Dynamic imaging of a bolus of contrast agent passing through the brain's vascular system has been used to study blood flow in a variety of approaches (

Axel, 1980;

Herfkens et al., 1982;

Miles, 1991;

Ostergaard et al., 1996a;

Ostergaard et al., 1996b;

Rempp et al., 1994). In dynamic susceptibility contrast-enhanced MRI (DSC-MRI) of Gd-DTPA bolus injection, relative cerebral blood flow (rCBF) can be measured using the strategy called “Early Time Points” method (ET) (

Buxton, 2002;

Klocke et al., 2001;

Kwong et al., 2011;

Lee et al., 2004). The concept of ET, an idea close to the microsphere perfusion model, proposes to calculate rCBF by measuring the amount of Gd-DTPA arriving within a time window before any contrast agent has a chance to leave the tissue. That time window is in the order of the tissue mean transit time τ, a quantity known to be a few seconds for gray matter in humans (

Calamante et al., 2000;

Ostergaard et al., 1996a). Given that basic assumption of

*before the contrast agent having a chance to clear from the tissue*, the quantity of Gd-DTPA present in the tissue will be proportional to local CBF. Since the basic assumption will be cited repeatedly, it will be named the ET's basic assumption for the rest of the manuscript.

Some of the differences of ET in comparison with the deconvolution method (

Ostergaard et al., 1996b) for perfusion analysis had been presented previously (

Kwong et al., 2011) so we will just give a brief summary here. Correction of bolus transit delay is a built-in component of ET. Blood flow results can be obtained from the early time points before τ so ET will not be affected by noise from the later part of the bolus time course which includes recirculation. Whatever noise that may be introduced by the deconvolution process itself is not a question for ET. The evaluation of rCBF by ET for any single patient is independent of the arterial input function (AIF) if a global AIF is assumed, an assumption broadly useful in clinical perfusion imaging. For cross subject comparison of rCBF, a reference based method (

Kimura and Kusahara, 2009) which sets a flow reference region of interest (ROI) on some referential tissue (e.g. white matter ROI) is one possible option for ET. When global AIF is not assumed, the local AIF question is technically challenging and beyond the scope of this study. The answer to cross subject comparison in ET and to some of the questions of local AIF is provided in a companion manuscript (

Kwong and Chesler, 2011).

While ET is useful for measuring rCBF, it should be mentioned that absolute CBF can in principle be obtained by ET just like the deconvolution method of Ostergaard et al. if a proportionality constant, required for absolute CBF estimation, can be obtained independently and separately from a different technique like PET. Absolute blood flow might also be obtainable by ET based on the well studied microsphere model if arterial blood samples can be obtained from the appropriate arteries.

Having provided a summary of some of the features of ET, we will present now the focus of this manuscript: the identification of useful reference time points to allow ET to correct the bolus transit delay for the purpose of carrying out rCBF analysis.

The correction of the transit delay of the contrast agent bolus for each voxel is required for ET to make relative cerebral blood flow (rCBF) maps. Intuitively, the time of arrival (TOA) of the contrast agent at the voxel can be used to correct the transit delay of the bolus. But TOA brings its own concerns. While the search for accurate TOA is an ongoing research topic studied elsewhere (

Kwong et al., 2011), TOA is recognized as being located at a low contrast-to-noise ratio (CNR) region of the bolus signal due to the small amount of contrast agent entering the voxel at TOA.

Fortunately, one does not need to know the time location of TOA of each voxel for ET to work. In this manuscript, we propose that relative TOA (rTOA) can also be used as a reference time point for ET. The requirement for a time location to be rTOA is that it retains the *same* time distance from TOA for every voxel. If rTOA can be found, it is equivalent to TOA as a reference time point for rCBF calculation by ET. Due to the low CNR location of TOA, there is a strong motivation for selecting rTOA at the high CNR region of the rising bolus concentration time curve. Our aim is to identify and investigate some of those candidate rTOA's. For convenience, the bolus concentration time course is named *C*(*t*). The signal intensity of *C*(*t*) is described in terms of ΔR2*, an MR susceptibility signal related unit which represents the contrast agent concentration.

A preferred candidate for rTOA would likely have the following properties: reasonably far away from the low CNR neighborhood of TOA, easy to evaluate, and retains the same time distance from TOA for every voxel without at the same time requiring the knowledge of the value of TOA. Those properties will be made clearer as our presentation proceeds. It turns out that there are many data points from the rising *C*(*t*) that are good candidates for rTOA.

Meeting ET's basic assumption is a requirement for rTOA, meaning that rTOA < τ. Since we do not usually have the knowledge of whether a particular early time point picked for rTOA meets ET's basic assumption in real world experimental data, we made the initial assumption in this manuscript that ET's basic assumption had been met by the rTOA candidates. Then we investigated the implication for those rTOA candidates when ET's basic assumption was violated.

Candidate rTOA's --TMD2, TMD1.5, TMD1, TMD0.5

Based on the requirements of reasonable CNR locations, simplicity in calculation and independence of knowledge of TOA, we will first propose a list of rTOA candidates with the justification to be given in later paragraphs. Relative to the unknown (and uncalculated) TOA, the first candidate is the time to reach the maximum derivative (TMD1) of the rising bolus time course. TMD1 is of course just the time position of the classical maximum gradient approach to evaluate rCBF (

Kimura and Kusahara, 2009;

Koenig et al., 1998;

Miles and Griffiths, 2003). The second candidate is the time to reach the maximum second derivative (TMD2). It turns out there are other choices as well. Intuitively, if the time to reach the maximum fractional derivative (TMD-FR) exists between TMD1 and TMD2, we can consider a TMD-FR value between 1 and 2 as a third rTOA candidate. For convenience, the fractional derivative selected between 1 and 2 is an arbitrary 1.5 and the time to reach its maximum value is TMD1.5. There is a motivation for considering fractional derivative. TMD1 has good CNR because it is farther away from TOA but TMD1 runs a higher risk of being longer than τ for some high flow values, thereby violating the ET's basic assumption. TMD2 has minimal risk of being longer than τ but it is also at a relatively low CNR region of

*C*(

*t*). Noise amplification can also occur with the second derivative due to the propagation of error if derivatives are calculated by finite difference, namely, taking the difference of neighborhood data points (

Appendix A). Since TMD-1.5 is chosen to be a time point between TMD1 and TMD2, it is expected be able to offer a good compromise. There is one more choice of rTOA to be considered. If good CNR is of paramount importance, it is also reasonable to examine lower order fractional derivative such as ½ with the time to reach the maximum derivative of the order ½ (TMD0.5) being the desirable rTOA. In choosing TMD0.5, the size of error generated by violation of ET's basic assumption, if there is any, needs to be considered to be tolerable for a particular research goal.

Beyond TMD0.5, the time to reach the maximum peak of

*C*(

*t*) will not meet ET's basic assumption as

*C*(

*t*) starts making a turn at its maximum peak. Had ET's basic assumption or the microsphere condition been always met,

*C*(

*t*) would keep on growing and then leveling off but would never turn back down to baseline. When

*C*(

*t*) of experimental data starts turning away from its peak signal towards baseline, it is an indication that the time of the peak of

*C*(

*t*) has gone beyond τ. That is why the approach used by earlier reports (

Klocke et al., 2001;

Lee et al., 2004) of first-pass magnetic resonance perfusion imaging which estimates rCBF using data including the peak of

*C*(

*t*) will not be explored here.

Attention needs to be paid to the properties of fractional derivatives (

Kilbas et al., 2006;

Oldham and Spanier, 1974;

Samko et al., 1993). It is well known that the local property of the fractional derivative at a time point

*t* involves boundary conditions and is not well defined like the local property of the integer derivative. However, TMD-FR can still be a candidate for rTOA as long as a local maximum of the fractional derivative can be found at TMD-FR. The important test is whether TMD-FR can retain the same time distance from TOA for every voxel. We will verify that rTOA property of TMD-FR with simulation data.

For convenience, we also set the following abbreviations: TMDX is the time to reach a maximum derivative of arbitrary order. MD1 is short for the maximum first derivative, MD2 for the maximum second derivative, and MD-FR for the maximum fractional derivative. MD1.5 and MD0.5 are specific for the maximum values of fractional derivative of 1.5 and of 0.5, respectively. MDX is the maximum value of a derivative of arbitrary order. D1 is short for the first derivative value and D1(t) is short for the first derivative time course. D2, D2(t), D1.5, D1.5(t), D0, D0.5(t) are similarly defined for the derivative orders of 2, 1.5, and 0.5. For derivative of an arbitrary order X, we also made the abbreviation DX for the derivative value and DX(t) for the time course. D-FR is short for fractional derivative.

shows schematically that for TMDX, TMD0.5 > TMD1 > TMD1.5 > TMD2 for a simulated bolus concentration time course.

Justification of the candidate rTOA's

The reason why TMD1, TMD2 and TMD-FR are good candidates for rTOA is based on two claims. First claim: the relative time between TOA and each of these rTOA candidates is independent of bolus transit delay. Second claim: the relative time between TOA and each of the rTOA candidates is independent of flow of each voxel. For example, the relative time between TOA and TMD2 is the same for all voxels, independent of whether one voxel has a flow level of 20ml/100mg/min and another voxel has a flow level of 50ml/100mg/min. Of course, the second claim is true only if ET's assumption is met (e.g. TMD2 < τ).

The first claim is obvious as the relative time (from TOA) to reach a maximum parameter (e.g. MDX) of

*C*(

*t*) does not depend on the value of TOA. No knowledge of TOA is required. As a corollary, any time course parameters, e.g.

*C*(

*t*), D2(t), etc. obtained at rTOA are independent of the bolus transit delay even though they are sensitive to the accuracy of rTOA estimation. With regard to the second claim that the relative time distance between TOA and each of the rTOA candidates is independent of flow of each voxel, a lengthier explanation, given below in

Appendix B, is offered.

Appendix B shows how an understanding of the way rCBF is calculated by ET leads to an explanation that TMDX relative to TOA is independent of flow.

Based on those two claims, TMD2, TMD1.5, TMD1, TMD0.5 are considered appropriate rTOA candidates which can be used as proper reference points by ET for rCBF calculation.

Out of all the rTOA candidates, TMD1 is of additional interest because it matches the well known maximum gradient-based method used in computed Tomography (CT) and MR perfusion analysis. Since TMD1 is located at the superior contrast to noise location of the tissue time course one might question the need to study TDM2 and TMD-FR at all. The basic concern is about the possibility of TMD1 > τ and thereby not meeting ET's basic assumption. It is important to recognize that the time taken to reach the maximum gradient

*is dependent on the shape of the external driving function AIF* which depends in turn on bolus volume, the rate of injection and the patient's cardiac output (

Miles and Griffiths, 2003). Contrast agent washout violating ET's basic assumption of no contrast agent departure had been reported for the maximum gradient condition, resulting in drop of estimated perfusion values (

Miles and Griffiths, 2003). On the other hand, TMD2 is much more likely to lie within the time window of τ because it is so close to TOA. But it may not be far enough from the influence of the low CNR of the TOA neighborhood. That is why we also explore the option of TMD-1.5 because the time to reach the maximum fractional derivative between the value of 1 and 2 should define an rTOA time point between TMD1 and TMD2.

In a flow ratio calculation, it should be mentioned that whether TMDX exceeds τ is determined by the τ of the higher flow level. For example, in a gray-white matter flow ratio, τ refers to τ of gray matter.

The presentation so far does not imply that there are always voxels where TMD1 is longer than τ. But until it can be thoroughly demonstrated that TMD1 is always shorter than τ under all kinds of imaging settings and a wide range of clinical conditions, it is useful to consider TMD2 and TMD1.5 as available backups.

Our manuscript is built on the premise that TMD2, TMD1.5, TMD1 and TMD0.5 are all currently useful and valid rTOA's to be considered for ET. While in principle we have almost an unlimited number of ways (

Kwong et al., 2011) of evaluating rCBF in ET (e.g. one can choose

*C*(

*t*) at TMD1 or

at TMD1.5), we will focus in this manuscript only on the study of the maximum values, namely the values of MD2, MD1.5, MD1 and MD0.5 for rCBF evaluation. For noiseless data, the same calculated rCBF is expected to be obtainable by ET no matter which type of approach we choose to use. For noisy real life experimental data, there are tradeoffs between using signal intensity of

*C*(

*t*) and the various values of MDX. There are practical advantages of using those maximum signals MD2, MD1.5, MD1 and MD0.5 at the respective rTOA's. The reason is that in noisy data, the inaccuracy in the estimation of the TMDX can derail the estimation of rCBF by signal intensity. The use of MDX at TMDX is considered more robust because the value of MDX remains relatively stable ranging from TMDX-1 to TMDX+1 while

*C*(

*t*) could vary significantly in the same time range. Error in estimation of TMDX could potentially compromise the accuracy of calculated rCBF more if

*C*(

*t*) instead of MDX is used for ET.

Work flow of this manuscript

Given that the use of MD1 is a well established perfusion protocol, The result of MD1 is a good reference for the results of MD2 and MD-FR. The work flow of this manuscript consists of first studying a set of noiseless simulation perfusion data to verify the theoretical properties of TMD2, MD2, TMD1.5, MD1.5, TMD1, MD1, TMD0.5 and MD0.5. The simulated data is a subset of the simulated perfusion data created using the algorithm of Wu et al. (

Wu et al., 2003a). We then investigate experimental monkey MRI data to see how well they matched the expected ideal ET behavior. In collecting experimental data, we imaged an anesthetized macaque in a clinical 3 Tesla MR scanner. Short TR image acquisition, which is the best way to collect many data points between TOA and τ to test and verify the ET model, is not yet an approved technology for clinical imaging,. That is the reason why we acquired monkey instead of human perfusion imaging data with short TR MRI. Proposals to manage the shortcomings of reduced brain coverage of short TR acquisition have been presented previously (

Kwong et al., 2011) and won't be investigated here.

A number of factors needs to be taken into account in comparing the perfusion results of simulation data and real world experimental data. Calculated perfusion results of simulation data using various rTOA's would be expected to match the pre-given perfusion values. The inevitable noise of experimental data (

Kwong et al., 2011) would introduce uncertainties into the results of rTOA's and MDX, degrading the accuracy of the estimated perfusion results. In addition, the assumption of

*a single transit time* in the residue function of the simulation data is bound to be just an approximation for experimental data because partial volume of gray, white matter and vascular tissue in a single voxel can give rise to a mixed compartment of tissue transit times. Despite that approximation, previous work showed that rCBF calculated by ET for experimental data yielded reasonable results consistent with what were reported in the literature (

Leenders et al., 1990;

Ostergaard et al., 1996a). As long as consistent perfusion results can be obtained, the new rTOA and perfusion information retrieved from simulation data by ET can serve as an approximate and useful model for the study of real world experimental data.

In this manuscript, we will investigate whether flow can be estimated by maximum values of higher order fractional derivatives instead of using the maximum derivative which runs a larger risk of violating ET's assumption. We will also study whether experimental rCBF calculated using MDX is consistent with what was previously reported in the literature. While we verify such consistency, any discrepancies observed between the ideal ET model and experimental data will serve to suggest ways to improve the modeling of experimental data and improve the calculated rCBF in future research.