Tracer Kinetic Modeling: Basics and Concepts

Abstract

In nuclear medicine studies, such as PET or SPECT, the tracer distribution changes over time depending on delivery, retention and clearance of the tracer in different organs or tissues. If dynamic data acquisition is performed, it is possible to analyse the kinetic behaviour of the tracer and determine quantitative parameters related to various physiological or biochemical processes. This type of information can be clinically relevant in several areas, such as cardiology, oncology and neurology. Mathematical models for the tracer behaviour can be used for estimating outcome measures such as flood flow, volume of distribution or binding potential. Usually an arterial input function is needed, which can be obtained by arterial sampling or sometimes directly from the dynamic PET or SPECT data. For some tracers, an indirect input function, obtained from a reference region, can be used. Simplified, model-free analysis techniques are also available. In summary, kinetic analysis requires defining an acquisition protocol, selecting an appropriate model and analysis technique, as well as the outcome measures desired.

Appendices

Appendix 1: Compartmental Models

Expressions for the impulse response functions for the 1-TC and 2-TC models are derived below. L{·} represents the Laplace transform, Laplace domain functions are identified with a tilde, and s is a complex Laplace domain variable.

1.1 1-TC Model

$$ {\displaystyle \begin{array}{c}\frac{\mathrm{d}}{\mathrm{d}t}{C}_{\mathrm{T}}(t)={K}_1{C}_{\mathrm{p}}(t)-{k}_2^{{\prime\prime} }{C}_{\mathrm{T}}(t)\\ {}\iff \\ {}L\left\{\frac{\mathrm{d}}{\mathrm{d}t}{C}_{\mathrm{T}}(t)\right\}=L\left\{{K}_1{C}_{\mathrm{p}}(t)-{k}_2^{{\prime\prime} }{C}_{\mathrm{T}}(t)\right\}\\ {}\iff \\ {}s{\hat{C}}_{\mathrm{T}}(s)-{C}_{\mathrm{T}}(0)={K}_1{\hat{C}}_{\mathrm{p}}(s)-{k}_2^{{\prime\prime} }{\hat{C}}_{\mathrm{T}}(s)\\ {}\Rightarrow \end{array}} $$

With initial condition, C_T(0) = 0:

$$ {\displaystyle \begin{array}{c}{\hat{C}}_{\mathrm{T}}(s)=\frac{K_1}{s+{k}_2^{{\prime\prime} }}{\hat{C}}_{\mathrm{p}}(s)\\ {}\iff \\ {}{C}_{\mathrm{T}}(t)={K}_1{\mathrm{e}}^{-{k}_2^{{\prime\prime} }t}\otimes {C}_{\mathrm{p}}(t)\\ {}\iff \end{array}} $$

(20.24)

Impulse response function:

$$ {H}_1(t)={K}_1{\mathrm{e}}^{-{k}_2^{{\prime\prime}}\;t} $$

(20.25)

1.2 2-TC Model

$$ {\displaystyle \begin{array}{c}\begin{array}{l}\frac{\mathrm{d}}{\mathrm{d}t}{C}_{ND}(t)={K}_1{C}_{\mathrm{p}}(t)-\left({k}_2^{\prime }+{k}_3^{\prime}\right){C}_{ND}(t)+{k}_4{C}_{\mathrm{S}}(t)\\ {}\frac{\mathrm{d}}{\mathrm{d}t}{C}_{\mathrm{S}}(t)={k}_3^{\prime }{C}_{ND}(t)-{k}_4{C}_{\mathrm{S}}(t)\end{array}\\ {}\iff \\ {}\left\{\begin{array}{l}s{\hat{C}}_{ND}(s)-{C}_{ND}(0)={K}_1{\hat{C}}_{\mathrm{p}}(s)-\left({k}_2^{\prime }+{k}_3^{\prime}\right){\hat{C}}_{ND}(s)+{k}_4{\hat{C}}_{\mathrm{S}}(s)\\ {}s{\hat{C}}_{\mathrm{S}}(s)-{C}_{\mathrm{S}}(0)={k}_3^{\prime }{\hat{C}}_{ND}(s)-{k}_4{\hat{C}}_{\mathrm{S}}(s)\end{array}\right.\end{array}} $$

With initial conditions, C_ND(0) = C_S(0) = 0:

$$ {\displaystyle \begin{array}{c}\Rightarrow \\ {}\left\{\begin{array}{l}\left(s+{k}_2^{\prime }+{k}_3^{\prime}\right){\hat{C}}_{ND}(s)={K}_1{\hat{C}}_{\mathrm{p}}(s)+{k}_4{\hat{C}}_{\mathrm{S}}(s)\\ {}\left(s+{k}_4\right){\hat{C}}_{\mathrm{S}}(s)={k}_3^{\prime }{\hat{C}}_{ND}(s)\end{array}\right.\end{array}} $$

$$ {\displaystyle \begin{array}{c}\iff \\ {}\left\{\begin{array}{l}{\hat{C}}_{ND}(s)={\left(s+{k}_2^{\prime }+{k}_3^{\prime }-\frac{k_3^{\prime }{k}_4}{s+{k}_4}\right)}^{-1}{K}_1{\hat{C}}_{\mathrm{p}}(s)\\ {}{\hat{C}}_{\mathrm{S}}(s)=\frac{k_3^{\prime }}{s+{k}_4}{\hat{C}}_{ND}(s)\end{array}\right.\end{array}} $$

$$ {\displaystyle \begin{array}{c}\Rightarrow \\ {}{\hat{C}}_T(s)\equiv {\hat{C}}_{ND}(s)+{\hat{C}}_{\mathrm{S}}(s)=\left(1+\frac{k_3^{\prime }}{s+{k}_4}\right){\hat{C}}_{ND}(s)\\ {}=\left(\frac{s+{k}_3^{\prime }+{k}_4}{s+{k}_4}\right)\left(\frac{s+{k}_4}{\left(s+{k}_2^{\prime }+{k}_3^{\prime}\right)\left(s+{k}_4\right)-{k}_3^{\prime }{k}_4}\right){K}_1{\hat{C}}_{\mathrm{p}}(s)\end{array}} $$

$$ {\displaystyle \begin{array}{c}=\left(\frac{s+{k}_3^{\prime }+{k}_4}{s^2+\left({k}_2^{\prime }+{k}_3^{\prime }+{k}_4\right)s+{k}_2^{\prime }{k}_4}\right){K}_1{\hat{C}}_{\mathrm{p}}(s)\\ {}\sim \sim \sim \end{array}} $$

(20.26)

Find poles:

$$ {\displaystyle \begin{array}{c}{s}^2+\left({k}_2^{\prime }+{k}_3^{\prime }+{k}_4\right)s+{k}_2^{\prime }{k}_4=0\\ {}\iff \\ {}s=-\frac{1}{2}\left({k}_2^{\prime }+{k}_3^{\prime }+{k}_4\mp \sqrt{{\left({k}_2^{\prime }+{k}_3^{\prime }+{k}_4\right)}^2-4{k}_2^{\prime }{k}_4}\right)\equiv -{\theta}_{1,2}\\ {}\sim \sim \sim \end{array}} $$

Partial fraction expansion:

$$ \frac{\phi_1}{s+{\theta}_1}+\frac{\phi_2}{s+{\theta}_2}={K}_1\frac{s+{k}_3^{\prime }+{k}_4}{s^2+\left({k}_2^{\prime }+{k}_3^{\prime }+{k}_4\right)s+{k}_2^{\prime }{k}_4} $$

(20.27)

$$ {\displaystyle \begin{array}{c}\iff \\ {}\left\{\begin{array}{l}{\phi}_1={K}_1\frac{k_3^{\prime }+{k}_4-{\theta}_1}{\theta_2-{\theta}_1}\\ {}{\phi}_2=-{K}_1\frac{k_3^{\prime }+{k}_4-{\theta}_2}{\theta_2-{\theta}_1}\end{array}\right.\end{array}} $$

$$ {\displaystyle \begin{array}{c}\sim \sim \sim \\ {}(20.26)+(20.27)\\ {}\Rightarrow \end{array}} $$

$$ {\hat{C}}_{\mathrm{T}}(s)=\left(\frac{\phi_1}{s+{\theta}_1}+\frac{\phi_2}{s+{\theta}_2}\right){\hat{C}}_{\mathrm{p}}(s) $$

(20.28)

$$ {\displaystyle \begin{array}{c}\iff \\ {}{C}_{\mathrm{T}}(t)=\left({\phi}_1{\mathrm{e}}^{-{\theta}_1t}+{\phi}_2{\mathrm{e}}^{-{\theta}_2t}\right)\otimes {C}_{\mathrm{p}}(t)\\ {}\iff \end{array}} $$

Impulse response function:

$$ {H}_2(t)={\phi}_1{\mathrm{e}}^{-{\theta}_1t}+{\phi}_2{\mathrm{e}}^{-{\theta}_2t} $$

(20.29)

Appendix 2: Reference Tissue Models

1.1 1-TC Model

$$ {\displaystyle \begin{array}{c}\left\{\begin{array}{l}\frac{\mathrm{d}}{\mathrm{d}t}{C}_{\mathrm{T}}(t)={K}_1{C}_{\mathrm{p}}(t)-{k}_2^{{\prime\prime} }{C}_{\mathrm{T}}(t)\\ {}\frac{\mathrm{d}}{\mathrm{d}t}{C}_{\mathrm{R}}(t){=}^R{K}_1{C}_{\mathrm{p}}(t){-}^R{k}_2^{\prime }{C}_{\mathrm{R}}(t)\end{array}\right.\\ {}\iff \end{array}} $$

From Eq. (20.24):

$$ {\displaystyle \begin{array}{c}\left\{\begin{array}{l}{\hat{C}}_{\mathrm{T}}(s)=\frac{K_1}{s+{k}_2^{{\prime\prime} }}{\hat{C}}_{\mathrm{p}}(s)\\ {}{\hat{C}}_{\mathrm{R}}(s)=\frac{{}^R{K}_1}{s{+}^R{k}_2^{\prime }}{\hat{C}}_{\mathrm{p}}(s)\end{array}\right.\\ {}\Rightarrow \end{array}} $$

$\left(\mathrm{with}\;{R}_1=\frac{K_1}{{}^R{K}_1}\right)$:

$$ {\displaystyle \begin{array}{c}{\hat{C}}_{\mathrm{T}}(s)={R}_1\frac{s{+}^R{k}_2^{\prime }}{s+{k}_2^{{\prime\prime} }}{\hat{C}}_{\mathrm{R}}(s)\\ {}={R}_1{\hat{C}}_{\mathrm{R}}(s)+{R}_1\frac{{}^R{k}_2^{\prime }-{k}_2^{{\prime\prime} }}{s+{k}_2^{{\prime\prime} }}{\hat{C}}_{\mathrm{R}}(s)\\ {}\Rightarrow \\ {}{C}_{\mathrm{T}}(t)={R}_1{C}_{\mathrm{R}}(t)+{R}_1\left({}^R{k}_2^{\prime }-{k}_2^{{\prime\prime}}\right){\mathrm{e}}^{-{k}_2^{{\prime\prime} }t}\otimes {C}_{\mathrm{R}}(t)\\ {}\iff \end{array}} $$

Impulse response function:

$$ {H}_{1\mathrm{R}}(t)={R}_1\delta (t)+{R}_1\left({}^R{k}_2^{\prime }-{k}_2^{{\prime\prime}}\right){\mathrm{e}}^{-{k}_2^{{\prime\prime} }t} $$

(20.30)

where δ(t) is the Dirac delta-function.

1.2 2-TC model

$$ \left\{\begin{array}{l}\frac{\mathrm{d}}{\mathrm{d}t}{C}_{ND}(t)={K}_1{C}_{\mathrm{p}}(t)-\left({k}_2^{\prime }+{k}_3^{\prime}\right){C}_{ND}(t)+{k}_4{C}_{\mathrm{S}}(t)\\ {}\frac{\mathrm{d}}{\mathrm{d}t}{C}_{\mathrm{S}}(t)={k}_3^{\prime }{C}_{ND}(t)-{k}_4{C}_{\mathrm{S}}(t)\\ {}\frac{\mathrm{d}}{\mathrm{d}t}{C}_{\mathrm{R}}(t){=}^R{K}_1{C}_{\mathrm{p}}(t){-}^R{k}_2^{\prime }{C}_{\mathrm{R}}(t)\end{array}\right. $$

From Eqs. (20.24) and (20.28):

$$ {\displaystyle \begin{array}{c}\left\{\begin{array}{l}{\hat{C}}_{\mathrm{T}}(s)=\left(\frac{\phi_1}{s+{\theta}_1}+\frac{\phi_2}{s+{\theta}_2}\right){\hat{C}}_{\mathrm{p}}(s)\\ {}{\hat{C}}_{\mathrm{R}}(s)=\frac{{}^R{K}_1}{s{+}^R{k}_2^{\prime }}{\hat{C}}_{\mathrm{p}}(s)\end{array}\right.\\ {}\Rightarrow \\ {}{\hat{C}}_{\mathrm{T}}(s)=\left(\frac{\phi_1}{s+{\theta}_1}+\frac{\phi_2}{s+{\theta}_2}\right)\frac{s{+}^R{k}_2^{\prime }}{{}^R{K}_1}{\hat{C}}_{\mathrm{R}}(s)\\ {}\iff \end{array}} $$

$\left(\mathrm{with}\;{R}_1=\frac{K_1}{{}^R{K}_1}\right)$:

$$ {\displaystyle \begin{array}{c}{\hat{C}}_{\mathrm{T}}(s)=\frac{R_1}{\theta_2-{\theta}_1}\left(\frac{k_3^{\prime }+{k}_4-{\theta}_1}{s+{\theta}_1}-\frac{k_3^{\prime }+{k}_4-{\theta}_2}{s+{\theta}_2}\right)\left(s{+}^R{k}_2^{\prime}\right){\hat{C}}_{\mathrm{R}}(s)\\ {}=\frac{R_1}{\theta_2-{\theta}_1}\left(\left({k}_3^{\prime }+{k}_4-{\theta}_1\right)\left(1+\frac{{}^R{k}_2^{\prime }-{\theta}_1}{s+{\theta}_1}\right)-\left({k}_3^{\prime }+{k}_4-{\theta}_2\right)\left(1+\frac{{}^R{k}_2^{\prime }-{\theta}_2}{s+{\theta}_2}\right)\right){\hat{C}}_{\mathrm{R}}(s)\\ {}={R}_1{\hat{C}}_{\mathrm{R}}(s)+\frac{R_1}{\theta_2-{\theta}_1}\left(\frac{\left({k}_3^{\prime }+{k}_4-{\theta}_1\right)\left({}^R{k}_2^{\prime }-{\theta}_1\right)}{s+{\theta}_1}-\frac{\left({k}_3^{\prime }+{k}_4-{\theta}_2\right)\left({}^R{k}_2^{\prime }-{\theta}_2\right)}{s+{\theta}_2}\right){\hat{C}}_{\mathrm{R}}(s)\\ {}\iff \end{array}} $$

$$ {\hat{C}}_{\mathrm{T}}(s)={R}_1{\hat{C}}_{\mathrm{R}}(s)+\left(\frac{\rho_1}{s+{\theta}_1}+\frac{\rho_2}{s+{\theta}_2}\right){\hat{C}}_{\mathrm{R}}(s) $$

(20.31)

where

$$ \left\{\begin{array}{l}{\rho}_1={R}_1\frac{\left({k}_3^{\prime }+{k}_4-{\theta}_1\right)\left({}^R{k}_2^{\prime }-{\theta}_1\right)}{\theta_2-{\theta}_1}\\ {}{\rho}_2=-{R}_1\frac{\left({k}_3^{\prime }+{k}_4-{\theta}_2\right)\left({}^R{k}_2^{\prime }-{\theta}_2\right)}{\theta_2-{\theta}_1}\end{array}\right. $$

$$ {\displaystyle \begin{array}{c}\sim \sim \sim \\ {}(20.31)\\ {}\Rightarrow \\ {}{C}_{\mathrm{T}}(t)={R}_1{C}_{\mathrm{R}}(t)+\left({\rho}_1{\mathrm{e}}^{-{\theta}_1t}+{\rho}_2{\mathrm{e}}^{-{\theta}_2t}\right)\otimes {C}_{\mathrm{R}}(t)\\ {}\iff \end{array}} $$

Impulse response function:

$$ {H}_{2\mathrm{R}}(t)={R}_1\delta (t)+{\rho}_1{\mathrm{e}}^{-{\theta}_1t}+{\rho}_2{\mathrm{e}}^{-{\theta}_2t} $$

(20.32)

Appendix 3: Logan Graphical Analysis

1.1 1-TC Model

$$ {\displaystyle \begin{array}{c}\frac{\mathrm{d}}{\mathrm{d}t}{C}_{\mathrm{T}}(t)={K}_1{C}_{\mathrm{p}}(t)-{k}_2^{{\prime\prime} }{C}_{\mathrm{T}}(t)\\ {}\iff \\ {}{C}_{\mathrm{T}}(t)-{C}_{\mathrm{T}}(0)={K}_1{\int}_0^t{C}_{\mathrm{p}}\left(\tau \right)\mathrm{d}\tau -{k}_2^{{\prime\prime} }{\int}_0^t{C}_{\mathrm{T}}\left(\tau \right)\mathrm{d}\tau \end{array}} $$

with C_T(0) = 0:

$$ {\displaystyle \begin{array}{c}\iff \\ {}\frac{\int_0^t{C}_{\mathrm{T}}\left(\tau \right)\mathrm{d}\tau }{C_{\mathrm{T}}(t)}=\frac{K_1}{k_2^{{\prime\prime} }}\frac{\int_0^t{C}_{\mathrm{p}}\left(\tau \right)\mathrm{d}\tau }{C_{\mathrm{T}}(t)}-\frac{1}{k_2^{{\prime\prime} }}\\ {}\Rightarrow \end{array}} $$

$$ \frac{\int_0^t{C}_{\mathrm{T}}\left(\tau \right)\mathrm{d}\tau }{C_{\mathrm{T}}(t)}={V}_{\mathrm{T}}\frac{\int_0^t{C}_{\mathrm{p}}\left(\tau \right)\mathrm{d}\tau }{C_{\mathrm{T}}(t)}+\mathrm{const}. $$

(20.33)

1.2 2-TC Model

$$ {\displaystyle \begin{array}{c}\left\{\begin{array}{l}\frac{\mathrm{d}}{\mathrm{d}t}{C}_{ND}(t)={K}_1{C}_{\mathrm{p}}(t)-\left({k}_2^{\prime }+{k}_3^{\prime}\right){C}_{ND}(t)+{k}_4{C}_{\mathrm{S}}(t)\\ {}\frac{\mathrm{d}}{\mathrm{d}t}{C}_{\mathrm{S}}(t)={k}_3^{\prime }{C}_{ND}(t)-{k}_4{C}_{\mathrm{S}}(t)\end{array}\right.\\ {}\iff \\ {}\left\{\begin{array}{l}{C}_{ND}(t)-{C}_{ND}(0)={K}_1{\int}_0^t{C}_{\mathrm{p}}\left(\tau \right)\mathrm{d}\tau -\left({k}_2^{\prime }+{k}_3^{\prime}\right){\int}_0^t{C}_{ND}\left(\tau \right) d\tau +{k}_4{\int}_0^t{C}_{\mathrm{S}}\left(\tau \right) d\tau \\ {}{C}_{\mathrm{S}}(t)-{C}_{\mathrm{S}}(0)={k}_3^{\prime }{\int}_0^t{C}_{ND}(t)\mathrm{d}\tau -{k}_4{\int}_0^t{C}_{\mathrm{S}}\left(\tau \right)\mathrm{d}\tau \end{array}\right.\end{array}} $$

with C_ND(0) = C_S(0) = 0:

$$ {\displaystyle \begin{array}{c}\iff \\ {}\left\{\begin{array}{l}{C}_{\mathrm{T}}(t)={K}_1{\int}_0^t{C}_{\mathrm{p}}\left(\tau \right)\mathrm{d}\tau -{k}_2^{\prime }{\int}_0^t{C}_{ND}\left(\tau \right)\mathrm{d}\tau \\ {}{C}_{\mathrm{S}}(t)={k}_3^{\prime }{\int}_0^t{C}_{ND}\left(\tau \right)\mathrm{d}\tau -{k}_4{\int}_0^t{C}_{\mathrm{S}}\left(\tau \right)\mathrm{d}\tau \end{array}\right.\end{array}} $$

$$ {\displaystyle \begin{array}{c}\iff \\ {}\left\{\begin{array}{l}\frac{\int_0^t{C}_{ND}\left(\tau \right)\mathrm{d}\tau }{C_{\mathrm{T}}(t)}=\frac{K_1}{k_2^{\prime }}\frac{\int_0^t{C}_{\mathrm{P}}\left(\tau \right)\mathrm{d}\tau }{C_{\mathrm{T}}(t)}-\frac{1}{k_2^{\prime }}\\ {}\frac{\int_0^t{C}_{\mathrm{S}}\left(\tau \right)\mathrm{d}\tau }{C_{\mathrm{T}}(t)}=\frac{k_3^{\prime }}{k_4}\frac{\int_0^t{C}_{ND}\left(\tau \right)\mathrm{d}\tau }{C_{\mathrm{T}}(t)}-\frac{1}{k_4}\frac{C_{\mathrm{S}}(t)}{C_{\mathrm{T}}(t)}\end{array}\right.\end{array}} $$

$$ {\displaystyle \begin{array}{c}\Rightarrow \\ {}\frac{\int_0^t{C}_{\mathrm{T}}\left(\tau \right)\mathrm{d}\tau }{C_{\mathrm{T}}(t)}\equiv \frac{\int_0^t{C}_{ND}\left(\tau \right)+{C}_{\mathrm{S}}\left(\tau \right)\mathrm{d}\tau }{C_{\mathrm{T}}(t)}=\frac{K_1}{k_2^{\prime }}\frac{\int_0^t{C}_{\mathrm{P}}\left(\tau \right)\mathrm{d}\tau }{C_{\mathrm{T}}(t)}-\frac{1}{k_2^{\prime }}+\frac{k_3^{\prime }}{k_4}\left(\frac{K_1}{k_2^{\prime }}\frac{\int_0^t{C}_{\mathrm{P}}\left(\tau \right)\mathrm{d}\tau }{C_{\mathrm{T}}(t)}-\frac{1}{k_2^{\prime }}\right)-\frac{1}{k_4}\frac{C_{\mathrm{S}}(t)}{C_{\mathrm{T}}(t)}\\ {}=\frac{K_1}{k_2^{\prime }}\left(1+\frac{k_3^{\prime }}{k_4}\right)\frac{\int_0^t{C}_{\mathrm{P}}\left(\tau \right)\mathrm{d}\tau }{C_{\mathrm{T}}(t)}-\frac{1}{k_2^{\prime }}\left(1+\frac{k_3^{\prime }}{k_4}\right)-\frac{1}{k_4}\frac{C_{\mathrm{S}}(t)}{C_{\mathrm{T}}(t)}\end{array}} $$

$$ {\displaystyle \begin{array}{c}\iff \\ {}\frac{\int_0^t{C}_{\mathrm{T}}\left(\tau \right)\mathrm{d}\tau }{C_{\mathrm{T}}(t)}={V}_{\mathrm{T}}\frac{\int_0^t{C}_{\mathrm{P}}\left(\tau \right)\mathrm{d}\tau }{C_{\mathrm{T}}(t)}-\frac{1}{k_2^{{\prime\prime} }}-\frac{1}{k_4}\frac{C_{\mathrm{S}}(t)}{C_{\mathrm{T}}(t)}\\ {}\Rightarrow \end{array}} $$

with C_S(t)/C_T(t) = constant (pseudo-equilibrium):

$$ \frac{\int_0^t{C}_{\mathrm{T}}\left(\tau \right)\mathrm{d}\tau }{C_{\mathrm{T}}(t)}={V}_{\mathrm{T}}\frac{\int_0^t{C}_{\mathrm{P}}\left(\tau \right)\mathrm{d}\tau }{C_{\mathrm{T}}(t)}+\mathrm{const}. $$

(20.34)