Benjamin Andrew Ratliff^{1,2}, Diego Hernando^{2,3}, Curtis Wiens^{2}, Changqing Wang^{2,4,5}, Rao Watson^{6}, Rashmi Agni^{6}, Claude B Sirlin^{7}, and Scott B Reeder^{1,2,3,8,9}

The purpose of this work was to quantify the size and clustering of fat droplets using liver biopsy, as part of a long-term effort to characterize the relationship between tissue microstructure and quantitative MRI signals in fat-containing tissue. Three H&E stained liver core biopsies with varying fat-fractions were analyzed using segmentation software in order to generate probability density functions for fat droplet size and location. This work demonstrates that fat droplet distribution in the liver can be modeled statistically to determine the size and location distribution of fat droplets, potentially enabling characterization of the MR signal observed from fatty liver.

This study was IRB approved and HIPAA compliant. Subjects scheduled to undergo bariatric weight loss surgery (WLS), with a high pretest probability of nonalcoholic fatty liver disease (NAFLD), were recruited from two bariatric centers. Subjects underwent biopsy from the left lobe of the liver at the time of weight loss surgery.

A total of 3 H&E stained core liver biopsies were chosen randomly from patients with low (n=1), medium (n=1), and high (n=1) fat content. Each core liver biopsy was segmented into six areas of interest for analysis.

High-resolution digitized histological slides were processed to determine the steatosis area-fraction along with the fat droplet distribution (coordinates of the droplet center, as well as diameter of each droplet) from the biopsy specimens. Semi-automated threshold macro segmentation of the fat droplets to determine these distributions was determined using Image Pro Plus 7.0 software (Media Cybernetics, Inc., Rockville, MD). Further, the coordinates of the central veins were determined manually using the same software.

The coordinate and diameter data collected from
the digitized slides was imported into Matlab2014b (The MathWorks, Inc.,
Natick, MA). For each segment of every
histology slide, histograms of the droplet size, nearest neighbor distance, and
distance to the central vein were generated and fit to both gamma and
exponential density functions. The Gamma
distribution function^{3} is given by,

$$GDF(x)=\frac{1}{\beta+\Gamma(\gamma)}\times\left(\left(\frac{x-\mu^{(\gamma -1)}}{\beta}\right)\times exp\left(-\frac{x-\mu}{\beta}\right)\right)$$

where β > 0 (scale parameter), γ > 0 (shape parameter), x ≥ μ (location parameter) and Γ(γ) is the function given by,

$$\Gamma(z)=\int_{0}^{\infty }{t^{z-1}e^{-t}dt}$$

Alternatively, the exponential density function is given by,

$$PDF(x)=\lambda e^{-\lambda x}$$

where 1/λ is the events per unit time is both the mean and standard deviation of the distribution.

Using these two probability density functions along with a Monte Carlo simulation, 3D versions of the liver can be generated, to facilitate Bloch equation simulations to predict MR signal characteristics in fatty liver (Figure 5).

* Fat Droplet Size Distribution*: Density function
analysis determined that the best fit the fat droplet size distribution based
on the diameter is provided by the gamma distribution (Figure 2). Droplet size
distribution over a range of fat concentrations was not significant between the
three subjects. The fitting results for droplet size to the gamma distribution
were as follows: β

* Fat Droplet Location:* Fat droplet location
distribution was split into distance to nearest neighbor fat droplet and
distance to nearest central vein. An
exponential distribution fit was determined to provide the best fit for the
nearest neighbor distribution (Figure 3), while the gamma distribution fit was
determined to provide the best fit for the central vein distribution (Figure
4). The nearest neighbor distribution showed that with a decrease in fat
fraction, the average distance to the nearest neighbor is higher due to an
increase in the λ parameter from the exponential fit, with the
following fitting results: λ

The
central vein distributions also showed that as the fat fraction decreases, the
fat droplets tend to clump less, with the following fits to the gamma
distribution function: β_{large}=3.76±0.24; γ_{large}=88.6±18.3, β_{medium}=2.09±0.34; γ_{medium}=184.9±63.6, β_{small}=1.89±0.16, γ_{small}=300.9±129.6.

1. Hernando D, Haufe WM, Hooker CA, et al. Relationship between liver proton density fat fraction and R2* in the absence of iron overload. In Proceedings of the 23nd Annual Meeting of ISMRM, Toronto, Canada, 2015. Abstract 4118.

2. Ghugre NR, Gonzalez-Gomez I, Shimada H, et al. Quantitative analysis and modelling of hepatic iron stores using stereology and spatial statistics. Journal of Microscopy 2010; 238(3): 265-274.

3. Beyer, WH. CRC Standard Mathematical Tables. CRC Press, Boca Raton, FL, 1987.

**Figure
1: Methods of Analyzing Biopsy**

The images below show the methodologies used to generate the probability density functions for size and location. The process begins with the digitized biopsy slide. A semi-automated threshold macro is used to generate a segmented biopsy. Data is gathered using Image Pro Plus 7.0, and a Matlab generated biopsy is created. From here, the size histogram along with the nearest neighbor histogram and central vein histogram are generated.

**Figure
2: Particle Size Analysis**

The images below show the gamma distribution functions for one case in each fat fraction range. It was determined that there was a insignificant difference in the probability distribution functions between the three fat fraction cases.

**Figure
3: Nearest Neighbor (NN)
Analysis**

The
images below show the exponential distribution functions for the** **nearest
neighbor distribution for one case in each fat fraction range. It was
determined that there was a significant difference in the probability
distribution functions between the three fat fraction cases, which means that
the location density functions change with respect to the fat fraction.

**Figure
4: Central Vein
(CV) Distance
Analysis**

The images below show the gamma distribution functions for the central vein distribution for one case in each fat fraction range. It was determined that there was a significant difference in the probability distribution functions between the three fat fraction cases, which means that the location density functions change with respect to the fat fraction.

**Figure
5: 2D Generated Liver Slice**

The image below shows a 2D generated liver slice based on the fat droplet size and nearest neighbor location distributions. This liver slice is only a portion of the larger 3D generated case, which also follows the same distributions.