# Identification of Laminar Composition in Cerebral Cortex Using Low-Resolution Magnetic Resonance Images and Trust Region Optimization Algorithm

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## Abstract

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_{1}relaxation at the sub-voxel level. This work proposes a new approach for their estimation. The approach is validated using simulated data. Sixteen MRI experiments were carried out on healthy volunteers. A modified echo-planar imaging (EPI) sequence was used to acquire 105 individual volumes. Data simulating the images were created, serving as the ground truth. The model was fitted to the data using a modified Trust Region algorithm. In single voxel experiments, the estimation accuracy of the T

_{1}relaxation times depended on the number of optimization starting points and the level of noise. A single starting point resulted in a mean percentage error (MPE) of 6.1%, while 100 starting points resulted in a perfect fit. The MPE was <5% for the signal-to-noise ratio (SNR) ≥ 38 dB. Concerning multiple voxel experiments, the MPE was <5% for all components. Estimation of T

_{1}relaxation times can be achieved using the modified algorithm with MPE < 5%.

## 1. Introduction

_{1}-weighted images at 3 T. Due to the sub-millimeter resolutions possible at 7 T, the focus of the research community shifted to image acquisition at the highest field strengths. Multiple Brodmann areas of the cortex were measured using a magnetization-prepared fluid-attenuated inversion recovery sequence. The result was several intensity profiles, which exhibited a multiple-layer appearance similar to the patterns of the cortical lamination [13]. Different contrasts resulting from a modified magnetization-prepared rapid acquisition GE sequence were combined to create intercortical maps related to myelin content. Subsequent clustering yielded a delineation of the auditory area [14]. Laminar profiles resembling the lines of Baillarger were also revealed in the images resulting from a modified T

_{1}-weighted MPRAGE sequence [15]. Magnetization-prepared sequences of two rapid acquisition gradient-echoes (MP2RAGE) were used to acquire high-resolution T

_{1}-weighted images. The cortical gray matter was segmented out of the volume and then segmented further, revealing four cortical layers [16]. A conceptually different approach was used to visualize cortical layers without the necessity of sub-millimeter image resolution. A fast spin-echo (SE) sequence with several different IR times at 3 T captured several images with corresponding contrasts. The dataset was then fitted to an exponential decay function to estimate the T

_{1}relaxation times individually for each voxel. The estimated values served as the basis for the classifications of individual voxel into five or six groups, corresponding to the cortical layers [17]. Using a similar imaging protocol, a series of low-resolution echo-planar images (3 mm) were acquired with the contrast based on a set of varying IR times. A modified fitting procedure allowed for the estimation of multiple T

_{1}relaxation times related to individual voxel components, thus capturing several layers within a single voxel [18]. The above-mentioned imaging procedure was also made to better reflect the natural curvature of the cerebral cortex. This was accomplished via sub-sampling of individual voxels and their mapping onto a grid of virtual spheres, spanning the cortical gray matter [19]. The works presented so far show two emerging pathways in the imaging of whole-brain cortical lamination. The first approach is focused on acquisitions of high-resolution images at higher field strengths (7 T respectively) [14,15,16]. Although utilized in a variety of research endeavors, this approach is not without limitations, the most notable being the partial volume effect (PVE). This is the occurrence of multiple tissue types within a single voxel, which manifest in the obtained voxel intensity [20]. In the context of cortical laminations, this effect persists even at 7 T [18]. An alternative approach to imaging the cortical layers is based on the acquisition of a multitude of images—surprisingly—with lower resolutions at lower field strengths. The low-resolution images are subjected to a complex modeling and visualization pipeline resulting in high-detail maps of cortical lamination. This approach is limited due to the need for estimation of T

_{1}relaxation times, the process of which is a tradeoff between computational complexity, time constraints, and estimation accuracy [18,19].

_{1}values of several components within a single voxel image using the pulse sequence proposed in [18]. A dataset with known values of T

_{1}times is generated to assess the validity of the method. This is achieved via simulations of MRI images and individual voxels. Simulations are carried out using signal equations and an established simulator MRiLab, with a custom sequence and an imaging phantom.

_{1}relaxation times as an optimization problem and describe the chosen algorithm. Later, we focus on the description of the experimental and simulated data. In the Results section, we present the outcomes of the optimization algorithm for various levels of noise and types of simulated data. The Discussion compares the results with results of similar research endeavors in estimating T

_{1}relaxation and concludes the paper.

## 2. Materials and Methods

#### 2.1. Fitting Problem

_{1}but is also influenced by other relaxation mechanisms. A more generalized form of Equation (1) can be used to estimate the T

_{1}relaxation times of a single signal source. This is usually a single voxel of an MR image, commonly used for T

_{1}mapping, as evidenced by the state-of-the-art method [21].

_{1}relaxation time for the j-th component, and n denotes the number of components per voxel. While the parameter T

_{1}uniquely identifies the cortical component, the parameter M

_{0}is proportional to the relative representation of the cortical component within the voxel.

#### 2.2. Experimental MRI Data

- Low-resolution modified echo-planar sequence with the following parameters: TR/TE = 1200/39 ms, 105 inversion times from the interval of 50–3000 ms, with the resolution of 3 × 3 × 3 mm
^{3}. The size of the image obtained from this sequence was 64 × 64 × 42 voxels. - High-resolution MPRAGE sequence with the following parameters: TR/TE = 2150/2.5 ms, TI = 1100 ms, with the resolution of 1 × 1 × 1 mm
^{3}. The size of the image obtained from this sequence was 160 × 256 × 256 voxels.

#### 2.3. Simulated Data

_{1}were chosen based on the whole-brain estimates, using the method presented in [21]. A single T

_{1}value was obtained for each voxel of the whole image. The histogram of the relaxation times is presented in Figure 2.

_{1}times of 700 ms and 1000 ms. These represent the T

_{1}relaxation times of white (700 ms) and gray (1000 ms) matter at 3 T, similar to the values found in the literature [27,28].

_{0}were randomly generated while the values of parameters T

_{1}(700, 800, 1100, 1200, 1500, 1700, and 2000 ms) remained constant. Parameters M

_{0}were generated from a uniform distribution to ensure a minimum representation of 5% of each component in the voxel. To make the simulated data more closely resemble the outcome of the MRI experiments, a noise component with the Gaussian distribution with the expected value equal to zero and increasing variance (${\sigma}^{2}=\left\{0,0.1,1,5,10,25,50,100\right\}$) was added. This process resulted in eight datasets with varying levels of noise. An illustration describing the generation of the simulated data is presented in Figure 3.

_{11}= 700 ms, T

_{21}= 80 ms, ρ

_{1}= 0.4 T

_{12}= 800 ms, T

_{22}= 90 ms, ρ

_{2}= 0.6). Given the limitations of the simulator, only two components per voxel could be simulated. The size of the virtual object was chosen based on the properties of the virtual objects supplied with the simulator. Using the defined virtual object and imaging protocol, a series of MRI experiments with different inversion recovery times from 50 ms to 960 ms were simulated. The result was a dataset consisting of 70 images (64 × 64 pixels).

#### 2.4. Modified Trust Region Algorithm

## 3. Results

^{−27}for 100 starting points) with a standard deviation of 0.288 (1.46e

^{−27}for 100 starting points). This contrasts with the values from Table 1, which show an error for individual coefficients up to several hundred percent. Table 1 also shows the difference between the numbers of starting points used to initiate the optimization and the relative error of coefficients. In the case of a single starting point, the maximum relative error for M

_{0}was 604% (ground truth = 61.8, estimated = 435.1), while the optimization with 100 starting points resulted in 0% relative error.

_{0}and the estimates of T

_{1}relaxation times in the literature [28] (fat ${T}_{1}=250\text{}\mathrm{ms}$, CSF ${T}_{1}=4000\text{}\mathrm{ms}$). The results are presented in Table 2.

_{0}. The mean relative error for the coefficients M

_{0}at the SNR level of 45 dB is 28%, while the maximum relative error of a single coefficient is 109%. The relative error of the coefficients T

_{1}is lower than the relative error of the coefficients M

_{0}. At the SNR level of 31 dB, the mean relative error of coefficients T

_{1}is 11%, while the maximum relative error of a single coefficient is 26%.

_{0}and estimates of T

_{1}relaxation times in the literature for parameter T

_{1}. The histogram of the estimated coefficients T

_{1}for all voxels representing the virtual object is presented in Figure 4.

_{1}for the first component is less precise than the estimation for the second component.

## 4. Discussion

_{1}values of each cortical component/layer.

_{1}mapping. It relies on the Levenberg–Marquardt algorithm to solve the underlying optimization problem and estimate the relaxation times [21]. The method has a simplified form used when only magnitude data are available, a procedure named polarity restoration. This process stands for the inversion of select data, circumventing the restrictions posed by the magnitude-only data on the objective function. The function itself has only two parameters per voxel, which leads to a 2D search for the optimal solution. This is further eased by restricting the possible values of T

_{1}times to whole numbers between 1 and 5000 ms. As a direct result, a grid search coupled with a 1D search of the L–M algorithm can be used, increasing the precision of the resulting estimate.

_{1}times within a predefined interval would also cause a substantial increase in the estimation time. The size of the search grid would expand to 5000

^{7}points, and a 7D search would still need to be performed from all the points.

_{1}) increased the complexity of the model and likely introduced nonoptimality traps. To circumvent this limitation, we modified the algorithm in question. The search space for each variable was bounded, and the optimization procedure was initiated from multiple starting points, which resulted in the ideal fit of the model.

_{0}than the coefficients T

_{1}. It could be concluded that the method used is more efficient in the distinction of individual exponential curves (parameter T

_{1}) than in estimating their relative representation (proportional to M

_{0}) within the voxels.

_{1}coefficients estimates under 5%. Estimates of the M

_{0}coefficients could not be fully examined as there is no precise relationship to the proton density ρ used to create the virtual object.

_{1}values.

_{1}relaxation properties. With this approach, it is possible to acquire a proportional representation of the cortical layers in the domain of T

_{1}relaxation, not the spatial domain. However, at this stage, the method is still highly experimental. Its introduction into preclinical studies will not be feasible without first assessing the ability of the method to reliably estimate the proportional representation of the layers. Further steps prior to clinical studies of a diagnostic application may include (i) evaluation of correlations between the identified MR imaging parameters of the cortical layers and selected variables resulting from histology of animal models, (ii) comparison with other, established methods such as voxel-based morphometry, deformation-based morphometry, or diffusion kurtosis imaging. Given the possible clinical application in diagnostics, it is difficult to imagine this modality being used in isolation. Rather, it will be part of a multimodal approach to cortical pathology imaging, as it has the potential to provide complementary information on the internal cortex arrangement in pathologies that have a complex morphological correlate involving changes in elemental complexity, such as neuronal atrophy, dendritic tree reductions with increased density of neuronal bodies, migration of activated microglia, etc. We currently employ an onsite modified pulse sequence to ensure full control of the sequence parameters, over all TIs, to ensure reliable intra-voxel multiple T

_{1}fitting. We are aware of several limitations related to the parameters of a real non-ideal pulse sequence, such as nonzero TE and finite TR. At this stage, we aim for replicable imaging with sufficient sensitivity to identify laminar cortical layer composition, not a quantitative measurement.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Illustration of the data acquisition approach proposed by Lifshits et al. [18]. The experimental data consist of a series of EPI images with different times of inversion. A one-dimensional signal is constructed for every image voxel, dependent on the TI time. The signal is then decomposed into several curves, each representing a voxel component with specific values of M

_{0}and T

_{1}. In this way, a cortical composition of a single voxel series can be decomposed into multiple signals in the T

_{1}relaxation domain.

**Figure 3.**Generating simulated data. (

**A**) Data simulating the signal of a single-voxel series. Individual signals with chosen parameters M

_{0}and T

_{1}are linearly combined, and a noise component of varying power is added. The result is a signal resembling a magnitude series of a single voxel from the experimental data. (

**B**) Simulation of a 2D image. A numerical phantom with two components per voxel is constructed. It is then subject to the experimental EPI sequence, resulting in a series of images.

N_{starting points} = 1 | N_{starting points} = 100 | |||||
---|---|---|---|---|---|---|

Min. Error [%] | Mean Error [%] | Max. Error [%] | Min. Error [%] | Mean Error [%] | Max. Error [%] | |

M_{0} | 0.00 | 44.60 | 604.00 | 0.00 | 0.00 | 0.00 |

T_{1} | 0.00 | 6.11 | 36.00 | 0.00 | 0.00 | 0.00 |

Noise Variance | SNR [dB] | Min. M_{0} Error[%] | Mean M_{0} Error[%] | Max. M_{0} Error[%] | Min. T_{1} Error[%] | Mean T_{1} Error[%] | Max. T_{1} Error[%] |
---|---|---|---|---|---|---|---|

0 | Inf | 0 | 0 | 0 | 0 | 0 | 0 |

0.1 | 61 | 0 | 2 | 4 | 0 | 0 | 0 |

1 | 51 | 2 | 5 | 14 | 0 | 0 | 1 |

5 | 45 | 2 | 28 | 109 | 0 | 2 | 4 |

10 | 41 | 3 | 18 | 48 | 0 | 1 | 3 |

25 | 38 | 3 | 28 | 86 | 0 | 2 | 5 |

50 | 34 | 7 | 80 | 268 | 1 | 10 | 25 |

100 | 31 | 18 | 60 | 131 | 2 | 11 | 26 |

T_{1} | Min. Error [%] | Mean Error [%] | Max. Error [%] |
---|---|---|---|

700 | 0.05 | 4.86 | 8.65 |

800 | 0.12 | 2.98 | 8.59 |

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**MDPI and ACS Style**

Jamárik, J.; Vojtíšek, L.; Churová, V.; Kašpárek, T.; Schwarz, D.
Identification of Laminar Composition in Cerebral Cortex Using Low-Resolution Magnetic Resonance Images and Trust Region Optimization Algorithm. *Diagnostics* **2022**, *12*, 24.
https://doi.org/10.3390/diagnostics12010024

**AMA Style**

Jamárik J, Vojtíšek L, Churová V, Kašpárek T, Schwarz D.
Identification of Laminar Composition in Cerebral Cortex Using Low-Resolution Magnetic Resonance Images and Trust Region Optimization Algorithm. *Diagnostics*. 2022; 12(1):24.
https://doi.org/10.3390/diagnostics12010024

**Chicago/Turabian Style**

Jamárik, Jakub, Lubomír Vojtíšek, Vendula Churová, Tomáš Kašpárek, and Daniel Schwarz.
2022. "Identification of Laminar Composition in Cerebral Cortex Using Low-Resolution Magnetic Resonance Images and Trust Region Optimization Algorithm" *Diagnostics* 12, no. 1: 24.
https://doi.org/10.3390/diagnostics12010024