Evaluation of Liver Function Within MR Imaging

Fig. 1

Linear two-compartment model of gadolinium-based extracellular fluid contrast agent. C _a(t), concentration of the contrast agent in systemically circulating blood at time t; C _t(t), concentration of the contrast agent in tissue extracellular space at time t; τ, time required for the contrast agent to arrive at tissue via the systemic circulation; K1, rate coefficient for diffusion of contrast agent from systemic circulation to tissue; and k2, rate coefficient for diffusion of contrast agent from tissue to systemic circulation. The contrast agent, administered intravenously into systemically circulating blood, reaches a tissue after a period τ, diffuses into the tissue at a velocity proportional to the concentration in systemically circulating blood C _a(t) and the diffusion rate coefficient K1, and diffuses out of the tissue at a velocity proportional to the concentration in the tissue extracellular space C _t(t) and the diffusion rate coefficient k2. The contrast agent is not incorporated into the cell and is excreted from the kidney into urine

$\mathrm{d}{C}_{\mathrm{t}}(t)/\mathrm{d}t=\mathrm{K}1\times {C}_{\mathrm{a}}\left(t-\tau \right)-\mathrm{k}2\times {C}_{\mathrm{t}}(t)$

Then, the T1 relaxation rate of the tissue after imaging (1/T1) can be expressed by the sum of the T1 relaxation rate inherent to the tissue (1/T1₀) and the T1-shortening caused by the contrast agent present in the extracellular fluid (1/T1_ECF) [5].

$1/\mathrm{T}1=1/\mathrm{T}{1}_0+1/\mathrm{T}{1}_{\mathrm{ECF}}$

$1/\mathrm{T}{1}_{\mathrm{ECF}}=\mathrm{R}1\times {C}_{\mathrm{t}}(t)$

where R1 is an indicator termed “T1 relaxivity,” which reflects the degree of T1-shortening of the contrast agent; the enhancement effect on the T1-weighted image is stronger when R1 is larger. For example, in the case of the spin echo method, the following equation describes the relationship between the T1 relaxation rate (1/T1) and the MR signal strength (S) [8]:

$S=\kappa \times \rho \times \left(1-2{\mathrm{e}}^{-\left(\mathrm{TR}-\mathrm{TE}/2\right)\mathrm{T}1}+{\mathrm{e}}^{-\mathrm{TR}/\mathrm{T}1}\right)\times {\mathrm{e}}^{-\mathrm{TE}/\mathrm{T}2}$

κ is a proportionality factor, ρ is proton density, TR is repetition time, and TE is echo time. Therefore, there is no linear proportional relationship between MRI contrast agent concentration and the MR signal intensity similar to that between the iodine contrast agent concentration and CT value. However, because there is an exponential correlation between the MRI contrast agent concentration and the signal intensity up to a certain concentration (about 5 mmol/L in a 1.5 T MR apparatus) [9], the enhancement effect on MR images is considered to mostly reflect the concentration of the contrast agent. As C _a(t) and C _t(t) can be determined on MR images in this way, it is possible to determine the parameters τ, K1, and k2 by curve fitting, using the time-intensity curves (TIC) of the systemic circulation and tissue obtained by multiphase imaging [10].

In the following subsections, we discuss the significance of each parameter determined in the linear two-compartment model of the Gd-based extracellular fluid contrast agent, in terms of diagnostic imaging.

τ is the time required for the contrast agent to arrive at tissue via the systemic circulation. The contrast agent arrives more slowly at a tissue with large τ than at a tissue with small τ. A difference in τ is observed as a difference in the upward flexion point of the TIC (Fig. 2) [7].

Fig. 2

Influence of τ on time-intensity curve. Shown are time-intensity curves obtained as K1 and k2 were kept constant and τ was altered. The upward flexion point of the time-intensity curve is delayed when τ is large, compared to when τ is small

K1 is the rate coefficient for diffusion of the contrast agent from the systemic circulation into the tissue. Therefore, it can be said that within a particular period, more contrast agent flows into a tissue with larger K1 than into a tissue with smaller K1. The diffusion of the contrast agent into the tissue is influenced both by blood flow rate and by blood vessel permeability [6]. Differences in K1 are observed as differences in the peak height of the TIC during early-phase imaging (Fig. 3) [11]. The usefulness of K1 as a factor for estimating the outcome of patients with hepatocellular carcinoma, and also as an indicator for evaluation of therapeutic effects, has been reported [12].

Fig. 3

Influence of K1 on time-intensity curve. Shown are time-intensity curves obtained as τ and k2 were kept constant and K1 was altered. The peak of the time-intensity curve is higher when K1 is large, compared to when K1 is small

k2 is the rate coefficient for diffusion of the contrast agent from the tissue into the systemic circulation. It is also expressed as the inverse of mean transit time, and it can be said that the contrast agent diffuses more rapidly out of a tissue with larger k2 than out of a tissue with smaller k2 [6]. Differences in k2 are observed as differences in the slope of the TIC between early- and late-phase imaging (Fig. 4) [11]. Similarly to K1, k2 has also been reported to be useful as a factor for estimating the outcome of patients with hepatocellular carcinoma and as an indicator for evaluating therapeutic effects [12].

Fig. 4

Influence of k2 on time-intensity curve. Shown are time-intensity curves obtained as τ and K1 were kept constant and k2 was altered. The time-intensity curve declines more rapidly when k2 is large, compared to when k2 is small

K1/k2

K1/k2 is also expressed as distribution volume (V _d), which is defined as the volume of all tissues and organs in the body where the drug is distributed, assuming that it is distributed at the concentration identical to that in the systemically circulating blood [3]. It can be said that a tissue with large K1/k2 contains more contrast agent in the extracellular space than a tissue with small K1/k2. However, note should be taken of the fact that K1/k2 does not indicate the actual volume of the tissue where the contrast agent is distributed. Differences in K1/k2 are observed as differences in the height of the TIC during late-phase imaging and reflect the contrast-enhancement effect in the equilibrium-phase imaging (Fig. 5) [11].

Fig. 5

Influence of magnitudes of K1 and k2 on time-intensity curve. Shown are time-intensity curves obtained as τ and K1/k2 were kept constant and K1 and k2 were altered. Various peak heights and decay patterns are observable, depending on the values of K1 and k2, but the contrast enhancement is constant in late-phase imaging

Two-in-one-outlinear compartment model

Because hepatic blood flow is supplied by both the hepatic artery and the portal vein, a 2-in-1-outlinear compartment model should be considered, in order to evaluate hepatic blood flow dynamics in detail [7]. Such a model takes into the account the influence of portal blood flow in a linear two-compartment model:

$\mathrm{d}{C}_{\mathrm{t}}(t)/\mathrm{d}t=\mathrm{K}{1}_{\mathrm{a}}\times {C}_{\mathrm{a}}\left(t-{\tau}_{\mathrm{a}}\right)+\mathrm{K}{1}_{\mathrm{p}}\times {C}_{\mathrm{p}}\left(t-{\tau}_{\mathrm{p}}\right)-\mathrm{k}2\times {C}_{\mathrm{t}}(t)$

where C _a(t), C _p(t), τ _a, τ _p, K1_a, and K1_p are arterial and portal concentrations of the contrast agent, time to arrival, and transfer rate coefficient. It is possible to determine the arterial blood flow rate [K1_a/(K1_a + K1_p)], V _d [(K1_a + K1_p)/k2], and other parameters by analyzing these values, which have been used for evaluation of liver fibrosis [3].

Quantitative Evaluation of Liver Function Using Gadoxetate Disodium

Gadoxetate disodium is a tissue-specific Gd contrast agent consisting of a Gd-based extracellular fluid contrast agent, Gd-DTPA, linked to a lipophilic branched-chain EOB moiety. When administered intravenously, this compound is transferred from the systemic circulation into the tissue, distributed into the extracellular fluid, and incorporated specifically into hepatocytes by the organic anion transporter peptide (OATP) expressed on the hepatocyte membrane. About 40 % of the administered agent is excreted into bile, while about 60 % of it is excreted into urine. Thus, the pharmacokinetics of gadoxetate disodium and the resulting imaging findings are easily explained by a nonlinear compartment model, in which the model of the extracellular fluid Gd-based contrast agent is modified to take into account incorporation into cells (Fig. 6) [13]:

$\mathrm{d}{C}_{\mathrm{t}}(t)/\mathrm{d}t=\mathrm{K}1\times {C}_{\mathrm{a}}\left(t-\tau \right)-\left\{\mathrm{k}2+v(t)\right\}\times {C}_{\mathrm{t}}(t)$

where v(t) is the nonlinear rate coefficient at time t, representing the incorporation from the tissue extracellular space into cells. v(t) can be expressed by the following Michaelis-Menten equation:

$v(t)={V}_{\max }/\left\{{K}_{\mathrm{m}}+{C}_{\mathrm{t}}(t)\right\}$

where V _max is the maximum rate of incorporation, and K _m is the Michaelis-Menten constant, which is defined as the substrate concentration giving an incorporation rate of half V _max. Normally, gadoxetate disodium-enhanced MRI is carried out within 20 min after administration of the contrast agent; if it is postulated that excretion of the contrast agent into bile is negligibly small, then the intracellular concentration of the contrast agent at time t can be expressed by ∫ {v(t) × C _t(t)} • dt. In other words, gadoxetate disodium uptake into hepatocytes is governed by the amount of the contrast agent delivered to the tissue extracellular space and the uptake rate into hepatocytes, thus reflecting both blood flow and transporter function. Furthermore, in view of the fact that R1 of EOB differs between extracellular fluid (R1_{extracellular fluid} = 8.7 L mmol⁻¹ s⁻¹) and hepatocytes (R1_HC = 16.6 L mmol⁻¹ s⁻¹) [21], the T1 relaxation rate in tissue (1/T1) at time t can be expressed by the sum of the T1 relaxation rate specific to the tissue (1/T1₀), the T1-shortening by the contrast agent present in the extracellular fluid (1/T1_{extracellular fluid}) and the T1-shortening of the contrast agent present in the hepatocytes (1/T1_HC):

Fig. 6

Nonlinear two-compartment model of hepatocyte-specific gadolinium contrast agent. C _a(t): concentration of the contrast agent in systemically circulating blood at time t; C _t(t): concentration of the contrast agent in tissue extracellular space at time t; τ

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