A novel mathematical approach to estimate the hepatic, cerebral and renal blood ow from dynamic nuclear imaging studies independently of the used radio tracer

The blood ow (BF) is a critical determinant of organ functionality. Its assessment in the course of routine nuclear medicine examinations, including planar scintigraphy and positron emission tomography (PET), can be relevant for the diagnosis and monitoring of various conditions. The aim of this study was to investigate a new mathematical approach developed to estimate the organ BF from dynamic imaging data and to analyze if this method can be applied independent of the used radio tracer or imaging modality. The new approach uses the early phase of time activity curves extracted from animal and human dynamic scintigraphy and PET scans. Independence of tracer characteristics was evaluated with major oxygen-dependent organs (kidneys, liver, brain) of a mouse model. The approach was also applied on renal scans with two different imaging modalities from a representative cohort of 32 human subjects and compared to reference values.


Results
The mean organ-speci c BF determined in the mouse model revealed no signi cant differences between the administered radiotracers and all calculated values corresponded to normal values (kidneys: 1.0-1.1 ml/min, liver: 1.4-1.6 ml/min, brain: 0.2 ml/min). In the human study cohort, the renal BFs from the two performed imaging modalities showed a good correlation (r = 0.61, p = 0.001) and a small signi cant difference (p = 0.047) among each other and good correlations to the reference value obtained from blood sampling (r = 0.79 and r = 0.52).

Conclusions
A mathematical approach was developed to assess the organ BF solely from dynamic imaging scans without the necessity of additional measurements. Preliminary data suggests that several radiotracers might be feasible to estimate the BF in major oxygen-dependent organs.

Background
An adequate supply of oxygen and nutrients via the blood is of fundamental importance for the maintenance of the structural integrity and function of organs [1]. In healthy individuals, the organ blood ow (BF) is tightly regulated at the local level in response to varying metabolic or functional demands and remains relatively unaffected by changes in the arterial blood pressure [2]. However, many physiologic and pathophysiologic processes can cause a dysregulation of this homeostasis and result in conditions, such as portal or renovascular hypertension, with signi cant systemic manifestations and an increased risk for secondary organ injury [3,4]. Interestingly, the modi cation of the microvascular perfusion may also be an important mechanism of action of pharmacological interventions: sodium-glucose linked transporter 2 (SGLT2) inhibitors were shown to signi cantly reduce the effective renal plasma ow and renal BF, thus a mechanism which might contribute to the nephroprotective effects of this substance class observed in patients with diabetic kidney disease [5]. This suggests that the quanti cation of the BF to the organs might represent a yet underestimated tool to provide a more detailed insight into the pathophysiology of speci c disease processes and to assess in vivo pharmaceutical effects, in particular within interventional studies.
In practice, various methods are available to determine the BF of organs. For the hepatic blood ow (HBF), methods vary from highly invasive such as plethysmography to more convenient methods such as ultrasound or magnetic resonance imaging (MRI) [4,6,7]. The renal blood ow (RBF), which might serve as an important parameter for the functional assessment of native and transplanted kidneys [8,9], can be determined with several minimal to non-invasive nuclear medicine methods, including planar dynamic scintigraphy and, as recently shown, also with combined positron emission tomography and MRI (PET/MRI) scans [8,[10][11][12][13][14][15][16][17][18]. A non-invasive, but rather sophisticated method to assess the cerebral blood ow (CBF) was described using the rst minutes of a dynamic PET scan [19].
Despite of this variety of available techniques, only few studies have systematically characterized the impact of therapeutic regimes or certain pathophysiological conditions on the organ BF, perhaps due to the limited applicability. This reinforces the need for a simple, non-invasive approach to estimate the BF in different organs in the course of routine examinations.
In this study, we describe a novel mathematical method, which was developed to estimate the organ BF by analyzing tracer kinetics in the early phase of time activity curves (TACs) extracted from dynamic imaging scans. Notably, this approach was designed to be applicable to different image modalities and tracers. Thus, it was evaluated with planar scintigraphy and PET scan data using several radiotracers: the

Material And Methods
Description of the new approach to determine the organ blood ow We demonstrate a method, in which the organ BF is obtained by using (a) the total radiotracer activity A(t) = V c(t) in the organ, where c(t) is represented by the corresponding TAC and V denotes the volume of the organ, as well as (b) the measured radiotracer concentration b(t) in a supplying blood vessel, if possible the artery directly attached to the organ or the aorta / left ventricle, with an accordingly applied time shift. The total radiotracer activity in the organ obeys the balance equation where J in (t) refers to the radiotracer current owing into the organ through an artery and J out (t) to the radiotracer current leaving the organ through a vein. We emphasize that the functions in equ. (1) should be regarded as the time average over a certain time bin length (in our case: 3 s, 5 s and 10 s, respectively), being higher than the typical pulsation period of the blood ow. As a consequence, the incoming current can be written as Assuming that the entrance of the radiotracer into the organ starts at t = 0 (i.e. A(t) = J in (t) = 0 for t < 0), integration of equ. (1) together with equ. (2) and the de nition of A(t) lead to the relation where A in and A out can be visualized as the total number of radiotracer particles having entered or left the organ until time t; Vc(t) can be interpreted as the number of radiotracer particles present in the organ at time t. The rst few seconds are characterized by a steep rise of the incoming current J in (t) followed by a subsequent decline, whereas the output current J out (t) starts only with some time delay at t = t crit .
Concentrating on this initial phase (0 < t < t crit ), equ. (3) with t = t crit allows to express BF as being our desired relation. The only remaining task is the determination of t crit . Because of equ. (1), the position t = T of the maximum of A(t) is characterized by J in (T) = J out (T) and we infer 0 < t crit < T. Although we have no direct information on the output current itself, the turning point of A(t) before its maximum serves as an excellent estimate for the value of t crit .
In other words, it is assumed that the total number of radiotracer particles entering an organ via the BF must be equal to the total number of particles in the organ Vc(t) until t crit is reached. With a precise determination of the volume V and a su cient time resolution in the measurement of the TACs b(t) and c(t), the BF can be determined by simply nding the turning point of c(t) and inserting all values into equ. (4).
In Fig. 1, typical total TACs (i.e. Vc(t)) of the kidneys with three radiotracers are shown (gray curves), as well as the according A in (t) with the BF calculated from equ. (4). t crit was taken as the turning point of the rising total TAC, which is indicated with an arrow in the small images. During all scans, mice were kept under anesthesia with a mixture of iso urane (2%) and O 2 (2.5 l/min) continuously administered via a mask and kept at a body temperature of 37 °C. The heart rate and breathing frequency were monitored. Anaesthetized animals were positioned in the animal Inveon PET/SPECT/CT scanner (Siemens Medical Solutions, Knoxville, TN) and the according radiotracer was injected via the lateral tail vein. All dynamic scans started with the injection and lasted for 45 min. For the PET scans, an additional CT image was obtained and used for attenuation correction and co-registration. The latter were reconstructed by Fourier re-binning followed by 2-dimensional ltered back projection with a ramp lter. CT raw data were reconstructed with a Feldkamp algorithm using a Shepp-Logan lter followed by standard mouse beam-hardening correction and noise reduction (matrix size: 1024×1024; effective pixel size: 97.56 µm). The standard data correction protocol (normalization, attenuation, decay correction and injection decay correction) was applied to the PET data. All dynamic data sets were rebinned into a dynamic sequence of 8 x 3 s, 9 x 5 s, 13 x 10 s and 11 x 222 s. Volumes of interests (VOIs) of the liver, the brain, the kidneys and the aorta were delineated by hand on the CT images and transferred to the individual time frame PET images using PMOD software (version 3.8, Pmod Ltd, Zurich, Switzerland). With the same software, regions of interest (ROIs) were delineated in the planar scintigraphy images around the kidneys, the perirenal background and the heart.
The normal values for renal blood ow (RBF), hepatic blood ow (HBF) and cerebral blood ow (CBF) were compared to normal values found in literature [20][21][22][23] accounting for RBF-norm with 0.8 ml/min, HBF-norm with 1.8 ml/min and CBF-norm around 0.26 ml/min. They are denoted as RBF-norm, HBF-norm and CBF-norm The planar scintigraphy scans were performed according to the EANM guidelines [23] after the injection of 80 MBq [ 99m Tc]MAG3 and lasted for 20 minutes. The images were acquired with a large eld of view gamma camera (low-energy, high-resolution collimator, 64 × 64 matrix, frame rate 10 s/frame, energy window 140 keV with 20% width). In case of 24 subjects, a standard was additionally measured with the gamma camera for 10 seconds. Using Hermes software, ROIs were drawn around the kidneys, the perirenal background (see g. 1) as well as in the left ventricle.
From all subjects, one blood sample was drawn before the rst scan and used to determine the hematocrit value (Hct); a second blood sample was drawn 41 ± 2 minutes after each scintigraphy, which was measured, together with a standard, in a gamma counter.
For the human data set, two reference values were available. The blood-sample derived method RBFblood, originally developed by Tauxe et al. [10], is based on the renal clearance of a radiotracer by measuring drawn blood samples after its venous injection together with a standard in a gamma counter.
In this study, the effective renal plasma ow (eRPF) was determined from the [ 99m Tc]MAG3 clearance according to a re ned method by Russell et al. [8,12], from which RBF-blood was then calculated by dividing eRPF by (1 -Hct).
The image derived, graphical methods are based on the idea that the renal [ 99m Tc]MAG3 uptake within the rst few minutes can be converted to the eRPF provided, also a standard is measured. The graphical eRPF was calculated for each kidney from the renal uptake between minute 2 and 3 after injection according to the formula of Arroyo et al. [12,13]. The resulting eRPF was divided by (1 -Hct) and summed for both kidneys in order to obtain RBF-graph. Note that in case of 8 subjects, no standard was measured with the gamma camera, omitting the determination of RBF-graph. All human RBF values were normalized to a body surface are of 1.73 m².
A summary of all available data sets is given in table 1. Time activity curves and organ volumes The radiotracer concentrations over time, i.e. the time activity curves (TACs) of all animal and human scans were exported in units of kilobecquerel per milliliter (kBq/ml). TACs from planar scintigraphy images were corrected for the perirenal background. Via linear interpolation between the measured concentration points, the time binning of the TACs was reduced to one second. The organ TACs were smoothed with a Savitzky-Golay lter, the TACs from the blood pool (heart or aorta) were tted with a three-exponential curve starting from their according peak.
For human PET/MRI data, kidney volumes were measured using the MRI sequences. Animal organ volumes from the PET data were measured in the co-registered CT images.
In order to assess the renal volumes from the measured kidney areas in the images from the human and animal renal planar scintigraphy, an isotropic scaling was performed, i.e. a transformation of an area to a

Results
All obtained BF mean values and ranges from minimum to maximum are summarized in Table 2. A summary of human subject demographics (female: n = 10 female, male: n = 22) is shown in Table 3. Table 2 Mean values (ranges in parenthesis) of all blood ow (BF) values: renal blood ow (RBF), hepatic blood ow (HBF) and cerebral blood ow (CBF). The epithet "new" indicated that the according BF was calculated with the new approach. The normal values, indicated with the epithet "norm", were taken from refs. [20][21][22][23]. volume by multiplying the area with its square root and an appropriate factor. For humans, the kidney was considered as an ellipsoid with an averaged thickness between 4 and 5 cm and a height of 11 to 13 cm. Therefore, the appropriate factor was set to 0.4. For mice, the same renal dimension ratios were assumed, resulting in the same factor. Statistical analysis Mean, minimum and maximum values were calculated for each BF value. For the comparison of the blood ow of a certain organ between different radiotracers, an unpaired Student's t-test was performed. For comparisons between new and reference values of the same organ, the Pearson correlation coe cient r and the Student's paired t-test was applied. A p value < 0.05 was considered as statistically signi cant. All according calculations were performed with LibreO ce version 5.3.7.2. In case of human kidney data, reference values RBF-blood and RBF-graph were available. Both RBF-new-FDG and RBF-new-MAG3 had good correlations to both reference methods (see Table 4 and Fig. 3). RBFnew-FDG showed signi cant differences to RBF-blood, but not to RBF-graph, which was opposite for RBFnew-MAG3. Note that both reference methods, RBF-blood and RBF-graph, also showed a signi cant difference of 12% and a correlation of r = 0.57 between each other. Table 4 Comparison of renal blood ow (RBF) obtained with the new method from two radiotracers ([ 18 F]FDG and [ 99m Tc]MAG3) with two reference methods obtained from blood sample measurements (RBF-blood) and with an image derived graphical method (RBF-graph): correlation coe cient r (with according p value) and difference in percent (with according p value).

Discussion
A comprehensive mathematical treatise about the estimation of the organ blood ow independent of the radiotracer or the organ was already done by Bassingthwaighte and Holloway [23], based on the measurement of both arterial in ow and venous out ow, whereas the latter is hampered by an insu cient determination of the venous radiotracer concentration in functional imaging data. Using solely the in ow and the measured tissue concentration, a solution was presented for the CBF by Raichele et al. [19], which starts with similar consideration as our approach, but is more sophisticated and dependent on a not analytically solvable integration. While many methods for the blood ow determination of organs are invasive or impracticable, several non-invasive methods based on imaging modalities were developed. However, most of them are speci c to an organ of interest or a radiotracer, e.g. using certain MRI sequences for the HBF [7], dynamic contrast enhanced CT measurements for the CBF [26] or image derived, graphical methods from scintigraphy images for the RBF [14][15][16][17].
In this study, we evaluated a new approach allowing to determine theoretically any organ blood ow from dynamic imaging scans independently of the used radiotracer, organ or modality. The presented approach uses the early phase of dynamic imaging scans and is non-invasive, provided the dynamic scan is reconstructed with a su cient high time binning and the blood pool concentration can be measured in the obtained images, respectively. The latter is possible if the eld of view of the scanner covers the upper part of the aorta, which was shown to have minor partial volume and motion effects [18,27]. In case of planar scintigraphy, performed e.g. for a renal scan, the blood concentration usually can be measured from the heart, since the according radiotracers show no uptake in the myocardium.
The reliability of the new approach was rst evaluated rst on PET/CT and scintigraphy scans of animals in three major oxygen-dependent organs. The results support our assumption that the early phase of a dynamic scan is solely dependent on the organ blood ow and not on the radiotracer pharmacokinetics, since the hereby calculated renal blood ow RBF was almost equal between three different radiotracers scans used higher time frames of 10 sec, leading to a probably insu cient low number of data points in the early phase used for RBF. Second, planar imaging methods need a background subtraction leading to errors [30]. Third, no accompanying imaging method for anatomical information was performed, making the delineation of the heart precarious. Finally, the new approach relies on a determination of organ volumes, which needs to be estimated from planar areas in case of renal scintigraphy, which we did with an isotropic scaling. The isotropic scaling obviously leads to negligible differences in case of the planar animal scintigraphy data using genetically almost identical mice, but most certainly not for the human data. However, RBF-new-MAG3 and the reference method RBF-graph showed very similar differences and correlations to the probably most reliable reference method RBF-blood, thus RBF-new-MAG3 might be considered as a comparable convenient method to determine the RBF from planar renal scintigraphy, with the additional advantage of omitting a standard measurement.

Limitations
The blood TACs were image derived and not corrected for partial volume or motion effects. Although these effects have probably a minor impact when derived from the aorta, omitting an appropriate correction might also explain deviations to reference or normal values. Also, no reference methods were available for the animal data. However, a further separate according animal study would have been beyond of the scope of this proof-of-concept paper.
Furthermore, this study not nearly covers all available imaging methods, organs or radiotracers. The new approach therefore needs further investigation in other settings, which will be done in our clinic in the course of future dynamic studies.
With the presented new approach, in principle a scan lasting for a few minutes might be su cient to determine an organ blood ow. While the clinical practicability of such a short scan is questionable, the new approach might be used in the course of dynamic scans allowing an additional assessment of those organ blood ows which are in the eld of view.

Conclusions
A new mathematical approach was presented using the early phase of dynamical functional imaging.
Provided the blood and tissue concentrations were obtained with su cient time resolution, it was shown that this approach allows to asses at least the renal, hepatic and cerebral blood ow independently of the administered tracer or imaging modality, and being in good agreement with normal and reference values. Availability of data and material: The datasets generated and/or analysed during the current study are not publicly available due to the size of dynamic scan data, but are available from the corresponding author on reasonable request.

Abbreviations
Competing interests: The authors declare that they have no competing interests.   Top: human dynamic positron emission tomography and magnetic resonance imaging (PET/MRI) scan. Time activity curves (TACs) were obtained from volumes of interest (VOIs), which were delineated from the left and right kidney (green shapes), as well as from the aorta abdominalis (blue shape). Bottom: human dynamic renal scintigraphy. TACs were obtained from regions of interest (ROIs), drawn around the left and the right kidney as well as the left ventricle.