Quasi‐Diffusion Imaging: Application to Ultra‐High <i>b</i>‐Value and Time‐Dependent Diffusion Images of Brain Tissue

We demonstrate that quasi-diffusion imaging (QDI) is a signal representation that extends towards the negative power law regime. We evaluate QDI for in vivo human and ex vivo fixed rat brain tissue across <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>b</mml:mi></mml:mrow> <mml:annotation>$$ b $$</mml:annotation></mml:semantics> </mml:math> -value ranges from 0 to 25,000 s mm^-2, determine whether accurate parameter estimates can be acquired from clinically feasible scan times and investigate their diffusion time-dependence. Several mathematical properties of the QDI representation are presented. QDI describes diffusion magnetic resonance imaging (dMRI) signal attenuation by two fitting parameters within a Mittag-Leffler function (MLF). We present its asymptotic properties at low and high <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>b</mml:mi></mml:mrow> <mml:annotation>$$ b $$</mml:annotation></mml:semantics> </mml:math> -values and define the inflection point (IP) above which the signal tends to a negative power law. To show that QDI provides an accurate representation of dMRI signal, we apply it to two human brain datasets (Dataset 1: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mn>0</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>b</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>15,000</mml:mn></mml:mrow> <mml:annotation>$$ 0\le b\le \mathrm{15,000} $$</mml:annotation></mml:semantics> </mml:math> s mm^-2; Dataset 2: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mn>0</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>b</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>17,800</mml:mn></mml:mrow> <mml:annotation>$$ 0\le b\le \mathrm{17,800} $$</mml:annotation></mml:semantics> </mml:math> s mm^-2) and an ex vivo fixed rat brain (Dataset 3: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mn>0</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>b</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>25,000</mml:mn></mml:mrow> <mml:annotation>$$ 0\le b\le \mathrm{25,000} $$</mml:annotation></mml:semantics> </mml:math> s mm^-2, diffusion times <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mn>17.5</mml:mn> <mml:mo>≤</mml:mo> <mml:mo>∆</mml:mo> <mml:mo>≤</mml:mo> <mml:mn>200</mml:mn></mml:mrow> <mml:annotation>$$ 17.5\le \Delta \le 200 $$</mml:annotation></mml:semantics> </mml:math> ms). A clinically feasible 4 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>b</mml:mi></mml:mrow> <mml:annotation>$$ b $$</mml:annotation></mml:semantics> </mml:math> -value subset of Dataset 1 ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mn>0</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>b</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>15,000</mml:mn></mml:mrow> <mml:annotation>$$ 0\le b\le \mathrm{15,000} $$</mml:annotation></mml:semantics> </mml:math> s mm^-2) is also analysed (acquisition time 6 min and 16 s). QDI showed excellent fits to observed signal attenuation, identified signal IPs and provided an apparent negative power law. Stable parameter estimates were identified upon increasing the maximum <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>b</mml:mi></mml:mrow> <mml:annotation>$$ b $$</mml:annotation></mml:semantics> </mml:math> -value of the fitting range to near and above signal IPs, suggesting QDI is a valid signal representation within in vivo and ex vivo brain tissue across large <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>b</mml:mi></mml:mrow> <mml:annotation>$$ b $$</mml:annotation></mml:semantics> </mml:math> -value ranges with multiple diffusion times. QDI parameters were accurately estimated from clinically feasible shorter data acquisition, and time-dependence was observed with parameters approaching a Gaussian tortuosity limit with increasing diffusion time. In conclusion, QDI provides a parsimonious representation of dMRI signal attenuation in brain tissue that is sensitive to tissue microstructural heterogeneity and cell membrane permeability.