We start with the standard Patlak formula but avoid division by

*c*
_{
a
}(

*t*):

^{a}
$\begin{array}{c}{c}_{t}\left(t\right)={K}_{m}\xb7{\int}_{0}^{t}{c}_{a}\left(s\right)\mathit{\text{ds}}+{V}_{r}\xb7{c}_{a}\left(t\right),\end{array}$

(5)

where

*K*
_{
m
} is the metabolic trapping rate, defined by

${K}_{m}=\frac{{K}_{1}{k}_{3}}{{k}_{2}+{k}_{3}}$

and

*V*
_{
r
} is the apparent volume of distribution defined by

${V}_{r}=\frac{{K}_{1}{k}_{2}}{{({k}_{2}+{k}_{3})}^{2}}=\frac{{K}_{1}}{{k}_{2}}\xb7{\left(\frac{{k}_{2}}{{k}_{2}+{k}_{3}}\right)}^{2}.$

(6)

Equation 5 is valid for times

*t* >

*T*^{∗} where

*T*^{∗} ≈ 20 to 30 min p.i.. Utilization of this equation for

*K*
_{
m
} determination requires measurements of the TRF only for

*t* >

*T*^{∗} but measurement of the complete AIF starting at time zero. We now want to eliminate the dependency on measurements prior to

*T*^{∗}. By taking the time derivative at some time point

*t* >

*T*^{∗}, it follows directly from Equation

5 that

${\u010b}_{t}\left(t\right)={K}_{m}\xb7{c}_{a}\left(t\right)+{V}_{r}\xb7{\u010b}_{a}\left(t\right)$

or after division by

*c*
_{
a
}(

*t*) (suppressing the

*t* argument)

$\frac{{\u010b}_{t}}{{c}_{a}}={K}_{m}+{V}_{r}\xb7\frac{{\u010b}_{a}}{{c}_{a}}.$

(7)

Focusing on some specific time point

*t* =

*t*
_{0}, we use the Taylor expansion of

*c*
_{
a
}(

*t*) around

${t}_{0}\phantom{\rule{0.3em}{0ex}}\left({c}_{a}^{\left(n\right)}\right({t}_{0})$:

*n* th derivative at

*t* =

*t*
_{0}):

${c}_{a}\left(t\right)=\sum _{n=0}^{\infty}\frac{{c}_{a}^{\left(n\right)}\left({t}_{0}\right)}{n!}{(t-{t}_{0})}^{n}.$

(8)

Introducing the parameters

*τ*
_{
n
} defined by

${(-{\tau}_{n})}^{n}={c}_{a}^{0}/{c}_{a}^{\left(n\right)}\left({t}_{0}\right)$, Equation

8 can be rewritten as

$\begin{array}{ll}\phantom{\rule{1em}{0ex}}{c}_{a}\left(t\right)& ={c}_{a}^{0}\xb7\sum _{n=0}^{\infty}\frac{{(-1)}^{n}}{n!}{\left(\frac{t-{t}_{0}}{{\tau}_{n}}\right)}^{n}\\ ={c}_{a}^{0}\xb7\left(1-\frac{t-{t}_{0}}{{\tau}_{1}}+\frac{1}{2!}{\left(\frac{t-{t}_{0}}{{\tau}_{2}}\right)}^{2}-\dots \right),\end{array}$

(9)

where *τ*
_{0} is always equal to one. The parameters *τ*
_{
n > 0} are constructed in such a way that for a mono-exponential decrease of *c*
_{
a
}(*t*) near *t*
_{0}, we obtain *τ*
_{
n > 0} = *τ*
_{
a
}, where *τ*
_{
a
} is the time constant of the exponential. Actually, it is known that starting rather early after bolus injection (*t* > 20 min), *c*
_{
a
}(*t*) can be reasonably well described by a slow mono-exponential decrease with a time constant *τ*
_{
a
} ≈ 100 min (in the present study, we found an average value of *τ*
_{
a
} = 99 min, while a value of *τ*
_{
a
} = 80 min was reported in [18]).

Inserting the Taylor expansion from Equation

9 into Equation

7, we get (

${\u010b}_{t}^{0}={\u010b}_{t}\left({t}_{0}\right)$)

$\frac{{\u010b}_{t}^{0}}{{c}_{a}^{0}}={K}_{m}-\frac{{V}_{r}}{{\tau}_{1}}.$

(10)

In order to derive

*K*
_{
m
} from this equation, we need to reliably estimate

${\u010b}_{t}^{0}/{c}_{a}^{0}$ as well as to have knowledge of 1 /

*τ*
_{1} (the fractional rate of decrease of the AIF at

*t* =

*t*
_{0}). Obviously, direct determination of the time derivative

${\u010b}_{t}^{0}$ at

*t* =

*t*
_{0} is not feasible in real (noisy) data. On the other hand, it is not clear whether the average slope over a necessarily rather large neighborhood (required for reasons of limited time resolution and count rate statistics) is an acceptable approximation of

${\u010b}_{t}^{0}$ (since the slope changes over time). For investigation of this question, we compute from Equation

5 the difference

${c}_{t}^{+}-{c}_{t}^{-}={c}_{t}\left({t}_{0}+\frac{\mathrm{\Delta t}}{2}\right)-{c}_{t}\left({t}_{0}-\frac{\mathrm{\Delta t}}{2}\right)$ for two time points lying symmetrically around

*t*
_{0} at a finite (possibly large) distance

*Δ* *t*
$\phantom{\rule{-12.0pt}{0ex}}\Delta {c}_{t}\left(\mathrm{\Delta t}\right)={c}_{t}^{+}-{c}_{t}^{-}={K}_{m}\xb7{\int}_{{t}_{0}-\frac{\mathrm{\Delta t}}{2}}^{{t}_{0}+\frac{\mathrm{\Delta t}}{2}}{c}_{a}\left(s\right)\mathit{\text{ds}}+{V}_{r}\xb7\left({c}_{a}^{+}-{c}_{a}^{-}\right)$

(11)

with ${c}_{a}^{\pm}={c}_{a}\left({t}_{0}\pm \frac{\mathrm{\Delta t}}{2}\right)$.

Replacing all occurrences of

*c*
_{
a
}(

*t*) in Equation

11 by the Taylor series in Equation

9 (neglecting fourth and higher order terms) and executing the integration separately for each term of the series yield after some straightforward but lengthy calculations the following equation:

$\begin{array}{ll}\phantom{\rule{.8em}{0ex}}\Delta {c}_{t}=& \left({K}_{m}\left[1+\frac{1}{24}{\left(\frac{\mathrm{\Delta t}}{{\tau}_{2}}\right)}^{2}\right]\right.\\ \left.-\frac{{V}_{r}}{{\tau}_{1}}\left[1+\frac{1}{24}\frac{{\tau}_{1}}{{\tau}_{3}}{\left(\frac{\mathrm{\Delta t}}{{\tau}_{3}}\right)}^{2}\right]\right){c}_{a}^{0}\xb7\mathrm{\Delta t}\end{array}$

(12)

The detailed derivation of Equation 12 is presented in an additional file (see Additional file 1). The factors in square brackets deviate only minimally from one up to even quite large values of *Δ* *t*. For the sake of simplicity, we will demonstrate this only for the well-established approximately mono-exponential behavior of *c*
_{
a
}(*t*) at later times but emphasize that the conclusions remain the same when using other reasonable parametrizations of the observed shape of the AIF at later times (e.g., by an inverse power law).

As already pointed out, for a mono-exponential decrease of

*c*
_{
a
}(

*t*), all

*τ*
_{
n > 0} coincide with the time constant

*τ*
_{
a
} of the exponential. Consider, then, choosing

*Δ* *t* = 60 min in Equation

12. Since

*τ*
_{
a
} ≈ 100 min, we have for both square brackets 1 + 1 / 24 · 0.6

^{2} = 1.015. It is, therefore, permissible to replace both square brackets by one. This yields

$\Delta {c}_{t}\approx \left[{K}_{m}-\frac{{V}_{r}}{{\tau}_{a}}\right]{c}_{a}^{0}\xb7\mathrm{\Delta t.}$

Thus,

*Δ* *c*
_{
t
} is to a very good approximation proportional to

*Δ* *t*.

*Δ* *t* can become quite large, e.g.,

*Δ* *t* = 1 h, as long as the lower bound

${t}_{0}-\frac{\mathrm{\Delta t}}{2}$ remains larger than

*T*^{∗}. Defining the secant slope

*m*
_{
t
} between

${c}_{t}^{-}$ and

${c}_{t}^{+}$
${m}_{t}=\frac{\Delta {c}_{t}}{\mathrm{\Delta t}}$

and introducing the rate constant

*K*
_{
s
}
${K}_{s}=\frac{{m}_{t}}{{c}_{a}^{0}}$

for the ratio of the secant slope and the blood concentration at

*t*
_{0}, we get

${K}_{s}=\frac{{m}_{t}}{{c}_{a}^{0}}={K}_{m}-\frac{{V}_{r}}{{\tau}_{a}}$

(13)

or

${K}_{m}={K}_{s}+\frac{{V}_{r}}{{\tau}_{a}}.$

(14)

Comparison of Equation

13 with Equation

10 yields the important result

${m}_{t}={\u010b}_{t}^{0}.$

In other words, the secant slope is to a very good approximation equal to the instantaneous slope at *t*
_{0} and thus can be used instead. This in turn implies that the average slope of the TRF (derivable, e.g., by a least squares fit of a straight line in the considered time window), too, is very nearly identical to *m*
_{
t
}. Note that these conclusions are valid even if ${\u010b}_{t}\left(t\right)$ varies considerably over the considered time interval (see Figure 1). Formally, this result is identical to stating that a second-order Taylor expansion of *c*
_{
t
}(*t*) around *t*
_{0} turns out to be sufficiently accurate within *t*
_{0} ± *Δ* *t* / 2.

The quantitative relation between *K*
_{
s
} and *K*
_{
m
} is investigated in Figure 6. For this figure, we computed *K*
_{
m
} and *V*
_{
r
} over a range of sensible choices for the transport constants *K*
_{1}, *k*
_{2}, and *k*
_{3}. The resulting *K*
_{
m
} and *V*
_{
r
} (top row of Figure 6) are used to compute *K*
_{
s
} from Equation 13 for a realistic value of *τ*
_{
a
} (we chose *τ*
_{
a
} = 99 min). The bottom row in Figure 6 compares the true *K*
_{
m
} to *K*
_{
s
}.

As can be seen (bottom right), the fractional deviation of *K*
_{
s
} from *K*
_{
m
} becomes large only when *k*
_{3} is very small (i.e., when there is virtually no trapping). Overall *K*
_{
s
} is a negatively biased estimator of *K*
_{
m
}, but an approximate correction of the bias is possible considering the following.

According to Equation

14, the

*V*
_{
r
} and

*K*
_{
m
} -

*K*
_{
s
} maps in Figure

6 differ only by a constant factor

*τ*
_{
a
} (and a conversion factor of 100 due to the chosen units of ml/min/100 ml for

*K*
_{
m
} and

*K*
_{
s
}). Moreover,

*V*
_{
r
} does vary only modestly in comparison to the individual rate constants and to

*K*
_{
m
} (except when

*k*
_{3} becomes distinctly larger than

*k*
_{2}, but this is not observed in real data). Therefore,

*K*
_{
m
} -

*K*
_{
s
} does not vary much across the relevant part of the

*k*
_{2} /

*k*
_{3} plane. We, therefore, hypothesize that the difference

*K*
_{
m
} -

*K*
_{
s
} can be actually treated to be approximately constant. Consequently, we propose to estimate

*K*
_{
m
} using only late time measurements of

*c*
_{
a
}(

*t*) and

*c*
_{
t
}(

*t*) as follows:

- 1.
Determine the secant TRF slope *m* _{
t
} in the time interval ${t}^{\pm}={t}_{0}\pm \frac{\mathrm{\Delta t}}{2}$ from a dual time point measurement of *c* _{
t
}(*t*) starting at sufficiently late times after injection, typically *t* > (20-30) min.

- 2.
Estimate ${c}_{a}^{0}={c}_{a}\left({t}_{0}\right)$ and *τ* _{
a
} from the exponential connecting the two time points *t* ^{ - }, *t* ^{+}.

- 3.
Compute ${K}_{s}={m}_{t}/{c}_{a}^{0}$.

- 4.
Compute a correction term ${K}_{0}={\stackrel{\u0304}{V}}_{r}/{\tau}_{a}$ using the individually determined *τ* _{
a
} and a fixed value ${\stackrel{\u0304}{V}}_{r}$ for the distribution volume. In the absence of any specific information regarding *V* _{
r
} in the investigated tumor entity, we propose to use the average *V* _{
r
} determined in this study, i.e., ${\stackrel{\u0304}{V}}_{r}=0.53$ ml/ml.

- 5.
Finally, compute the corrected

*K* _{
s
}, i.e.,

${K}_{s}^{\left(c\right)}={K}_{s}+{K}_{0}={K}_{s}+{\stackrel{\u0304}{V}}_{r}/{\tau}_{a}$

(15)

as a quantitative estimate of the true *K*
_{
m
}.

According to Equations 14 and 15, ${K}_{s}^{\left(c\right)}$ is equal to *K*
_{
m
} if ${V}_{r}={\stackrel{\u0304}{V}}_{r}$ (irrespective of the values of *K*
_{1} - *k*
_{3} yielding this *V*
_{
r
} value). Therefore, ${K}_{s}^{\left(c\right)}$ remains a very good approximation of *K*
_{
m
} as long as *V*
_{
r
} does not deviate too much from the assumed value. This behavior is illustrated in Figure 7.