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Magnetic resonance imaging based on steady-state free precision (SSFP) sequences is a fast method to acquire

Magnetic resonance imaging is a powerful tool for

Moreover, quantitative blood oxygenation level dependent (qBOLD) measurements in functional magnetic resonance imaging (fMRI) might be a relevant application for SSFP sequences. So far, qBOLD measurements are based on FLASH-like MR sequences and evaluated with analytical solutions of the Bloch- or the Bloch–Torrey-equation [

For nonhomogeneous tissues, susceptibility differences between different compartments cause magnetic field inhomogeneities, and the gradient echo and spin echo signal decay cannot be described sufficiently by monoexponential decays. Typical examples are vessel networks in the brain [

As visualized in Figure

For negligible diffusion effects, this modulation depends on the response function

In general, the lineshape

is assumed to describe the resulting Larmor frequency distribution of the underlying tissue [

By virtue of Equation (

which results in a purely monoexponential signal decay

Free induction decay for dephasing in the absence and presence of microscopic magnetic field inhomogeneities. In the case of homogeneous tissue (a), the intrinsic transverse relaxation leads to purely monoexponential signal decay with the intrinsic transverse relaxation time

In this work, the SSFP signal of important tissue geometries is derived. Therefore, the underlying local Larmor frequencies as well as the resulting lineshapes for the important cases of dephasing in a constant gradient as well as dephasing in a two-dimensional and three-dimensional dipole field (see Figure

Larmor-frequencies for different local magnetic field inhomogeneities. For dephasing in an imaging gradient or diffusion weighted gradient (a) the local resonance frequency is proportional to the spatial coordinate (b) according to Equation (

Externally applied magnetic field gradients or internal gradients due to susceptibility differences cause a local magnetic field which is associated with a local Larmor frequency in which the dephasing occurs. The most important case is the dephasing in a constant gradient

where

Cylindrical objects as for example blood filled vessels in the myocardium or in skeletal muscle (see Figure

where

Spherical magnetic field inhomogeneities, such as magnetically labeled cells or alveoli of the lung (see Figure

which depends on the distance

If the dephasing is restricted to the surface of a spherical object with radius

As visualized in Figure

If the movement of the spin-bearing particles is restricted to the alveolar surface, the extended alveolar surface model agrees with the alveolar surface model. However, it yields more realistic results for particles being not exactly restricted but diffusing close to the alveolar surface [

As in detail shown in Appendix

A monoexponential magnetization decay with relaxation time

For dephasing in the constant gradient

and all possible Larmor frequencies take the same probability

Lineshapes for a Lorentzian frequency distribution (a) according to Equation (

The lineshape caused by a two-dimensional dipole field around a cylindrical object (see Figure

Details of the derivation can be found in [

The asymmetric lineshape caused by the three-dimensional dipole field (see Equation (

Details of the derivation can be found in [

In the limit of restricting water molecules to the surface of the alveoli, the lineshape is given by (see also the corresponding local Larmor frequency given in Equation (

This expression also follows from the lineshape caused by a spherical object given in Equation (

In the extended alveolar surface model, the lineshape was obtained as [

where the function

A pulse sequence similar to the previously described FLASH-sequence is the balanced SSFP-sequence [

with the parameters

and the abbreviations

For further analysis it is advantageous, to expand the response function in a Fourier series [

where the Fourier coefficients are given as

with the series expansion point

and the step function of the discrete variable

In Figure

Fourier coefficients

Finally, the total signal that is measured depends on the intrinsic

Thus, for detailed understanding of SSFP signal formation, it is mandatory to consider the exact lineshape as presented in the last subsection.

Introducing the Fourier series of the response function (

In the following subsections, this general expression is evaluated for the specific field inhomogeneities as described in Material and Methods.

To obtain the SSFP signal for the Lorentzian lineshape given in (

with the prefactor

as well as the initial signal

Measuring the SSFP signal allows fitting the relaxation times

The series expansion point

The definition of the series expansion point in Equation (

In case of a Lorentzian lineshape, the parameters

In the case of constant gradients, the time evolution of the magnetization according to Equation (

where

Since the lineshape for dephasing in a constant gradient is a symmetric function (see Equation (

To find the SSFP signal for dephasing on the alveolar surface, the relavant magnetization given in Equation (

The initial signal follows as

Similar expressions can also be derived for the extended alveolar surface model.

So far, the SSFP signal was usually described under the assumption of Lorentzian lineshapes. A comparison of both lineshapes is shown in Figure

Transverse components

Comparison between Lorentzian lineshape and alveolar surface model. The initial signal is shown in dependence on repetition time

To validate the theoretical findings, numerical simulations were performed. A steady-state signal can be measured in the equilibrium state after applying a certain number of rf pulses. The signal after the

where the matrix

Furthermore, it is convenient to introduce the column vectors

The steady state is characterized by the condition

where

This integral is numerically calculated for the constant gradient lineshape, the alveolar surface model as well the Lorentzian lineshape and agrees with the theoretically derived signals in all models. Moreover, Equation (

Measurements were performed on a 7 Tesla Bruker Biospec 70/30 (Ettlingen, Germany). A

Measured SSFP signal (black dots) in a constant gradient field compared with the theoretical predictions assuming a boxcar lineshape (see Equation (

To estimate the impact of the presented results on the relaxation time determination, a simple numerical model is analyzed: we assume tissue that produces a boxcar lineshape with a width of

Bias of the

In this work, more realistic lineshape models were analyzed to overcome the drawbacks of the Lorentzian lineshape model. The SSFP signal in the constant gradient model

To assess the expected precision and accuracy of determining the parameters in this model, the SSFP signal for a constant gradient was calculated according to Equation (

Expected accuracy and precision of

In this work, the signal formation in local magnetic field inhomogeneities is recapitulated. Important local magnetic field inhomogeneities for magnetic resonance imaging are the constant gradient and dipole fields caused by cylindrical or spherical objects. For these cases, the free induction decay, the corresponding lineshape as well as the SSFP signal are analyzed.

The lineshape for a constant gradient and for cylindrical objects are symmetric. Consequently, the free induction decay as well as the SSFP signal become purely real meaning that only the transverse component

Lung tissue consists of very densely packed air-filled alveoli [

The initial SSFP signal

The analytical description of the SSFP signal derived in this work is validated with experimental measurements for a constant field gradient, see Section

As a consequence of these findings, the determination of the relaxation time

In Figure

This also indicates that SSFP sequences in qBOLD imaging might be interesting as they are very sensitive towards changes in the lineshape. The lineshape in blood vessel networks, on the other hand, is sensitive towards changes of physiological parameters like the blood volume fractions or the oxygenation level.

In Figure

In this work, the SSFP signal is analyzed for nonhomogeneous tissue. For a purely monoexponential free induction decay, the lineshape is Lorentzian and the SSFP signal can be calcuated by utilizing the Fourier coefficents. However, even for nonexponential signal decay that may occur due to two-dimensional or three-dimensional dipole fields in muscle or lung tissue, the SSFP signal can analytically expressed in terms of the Fourier coefficients. The relaxation times can also be determined for non-Lorentzian lineshapes by utilizing the correct lineshape model.

For analysis of the local magnetization, it is advantageous to combine the

In general, the time evolution of this local transverse magnetization density is then governed by the Bloch equation

where

The total magnetization can be obtained by an integration of the local magnetization density over the voxel with volume

with the total initial magnetization

which will produce the transverse magnetization

Since the total magnetization is in general complex-valued

To obtain a purely real lineshape, it is necessary to continue the magnetization for negative times in the form of

Thus, the lineshape can be written as

which requires only the knowledge of the magnetization for positive times. Introducing the expression for the total magnetization in Equation (

where the Fourier representation of Diracs delta distribution was used (note that a sign error occurred in Equation (7) in [

The magnetization can be expressed as the inverse Fourier transform of the lineshape according to Equation (

The magnetization

The magnetization around a two-dimensional dipole field that occurs

with the generalized hypergeometric function

An alternative expression for the magnetization decay is given in Equations (39) and (40) in [

The magnetization around a three-dimensional dipole field (as present around magnetized particles or alveoli in the lung) reads:

with the function

where

In the limit of restricting water molecules to the surface of the alveoli, the time evolution of the magnetization is given by:

where

In the extended alveolar surface model, the magnetization reads:

with

In general, the magnetization

The range correctly described by the mean time approximation depends on the actual form of the magnetization

where the pathway of integration is visualized in Figure

Integration pathway according to the right hand side of Equation (

The real part of the relaxation time

Appyling Equation (

The transverse relaxation time for a cylinder

Within the static dephasing limit, where diffusion is neglected, the lineshape in the extended alveolar surface model is given in Equation (

A comparison of the lineshapes for the exact three-dimensional dipole field and the extended alveolar surface model is shown in Figure

Comparison of lineshapes in the static dephasing for the exact dipole field

If diffusion effects are included, an additional diffusion term has to be added to the original Bloch equation (

Analytical solutions for this partial differential equation can be given for diffusion in a constant gradient as given in Equation (

with eigenvalues

where

Eigenvalues

One important issue to be mentioned is the range of possible resonance frequencies for dephasing with and without diffusion. In the static dephasing regime (

Since the imaginary parts of the eigenvalues

The transverse relaxation time in the alveolar surface model for nonvanishing diffusion effects

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

C. H. Ziener and M. Uhrig are contributed equally.

This work was supported by grants from the Deutsche Forschungsgemeinschaft (Contract Grant number: DFG ZI 1295/2-1 and DFG KU 3555/1-1). T. Kampf and M. Pham were supported by the Heidelberg Pain Consortium (SFB 1158). V. J. F. Sturm was supported by the Deutsche Forschungsgemeinschaft (SFB 1118/2).

^{∗}: a theoretical approach for the vasculature of myocardium