Non-invasive estimation of hepatic blood perfusion from H₂ ¹⁵O PET images using tissue-derived arterial and portal input functions

Abstract

Purpose

The liver is perfused through the portal vein and the hepatic artery. When its perfusion is assessed using positron emission tomography (PET) and ¹⁵O-labeled water (H₂ ¹⁵O), calculations require a dual blood input function (DIF), i.e., arterial and portal blood activity curves. The former can be generally obtained invasively, but blood withdrawal from the portal vein is not feasible in humans. The aim of the present study was to develop a new technique to estimate quantitative liver perfusion from H₂ ¹⁵O PET images with a completely non-invasive approach.

Methods

We studied normal pigs (n = 14) in which arterial and portal blood tracer concentrations and Doppler ultrasonography flow rates were determined invasively to serve as reference measurements. Our technique consisted of using model DIF to create tissue model function and the latter method to simultaneously fit multiple liver time–activity curves from images. The parameters obtained reproduced the DIF. Simulation studies were performed to examine the magnitude of potential biases in the flow values and to optimize the extraction of multiple tissue curves from the image.

Results

The simulation showed that the error associated with assumed parameters was <10%, and the optimal number of tissue curves was between 10 and 20. The estimated DIFs were well reproduced against the measured ones. In addition, the calculated liver perfusion values were not different between the methods and showed a tight correlation (r = 0.90).

Conclusion

In conclusion, our results demonstrate that DIF can be estimated directly from tissue curves obtained through H₂ ¹⁵O PET imaging. This suggests the possibility to enable completely non-invasive technique to assess liver perfusion in patho-physiological studies.

Appendix

A model function for AIF was created by assuming a two-compartment model in which the tracer is administered in a rectangular form and diffuses bi-directionally between arterial and interstitial space in whole body peripheral tissue compartments. Differential equations for the model function (C _A(t)) can be expressed as

$$\frac{{\operatorname{d} C_{\text{A}} \left( t \right) }}{{\operatorname{d} t}} = {\text{ }}\frac{{\operatorname{d} F }}{{\operatorname{d} t}} - K_{\text{e}} C_{\text{A}} \left( t \right) + K_{\text{i}} C_{{\text{WB}}} \left( t \right)$$

(9)

$$\frac{{\operatorname{d} C_{{\text{WB}}} \left( t \right)}}{{\operatorname{d} t}} = K_{\text{e}} C_{\text{A}} \left( t \right) - K_{\text{i}} C_{{\text{WB}}} \left( t \right)$$

(10)

$$\begin{array}{*{20}c} {\frac{{\operatorname{d} F }}{{\operatorname{d} t}} = A}{{\left( {t_{1} \leqslant t \leqslant t_{2} } \right)}} \\ 0{{\left( {{\text{elsewhere}}} \right)}} \\ \end{array} $$

(11)

where t ₁ and t ₂ assumes the appearance time of administered tracer, and t ₂ − t ₁ represents the administration duration; A corresponds to the given amount of tracer. The equation F (Eq. 11) represents the bolus administration of tracer in the rectangular form with duration t ₂ − t ₁. C _WB(t) is the expected tracer concentration in interstitial spaces in whole body peripheral tissues; K _e and K _i are bidirectional tracer diffusion rates between blood and peripheral tissue compartments, respectively. Solving Eq. 10 for C _WB gives

$$C_{{\text{WB}}} \left( t \right) = K_{\text{e}} e^{ - K_{\text{i}} \cdot t} \int_0^t {C_{\text{A}} \left( \tau \right)e^{K_{\text{i}} \cdot \tau } \operatorname{d} \tau } {\text{.}}$$

(12)

Sum of Eqs. 9 and 10 is

$$\frac{{\operatorname{d} \left( {C_{\text{A}} \left( t \right) + C_{{\text{WB}}} \left( t \right)} \right) }}{{\operatorname{d} t}} = {\text{ }}\frac{{\operatorname{d} F }}{{\operatorname{d} t}}{\text{.}}$$

(13)

Thus,

$$\begin{aligned} & C_{{\text{A}}} {\left( t \right)} + C_{{{\text{WB}}}} {\left( t \right)} = {\text{ }}F \\ & \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\begin{array}{*{20}c} { = 0}{{\left( {t < t_{1} } \right)}} \\ { = A{\left( {t - t_{1} } \right)}}{{\left( {t_{1} \leqslant t \leqslant t_{2} } \right)}} \\ { = A{\left( {t_{2} - t_{1} } \right)}}{{\left( {t > t_{2} } \right)}} \\ \end{array} \\ \end{aligned}$$

(14)

Substitution of C _WB from Eq. 12 into 14 after multiplying $e^{K_{\text{i}} \cdot t}$ gives

$$e^{{K_{{\text{i}}} \cdot t}} C_{{\text{A}}} {\left( t \right)} + K_{{\text{e}}} {\int_0^t {C_{{\text{A}}} {\left( \tau \right)}e^{{K_{{\text{i}}} \cdot \tau }} \operatorname{d} \tau } } = e^{{K_{{\text{i}}} \cdot t}} F$$

(15)

Differentiation with respect to t after arranging gives

$$\frac{{\operatorname{d} C_{{\text{A}}} {\left( t \right)}}}{{\operatorname{d} t}} = \alpha F + \frac{1}{{K_{{\text{e}}} }}\frac{{\operatorname{d} F }}{{\operatorname{d} t}} - K_{{\text{e}}} {\left( {1 + \alpha } \right)}C_{{\text{A}}} {\left( t \right)}$$

(16)

where α = K _i/K _e . Thus,

$$C_{{\text{A}}} {\left( t \right)} = K_{{\text{e}}} e^{{ - K_{{\text{e}}} \cdot (1 + \alpha ) \cdot t}} {\int_0^t {{\left( {\alpha F + \frac{1}{{K_{{\text{e}}} }}\frac{{\operatorname{d} F}}{{\operatorname{d} t}}} \right)}e^{{K_{{\text{e}}} \cdot {\left( {1 + \alpha } \right)} \cdot \tau }} \operatorname{d} \tau } }$$

(17)

Solving Eq. 17, we obtain

$$\begin{array}{*{20}c} {C_{{\text{A}}} {\left( t \right)} = 0}{{\left( {t < t_{1} } \right)}} \\ { = \frac{A}{{K_{{\text{e}}} ^{2} {\left( {1 + \alpha } \right)}^{2} }}{\left( {K_{{\text{e}}} \alpha {\left( {1 + \alpha } \right)}{\left( {t - t_{1} } \right)} + 1 - e^{{K_{{\text{e}}} \cdot {\left( {1 + \alpha } \right)} \cdot {\left( {t_{1} - t} \right)}}} } \right)}}{{\left( {t_{1} \leqslant t \leqslant t_{2} } \right)}} \\ { = \frac{A}{{K_{{\text{e}}} ^{2} {\left( {1 + \alpha } \right)}^{2} }}\left. {{\left( {K_{{\text{e}}} \alpha {\left( {1 + \alpha } \right)}{\left( {t_{2} - t_{1} } \right)} + e^{{K_{{\text{e}}} \cdot {\left( {1 + \alpha } \right)} \cdot {\left( {t_{2} - t} \right)}}} - e^{{K_{{\text{e}}} \cdot (1 + \alpha ) \cdot (t_{1} - t)}} } \right)}} \right)}{{\left( {t > t_{2} } \right)}} \\ \end{array} $$

(18)

The first term in the second equation for t ₁ < t <t ₂, i.e., K _e α(1 + α)(t ₁ − t ₂), would complicate further calculations (such as tissue response and portal input); thus, this term was omitted, and the model function (Eq. 18) was modified to set the C _A value as 0 at t = t ₁, as continuous at t = t ₂, and as non-zero value at the equilibrium, i.e., at t = ∞. Thus, the following equation was derived:

$$\begin{array}{*{20}c} {C_{{\text{A}}} {\left( t \right)} = 0.}{{\left( {t < t_{1} } \right)}} \\ { = \frac{A}{{K_{{\text{e}}} ^{2} {\left( {1 + \alpha } \right)}^{2} }}{\left( {1 - \exp {\left( {K_{{\text{e}}} {\left( {1 + \alpha } \right)}{\left( {t_{1} - t} \right)}} \right)}} \right)}}{{\left( {t_{1} \leqslant t \leqslant t_{2} } \right)}} \\ { = \frac{A}{{K_{{\text{e}}} ^{2} {\left( {1 + \alpha } \right)}^{2} }}{\left( {\exp {\left( {K_{{\text{e}}} {\left( {1 + \alpha } \right)}{\left( {t_{1} - t_{2} } \right)}} \right)} + \exp {\left( {K_{{\text{e}}} {\left( {1 + \alpha } \right)}{\left( {t_{2} - t} \right)}} \right)} - 2 \cdot \exp {\left( {K_{{\text{e}}} {\left( {1 + \alpha } \right)}{\left( {t_{1} - t} \right)}} \right)}} \right)}}{{\left( {t > t_{2} } \right)}} \\ \end{array} $$

(19)