Fig 1.
NLME, the mechanistic model framework, and model.
(A) The standard-two-stage (STS) method. The model is parametrized for each data set separately, and then the parameter values are combined to derive population parameter distributions. (B) The ‘non-linear mixed effect’ (NLME) method. The shapes of the population parameter distributions are first postulated, then distributions are parametrized to all datasets, and finally each patient is parametrized following the population parameter distributions with a joint-likelihood function. This allows NLME to use the global information obtained from an entire cohort, which is utilized to improve model parametrization for each individual subject. (C) The framework consists of gadoxetate-enhanced images, which are processed to obtain gadoxetate concentrations in the liver. A mechanistic systems pharmacology model, describing how gadoxetate is taken up and excreted, is fitted to the data using NLME parameterization to obtain reliable pharmacokinetic parameters, which can be used as biomarkers for liver function. (D) Schematic diagram of the mechanistic model for quantification of liver transporter function. Rounded rectangles represent compartments in the model, with arrows indicating gadoxetate fluxes between blood plasma and extracellular extravascular space (EES; kdiff), elimination via the kidneys to urine (CLr), uptake into hepatocytes (kph), back-flux from hepatocytes into blood plasma (khp), and excretion from hepatocytes into bile (khb). Gadoxetate injection into the blood-plasma compartment is indicated in blue. Gray areas show the signal part of the model in which compartmental gadoxetate concentrations are combined to predict the information in the gadoxetate-enhanced MRI time series.
Table 1.
Demographic and clinical data from of the final study population (N = 91).
Fig 2.
The mechanistic model framework predicts observed gadoxetate levels in chronic liver disease.
(A, B) Observations and model-based predictions of gadoxetate levels, indicated by the changes in R1 relaxation rate (ΔR1, which is directly proportional to the gadoxetate concentration) in two patients, one with no fibrosis (F0) and one with histologically proven cirrhosis (F4). (C, E) The model is validated by predicting the gadoxetate concentration in blood and biopsy samples, which was acquired after the MRI examination. (D, F) Correlation between the measured and model-predicted gadolinium concentrations. The solid line is a linear regression and the dotted lines are 95% confidence intervals.
Fig 3.
The non-linear mixed effects (NLME) model parametrization enables shorter gadoxetate MRI examinations.
The models were parametrized with data up to 10 min after gadoxetate injection and were validated against the remaining data. (A) shows an example of simulation after NMLE parameterization, (B) shows a simulation of the same patient after STS parameterization. The natural logarithms of the model parameter values hepatocyte uptake rate (kph; C-D), hepatocyte elimination rate (khb; E-F), and hepatocyte to plasma flux (khp; G-H), estimated with the NLME method using the full data set, were compared with the parametrization using data up to 10 minutes. Significant differences were observed for khp. The solid line is a linear regression and the dotted lines are 95% confidence intervals.
Fig 4.
Effects of fibrosis on hepatic function.
Liver (A) and spleen (B) time series showing average induced change in R1 relaxation rate (ΔR1) in patients with different levels of fibrosis. Error bars indicate the standard error of the mean. In (C, D), liver function parameters are shown for each level of fibrosis. Horizontal lines indicate significant differences (ANOVA, Tukey’s post-test: * <0.05; ** <0.01; *** <0.001).
Table 2.
Confusion matrix for the ability of kph to identify patients with advanced fibrosis.
Fig 5.
Gadoxetate MRI time series of the liver.
(A-G) Representative placement of seven regions of interest (yellow polygons) within the liver, of which four (A-D) were placed in the right liver lobe and three (E-G) were placed in the left liver lobe. This set shows an entire time series in a single patient, from before gadoxetate injection (A) to 30 min after gadoxetate injection (G). The arterial phase (B) typically occurs 30 s after gadoxetate injection, and the portal-venous phase (C) typically occurs 1 min after gadoxetate injection.
Fig 6.
Estimation of the lower limit of data uncertainty.
Histogram of the average uncertainty of the normalized signal intensity in the liver and spleen regions of interest, with one entry per patient. The blue bell curve shows the fitted normal distribution, which indicates a lower limit of uncertainty of 0.18.