-grid finite difference (SGFD) method. The SGFD method allows the mechanical properties to be defined on a pixel-basis for shear wave motion simulation. Segmentation techniques enabled the isolation of four histological constituents: glomeruli, tubules, interstitium, and fluid. Baseline ex vivo Kelvin-Voigt mechanical properties for each constituent were drawn from established literature. The parametric evaluation was then performed by altering the baseline values individually. Shear wave velocity dispersion curves were measured with the generalized Stockwell transform in conjunction with slant frequency-wavenumber analysis (GST-SFK) algorithm. By fitting the curve within the 100–400 Hz range to the Kelvin-Voigt model, the rheological parameters, shear elasticity (μ₁) and viscosity (μ₂), were estimated. A time-to-peak algorithm was used to estimate the group velocity. The resultant in silico models emulated the heterogeneity of kidney cortex within the shear wave speed (SWS) reconstructions.

Results:

The presence of inflammation showed considerable spatial composition disparities compared to normal cases, featuring a 23% increase in interstitial area and a 19% increase in glomerular area. Concomitantly, there was a reduction of 12% and 47% in tubular and fluid areas, respectively. Consequently, mechanical changes induced by inflammation predominate in terms of rheological differentiation, evidenced by increased elasticity and viscosity. Mild tubular atrophy showed significant elevation in group velocity and μ₁. Conversely, mild and moderate fibrosis exhibited negligible alterations across all parameters, compatible with relatively limited morphological impact.

Conclusions:

This proposed model holds promise in enabling patient-specific simulations of the kidney cortex, thus facilitating exploration into how pathologies altering cortical morphology correlates to modifications in SWE-derived rheological measurements. We demonstrated that inflammation caused substantial changes in measured mechanical properties.

1. Introduction

Chronic kidney disease (CKD) is defined by a progressive decrease in renal filtration. In later stages of failure, when the kidneys are no longer capable of sustaining homeostasis, hemodialysis or kidney transplant are necessary for patient survival and quality of life improvement. Although patient survival rates have increased with the improvement of transplant protocols, improving allograft longevity is still challenging and essential, as transplanted patients with allograft failure are required to return to dialysis and transplant waiting lists [1]. To monitor allograft health, and adjust the course of treatment accordingly, several biomarkers are commonly used to evaluate kidney function, such as serum creatinine and estimated glomerular filtration rate (eGFR) [2]. Although important surrogates of kidney health, biopsy is still the gold standard for allograft assessment, because it can detect microscopic tissue alterations even at an early stage, before kidney function is affected significantly.

The most common injuries used for allograft prognosis from biopsy samples, as described by the Banff criteria [3], [4], are interstitial inflammation, interstitial fibrosis and tubular atrophy (Fig. 1) [5], [6]. The Banff scores can range from 0, indicating no presence of disease, to 4, with increasing severity. In spite of their usefulness in the clinical practice, protocol biopsies cannot be performed frequently, because the invasive nature of the procedure can cause complications such as bleeding, fistulas, infections and even death [7], [8].

Figure 1 –

Masson’s trichrome biopsy core images of four different subjects. The top left image (a) exemplifies a normal core, without injury. Inflammation (b) is characterized by high concentration of interstitial infiltrates (e.g., lymphocytes and macrophages). Interstitial fibrosis (c) is characterized with the accumulation of collagen and related molecules in the interstitium. Tubular atrophy (d) can be characterized by tubules with narrow lumen and thick basement membrane. The aforementioned features are indicated by arrows.

Over the last two decades the use of elasticity imaging techniques has been evaluated to discriminate between normal and abnormal tissue [9], [10]. Different pathological processes can physiologically and morphologically modify the structure of organs, and therefore, possibly modify their rheological characteristics [11]–[15]. The shear wave elastography (SWE) method relies on exciting shear waves with a focused ultrasound beam, which produces acoustic radiation force. The wave propagation is then measured with high frame rate ultrasound imaging. The wave motion is processed to assess a variety of shear wave characteristics, such as shear wave group velocity, phase velocity, and attenuation [16]–[20].

In addition to the more established elastic properties (e.g., shear wave group velocity, shear and Young’s moduli), the viscoelasticity of human tissue may also change with disease [21]–[25]. The viscosity of a given material dictates time-dependent deformation properties, which can be estimated using the shear wave velocity dispersion, i.e., the change in shear wave velocity with respect to frequency. The velocity dispersion curves can be obtained by applying a two-dimensional Fourier transform (2D-FT), in space and time, to the shear wave motion data. It is possible then to evaluate the dispersion curves with the use of model-free parameters such as specific phase velocities and attenuation [26]–[28], or perform fitting to rheological models, such as the Kelvin-Voigt model, to estimate the shear elasticity and viscosity of the material [29].

It has been shown that various elastography measurements, such as shear modulus and shear wave group velocity, correlate to different levels of interstitial inflammation, fibrosis and tubular atrophy [20], [30]–[32]. However, there is still a gap in knowledge of how the morphological alterations in the kidney microstructure can alter the macroscopic shear wave propagation for different injuries.

Viscoelastic material simulation through finite difference (FD) numerical techniques have been widely adopted in geophysical and biomedical applications due to their capabilities of simulating wave propagation in complex discretized media at feasible computational cost [33]–[35]. The staggered-grid finite difference (SGFD) scheme has been shown to successfully implement the Kelvin-Voigt material model for mimicking human tissue [36], [37]. The use of the staggered grid method for spatial discretization enhances the stability, efficiency and accuracy of FD models for nearly incompressible materials [34], [38]. In this study, heterogeneous models based on kidney allograft biopsy histology were simulated using SGFD to investigate the interaction between the inflammation, fibrosis, and tubular atrophy morphological alterations that are signs of chronic allograft rejection (Fig. 1) and the rheological alterations as measured with SWE.

2. Methods

2.1. Patient cohort

The data for this study were obtained from a cohort of patient data and tissue samples. The patients’ biopsies were collected at routine allograft surveillance visits at Mayo Clinic from February 2016 to June 2019. Our SWE study followed a protocol approved by the Mayo Clinic Institutional Review Board (11–003249). All subjects provided written informed consent. In addition to the biopsy Banff scores annotation and core digitization, each patient was also scanned using the SWE capabilities of a Logiq E9 ultrasound system (General Electric Healthcare, Wauwatosa, WI, USA) and C1–6 curvilinear array transducer. The curvilinear array is the standard transducer used for scanning both native and transplanted kidneys. It should be noted that transplant kidneys are generally placed at the abdominal lower quadrants shallow under the body wall (1–4 cm) compared to native kidneys (4–8 cm), regardless there are situations where the deeper penetration of the curvilinear array is necessary, when there is significant adipose tissue and bowel occlusion that may cause the kidney to be deep-seated or obscured. The in-phase and quadrature (IQ) data were stored for post-processing and viscoelastic parameter estimation. The 12 subjects were selected based on having low IQR/median values of measured parameters available in the dataset, to minimize intra-operator variability. The subjects’ demographics are displayed in Table 1.

Table 1 -.

Study cohort demographics

	Patients (n=12)
	Number	%
Sex
Male	5	42
Female	7	58
	Mean (Std. Dev)	Range
Age, yrs	43.7 (13.2)	24–60
BMI, kg/m2	25.0 (4.0)	20.7–35.7
Months after transplant	18.3 (20.3)	4–60

2.2. Biopsy slides segmentation

The biopsy sections stained with Masson’s trichrome were used for all analyses due to higher color contrast between the interstitial and tubule components. Biopsy slides were then digitized at 100x magnification into images with a resolution of 0.87 μm. Segmentation was then performed in MATLAB (MathWorks, Natick, MA, USA) leveraging the RGB component differences between interstitial space, tubules, and fluid. Using the RGB components presented difficulties for discerning the glomeruli from the interstitial and tubular space, and therefore, a user interface was implemented to manually segment the glomeruli due to its complex morphology.

An example of a histological sample and its respective segmented image is displayed in Figures 2a and 2b. The microscope used was not capable of digitizing the entire core into a single image at 100x magnification, therefore, the multiple segmented images of variable sizes from a single core were concatenated. The whole core segmented images were then randomly selected, flipped horizontally and vertically, and concatenated dynamically to create 12 composite 1.0 × 2.0 cm 2D simulations masks (Fig. 2c), one for each subject. The diagnostic scores and relative morphological component area, for each subject analyzed are listed in Table 2. The area percentages for each component were computed and reported for glomeruli (G), interstitial space (I), tubules (T), and fluid (Fl).

Figure 2 –

(a) Original 100x histology image, (b) segmented version of digitized histology, (c) 2.5 × 5 mm sample of 1.0 × 2.0 cm composite mask used for SGFD simulation. The black lines divide the different cores composing the image.

Table 2 –

Diagnostic category and spatial composition

Diagnostic Category/Patient	Banff Score			Percent Composition
	Inflammation	Fibrosis	Tubular atrophy	G	T	I	Fl
Normal 1	0	0	0	5.6%	46.1%	38.3%	10.0%
Normal 2	0	0	0	4.3%	38.0%	42.2%	15.5%
Normal 3	0	0	0	6.3%	33.1%	47.8%	12.7%
Inflammation 1	2	0	0	7.9%	42.4%	41.5%	8.2%
Inflammation 2	1	0	0	5.7%	40.0%	46.0%	8.3%
Inflammation 3	2	0	0	5.5%	20.7%	69.9%	4.0%
Fibrosis 1*	0	1	1	6.1%	40.1%	42.9%	10.9%
Fibrosis 2*	0	2	2	3.2%	55.4%	33.5%	8.0%
Fibrosis 3*	0	1	1	3.7%	25.5%	57.9%	12.9%
Tubular atrophy 1	0	0	1	5.8%	47.7%	37.1%	9.3%
Tubular atrophy 2	0	0	1	3.1%	41.7%	44.0%	11.1%
Tubular atrophy 3	0	0	1	4.1%	48.7%	41.6%	5.5%

G: Glomeruli, T: Tubules, I: Interstitial space, Fl: Fluid

^*All patients from the cohort that presented with interstitial fibrosis also presented with tubular atrophy.

2.3. Finite difference method

The media were assumed to be isotropic, linear, and incompressible [36]. All simulations were set with a density of 1000 kg/m³, compressional wave velocity of 1540 m/s, and ultrasound attenuation of 0.5 dB/cm/MHz. The simulations were implemented using the Kelvin-Voigt two-parameter model due to its tractability with respect to the number of simulations, simpler implementation, and ability to interpret the results [39]. The shear elasticity, μ₁, and shear viscosity, μ₂, of each kidney cortex component was determined using baseline values found in the literature [40]–[44]. Each pixel in the simulation mask had a unique definition of μ₁ and μ₂ depending on the component represented by that pixel. From baseline values, two other values, above and below baseline, were assigned to each of the six component parameters based on literature experimental variability [40]–[44] (Table 3). Finally, the parametric study was composed by a total of 13 simulations (baseline plus twelve variations) for each of the 12 subjects evaluated, totalling 156 simulations.

Table 3 –

Parametric mechanical properties of tissue constituents

Cortex component	Parameter	Lower value	Baseline	Upper value
Glomeruli	Shear elasticity, μ1	2 kPa	4 kPa	6 kPa
	Shear viscosity, μ2	0.1 Pa·s	0.25 Pa·s	1 Pa·s
Tubules	Shear elasticity, μ1	1 kPa	1.5 kPa	2 kPa
	Shear viscosity, μ2	0.1 Pa·s	0.25 Pa·s	1 Pa·s
Interstitium	Shear elasticity, μ1	0.5 kPa	1 kPa	2 kPa
	Shear viscosity, μ2	0.5 Pa·s	1 Pa·s	2 Pa·s
Fluid	Shear elasticity, μ1	N/A	0 kPa	N/A
	Shear viscosity, μ2	N/A	0.1 Pa·s	N/A

The Navier’s equations were used to calculate shear wave propagation excited by an acoustic radiation force (ARF) push [45]. The push was generated using Field II [46]. The characteristics of the beams used to generate the ARF for the clinical ultrasound used are not disclosed by the manufacturer, so the simulated ARF was specified to completely overlap the observed experimental frequency response range and typical focal depth and axial range. For simplicity, the transducer model was based on a L7–4 linear array transducer (Philips Healthcare, Andover, MA), with an element pitch of 0.308 mm, kerf of 0.025 mm, height of 7 mm, and the frequency was set to 4 MHz. A 200 μs ARF was applied using 40 elements to simulate the push at a focal depth of 25 mm. The use of a linear instead of curvilinear array probe does not alter the shear wave processing pipeline, as the curvilinear radial coordinates are converted to Cartesian coordinates, while significantly simplifying the SGFD simulation.

Boundary conditions are crucial for effectively simulating wave propagation in viscoelastic models, the interaction of pressure and shear waves with the borders of the simulation can generate artifacts that would not be present in real applications. The use of perfectly matched layers (PML) was implemented for optimal absorption of pressure and shear waves at the boundaries of the material [47]. The spatial resolution was set to 0.05 mm for both axes (200 × 400 pixels), and the simulation time for the wave propagation record was set to 15 ms. The simulation sampling frequency was defined by the Courant-Friederichs-Lewy criterion (CFL), allowing for the simulation’s conditional stability and convergent behavior. The CFL determines the temporal step Δt based on the compressional and shear wave velocities, V_p and V_s, and the spatial step, Δx, in the x dimension (Eq. 1). In this study, a CFL of 0.2 was used for proper numerical stability.

$$\Delta t=\frac{CFL\ast \Delta x}{\sqrt{v_p^2+v_s^2}},$$ (1)

The SGFD simulations were implemented in MATLAB and processed using computing resources from the Minnesota Supercomputing Institute at the University of Minnesota. The simulation computing time was approximately 1–2 hours when queued for Nvidia V100 (Nvidia Corporation, Santa Clara, CA, USA) nodes, and approximately 3–4 hours if the Nvidia K40 cluster nodes were used. The shear motion data were resampled then saved with a sampling frequency of 8 kHz.

To evaluate the masks’ rheological heterogeneity, the shear wave group velocity map of the baseline simulation of each patient was calculated. The map was calculated using a window of 0.3 × 0.3 mm with a patch size of 0.15 mm [48]. The dispersion curves, from the simulations and patient scans, were retrieved using a generalized Stockwell transformation combined with a slant frequency-wavenumber analysis (GST-SFK) [49] from 100–400 Hz, due to its wider and more reliable frequency band when compared to the standard 2D Fourier transform approach [26], [36]. The dispersion curves were then fit to the Kelvin-Voigt model to estimate the shear elasticity, μ₁, and shear viscosity, μ₂.

$$c_p(\omega )=\sqrt{\frac{2\left({\mu }_1^2+{\omega }^2{\mu }_2^2\right)}{\rho \left({\mu }_1+\sqrt{{\mu }_1^2+{\omega }^2{\mu }_2^2}\right)}},$$ (2)

where the shear wave phase velocity (c_p) is described as a function of the angular frequency (ω), the media density p. Additionally, the group velocity was calculated using a time-to-peak algorithm [50].

To compare the simulation results with the measurements obtained in vivo, we used data from SWE evaluations that were performed using a Logiq E9 ultrasound system acquisitions performed on the same patients prior to the biopsy procedure. The Pearson correlation coefficient between in silico and in vivo was calculated for the median values of each patient and the aggregated biopsy finding (e.g., three patients).

3. Results

A total of 156 simulations were performed for the 12 subjects’ masks. The simulations’ mechanical heterogeneity can be observed in Fig. 3 in maps of the shear wave group velocity. The shear wave velocity maps illustrated the spatial variation of mechanical properties induced by the presence of the various different components of kidney morphology and the different mechanical properties, particularly as the shear wave propagates through glomerular structures. In some cases, due to the limitation of available biopsy cores for digitization, the shear wave map showed some level of vertical striation, as the biopsy core segments had to be repeated more often to generate the simulation mask, creating the pattern observed for tubular atrophy patients 1 and 3. The x-axis origin was positioned at the ARF push focus, therefore, the maps only cover about 15.5 mm. All shear waves showed significant attenuation after approximately 12 mm of lateral propagation, the higher velocity values estimated after shear wave dissipation (cyan to yellow hue) are, therefore, likely artifacts.

Figure 3 –

Shear wave group velocity maps calculated from baseline simulations of each subject. The shear wave velocity maps were able to capture the heterogeneity of the simulated media. The striated pattern might be caused by the limited number of cores available for digitization, and edge discontinuities between the individual core images.

Group velocity, μ₁ and μ₂ values were calculated using time-of-flight and Kelvin-Voigt dispersion fitting, respectively, for a 2 mm axial region centered at push focus. For simplification, the three different subjects composing each diagnostic score under evaluation (normal, inflammation, fibrosis and tubular atrophy) were compiled into mean and standard deviation for each of the simulated variations from baseline listed in Table 3. All measurements showed low standard deviations, with a maximum of 0.16 Pa·s shear viscosity for the upper interstitial viscosity variation (μ₂ = 2 Pa·s). The results are displayed in Figure 4.

Figure 4 –

Compiled results for each patient, iteration, and biopsy finding. Each cluster of biopsy finding (e.g., fibrosis) is composed of the six mechanical parameters that compose the cortex morphology: elasticity and viscosity for the glomeruli, tubule and interstitial space. The x-axis shows the iterated component values (lower, baseline and upper), and the y-axis shows the measured rheological properties (shear elasticity, viscosity and group velocity). The vertical bars display the standard deviation across the three patients that composed each biopsy finding. Shear viscosity showed the most sensitivity to rheological alterations, while alterations to tubules and interstitial space altered the measurements the most.

For diagnosis comparison, the average measurements’ difference from normal within each diagnostic category were calculated and divided by the normal subjects’ average to obtain a percentage change from normal (Eq. 4). Inflammation showed the highest levels of viscoelastic change, with a mean change of over 20% in μ₂. Both inflammation and tubular atrophy showed slight changes from normal in μ₁ and group velocity, ranging from −1.83% (Inflammation shear elasticity) to 7.25% (Tubular atrophy shear elasticity). Inflammation showed more significant change for interstitial alterations specifically, with 12.37 to 39.09% μ₂ mean change. Interstitial fibrosis was the least affected injury in terms of elasticity, with 0.60% and 0.59% mean change for group velocity and μ₁, respectively. Viscosity evaluations showed larger change trends for interstitial fibrosis, within 3.04% to 10.19% change in μ₂. The mean percent change is displayed in Table 4.

Table 4 –

Mean percent change from normal cases (%)

Rheological Measurement	Glomeruli Elasticity	Glomeruli Viscosity	Tubular Elasticity	Tubular Viscosity	Interstitial Elasticity	Interstitial Viscosity
Inflammation
Shear Elasticity (μ1)	3.36	3.67	1.84	3.76	5.08	2.29
Shear Viscosity (μ2)	24.00	24.68	39.09	22.20	20.90	12.37
Group Velocity	3.43	3.50	4.32	3.33	4.30	3.31
Fibrosis
Shear Elasticity (μ1)	0.53	0.26	−0.99	1.53	1.77	0.44
Shear Viscosity (μ2)	6.16	5.93	10.19	5.56	3.91	3.04
Group Velocity	1.07	0.63	0.24	0.34	0.88	0.44
Tubular Atrophy
Shear Elasticity (μ1)	4.20	4.93	4.27	3.96	7.26	5.24
Shear Viscosity (μ2)	1.84	−0.41	3.35	3.70	−4.68	−2.86
Group Velocity	1.07	0.63	0.24	0.34	0.88	0.44

$$\mathit{\text{Injury}}\%\mathit{\text{Change}}=\frac{{\mathit{\text{Injury}}}_{\mathit{\text{avg}}}-{\mathit{\text{Normal}}}_{\mathit{\text{avg}}}}{{\mathit{\text{Normal}}}_{\mathit{\text{avg}}}}$$ (4)

The in silico and in vivo results are compared in Figure 4 and Table 5. Figure 5 shows scatter plots of the group velocity, μ₁, and μ₂ values along with a regression result. The results from the Pearson correlation calculations are provided in Table 5. The p-values for correlations between the in silico measurements and in vivo estimations were <0.001, aside from μ₁ with p-values of 0.70 and 0.09. For all parameters evaluated, the in vivo estimations showed significantly higher values, particularly for μ₂ with an average difference by a factor of 6.72. Group velocity, and μ₁ had an average difference by a factor of 1.14 to 2.32. Due to its intrinsic challenges (e.g., operator variability, kidney position variability), the in vivo measurements also showed significant higher levels of variability when compared to simulation.

Table 5 –

in silico/in vivo comparison

Patient-wise analysis
Diagnostic Category/Patient	Group Velocity (m/s)		Shear Elasticity (kPa)		Shear Viscosity (Pa·s)
	In silico	In vivo	In silico	In vivo	In silico	In vivo
Normal 1	1.13	1.92	1.07	1.16	0.40	2.48
Normal 2	1.05	2.62	1.02	1.93	0.38	2.89
Normal 3	1.05	2.41	1.03	1.06	0.44	2.83
Inflammation 1	1.14	2.84	1.09	0.85	0.44	2.80
Inflammation 2	1.11	2.62	1.06	0.88	0.45	3.00
Inflammation 3	1.09	2.52	1.09	1.73	0.56	3.25
Fibrosis 1	1.09	2.08	1.05	1.20	0.43	2.92
Fibrosis 2	1.14	2.72	1.09	1.24	0.37	2.68
Fibrosis 3	1.02	2.69	0.98	1.37	0.46	3.31
Tubular atrophy 1	1.13	2.82	1.08	1.16	0.39	3.10
Tubular atrophy 2	1.09	2.86	1.06	1.14	0.41	3.32
Tubular atrophy 3	1.14	2.54	1.13	0.82	0.42	2.60
Pearson coefficient	0.02		−0.41		0.47
P-value	<0.001		0.70		<0.001
Diagnostic category group analysis
Normal	1.08 ± 0.04	2.33 ± 0.30	1.04 ± 0.02	1.38 ± 0.39	0.41 ± 0.02	2.73 ± 0.18
Inflammation	1.11 ± 0.02	2.66 ± 0.13	1.08 ± 0.01	1.15 ± 0.41	0.48 ± 0.05	3.02 ± 0.18
Fibrosis	1.08 ± 0.05	2.50 ± 0.29	1.04 ± 0.04	1.27 ± 0.07	0.42 ± 0.04	2.97 ± 0.26
Tubular atrophy	1.12 ± 0.02	2.74 ± 0.14	1.09 ± 0.03	1.04 ± 0.15	0.40 ± 0.01	3.01 ± 0.30
Pearson coefficient	0.95		−0.93		0.45
P-value	<0.001		0.09		<0.001

Figure 5 –

Scatter plot of in vivo and in silico elastography estimations, linear regression is also displayed. The color of the dots decodes the patient respective diagnostic category.

Although group velocity did not show correlation within the patient-wise median analysis, mean group velocity for each biopsy finding showed high correlation between the in silico and in vivo cases, with Pearson coefficient of 0.95. Therefore, group velocity variability impaired the patient-wise collinearity, but did not confound the diagnostic trend, where patients with inflammation and tubular atrophy tended to have higher group velocity estimations.

Shear viscosity (μ₂) showed similar correlation results with diagnostic category group Pearson correlation of approximately 0.45. Patients with inflammation showed higher levels of shear wave velocity dispersion than normal subjects, whereas for fibrosis and tubular atrophy alterations were not as prominent.

Although μ₁ showed the lowest scale factor between in silico and in vivo results, of approximately 1.14, μ₁ was the only measurement to show negative correlation, both among patients and diagnostic category groups.

4. Discussion

The results shown are encouraging and successfully demonstrated the capabilities of biopsy-based SGFD simulations of shear wave motion. The shear wave group velocity maps captured the heterogeneity of the kidney cortex microstructure created by the biopsy masks. The theoretical spatial resolution of SWE is dictated by the shear wave wavelength,λ = c/f. Given a shear wave speed of 1–3 m/s and the upper frequency of 400 Hz, the theoretical SWE resolution is in the order of 2.5–7.5 mm. The average glomeruli diameter in humans is 0.2 mm [51], and, consequently, it cannot be fully characterized by SWE. Notwithstanding, the effect of the presence of such structures can be detected as they alter the shear wave propagation characteristics (Figure 3).

Over the past decade, our research team has extensively utilized the Kelvin-Voigt (KV) model for the characterization of kidney tissue, focusing on a frequency range typically spanning 100–400 Hz. Although more general and accurate models might be available [39], [52], the application of the KV model to fit phase velocity dispersion curves within this range has demonstrated satisfactory agreement in numerous instances [25], [53], [54]. Various factors were also considered for the selection of the KV model for simulation purposes, including the practicality of characterizing tissue using two parameters for each component, as opposed to a three-parameter model, lower computational cost, enhanced numerical stability, and it also contributed for interpretation accessibility [38], [39].

Although the use of simulations might enable analysis with higher spatial resolution (0.05 mm), the SWE frequency response is also dictated by the ARF pulse profile, and tissue attenuation characteristics. In the interest of comparing in vivo and in silico testing, the 100–400 Hz SWE range was used for both, even though simulations can be configured to reach higher frequencies, and therefore, higher SWE resolution. In future work, the authors intend to investigate the effects of the use of higher frequency ranges (e.g., 100–1000 Hz) to SWS reconstructions and dispersion curve fitting in heterogeneous media. Although the ARF characteristics between the numerical and experimental ARF cannot be perfectly matched, the frequency response of both were reliable within the dispersion frequency range and focal depth and axial extent, therefore, little to no discrepancy between estimations is expected due to transducer and ARF beam mismatch.

The average morphological composition for each diagnostic category was calculated and displayed in Table 6. Inflammation differed the most from the other cases with higher interstitial and glomerular area (52.4% and 6.4%, respectively), and lower tubular and fluid relative area (34.1% and 6.8%, respectively). Tubular atrophy showed the highest tubular area, at 46%, and lower fluid composition, in agreement with the thickening of tubular basement membranes in addition to the tubular flow constriction and compensatory tubular hypertrophy [55], [56]. Normal and interstitial fibrosis showed similar component distributions with an average deviation of ±1.6%.

Table 6 –

Average spatial composition by diagnostic category

Category	Avg G%	Avg T%	Avg I%	Avg Fl%
Normal	5.4	39.1	42.7	12.8
Inflammation	6.4	34.4	52.4	6.8
Fibrosis	4.3	40.3	44.8	10.6
Tubular atrophy	4.4	46.0	40.9	8.7

G: Glomeruli, T: Tubules, I: Interstitial space, Fl: Fluid

Inflammation was found to alter the kidney composition the most, and therefore, change the cortex mechanical properties the most. This finding agreed with the in vivo study of the evaluated subjects, showing significantly higher values for all measurements, when comparing normal to inflammation patients [57]. The increase in interstitial space, coincident with the reduction of fluid and tubular space is compatible with higher tissue stiffness, and viscosity, and therefore compatible with the elastography findings. Interstitial fibrosis and tubular atrophy did not appear to have such differentiation potential. The lack of patients with exclusive interstitial fibrosis in the subject pool for this study may have confounded fibrosis characterization because fibrosis positive patients also presented with tubular atrophy. At the time of study, no patient in the cohort presented later stages of disease (Banff scores of 3–4). Regardless, kidney function tests, such as serum creatinine, can often detect more advanced stages, and therefore, it was not in the scope of this study. While obtaining higher Banff scores could be beneficial to the overall morphological analysis, subtle levels of fibrosis and inflammation are more likely to be overlooked by common kidney function tests. Hence, noninvasive tests to detect the earlier stages of disease could hold greater clinical significance.

The simulations showed, for the most part, consistent correlation with the in vivo studies performed. Although helpful for the understanding of the morphological alterations and its consequences, the simulation setting proposed in this study inherits several limitations that impair its capacity of numerically matching the in vivo assessments. The only parameter to show negative correlation between in vivo and in silico experiments was shear elasticity. The in vivo data showed approximately 5 times more viscosity than in silico, with Pearson coefficients above 0.45. The higher levels of dispersion, induced by higher levels of viscosity, tend to highly affect the shear elasticity estimation through Kelvin-Voigt model fitting. The increase of dispersion tends to decrease the dispersion curve zero intercept and, therefore, lower the shear elasticity estimation proportionally to the increase of viscosity.

One of the main challenges, and a focus for future investigations, is the effect of perfusion on the elastography assessment. Several groups investigated these phenomena [54], [58], nevertheless rheological parameters for each specific cortex component in vivo and under perfusion were not available at the time of this study. The values for the mechanical properties of each constituent in Table 3 were based on literature values from ex vivo tissue. As a result, they did not include perfusion effects and are likely lower than the effective values that may be found in an in vivo setting. Given the microscopic nature of such histological constituents, in vivo assessment of individual rheological characteristics can be challenging and, to the best of our knowledge, has not been done.

A future direction is the implementation of inverse problem techniques to numerically match the macroscopic findings and estimate the individual constituents’ properties in the kidney cortex. Inverse problems estimate the inputs based on the experimentally observed outputs. In recent years, numerous fields such as medical imaging, geophysics, and engineering have extensively employed inverse problem techniques to resolve critical issues, like image reconstruction and parameter estimation [59]–[61]. Inverse problem solving tends to have complex implementations and require extensive computational cost [62]. Prior information, such as the mutual dependence of rheological parameters to perfusion, can be embedded as boundary conditions for simplicity. Nevertheless, the effect of perfusion to each constituent is not known, and therefore, substantial work is necessary to further describe its mechanical association.

The 2D simplification of the simulations reported in this study may also include different bias levels due to out of plane movement that was not taken under consideration, it is also important to note the inherent two-dimensional nature of histology analysis, therefore, confounding 3D implementations. The repetition of core images and boundary discontinuities can also introduce bias and artifacts to the simulations. The core repetition artifacts were mitigated by the use of image augmentation techniques such as rotation and mirroring, additionally, the core image boundary discontinuities are smoothed by the reduction in resolution necessary for SGFD simulation. Ideally, the entire mask should be composed by a single histology image and the SGFD simulation performed at full digitization resolution. Unfortunately, it is not feasible to produce 1.0 × 2.0 cm samples from in vivo patient studies and, additionally, computational costs of finite difference simulation at full histology resolution are also impractical.

Histopathology from biopsies relies on the diffuseness of injury to characterize an organ’s condition based on limited sample size. Kidney transplant biopsies are limited by the accessibility of the kidney, and most often a few samples are taken from the most superficial site available. The small core size, of approximately 1.0 cm long and 0.8–1.2 mm wide [6], [8], might not capture significant injury for proper tissue characterization. Therefore, the histology-based simulation might not be representative of the condition of the site evaluated by the SWE examination. Optimally, SWE and histology should be performed at the exact same location for proper comparison, a matter that will be addressed in subsequent studies. We envision SWE implementations can also be leveraged to indicate suspicious sites for biopsy, optimizing histopathology findings.

Despite the disparities between in silico and in vivo limitations, the results were encouraging and present SGFD heterogeneous simulations as a viable testing framework for better understanding how the pathological changes to organs’ micro-structure can alter the macro-structure rheological characteristics. Compared to interstitial fibrosis and tubular atrophy, interstitial inflammation was shown to alter the kidney cortex’s morphology the most and therefore its mechanical properties the most both in vivo and in silico, this finding is particularly important as inflammation is a recurring injury that gives rise to fibrosis and tubular atrophy, and consequently, one of the main targets for more frequent allograft assessment. We envision the methods proposed in this work may also enable new approaches for mechanical parameter estimation and interpretation to a wide variety of biomedical applications.

Highlights.

• Shear wave elastography is a non-invasive tool for kidney transplant monitoring.

• Patient kidney cortex models were created with distinctive pathological features.

• Shear wave motion simulations based on segmented kidney cortex biopsy sections.

• Fibrosis and tubular atrophy showed limited morphological and rheological impact.

• Inflammation showed considerable composition disparities compared to normal cases.