Improved correction for the tissue fraction effect in lung PET/CT imaging

Lung voxel air fraction correction (AFC) was first discussed by Lambrou et al (2011). In their paper, Lambrou et al assume that a lung voxel contains only two components, air and tissue, where tissue is the combination of both parenchyma (the non-blood cellular and extra cellular components that make up the lung) and blood. Under this assumption, the Hounsfield Units (HU) within a CT voxel can be considered as

$$\begin{eqnarray}&&\text{H}{{\text{U}}_{\text{l}}}={{V}_{\text{a}}}\,\text{H}{{\text{U}}_{\text{a}}}+{{V}_{\text{t}}}\,\text{H}{{\text{U}}_{\text{t}}} \end{eqnarray} \tag{ 1 }$$

where $\text{H}{{\text{U}}_{\text{l}}}$, $\text{H}{{\text{U}}_{\text{a}}}$ and $\text{H}{{\text{U}}_{\text{t}}}$ are the HU of the lung voxel (CT voxel value), air and tissue respectively and ${{V}_{\text{a}}}$ and ${{V}_{\text{t}}}$ are the fractional air and tissue volumes in the voxel respectively, where ${{V}_{\text{a}}}+{{V}_{\text{t}}}=1$. Therefore, once the CT has been filtered to the resolution of the PET image and downsampled, the PET voxel fractional air volume can be determined from simple rearrangement of equation (1):

$$\begin{eqnarray}&&{{V}_{\text{a}}}=\frac{\text{H}{{\text{U}}_{\text{l}}}-\text{H}{{\text{U}}_{\text{t}}}}{\text{H}{{\text{U}}_{\text{a}}}-\text{H}{{\text{U}}_{\text{t}}}}. \end{eqnarray} \tag{ 2 }$$

The PET concentration can then be corrected using equation (3):

$$\begin{eqnarray}&&{{C}_{\text{t}}}=\frac{{{C}_{\text{measured}}}}{1-{{V}_{\text{a}}}}. \end{eqnarray} \tag{ 3 }$$

Where ${{C}_{\text{measured}}}$ is the measured (non-corrected) tracer concentration and ${{C}_{\text{t}}}$ is the tracer concentration in tissue after AFC. Once the AFC has been applied, the corrected voxel values are those that would have been present without the air component (i.e. just blood and parenchyma). However, to obtain an accurate parenchyma fraction, as is required, a further correction for blood is needed.

Equation (1) can be expanded to include the blood component as follows:

$$\begin{eqnarray}&&\text{H}{{\text{U}}_{\text{l}}}={{V}_{\text{a}}}\,\text{H}{{\text{U}}_{\text{a}}}+{{V}_{\text{b}}}\,\text{H}{{\text{U}}_{\text{b}}}+{{V}_{\text{pa}}}\,\text{H}{{\text{U}}_{\text{pa}}} \end{eqnarray} \tag{ 4 }$$

where HU_b and HU_pa represent the HU of blood and parenchyma respectively (which are approximately equivalent (Hounsfield 1980)) and ${{V}_{\text{b}}}$ and ${{V}_{\text{pa}}}$ are the fractional blood and parenchyma volumes in the voxel respectively.

Image concentration can be corrected for the parenchyma fraction as before with ${{V}_{\text{pa}}}$ (where ${{V}_{\text{pa}}}=1-{{V}_{\text{a}}}-{{V}_{\text{b}}}$) replacing $1-{{V}_{\text{a}}}$ in equation (3). However, the significant amount of blood in the lung voxel also makes it vital to correct for any PET signal that may originate from blood. Therefore the complete correction for the concentration becomes:

$$\begin{eqnarray}&&{{C}_{\text{pa}}}=\frac{{{C}_{\text{measured}}}-{{V}_{\text{b}}}\,{{C}_{\text{b}}}}{{{V}_{\text{pa}}}}=\frac{{{C}_{\text{measured}}}-{{V}_{\text{b}}}\,{{C}_{\text{b}}}}{1-{{V}_{\text{a}}}-{{V}_{\text{b}}}} \end{eqnarray} \tag{ 5 }$$

where ${{C}_{\text{b}}}$ is the blood concentration, determined from an aorta or ventricle region of interest (ROI). In this paper we demonstrate that equation (5) is far more appropriate than equation (3) for comparing between patients and between patients and controls. The difficulty is that the static PET/CT does not provide information on ${{V}_{\text{b}}}$, unlike ${{V}_{\text{a}}}$ which can easily be calculated from the CT image. However, ${{V}_{\text{b}}}$ can be estimated directly from kinetic analysis if it is included in the model fit (section 3.4).

Standard kinetic modelling methods are formulated assuming that the voxel or ROI contains only parenchyma and blood, i.e. ${{V}_{\text{b}}}+{{V}_{\text{pa}}}=1$. The voxel concentration can then be assumed to be:

$$\begin{eqnarray}&&{{C}_{\text{v}}}={{V}_{\text{b}}}{{C}_{\text{b}}}+\left(1-{{V}_{\text{b}}}\right){{C}_{\text{pa}}} \end{eqnarray} \tag{ 6 }$$

where ${{C}_{\text{v}}}$, ${{C}_{\text{b}}}$ and ${{C}_{\text{pa}}}$ are the activity concentrations in the voxel (or ROI), blood and parenchyma respectively. Equation (6) allows the simultaneous fit of ${{V}_{\text{b}}}$ and the kinetic parameters of the chosen model for ${{C}_{\text{pa}}}$ (Tsoumpas and Thielemans 2009). Now, consider the lung case where air is present such that ${{V}_{\text{b}}}+{{V}_{\text{pa}}}\ne 1$ but ${{V}_{\text{b}}}+{{V}_{\text{pa}}}+{{V}_{\text{a}}}=1$, then the appropriate air and blood corrected (ABC) model would be:

$$\begin{eqnarray}&&{{C}_{\text{v}}}={{V}_{\text{b}}}{{C}_{\text{b}}}+\left(1-{{V}_{\text{b}}}-{{V}_{\text{a}}}\right){{C}_{\text{pa}}}+{{V}_{\text{a}}}{{C}_{\text{a}}} \end{eqnarray} \tag{ 7 }$$

where ${{C}_{\text{a}}}$ is the tracer concentration in air. ${{C}_{\text{a}}}$ will be zero unless the tracer has been administered using the inhalation method.

To correctly determine the kinetic parameters, one of three methods can be utilised: (1) use a model that follows general equation (6) and then correct the estimated parameters affected by the TFE by multiplication of $\left(1-{{V}_{\text{b}}}\right)/\left(1-{{V}_{\text{b}}}-{{V}_{\text{a}}}\right)$, (2) directly fit a kinetic model to the measured data using equation (7), (3) precorrect the dynamic images for ${{V}_{\text{a}}}$ (as in section 2.1) and then fit the data as follows:

$$\begin{eqnarray}&&C_{\text{v}}^{\prime}=V_{\text{b}}^{\prime}{{C}_{\text{b}}}+\left(1-V_{\text{b}}^{\prime}\right){{C}_{\text{pa}}} \end{eqnarray} \tag{ 8 }$$

where $C_{\text{v}}^{\prime}={{C}_{\text{v}}}/\left(1-{{V}_{\text{a}}}\right)$ and

$$\begin{eqnarray}&&V_{\text{b}}^{\prime}=\frac{{{V}_{\text{b}}}}{1-{{V}_{\text{a}}}} \end{eqnarray} \tag{ 9 }$$

which is the blood volume as a fraction of the tissue.

To determine which of the estimated kinetic parameters will change after the corrections to equation (6) have been applied, consider the corrected general equation for a compartment model (Gunn et al 2002):

$$\begin{eqnarray}&&C_{\text{v}}^{\prime}(t)=V_{\text{b}}^{\prime}{{C}_{\text{b}}}(t)+\left(1-V_{\text{b}}^{\prime}\right)\underset{i=1}{\overset{n}{\mathop \sum}}\,{{\phi}_{i}}{{\text{e}}^{-{{\theta}_{i}}t}}\otimes {{C}_{\text{p}}}(t) \end{eqnarray} \tag{ 10 }$$

where, n is the number of compartments in the model, ${{\theta}_{i}}$ is the clearance from compartment i, ${{\phi}_{i}}$ the applied weighting and ${{C}_{\text{p}}}$ the concentration in the plasma. From this equation it can be seen that it is the ${{\phi}_{i}}$ parameters that will be affected by the TFE. As ${{K}_{1}}=\sum_{i=1}^{n}{{\phi}_{i}}$, the delivery of the tracer to the tissue, it is only K₁ and any parameter containing K₁ (e.g. the influx rate constant (K_i)) that will require corrections.

As K_i is affected by the TFE, all of the above theory is also applicable to Patlak analysis (Patlak et al 1983) where the corrected curve equation would be:

$$\begin{eqnarray}&&\frac{C_{\text{v}}^{\prime}-V_{\text{b}}^{\prime}{{C}_{\text{b}}}}{{{C}_{\text{p}}}\left(1-V_{\text{b}}^{\prime}\right)}={{K}_{i}}\frac{\int_{0}^{t}{{C}_{\text{p}}}(t)\text{d}t}{{{C}_{\text{p}}}}+D \end{eqnarray} \tag{ 11 }$$

where D is the intercept of the tissue fraction corrected Patlak curve. Unfortunately, as is the case with static image analysis, equation (11) requires knowledge of $V_{\text{b}}^{\prime}$, a parameter not normally determined during Patlak analysis. However, $V_{\text{b}}^{\prime}$ can be determined from the early part of the time activity curve (TAC), see section 3.4.

Six patients, three males and three females aged $70\pm 5$ yrs all with diagnosed IPF underwent ¹⁸F-FDG dynamic PET/CT using a GE Discovery VCT PET/CT scanner (Teras et al 2007). All patients were imaged supine immediately post-injection using $200\pm 26$ MBq ¹⁸F-FDG with the protocol outlined in table 1. Cine-CT imaging was acquired with a duration equivalent to the patients complete breathing cycle plus one second. Five of the six patient scans were obtained from an ongoing study (NCT01725139) designed to explore a number of doses of GSK2126458 for engagement of pharmacology in subjects with IPF after short term dosing. The remaining patient's scan was part of an ongoing study (MREC 06/Q0505/22) designed to explore the prognostic value of PET/CT in patients with interstitial lung disease. Institutional Review Board permission and informed patient consent were obtained for both studies and MHRA approval was obtained for study NCT1725139.

Table 1. Patient acquisition protocol.

Time post injection	Acquisition
	Cine CT (1 PET bed) cine duration 4.5–6 s depending on respiratory rate
0 min	Patient injection
0 min	20 min dynamic PET (1 bed) $6\times 10$ s, $3\times 20$ s, $3\times 60$ s, $5\times 120$ s, $1\times 300$ s
20 min	Patient break (free to move)
30 min	Cine CT (1 PET bed) cine duration same as in previous acquisition
35 min	20 min dynamic PET (1 bed) $4\times 300$ s

To investigate the errors associated with not accounting for air and blood within a voxel, four simulated TACs were created based on real patient parameters. These TACs were constructed to represent best and worst case scenarios for investigating the difference between normal and fibrotic regions. A best case scenario is where the difference in the fraction of parenchyma in the fibrotic and normal appearing tissue is as low as possible. This would give rise to similar PET voxel signals in both cases and would occur when the normal appearing regions have the lowest possible blood and air fractions (greatest possible parenchyma fraction) and the fibrotic regions have the highest air and blood fractions. Conversely, the worst case scenario is where the difference in the fibrotic and normal appearing tissue parenchyma fractions is high. Under these circumstances, the average PET voxel values in each region would appear to have the greatest contrast.

To achieve realistic values for the four TAC simulations, we used results obtained from patient analysis (section 3.4). ${{V}_{\text{a}}}$ and ${{V}_{\text{b}}}$ were calculated for all six patients to determine the highest and lowest air and blood fractions in the fibrotic and normal appearing regions (see table 2). An example patient derived input function (IF) was also determined. Additionally a set of kinetic parameters (K₁, k₂ and k₃) from a single patient were estimated for the whole lung TAC using the recommended FDG kinetic model, the 2 tissue irreversible compartment model (2-ICM) (Sokoloff et al 1977).

Table 2. Patient derived air, blood and parenchyma fractions in fibrotic and normal appearing regions for the best and worst case scenarios.

	Air fraction ${{V}_{\text{a}}}$	Blood fraction ${{V}_{\text{b}}}$	Parenchyma fraction ${{V}_{\text{pa}}}$
Best case 'Fibrotic'	0.43	0.18	0.39
Best case 'Normal'	0.66	0.14	0.20
Worst case 'Fibrotic'	0.37	0.15	0.48
Worst case 'Normal'	0.75	0.19	0.06

Having estimated a patient's whole lung kinetic parameters, the four simulated noise free TACs were created where only the air and blood fractions were varied using:

$$\begin{eqnarray}&&\text{TAC}(t)={{V}_{\text{b}}}\,\text{IF}(t)+\left(1-{{V}_{\text{a}}}-{{V}_{\text{b}}}\right)\text{TrueCurve}(t) \end{eqnarray} \tag{ 12 }$$

where TrueCurve is the simulated parenchyma response using the 2-ICM with the known patient derived kinetic parameters.

For each TAC the kinetic parameters were estimated using the 2-ICM and with Patlak analysis. The percentage difference between the true and estimated parameters with no corrections (NC), blood fraction correction (BFC) only and AFC only were determined in order to calculate the errors caused by not applying the full ABC correction method. The percentage difference between parameters determined in the normal and fibrotic regions were also calculated in each case as may be considered when only relative and not absolute comparison is required.

The standardised uptake value (SUV) is the most common parameter used in the clinical setting (Thie 2004). It is a measure of concentration normalised to the injected activity (A) and the patients weight (W):

$$\begin{eqnarray}&&\text{SUV}=\frac{{{C}_{\text{v}}}}{A/W} \end{eqnarray} \tag{ 13 }$$

Errors in SUV associated with not accounting for air and blood within a voxel were investigated. As in section 3.2, best and worst cases were created. In addition, two patient representations were determined (based on the data from the six patients), each with different blood tracer concentrations. This is because SUV, unlike the kinetic parameters, is affected by the concentration in blood as well as the ${{V}_{\text{b}}}$, introducing a further source of error specific to comparisons between patients and between patients and controls.

For the simulations, the final time point of the TrueCurve derived in section 3.2 was used to represent the true whole lung parenchyma concentration (and hence SUV) in a static image. The best and worst case concentrations for the fibrotic and normal appearing regions were achieved by applying the simple re-arrangement of equation (5) and using the ${{V}_{\text{a}}}$ and ${{V}_{\text{b}}}$ values in table 2. For the concentration in blood, the mean and standard deviation of the aorta concentrations at the final time point for all six patients was determined. The two ${{C}_{\text{b}}}$ values used were the mean ±1 standard deviation.

To determine the maximum and minimum errors that may be expected if the full ABC correction is not applied, the percentage differences between the NC, BFC and AFC and the true parenchyma concentration, between the values determined from the different blood concentrations and between fibrotic and normal appearing tissues were calculated.

PET images were reconstructed using FBP with an average of the Cine-CT acquisition for attenuation correction. The average Cine-CT was used to minimise any CT-PET mismatch due to respiratory motion (Pan et al 2005). As the patients were allowed to move between the two study sections, the two average Cine-CT images were registered using a non-rigid technique with the NiftyReg software (Ourselin et al 2001, Modat et al 2010) and the transformation fields applied to the appropriate PET images. The PET images then underwent AFC with the correction factors derived from the average Cine-CT as described in section 2.1. The concentration of activity in blood at each time point was determined using a 3D ROI drawn manually on the aorta. The aorta was identified on fused PET and CT (using early PET frames). A previously defined model (Sari et al 2014) was then used to fit these data points to produce an input function (IF).

${{V}_{\text{b}}}$ was determined on a voxel basis using the first minute of the initial dynamic acquisition. High voxel TAC noise rendered full compartment modelling impractical with poor fit stability even after image smoothing. ${{V}_{\text{b}}}$ was therefore considered as the scaling factor between the TAC and the IF allowing a model fitting ${{V}_{\text{b}}}$ and time delay (between the TAC and IF) to be used. With image smoothing using a Gaussian filter with a 12 mm FWHM, this method resulted in fast processing with high stability. To calculate $V_{\text{b}}^{\prime}$, the same method was adopted with AFC applied to the images prior to smoothing to ensure correct match between PET and CT.

For each patient, the average cine-CT was used to segment the lungs using a thresholding technique to create a whole lung mask (WLMask). The fractional parenchyma volume maps were reviewed and any voxels with a value of  <5% were removed from the WLMask to ensure voxels containing mostly air (e.g. bullae) or mostly blood (i.e. blood vessels) were not analysed. Regions within this WLMask were then further segmented to separate regions of higher density (HDMask) representative of fibrosis and lower density (LDMask) representative of normal appearing tissue. For each patient, SUV and voxel based Patlak analysis were applied to the regions within each of the HD and LD masks. AFC and ABC were applied to the patients as described in section 2 and the masks were used to compare average regional results before and after corrections. In all cases, kinetic analysis was performed using in-house validated software.

Table 3 displays the results of ignoring the TFE for the worst and best case scenarios. The 'True' values represent the correct parameters after ABC (i.e. with no air or blood present). The results confirm that the only parameters affected by the TFE are K₁ and K_i both in the case of 2-ICM and Patlak analysis. The TFE causes the affected kinetic parameters to be underestimated and this error reduces with BFC and AFC. It can also be seen that the determined ${{V}_{\text{b}}}$ is not affected by the presence of air (table 2).

Table 3. Determined kinetic parameters with no corrections applied.

	K1	K2	K3	Ki2	${{K}_{i\text{P}}}$	${{V}_{\text{b}}}$
True	0.17	0.51	0.0203	0.0065	0.0065	*
Best case
Fibrosis	0.07	0.51	0.0203	0.0025	0.0025	0.18
Normal	0.03	0.51	0.0203	0.0013	0.0013	0.14
Worst case
Fibrosis	0.08	0.51	0.0203	0.0031	0.0031	0.15
Normal	0.01	0.51	0.0203	0.0003	0.0003	0.19

^*See table 2 for the true ${{V}_{\text{b}}}$ for each case. K_i2 = influx rate determined from the 2-ICM. ${{K}_{i\text{P}}}$ = influx rate determined from Patlak analysis. Note these two methods should (and do) produce the same result.

Table 4 lists the percentage differences between the true and calculated K_i after each correction. The TFE causes the affected kinetic parameters to be underestimated by 96–52% without any corrections and this error reduces with both BFC and AFC.

Table 5 is a comparison between the normal and fibrotic regions in each case. As the TACs were all determined from the same parameters, then the true K_i should be identical in all cases. As can be seen, not correcting for the presence of air and blood results in a significant error with K_i appearing between 49% and 92% different.

The average whole lung parenchyma concentration in a static image, as determined from the simulated ABC TAC, was found to be 3.4 kBq ml⁻¹. The mean blood concentration for the six patients was found to be $3.9\pm 1.2$ kBq ml⁻¹ therefore the blood concentrations used to represent two patients were ${{C}_{\text{b}1}}$ = 5.1 kBq ml⁻¹ and ${{C}_{\text{b}2}}=2.8$ kBq ml⁻¹.

Table 6 lists the concentrations and their percentage differences (in brackets) from the true result in each case, with and without corrections. As can be seen, if the correction for blood fraction is applied without correcting for air, then the parameter estimate accuracy is decreased in comparison to no correction at all. As with the kinetic parameters, all concentrations are underestimated with an error range of 81% to 29% when no corrections have been applied. After AFC this error still ranges from  −4% to 41%. AFC without BFC can also be seen to overestimate the concentration when ${{C}_{\text{b}}}$ is greater than the concentration in the parenchyma. Complete correction is only obtained after ABC.

Table 6. Determined average parenchyma static concentrations before and after corrections for each region and case and their percentage difference to the true concentration, 3.4 kBq ml⁻¹.

	NC		BFC		AFC
	${{C}_{\text{b}1}}$	${{C}_{\text{b}2}}$	${{C}_{\text{b}1}}$	${{C}_{\text{b}2}}$	${{C}_{\text{b}1}}$	${{C}_{\text{b}2}}$
Best Case
Fibrosis	2248	1827	1328	1328	3944	3206
	(−34%)	(−46%)	(−61%)	(−61%)	(−16%)	(−6%)
Normal	1397	1070	681	681	4108	3146
	(−59%)	(−69%)	(−80%)	(−80%)	(−21%)	(−8%)
Worst Case
Fibrosis	2401	2050	1634	1634	3811	3255
	(−29%)	(−40%)	(−52%)	(−52%)	(12%)	(−4%)
Normal	1108	664	136	136	4816	2886
	(−67%)	(−81%)	(−96%)	(−96%)	(41%)	(−15%)

Note: ${{C}_{\text{b}1}}$ and ${{C}_{\text{b}2}}$ are the blood concentrations and are equal to 5.1 kBq ml⁻¹ and 2.8 kBq ml⁻¹ respectively.

The percentage difference between the two patient simulations was found to range between 17% and 67% without any corrections. As the only variation is the blood concentration, BFC alone causes the concentrations in both patients to be equal in each case (although underestimated by up to 96% due to the presence of air).

Table 7 is the percentage difference between the normal and fibrotic regions for a relative comparison. In the best case situation, after AFC, the concentrations are within 5% and therefore could be considered as approximately equivalent, however, this is not true for the worst case scenario where the difference is still as high as 26%.

Table 7. Percentage difference between measured concentrations in the two regions (normal/fibrotic) in each case before and after corrections.

	NC		BFC		AFC
	${{C}_{\text{b}1}}$	${{C}_{\text{b}2}}$	${{C}_{\text{b}1}}$	${{C}_{\text{b}2}}$	${{C}_{\text{b}1}}$	${{C}_{\text{b}2}}$
Best Case	−38%	−41%	−49%	−49%	4%	−2%
Worst Case	−54%	−68%	−92%	−92%	26%	−11%

Figure 1 displays the fibrotic (HD) and normal appearing (LD) Patlak derived influx rate constant (${{K}_{i\text{P}}}$) for the NC, AFC and ABC cases for all six patients. It can be seen that in all patients the HD regions have a higher ${{K}_{i\text{P}}}$ than that of the LD region in the NC case (p   =   0.01). After AFC, the regions become comparable with a p value of 0.35 and after ABC, the LD regions are significantly increased with respect to the HD regions (p   =   0.01). Figure 2 is the SUV equivalent to figure 1 where it can be seen that the trend is similar to the ${{K}_{i\text{P}}}$ case with the HD regions appearing significantly higher than the LD regions in the NC case (p   =   0.0001), approximately equivalent after AFC (p   =   0.092) and significantly lower in the ABC case (p   =   0.0001).

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Figure 3 displays an example parametric ${{K}_{i\text{P}}}$ image for each correction method. In the top left (a) the CT image can be seen displaying obvious regions of fibrosis (white arrows) and a region of normal appearing tissue (black arrow). In the top right (b) is the parametric image with no corrections displaying an apparent high influx rate in regions of fibrosis in comparison to those of normal appearance. In the bottom left (c) is the AFC image where the influx rate appears more uniform throughout the lung and after ABC (d) the regions of fibrosis appear to have a lower influx rate than those of normal appearance. Figure 4 displays the SUV results in the same layout as figure 3.

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