Patent Publication Number: US-10788558-B2

Title: Quantification of the relative amount of water in the tissue microcapillary network

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is a continuation of U.S. application Ser. No. 14/410,549, filed on Dec. 22, 2014, which is a national phase under 35 U.S.C. § 371 of PCT International Application No. PCT/SE2013/050755 which has an International filing date of Jun. 24, 2013, which claims priority to Swedish patent application number SE 1250736-4, filed on Jun. 29, 2012 and benefit to U.S. patent application 61/665,998, filed on Jun. 29, 2012; the entire contents of each of which are hereby incorporated by reference. 
    
    
     BACKGROUND OF THE INVENTION 
     The present invention relates to a method for analyzing diffusion-weighted magnetic resonance (MR) images recorded with a variable amount of velocity compensation to quantify the amount and velocity of blood flowing in the tissue microvasculature. 
     Both molecular diffusion and perfusion, i.e., blood flowing in the orientationally disordered capillary network, lead to attenuation of the signal intensity in diffusion-weighted MR imaging, an effect known as “intravoxel incoherent motion” (IVIM). Pioneered by the work of Le Bihan (1), the pseudo-diffusion coefficient D* of the flowing water, the perfusion fraction f, and the diffusion coefficient D of the non-flowing extra- and intracellular water are estimated by biexponential analysis of diffusion-weighted MR images recorded as a function of the diffusion-weighting variable b. Such an analysis has recently been applied to breast cancer (2) and liver cirrhosis (3), indicating that the perfusion parameters (D* and f) could be useful for diagnosing pathological conditions in which the blood flow in the microvasculature is altered. The values of f range from 4% in brain to 25% in the pancreas (4). The analysis is hampered by the well-known problem of extracting exponential components with similar decay constants from noisy multi-exponential signal attenuation data (5). In order to obtain sufficient difference between D, which is independent of the diffusion time, and D*, which is approximately proportional to the diffusion time, diffusion-weighting is often performed at long echo times, e.g. 100 ms, thus leading to additional signal reduction and influence of noise due to nuclear spin-spin relaxation. 
     The signal attenuation originating from perfusion can partially be removed by employing diffusion-weighting gradient modulation schemes in which the phase shifts of spins flowing at a constant velocity are refocused (6-8). Images obtained by taking the difference of flow-compensated and non-compensated images yield information on capillary density (6, 7). Unfortunately, the image signal-to-noise ratio (SNR) is usually too low to accurately quantify pathologically induced changes of intravascular fractions using analysis methods based on difference images. 
     The inordinate sensitivity to noise of currently existing protocols for signal acquisition and analysis (biexponential fit to signal vs. b data or difference images of flow-compensated and non-compensated data) have so far prevented a widespread clinical use of the potentially informative perfusion parameters. Based on the considerations above, it would be desirable to have the means for obtaining these parameters with greater accuracy and less sensitivity to noise than possible with currently existing methods. 
     U.S. Pat. No. 7,336,072 a method for visualizing macroscopic flow in MRI is presented. The method provides analysis of data obtained by the flow compensated and non-compensated sequence. The information about macroscopic flow (velocity) is contained in the phase of the signal and it is extracted by the method disclosed in U.S. Pat. No. 7,336,072. Based on the signal phase information, the velocity filed is constructed to visualize macroscopic flow. Different visualization methods are presented in U.S. Pat. No. 7,336,072, e.g. using color coded maps or vector fields. For comprehensive flow image data reading, the velocity field is superimposed on an anatomical image. To identify regions with flow and stationary tissue, the magnitudes of the signals acquired by flow compensated and non-compensated sequences are subtracted. 
     SUMMARY OF THE INVENTION 
     The invention relates to a data analysis method and corresponding image acquisition protocol overcoming the previously mentioned problems. The present invention enables the effects of diffusion and perfusion on the pseudo-diffusion coefficient to be analyzed separately based on the data from experiments with variable degree of flow compensation. Varying the degree of flow compensation allows for a more robust quantification of dispersed flow. 
     According to a one aspect of the present invention there is provided a method for magnetic resonance (MR) imaging comprising: 
     acquiring at least two MR images with different motion-weighting originating from a RF and gradient pulse sequence causing signal attenuation from diffusion but not flow (flow-compensated data); 
     acquiring at least two MR images with different motion-weighting originating from a RF and gradient pulse sequence causing signal attenuation from diffusion and flow (non-compensated data); 
     performing a model fit to the flow-compensated and non-compensated data in which at least one of the adjustable parameters are constrained to be the same for both sets of data; and obtaining quantitative information on microscopic flow by extracting at least one parameter of the intravoxel incoherent motion (IVIM) effect from the model fit. 
     It should be mentioned that the extraction of the at least one parameter of course may be related to extracting several parameters of the intravoxel incoherent motion (IVIM) effect. Moreover, according to one specific embodiment the extraction is related to extracting the information about the fraction of the microcapillary water and the velocity dispersion or the pseudo-diffusion value, which are attributed to the intravoxel incoherent motion (IVIM), from the model fitted data set. 
     The method according to the present invention, when being compared to that disclosed in U.S. Pat. No. 7,336,072, is also based on the analysis of the data acquired by flow compensated and non-compensated sequences. However, the analysis method aims to quantify the intravoxel incoherent motion effect (IVIM effect). The aim of the method according to the present invention is to quantify the relative amount water in tissue micro-capillary network with higher precision and accuracy compared to the established approaches, i.e. bi-exponential and segmented fit of the attenuation data. The present method also allows for a quantification of the velocity dispersion within the micro-capillary network. In contrast, the method disclosed in U.S. Pat. No. 7,336,072 aims at quantification of velocity in macroscopic flow. Furthermore, the analysis according to the present invention is based on the signal magnitude data, while the macroscopic velocity filed extracted by the method in U.S. Pat. No. 7,336,072 is based on the signal phase data. The method according to the present invention is based on a constrained fit of the flow compensated and non-compensated data. No such data fitting method is presented in U.S. Pat. No. 7,336,072. As notable from above, the method according to the present invention, as disclosed in the present claim  1 , is very different from the method disclosed in U.S. Pat. No. 7,336,072. 
     As said above, according to the present invention, the effects of diffusion and perfusion on the pseudo-diffusion coefficient can be analyzed separately based on the data from experiments with variable degree of flow compensation. As a special case, data from flow-compensated and non-compensated experiments is considered. The inventors suggest a novel diffusion-perfusion experiment with variable degree of flow compensation and a novel joint analysis of the flow compensated and non-compensated data to quantify flow with improved accuracy and precision compared to the conventional methods. 
     Information about the probability distribution P(D,ν d   2 ) can be obtained by regressing the equation (18) below onto the signal attenuation data E(b,α) at variable b and α. 
     Pulse sequences that allow for independent adjustment of diffusion weighting, b, and velocity dispersion weighting, α, can be used to quantify the velocity dispersion by disentangling the diffusion and the velocity dispersion contributions to the total signal attenuation, characterized by the pseudo-diffusion coefficient, D*. In the diffusion-perfusion correlation experiment (D-P), where the experimental parameters b and α can be adjusted independently, the measurement of the correlation between the diffusion coefficient and the velocity dispersion allows associating the velocity dispersion components and the corresponding diffusion components when one or more diffusion or velocity dispersion components are present in the system. 
     Joint bi-exponential analysis of signal attenuation data acquired with both flow-compensated and non-compensated gradient modulation schemes yields estimates of the root-mean-square flow velocity ν and the fractional population f of water flowing in the microcapillary network, as well as the diffusion coefficient D of the “stationary” water in the surrounding tissue. The flow-compensated and non-compensated schemes are optimized for yielding maximum insensitivity and sensitivity to flow, respectively, for given constraints on hardware performance and nuclear relaxation times. 
     Further specific embodiments of the present invention are presented below and in the claims. The present invention has many possible application areas. For instance, the obtained quantitative information on microscopic flow may be used for diagnosing disease. Possible uses are as a method for diagnosing tumor vascular properties, such as blood volume fraction and/or microvascular flow velocity, e.g. by use of the parameters CBV (cerebral blood volume) and/or CBF (cerebral blood flow). Examples of indications to diagnose are breast cancer or liver cirrhosis. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  displays a schematic representation of the probability distribution P(D, ν d   2 ) for the example with two signal contributions described by Eq. (19) (see below). 
         FIG. 2  displays schematics of pulse sequences for motion weighted nuclear magnetic resonance (NMR) or magnetic resonance imaging (MRI). 
         FIG. 3  also displays schematics of pulse sequences for motion weighted nuclear magnetic resonance (NMR) or magnetic resonance imaging (MRI). 
         FIG. 4  shows commonly known pulse sequences for motion weighted nuclear magnetic resonance (NMR) or magnetic resonance imaging (MRI). 
         FIG. 5  shows simulated data for flow compensated and non-compensated pulse sequences. 
         FIG. 6  shows the accuracy and the precision of the perfusion fraction f as quantified with conventional protocols (“standard” and “segmented”), as well as according to the present invention. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     Perfusion of blood in tissue is a significant physiological parameter. Vascular blood volume and flow velocity are well established as indicative parameters in tumor diagnosis and therapy. Despite a lot of effort, non-invasive methods for probing perfusion are very slowly paving their ground against clinically valuable but invasive methods like MRI methods based on paramagnetic contrast agents or methods which, for example, make use of radioactive tracers (9). The non-invasive diffusion based methods are very promising but have so far not proven adequate in routine clinical practice due to their inordinate sensitivity to noise. It has been shown that the effects of randomly oriented flow contribute to the signal attenuation in ordinary diffusion experiments (6-8). Consequently, the effects of random flow can be observed as superimposed on the molecular diffusion effects, resulting in bi-exponential signal attenuation (1). This effect, known as the “intravoxel incoherent motion” (IVIM) is quantified in terms of the perfusion fraction, f, and the so called pseudo-diffusion coefficient, D*. 
     Conventional perfusion measurements, based on the uptake or washout of radioactive tracers, provide information about the cerebral blood flow (CBF) and cerebral blood volume (CBV). The perfusion fraction, f, due to the IVIM effect, is proportional to the CBV parameter, while the product f D* is proportional to the CBF parameter (10). The exact relations between CBF/CBV and the IVIM parameters, viz. f and D*, depend on tissue and microcirculation properties. Likewise, the parameters obtained by the new analysis protocol, according to the present invention, are closely related to CBV and CBF. It has been shown that in vivo evaluation of CBV and CBF parameters is possible also with the diffusion-weighted arterial spin labeling MRI technique based on the IVIM analysis (10, 11). With reference to the above described, the method according to the present invention may be directed to evaluating tumor systems. Therefore, according to one specific embodiment of the present invention, the quantitative information is related to tumor vascular properties, such as blood volume fraction and/or microvascular flow velocity, e.g. consisting of at least parameters CBV (cerebral blood volume) and/or CBF (cerebral blood flow). 
     In flow-compensated NMR diffusion experiments, on the other hand, the additional attenuation due to perfusion can be avoided, potentially allowing for the effects of flow and diffusion to be separated (7). A combination of the flow compensated and non-compensated PGSE NMR technique can be used to extract information about the non-local dispersion tensor, representing a fundamental description of dispersive flow in terms of temporal and spatial correlations (12). Although in tissue the pseudo-diffusion coefficient is an order of magnitude larger than the regular diffusion coefficient, the perfusion contributes only a small fraction (about 10%) to the observable signal, thus it is very hard to reliably detect. Analysis of the IVIM effect in diffusion weighted MRI, accounting for relaxation effects, indicates that perfusion fraction in pancreas is about 25% while only about 4% in brain. Monte Carlo error analysis shows that large signal-to-noise ratios are required for accurate estimation of the perfusion fraction (13). Despite this limitation, diffusion based detection of perfusion is tempting, since it allows non-invasively separating the contributions of blood flow and capillary volume in different physiologic and pathologic conditions. 
     In terms of temporal displacement correlations, two extreme cases of dynamics can be identified, i.e. the non-correlated diffusive motion and the coherent flow. Due to spatial variation of resonance frequency in inhomogeneous magnetic fields, the observable NMR signal carries information about mean properties of motion on the time-scale of the NMR experiment. In a spin-echo experiment, the echo attenuation is proportional to diffusion coefficient while the signal phase is proportional to the mean flow velocity. Flow compensated NMR diffusion experiments are designed to eliminate the phase shift due to flow. In some cases the dynamic properties of several sub-ensembles of nuclei affect the NMR signal. Such case may result, for example, if nuclei with different diffusive properties are probed, e.g. due to different molecular species or when nuclei remain isolated in morphologically different environments within the probed sample volume during the NMR experiment. The two component diffusion due to extracellular and intracellular water in tissue is a well-known example of the latter case. The dispersed flow, in which several sub-ensembles of nuclei with constant but diverse velocities coexist, is yet another example of several sub-ensemble contributions. Even though the velocities of these contributions may be constant on the experimental time-scale, their phase contributions would be incoherent due to velocity dispersion, bringing about an additional attenuation factor in experiments without flow compensation. This is known as the IVIM model, which describes the bi-exponential spin echo attenuation in terms of diffusivity of stationary tissue without perfusion and as an additional attenuation due to flow (perfusion) quantified as the pseudo-diffusion coefficient D*. The effects of diffusion and perfusion on the pseudo-diffusion coefficient can be analyzed separately based on the data from experiments with variable degree of flow compensation. Varying the degree of flow compensation allows for a more robust quantification of dispersed flow. As a special case, data from flow-compensated and non-compensated experiments is considered. The inventors suggest a novel diffusion-perfusion experiment (D-P experiment) with variable degree of flow compensation and a novel joint analysis of the flow compensated and non-compensated data to quantify flow with improved accuracy and precision compared to the conventional methods. 
     In an ideally homogeneous magnetic field, the spin phase is coherent and the magnetization is revolving with the Larmor frequency ω 0 =γB 0 , where γ is the nuclear gyromagnetic ratio and B 0  is the magnetic field strength. In the rotating frame, defined by the mean magnetic field B 0 , the observable NMR signal is given by spin ensemble average
 
 S ( t )= S   0     e   −jφ(t)   ,  (1)
 
where S 0  is the signal magnitude after excitation, comprising the effects of relaxation and initial spin distribution, and ϕ is the relative phase of individual spin contributions given by the varying magnetic field gradient G(t) and the relative spin displacement z(t) along the gradient as
 
                     φ   ⁡     (   t   )       =       -   γ     ⁢       ∫   0   t     ⁢       G   ⁡     (     t   ′     )       ⁢     z   ⁡     (     t   ′     )       ⁢       dt   ′     .                   (   2   )               
Note that displacements and velocities are related by
 
                     z   ⁡     (   t   )       =       ∫   0   t     ⁢       v   ⁡     (     t   ′     )       ⁢       dt   ′     .                 (   3   )               
The phase can be expressed with velocity, by introducing the dephasing factor q(t), as
 
                       q   ⁡     (   t   )       =     γ   ⁢       ∫   0   t     ⁢       G   ⁡     (     t   ′     )       ⁢     dt   ′             ,           (   4   )               
which vanishes at the time of spin-echo, t i.e. q(t E )=0.
 
At the time of spin-echo, t E , the relative phase from Eq. 2 is thus given by
 
                     φ   ⁡     (     t   E     )       =       ∫   0     t   E       ⁢       q   ⁡     (   t   )       ⁢     v   ⁡     (   t   )       ⁢     dt   .                 (   5   )               
The ensemble average of the exponent in Eq. 1 can be expressed as an exponent of averages in terms of cumulant series (14). In the Gaussian approximation, i.e. neglecting cumulant contributions of order higher than two, the observable signal at the time of spin-echo, t E , is given by
 
                       S   ⁡     (     t   E     )       =       ∑   k     ⁢           ⁢       S   k     ⁢     e         -   i     ⁢           ⁢       ϕ   k     ⁡     (     t   E     )         -       β   k     ⁡     (     t   E     )                 ,           (   6   )               
where the summation runs over all the sub-ensemble contributions with different weights S k , cumulative phase ϕ k  and attenuation β k  factors. Note that possible sub-ensemble differences in relaxation times are accounted for in the S k  coefficients. The cumulative phase is proportional to the sub-ensemble mean velocity,
 
                         ϕ   k     ⁡     (     t   E     )       =       〈     v   k     〉     ⁢   α       ,     
     ⁢   with           (   7   )               α   =       γ   ⁢       ∫   0     t   E       ⁢       G   ⁡     (     t   ′     )       ⁢     t   ′     ⁢     dt   ′           =     -       ∫   0     t   E       ⁢       q   ⁡     (     t   ′     )       ⁢       dt   ′     .                     (   8   )               
Note that for flow-compensated sequences α=0. The attenuation, on the other hand, is given by velocity fluctuations u k =ν k − ν k    and expressed in terms of the velocity autocorrelation function (VACF),  u k (t 1 )u k (t 2 ) , as
 
                     β   k     =       1   2     ⁢       ∫   0     t   g       ⁢       dt   1     ⁢       ∫   0     t   E       ⁢       dt   2     ⁢     q   ⁡     (     t   1     )       ⁢     〈         u   k     ⁡     (     t   1     )       ⁢       u   k     ⁡     (     t   2     )         〉     ⁢       q   ⁡     (     t   2     )       .                       (   9   )               
Note that for non-correlated velocity fluctuations the VACF can be approximated as a Dirac delta function with diffusion coefficient D k  as
 
   u   k ( t   1 ) u   k ( t   2 ) =2 D   k δ( t   2   −t   1 ),  (10)
 
yielding the attenuation
 
                       β   k     =         D   k     ⁢       ∫   0     t   E       ⁢         q   ⁡     (   t   )       2     ⁢   dt         =       D   k     ⁢   b         ,           (   11   )               
where b is the diffusion weighting factor.
 
     In case of flow dispersion, the averaging over sub-ensembles in Eq. 6 with different velocities gives rise to an additional attenuation term. The extent of phase coherence loss due to dispersed flow, leading to additional signal attenuation, depends on the observation time relative to flow velocity and to the characteristic length-scale, l, on which flow velocity changes take place. If flow velocity varies during the experimental time, e.g. due to variation in flow direction relative to the position/motion encoding gradients, the phase coherence is lost, leading to diffusion-like signal attenuation. The apparent diffusion coefficient due to flow with velocity ν can in such case be approximated as D ν =lv/6 (1). A presence of a net flow will however yield a cumulative phase shift when α&gt;0. Here we consider another extreme case, which assumes that different sub-ensembles have constant but different velocities during the experimental time. This corresponds to a capillary network model consisting of straight segments which are long enough so that the blood flow does not change direction during the experimental time (see FIG. 3b in Ref. 1). 
     For a set of sub-ensembles with common diffusion coefficient, D, the Eq. 6 can be written as 
                       S   ⁡     (     t   E     )       =         e     -   bD       ⁢       ∑   k     ⁢           ⁢       S   k     ⁢     e       -   i     ⁢           ⁢   α   ⁢     〈     v   k     〉               =       S   0     ⁢     e     -   bD       ⁢       ∫     -   ∞     ∞     ⁢       P   v     ⁢     e       -   i     ⁢           ⁢   α   ⁢           ⁢   v       ⁢   dv             ,           (   12   )               
where P(ν) is the velocity probability density. The function P(ν) can account for different velocity dispersion models corresponding to flow in tissue (1, 6). If for simplicity, a Gaussian distribution with the width  ν 2    is assumed,
 
                       P   v     =       1         2   ⁢   π       ⁢     〈     v   2     〉         ⁢     e     -       v   2       2   ⁢     〈     v   2     〉                 ,           (   13   )               
the Eq. 12 results in
 
                     S   ⁡     (     t   E     )       =       S   0     ⁢     e     -   bD       ⁢       e       -     1   2       ⁢     α   2     ⁢     〈     v   2     〉         .               (   14   )               
The total attenuation factor in Eq. 14 is often expressed in terms of pseudo-diffusion coefficient, D*,
 
 S ( t   E )= S   0   e   −bD* .  (15)
 
Comparison of Eqs. (14) and (15) gives the pseudo-diffusion coefficient as
 
                     D   *     =     D   +         α   2     b     ⁢       v   d   2     .                 (   16   )               
Note that the ratio α 2 /b depends on the gradient pulse sequence design. To maximize the effect of flow, manifested in the pseudo diffusion coefficient, the pulse sequence can be designed to maximize the ratio α 2 /b. In Eq. 16, we introduce ν d   2  as a measure of velocity dispersion, which scales with velocity depending on a particular velocity dispersion model. If a Gaussian velocity distribution is assumed, according to Eq. 13 and the subsequent Eq. 14, ν d   2  is given by
 
                     v   d   2     =         〈     v   2     〉     2     .             (   17   )               
Note that the scaling factor in Eq. 17, reflecting the relation of ν d   2  to the actual microscopic properties of flow, can be complex and it is irrelevant to the analysis according to the present invention. If for example, straight capillary segments with plug flow and an even angular distribution are assumed, the resulting echo-attenuation is given by a sinc function of plug flow velocity (6), which in the Gaussian approximation gives the scaling factor of ⅙, instead of ½ (see Eq. 17) and  ν 2    representing the square of the plug flow velocity. The Eq. (14) describes the case with a single D and  ν 2    contribution. When more diffusion or velocity dispersion components are present in the system, Eq. (14) can be generalized as the Laplace transform of the probability distribution P(D, ν d   2 ), where the experimental parameters b and α 2  are reciprocal to the system parameters D and ν d   2 . For a multi-component system, the normalized signal attenuation is given by
 
                     E   ⁡     (     b   ,   α     )       =         S   ⁡     (     b   ,   α     )         S   0       =       ∫   0   ∞     ⁢       ∫   0   ∞     ⁢       P   ⁡     (     D   ,     v   d   2       )       ⁢     e     -   bD       ⁢     e       -     α   2       ⁢     v   d   2         ⁢       dDdv   d   2     .                     (   18   )               
The probability distribution is normalized, so that
 
                 ∫   0   ∞     ⁢       ∫   0   ∞     ⁢       P   ⁡     (     D   ,     v   d   2       )       ⁢     dDdv   d   2           =   1.         
The P(D,ν d   2 ) is given by the inverse Laplace transform of the measured signal intensity E(b,α). The correlation between different diffusion components (D) and the velocity dispersion components (ν d   2 ) can be revealed by the P(D, ν d   2 ). Information about the probability distribution P(D,ν d   2 ) can be obtained by regressing the equation (18) onto the signal attenuation data E(b,α) at variable b and α.
 
     Pulse sequences that allow for independent adjustment of diffusion weighting, b, and velocity dispersion weighting, α, can be used to quantify the velocity dispersion by disentangling the diffusion and the velocity dispersion contributions to the total signal attenuation, characterized by the pseudo-diffusion coefficient, D*. In the diffusion-perfusion correlation experiment (D-P), where the experimental parameters b and α can be adjusted independently, the measurement of the correlation between the diffusion coefficient and the velocity dispersion allows associating the velocity dispersion components and the corresponding diffusion components when one or more diffusion or velocity dispersion components are present in the system. 
     Different signal contributions might exist in tissue, some affected by both perfusion and diffusion, some affected only by diffusion. For example, one may consider two signal contributions; one for which the attenuation is given by both the diffusion coefficient, D f , and the velocity dispersion ν d   2 , and another, for which the attenuation is given only by the diffusion coefficient D. The normalized signal intensity is in this case given by
 
 E ( b ,α)= fe   −bD     f     e   −α     2     ν     d       2   +(1− f ) e   −bD .  (19)
 
Here f is the fraction of the contribution with perfusion. The probability distribution P(D, ν d   2 ) for the example summarized by the Eq. 19 is schematically illustrated in  FIG. 1 . Note that the two contributions, represented as peaks at the (D f , ν d   2 ) and (D,0) coordinates on the P(D, ν d   2 ) contour plot, can only be resolved along the ν d   2  axis, which is reciprocal to α 2 . Thus varying a at different diffusion weighting b provides a means of resolving the velocity dispersion components and thus correlating the velocity dispersion components and the diffusion components. If the system can be described by two diffusion components and one velocity dispersion component, the diffusion coefficients D and D f , the velocity dispersion ν d   2  and the ratio f can be quantified by regressing the equation (19) onto the signal attenuation data.
 
     Several pulse sequences meet the criterion of allowing for independent variation of α and b to achieve the D-P experiment. While different gradient modulation schemes can be used, including different oscillating gradient waveforms, the examples shown in  FIGS. 2 and 3  make use of pulsed gradients. 
     The examples of the D-P pulse sequences shown in  FIG. 2  can be view as consisting of two consecutive motion sensitizing blocks, from which one is flow compensated (α=0) and the other is not flow compensated. In  FIG. 2.1 , the D-P sequence consists of two different double-PGSE blocks, each with its own gradient amplitudes, G d1  and G d2  and timing parameters, δ d1 , δ d2 , δ d1 , δ d2 . The first block is not flow compensated, the second block is flow compensated (α=0). α and b can be independently adjusted by varying two of the gradient amplitudes G d1 , G d2  and/or durations δ d1 , δ d2  and/or pulse separations Δ d1 , Δ d2 . The D-P sequence in  FIG. 2.2  consists of a single PGSE block and a double-PGSE block, each with its own gradient amplitudes, G s  and G d  and timing parameters, δ s , δ d , δ s , δ d . The first block is not flow compensated, the second block is flow compensated (α=0). α and b can be independently adjusted by varying two of the gradient amplitudes G s , G d  and/or durations δ s , δ d  and/or pulse separations Δ s , Δ d . A special case of this example is when the gradient pulse duration is identical for all the pulses, δ=δ s =δ d . In that case, the dephasing magnitude is given by q s,d =γG s,d δ, the velocity dispersion weighting α=q s Δ s  and the diffusion weighting 
             b   =         q   s   2     ⁡     (       Δ   s     -     δ   3       )       +     2   ⁢         q   d   2     ⁡     (       Δ   d     -     δ   3       )       .               
α and b can be adjusted independently by adjusting q s  and q d .
 
     The examples shown in  FIG. 3  can be viewed as a single block design where α and b can be adjusted independently. The D-P sequence in  FIG. 3.1  consists of two single PGSE blocks, each with its own gradient amplitudes, G 1  and G 2  and timing parameters, δ 1 , δ 2 , Δ 1 , Δ 2 . The entire block allows for variable flow compensation (α). α and b can be independently adjusted by varying two of the gradient amplitudes G 1 , G 2  and/or durations δ 1 , δ 2  and/or pulse separations Δ 1 , Δ 2 . A special case of this example is when the gradient pulse durations are identical for all the pulses, δ=δ 1 =δ 2 , and pulse separations are identical in the two PGSE blocks, Δ=Δ 1 =Δ 2 , so that the dephasing magnitude is given by q 1,2 =G 1,2 δ, velocity dispersion weighting by α=(q 1 −q 2 )Δ and the diffusion weighting by 
             b   =       (       q   1   2     +     q   2   2       )     ⁢       (     Δ   -     δ   3       )     .             
α and b can be adjusted independently by adjusting q 1  and q 2 . Yet another special case of the example shown in  FIG. 3.1  is the example shown in  FIG. 3.2 . In this example, the gradient pulse durations are short, δ=δ 1 =δ 2 →0, and gradient magnitudes are all identical, |G 1 |=|G 2 |=G, so that the dephasing magnitude is given by q=γGδ, and the pulse separations are varied so that their sum is a constant diffusion time, t d =Δ 1 +Δ 2 , and the difference, expressed as ε=Δ 1 −Δ 2 , can span the interval 0−t d . The velocity dispersion weighting is then given by α=qε and the diffusion weighting by b=q 2 t d . α and b can be adjusted independently by adjusting q and ε. In the D-P sequence example shown in  FIG. 3.3 , the gradient modulation consists of three gradient pulses with amplitudes G 1 , G 2  and G 3  and timing parameters δ 1 , δ 2 , δ 3 , Δ 1 , Δ 2 . The entire block allows for variable flow compensation (α). The echo condition
 
               q   ⁡     (     t   E     )       =         ∫   0     t   E       ⁢       G   ⁡     (     t   ′     )       ⁢     dt   ′         =   0           
constrains the relations between the three gradient pulse amplitudes G 1 , G 2  and G 2  and durations δ 1 , δ 2 , δ 3 . α and b can be independently adjusted by varying two of the gradient amplitudes G 1 , G 2  and G 3  and/or durations δ 1 , δ 2 , δ 3  and/or pulse separations Δ 1 , Δ 2 . A special case for this gradient modulation example is when the gradient pulse durations are identical for the three pulses, δ=δ 1 =δ 2 =δ 3 , and pulse separations are identical, Δ=Δ 1 =Δ 2 . In this case, the gradient amplitudes are related by G 2 =−(G 1 +G 3 ), the dephasing magnitude is given by q 1,3 =γG 1,3 δ, the velocity dispersion weighting by
 
               α   =       1   2     ⁢     (       q   1     -     q   3       )     ⁢   Δ       ⁢                 
and the diffusion weighting by
 
             b   =         1   6     ⁢     (       q   1   2     +     q   3   2       )     ⁢     (       3   ⁢   Δ     -     2   ⁢           ⁢   δ       )       -       1   3     ⁢     q   1     ⁢     q   3     ⁢     δ   .               
α and b can be adjusted independently by adjusting q 1  and q 3 .
 
     For implementation of the D-P experiment on a clinical scanner, the sequences  3 . 1  and  3 . 3  might be best suited since they can be conveniently implemented in combination with the different read out protocols. These sequences, particularly the one shown in  FIG. 3.3 , may, with an appropriate choice of sequence timing parameters, require minimum gradient switching and are thus favored by the gradient slew rate and amplitude limitations often encountered in clinical scanners. 
     The experiment may be performed also by using only two values of the parameter α, where one value set to zero and the other value is set to a value different from zero. Such experiment can be performed in two parts, by using a flow compensated sequence (α=0) and by using a non-compensated sequence. Several pulse sequences may be used as a flow compensated sequence or non-compensated sequence. Some examples of such sequences are given in the  FIG. 4 . 
     For a single pulsed gradient spin-echo (single-PGSE) sequence with a pair of narrow pulses with intensity G s  of duration δ s  and the separation between the leading edges of the pulses Δ s  (see  FIG. 4.1 ), the dephasing magnitude is given by
 
 q   s   =γG   s δ s   (20)
 
and the diffusion and velocity weighting factors are given by
 
 b=q   s   2 (Δ s −δ s /3)  (21)
 
and
 
α=− q   s Δ s .  (22)
 
The pseudo-diffusion coefficient (see Eq. 16) is given by
 
     
       
         
           
             
               
                 
                   
                     D 
                     * 
                   
                   = 
                   
                     D 
                     + 
                     
                       
                         
                           Δ 
                           s 
                           2 
                         
                         
                           
                             Δ 
                             s 
                           
                           - 
                           
                             
                               δ 
                               s 
                             
                             / 
                             3 
                           
                         
                       
                       ⁢ 
                       
                         
                           v 
                           d 
                           2 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
           
         
       
     
     While D is independent of Δ in case of free diffusion and inversely proportional to Δ for restricted diffusion, D* is roughly proportional to Δ due to the dominant effect of flow (see Eq. 23). Note that Eq. 23 implies that the echo attenuation due to flow can be maximized by using long pulse separation times, such that Δ s &gt;&gt;δ s . 
     Flow compensation can be achieved with any oscillating gradient sequence, for which the condition α=0 is true. Such sequences may employ harmonically oscillating gradients or any kind of gradient waveforms, which achieve diffusion weighting and flow compensation simultaneously. The flow compensated double-PGSE sequence ( FIG. 4.3 ) or a repeated flow compensated double PGSE, denoted n-PGSE in  FIG. 4.4 , where n=2, 4, 6 . . . is an even integer corresponding to the number of gradient pulse pairs in the sequence, is just an example of the flow-compensated sequence we use in this description. In the case of n-PGSE sequence, the diffusion weighting is given by
 
 b   n   =nq   n   2 (Δ n −δ n /3),  (24)
 
where q n , Δ, and δ n  denote the corresponding parameters for a double-PGSE sequence (see  FIG. 4.4 ).
 
     For equal diffusion weighting b in a single-PGSE and in an n-PGSE experiment, the condition 
                     q   n     =       q   s     ⁢         (       Δ   s     -       δ   s     /   3       )       n   ⁡     (       Δ   n     -       δ   n     /   3       )                     (   25   )               
must be fulfilled. If in addition the dephasing magnitudes and the pulse durations are chosen to be identical for the single-PGSE and for the n-PGSE, setting q n =q s  and δ n =δ s  in condition (25) implies
 
     
       
         
           
             
               
                 
                   
                     Δ 
                     n 
                   
                   = 
                   
                     
                       
                         Δ 
                         s 
                       
                       n 
                     
                     + 
                     
                       
                         
                           δ 
                           s 
                         
                         / 
                         3 
                       
                       ⁢ 
                       
                         
                           ( 
                           
                             1 
                             - 
                             
                               1 
                               n 
                             
                           
                           ) 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
           
         
       
     
     Along the lines of Eq. 6, the above description of signal attenuation (Eq. 14) could be expanded to include flowing and non-flowing populations with different diffusion coefficients. Here we consider a bi-modal attenuation, resulting from a flowing and a non-flowing population, assuming that molecular exchange between the two populations is negligible during experimental time. We denote f as the perfusion fraction, and 1−f as the diffusion fraction. The spin-echo intensity is in this case given by 
                       I   s     ⁡     (       q   s     ,     Δ   s     ,     δ   s       )       =       I     0   ⁢           ⁢   s       ⁢     ⌊         (     1   -   f     )     ⁢     e       -       q   s   2     ⁡     (       Δ   s     -       δ   s     /   3       )         ⁢   D         +       fe       -       q   s   2     ⁡     (       Δ   s     -       δ   s     /   3       )         ⁢     D   f         ⁢     e       -     q   s   2       ⁢     Δ   s   2     ⁢     v   d   2             ⌋               (   27   )                       ⁢   and                                   ⁢         I   n     ⁡     (       q   n     ,     Δ   n     ,     δ   n       )       =       I     0   ⁢   n       ⁢     ⌊         (     1   -   f     )     ⁢     e       -       nq   n   2     ⁡     (       Δ   n     -       δ   n     /   3       )         ⁢   D         +     fe       -       nq   n   2     ⁡     (       Δ   n     -       δ   n     /   3       )         ⁢     D   f           ⌋                 (   28   )               
for the non-compensated single-PGSE and for the flow-compensated n-PGSE, respectively. Note that if D and D f  are identical, the flow-compensated intensity Eq. 28 is described by a mono-exponential decay.
 
     The analysis, according to the invention, does not relate to any particular diffusion weighting or flow-compensation gradient sequence. Analysis of the non-compensated and flow-compensated data sets may require adjustment to how the timing parameters, diffusion weighting and dephasing factors are calculated in Eqs. 27 and 28. 
     Extraction of the D* or ν d   2  and f from the bi-exponential regression of Eq. 27 to the flow non-compensated (IVIM) dataset suffers from an inordinate sensitivity to noise. The flow compensated dataset provides additional information, by disentangling the effects of flow from the effects of diffusion. Here we suggest the simultaneous analysis of both flow-compensated and non-compensated datasets, which significantly improves the accuracy and precision of the estimated parameters of interest. By constraining the parameters D, D f , and f to be identical for both the flow compensated and non-compensated datasets, the sensitivity to noise of the estimated parameters is considerably reduced compared to the standard analysis of IVIM dataset.  FIG. 5  shows a comparison of accuracy and precision of the flowing fraction f estimates for the conventional protocols (“standard” and “segmented”) and according to the present invention. The analysis is applied to simulated data based on experimentally extracted parameters by Sigmund et al. (2). 
     The Monte Carlo error estimation was used in prooving the efficiency of the present invention. Random noise is added to simulated data, which is then regressed with different protocols to obtain the fit parameters. This procedure is repeated 1000 times for each signal-to-noise level to obtain mean values and error estimates for the fit parameters (15). To ensure a fair comparison of the different analysis protocols, the same amount of data points is used in the conventional as in the new protocol. In the new protocol, only the data points corresponding to every second (even) b-value from the original series of b-values (used in the conventional protocols) are used in the flow-compensated dataset and only data points at the odd b-values from the original series are used in the non-compensated dataset (compare  FIG. 5 a    and  FIG. 5 b   ). In  FIG. 6 a    one can see that the “standard” analysis results in an overestimated f values and the “segmented” analysis results in an underestimated f values, while the new protocol yields considerably improved accuracy in the entire range of noise levels. As shown in  FIG. 6 b   , the “segmented” protocol only slightly improves the precision over the “standard” protocol, while the new protocol yields significantly more precise estimates of f values over the entire range of noise levels. 
     DETAILED DESCRIPTION OF THE DRAWINGS 
       FIG. 1  displays a schematic representation of the probability distribution P(D, ν d   2 ) for the example with two signal contributions described by Eq. (19). The P(D, ν d   2 ) is given by the inverse Laplace transform of the measured signal E(b,α). The relation between E(b,α) and P(D, ν d   2 ) is given by Eq. (18). The contour lines connect points with equal probability density. The solid lines on top and on the right hand side of the contour plot represent the projections of the probability density function, i.e. the probability distribution of diffusion coefficients, D, and the probability distribution of velocity dispersions, ν d   2 , respectively. On the contour plot of the P(D, ν d   2 ), two contributions are identified as peaks at (D f , ν d   2 ) and (D,0). The two contributions cannot be resolved along the D-axis, which is reciprocal to the diffusion weighting b, while they can be resolved along the v d   2 -axis, which is reciprocal to α 2 . 
       FIG. 2  displays schematics of pulse sequences for motion weighted nuclear magnetic resonance (NMR) or magnetic resonance imaging (MRI). The magnification of the motion encoding block shows the effective gradient wave form and the dephasing for two examples of the diffusion-perfusion correlation experiment, which can be viewed as composed of two motion encoding blocks, where the diffusion weighting, b, and the velocity dispersion weighting, a, can be adjusted independently: 1. double-double PGSE, the first double-PGSE is non-compensated (signal attenuation from diffusion and perfusion) and the second double-PGSE is flow-compensated (α=0, signal attenuation from diffusion but NOT from perfusion); 2. single-double PGSE, the first single-PGSE is non-compensated (signal attenuation from diffusion and perfusion) and the second double-PGSE is flow-compensated (α=0, signal attenuation from diffusion but NOT from perfusion). 
       FIG. 3  displays schematics of pulse sequences for motion weighted nuclear magnetic resonance (NMR) or magnetic resonance imaging (MRI). The magnification of the motion encoding block shows the effective gradient wave form and the dephasing for three examples of the diffusion-perfusion correlation experiment, which can be viewed as composed of one motion encoding block, where the diffusion weighting, b, and the velocity dispersion weighting, a, can be adjusted independently: 1. double-PGSE with long gradient pulses, where the independent timing parameters δ 1 , δ 2 , Δ 1 , Δ 2  and the gradient magnitudes G 1  and G 2  allow for independent adjustment of the diffusion weighting, b, and the velocity dispersion weighting, α; 2. double-PGSE with short gradient pulses, where the independent timing parameters Δ 1 , Δ 2  and the dephasing magnitude q allow for independent adjustment of the diffusion weighting, b, and the velocity dispersion weighting, α; 3. three-pulse PGSE, where the independent timing parameters δ 1 , δ 2 , δ 3 , Δ 1 , Δ 2  and the gradient magnitudes G 1  and G 2  are constraint by the echo condition (at the end of the motion encoding the dephasing is equal to zero) and allow for independent adjustment of the diffusion weighting, b, and the velocity dispersion weighting, α. 
       FIG. 4  displays schematics of pulse sequences for motion weighted nuclear magnetic resonance (NMR) or magnetic resonance imaging (MRI). The magnification of the motion encoding block shows the effective gradient wave form for four commonly known motion encoding schemes: 1. single-PGSE, non-compensated (signal attenuation from diffusion and perfusion); 2. double-PGSE, non-compensated (signal attenuation from diffusion and perfusion); 3. double-PGSE, flow-compensated (signal attenuation from diffusion but NOT from perfusion). 4. n-PGSE, flow-compensated (signal attenuation from diffusion but NOT from perfusion). Each gradient pulse pair is characterized by its amplitude G, pulse duration δ, and separation between leading edges Δ. 
       FIG. 5  shows simulated MR signal vs. diffusion-weighting b for some of the pulse sequences in  FIG. 4 . The simulations were based on the following parameters estimated on invasive ductal carcinoma by Sigmund et al. (2): perfusion fraction f=10.5%, tissue diffisivity D=1.14 μm 2 /ms, pseudo-diffusivity of the perfusion fraction D*=14.9 μm 2 /ms. 
     (a) Data obtained with a conventional protocol (obtained from ref. (2)) using a non-compensated single PGSE motion encoding scheme yielding signal attenuation from diffusion and perfusion. Circles: simulated noisy data; solid line: result of a conventional least squares fit yielding estimates of the perfusion fraction f, pseudo-diffusivity of the perfusion fraction D*, and tissue diffusivity D; dashed line: signal attenuation of the diffusion fraction. The perfusion fraction equals the difference between the b=0 intercepts of the solid and dashed lines. 
     (b) Data obtained with a protocol according to the present invention in which results from non-compensated single PGSE and flow-compensated double PGSE protocols are analyzed simultaneously. Circles: simulated noisy non-compensated data; squares: simulated noisy flow-compensated data; solid and dash-dotted lines: result of a simultaneous least squares fit to the non-compensated and flow-compensated according to the present invention yielding estimates of the perfusion fraction f, ν d   2  for the perfusion fraction, and diffusivity of the diffusion fraction D; dashed line: signal attenuation of the diffusion fraction. The perfusion fraction equals the difference between the dashed and dash-dotted lines. 
       FIG. 6  shows the accuracy (a) and the precision (b) of the perfusion fraction f as quantified with conventional protocols (“standard” and “segmented”) as well as according to the present invention. The accuracy and precision are plotted as a function of the signal-to-noise ratio of the raw MR signal data. Note the better accuracy and precision of the present invention when compared with the conventional protocols over the whole range of signal-to-noise ratios. 
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