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rsc.li/soft-matter-journal
Soft Matter
rsc.li/soft-matter-journal
ISSN 1744-6848
PAPER
Karsten Baumgarten and Brian P. Tighe
Viscous forces and bulk viscoelasticity near jamming
Volume 13
Number 45
7 December 2017
Pages 8341-8662
Soft Matter
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Chen, L. Wang, D. Zhu, T. Liu, G. Li and X. Wang, Soft Matter, 2025, DOI: 10.1039/D4SM01194E.
Role of Data-driven Regional Growth Model in Shaping Brain Folding Patterns
Abstract
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Keywords
1. Introduction
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, et al.
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Figure 1. Developmental trajectory of surface area in each region.
, et al.
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, et al.
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2. Methods
2.1. Biomechanics in modeling brain folding
, et al.
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é
2.2. Symbolic regression for discovering growth models
Inspired by Darwinian principles of natural selection, symbolic regression autonomously
uncovers mathematical relationships exclusively from provided data without requiring prior
knowledge, thereby significantly enhancing the interpretability and flexibility of the model
discovery process.37 It has demonstrated promising applications in model characterization38-40 and
parameter calibration.41 Symbolic regression operates through a process known as genetic
programming (GP). During GP execution, functional expressions are efficiently formatted using a
binary-tree structure, which consists of nodes and branches.
A complete tree structure, as illustrated in Figure 2a, involves variables, mathematical
operators (either unary or binary), and constants. The evolution process experiences the genetic
operations of evaluation, selection, mutation, and crossover, while the latter two are essential for
updating the tree structure. Specifically, the mutation operation amplifies genetic
diversity by randomly altering some nodes in an expression tree, as exemplified in Figure 2b,
where a new offspring is generated by substituting the exponential operator (exp) with the
hyperbolic tangent (tanh). Conversely, the crossover operation allows the algorithm to create new
offspring by combining building blocks from different parent candidates, as demonstrated in
Figure 2c. This iterative process of evaluation, selection, mutation, and crossover continues until
the optimal expression is obtained or the maximum number of generations is reached, whichever
is reached first.37
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Figure 2. Structure and operations of expression tree
In this study, we employed symbolic regression to discover appropriate growth models for the
human brain cortex. The raw data includes the measured surface area of each parcellated region in
the developing brain cortex along with the corresponding gestational ages, as shown in Figure 3a.
To facilitate computational implementation, we first converted the data into unitless growth
ratio and virtual time using the following formulas:
where and are tangential and radial growth ratio, respectively. and denotes the
surface area and cortical thickness measured at the initial gestational age, (29 postmenstrual
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weeks), respectively. and corresponds to the maximum and minimal gestational age
within the measured data range, herein, their values are 29 postmenstrual weeks and 24 postnatal
months of age, respectively. Through the above conversion, the value of ranges from 0 to 1,
which serves as the input for the symbolic regression algorithm to find the optimal growth model
for each region, as illustrated in Figure 3a.
Figure 3. Schematic diagram of this research
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SymbolicRegression.jl
2.3. Computational modeling of a developing brain
The identified regional growth models (both tangential and radial) were used to construct the
growth tensor , which can be applied to simulate folding evolution using the FEM, as shown in
Figure 3b. The simulation results were then compared with the brain imaging data for model
validation, as illustrated in Figure 3c. To simulate the folding evolution of the developing brain,
we constructed a three-dimensional double-layer patch model based on the geometries of a human
brain at 29 postmenstrual weeks, as depicted in Figure 3b. Initially, the regional brain inner surface
(the interface between the gray matter and white matter) was extracted using the parcellation map
provided by Huang, et al.30. The extracted surface was first extended by 2-5 mm along the
boundarys local curvature. Laplacian smoothing was then applied to the boundary area using a
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smoothing parameter of 0.2, with the number of iterations ranging from 15 to 22 across all
extracted surfaces. During smoothing, the intermediate area remains fixed to preserve the integrity
of the initial geometry (see Figure S1 in the supplementary material). Subsequently, we
interpolated the extended surface with a flat plane of 80 mm × 80 mm and merged these two
surfaces using Boolean operations. The connecting areas were further smoothed to ensure a natural
transition of the curvature. This extension and interpolation ensured that the models dimensions
were large enough compared to the wavelength of folded patterns observed in experiments, thereby
preventing boundary effects.25 Additionally, the squared boundaries significantly simplified the
prescription of boundary conditions during modeling. We then shifted the interpolated surface
upwards by 2 mm to form the initial cortical layer and extended the squared boundary downwards
by 50 mm to generate the initial white matter layer. This design was based on experimental
observations in neonatal human brains, which indicate that the cerebral cortex is a thin layer with
a thickness of 2-3.5 mm, while the core has a much greater thickness of around 50 mm.43
Consequently, the base models dimensions were approximately 80 mm × 80 mm × 50 mm
(excluding cortical thickness), as illustrated in Figure 4a, where , , and
.
All simulations were performed using the commercial software ABAQUS (Dassault Systems,
Paris, France).44 Dynamic-explicit solver was employed due to its superior performance in solving
nonlinear, dynamic, and larger deformation problems.45, 46 Both the gray and white matters were
modeled as incompressible neo-Hookean materials, with elastic stiffness values of 0.31 kPa for
the cortical and 0.45 kPa for the white matter layer.
Orthotropic growth was defined for the
cortical layer, while isotropic growth was applied to the white matter layer. In our modeling
approach, growth was simulated using thermal expansion, considering the analogy between the
volumetric growth and the thermal expansion.48 The expansion ratio correlates to the growth
ratio as . Specifically, for the white matter layer, the expansion ratio was defined
as , while for the cortical layer, was applied to out-of-plane growth
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and to in-plane growth, as shown in Figure 4b. The customized growth models
( and ), derived from symbolic regression, were implemented into the finite element
algorithm through a user-defined subroutine VUEXPAN. Symmetric boundary conditions were
prescribed on the four sides of the model and the bottom surface of the white matter layer was
fixed. Free boundary conditions were applied to the top surface of the cortical layer, accompanied
by a self-contact constraint to prevent self-penetration. The total simulation time was set to 1 s.
the maximum time step was determined as
. Temperature
variation was applied using a sigmoidal smooth step function.
Figure 4. Geometric model for modeling regional brain growth
Structural meshing with the element type C3D8R was conducted for both the cortical and
white matter layer. To determine the appropriate mesh size, we conducted a mesh sensitivity
analysis with mesh size ranging from 0.3 mm to 0.8 mm (Supplementary Material, Figures S2 and
S3). Based on the mesh convergence analysiswhere simulation results with the coarsest mesh
closely matched those of the finest meshwe selected a mesh size of 0.5 mm for all models. This
results in 84,700 elements for the cortical layer and 278,300 elements for the white matter layer.
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During the simulations, we recorded the coordinates and displacements of the region of interest
(ROI) for each frame. The ROI was defined as the smallest square area encompassing the extracted
brain surface region. Although this definition introduces some redundant areas, which potentially
biases the quantitative measurements, it serves our primary goal: to compare the effectiveness of
the regional growth model with the widely used isotropic growth theory. The inclusion of these
redundant areas does not significantly impact this comparative analysis. Additionally, defining the
ROI in this manner simplifies the partitioning process in ABAQUS and facilitates area
reconstructions during postprocessing. All simulations were performed on a Dell workstation
equipped with a 16-core Intel(R) Xeon(R) CPU E52687 W @ 3.1 GHz, and 64 GB of memory.
2.4. Postprocessing and quantitative metrics
After the simulations, the recorded coordinates and displacements were first extracted from
the result file using Python and subsequently imported into MATLAB to reconstruct the deformed
surface. During reconstruction, the surface was interpolated five times to generate a sufficiently
smooth surface, and the original quadrilateral surface mesh was transformed into a triangular mesh,
facilitating the calculation of quantitative features such as curvatures, gyrification index, and sulcal
depth in MATLAB.
Curvatures: The curvature of a surface describes the degree to which it deviates from being
flat at a given point. Normal curvature is defined as the inverse of the radius of the best-
approximated curve from a surface normal slice in a given direction. Considering all directions,
we obtain the curvature matrix, typically represented by the Weingarten matrix. Its principal
decomposition gives the principal curvatures, which correspond to the maximum and minimum
values of the surfaces normal curvature in different directions ( and ). The average of two
principal curvatures denotes the mean curvature (
, while the product of the
principal curvatures yields the Gaussian curvature (). In this study, we focused on the
mean curvature due to its extensive application in brain cortical folding analysis.49-51
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The curvatures of the triangular mesh were calculated using the method introduced by Meyer,
et al.52, where finite volume discretization was employed to estimate the local integral of mean
curvature over the normal areas of triangular faces associated with each point. If the surrounding
faces are obtuse triangles, the barycentric area was calculated; otherwise, the Voronoi area was
used. However, the calculated mean curvature is dependent on the geometrys shape or size,
meaning its magnitude varies with brain scales. To address this, we further introduced a non-
dimensional measure of mean curvature using the method provided by Balouchzadeh, et al.53,
where the mean curvature is multiplied by a characteristic length ,
where is the surface area. The dimensionless mean curvature was calculated for each point to
provide a qualitative representation, while the absolute value of the dimensionless mean curvature
was averaged across all model points for quantitative comparison. In the remainder of the
manuscript, we use the term curvature to refer to dimensionless mean curvature for clarity.
Gyrification index. To quantitatively describe the folding complexity of the deformed brain
surface, we introduced a global folding metric: the three-dimensional gyrification index (GI). The
GI is defined as the ratio of the total cortical surface area to the area of convex hull that completely
encloses the convoluted surface 8,
To calculate the GI, we first defined a fully enclosed convex hull comprising all points of the
deformed cortical surface, then we discretized and filtered this surface to ensure it completely
encloses the deformed surface with minimum surface area. Finally, we measured the area of
discrete convex hull, which serves as the denominator in the GI calculation.
Sulcal depth: Sulcal depth (SulcDepth) is another quantitative measure capable of reflecting
the extent of the folding in brain regions. Although Numerous methods have been suggested for
computing sulcal depth,51, 54 a well-defined computation remains elusive. In this study, we adopted
the approach introduced by Wang, et al.49, calculating SulcDepth as the distance between the
deformed mesh surface and its convex hull, which was previously defined in calculating the GI.
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Specifically, for each vertex on the deformed surface, we first determined its projection point on
each discrete triangular surface of the convex hull. Subsequently, we computed the distance
between the vertex and its corresponding projection point, with the shortest distance serving as the
sulcal depth for that vertex. Given the convex nature of the enclosed hull, the shortest distance
always exists between the vertex and a consistent piece of the convex hull, as demonstrated in the
Result section. SulcDepth was calculated for each vertex for qualitative representation, and the
values were averaged across all model points to ensure quantitative comparisons.
Evaluation metrics: To evaluate the accuracy of simulation results for the aforementioned
quantitative features, including curvatures, gyrification index, and sulcal depth, we introduced the
following statistical metrics: , mean absolute percentage error (), and Pearson correlation
coefficient (). Here, measures the goodness-of-fit of the simulation results, formulated as
, where denotes the number of data points, represents
the actual feature values measured from real brain data, is the mean of actual values, and is
the predicted value from the simulation. An value close to 1 indicates a strong agreement
between the simulation results and real brain imaging data. , expressed as a percentage,
quantifies the average absolute error between predicted () and actual values (). It is calculated
using the formula:
Moreover, we introduced the Pearson correlation coefficient () to assess the strength and direction
of the linear relationship between the simulation results and real brain measures, with its formula
as:
where is the mean of actual values. All these evaluation metrics were calculated in Python using
the sklearn and scipy libraries. Noted, due to the discrepancy in the number of data points between
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the simulation results and the actual data, we applied an interpolation method to resample the
extracted simulation data, ensuring a consistent data size before computing the evaluation metrics.
2.5. Brain imaging data
.
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Figure 5. Raw data of cortical surface area and thickness for five selected regions
p
compare the effectiveness of the regional growth model
with classic growth theories. Therefore, we selected five representative regions that exhibit
significant distinctions in surface area and cortical thickness. These regions are slow-growing
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Region 1, medium-growing Regions 4, 9, and 11, and fast-growing Region 16, corresponding to
respectively. As illustrated in Figure 5, the development of the surface area and cortical thickness
from 29 postmenstrual weeks to 2 years of age exhibits significant differences among these regions,
with the p-values all smaller than 0.001. Here, we conducted paired Student t-test to compute the
significance of these differences, with the null hypothesis being no difference in surface area or
cortical thickness between the two regions.
3. Results
3.1. Regional growth models identified from symbolic regression
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postnatal
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Figure 6. Tangential growth models discovered for five selected regions
Figure 7. Radial growth models discovered for five selected regions
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, et al.
3.2. Regional growth models accurately simulate folding evolutions patterns
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Figure 8. Longitudinal brain developing patterns for five regions
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Figure 9. Simulated folding patterns vs realistic brain images
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postnatal-
Figure 10. Mean curvature and sulcal depth
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Figure 11. Quantitative comparison of mean curvature, sulcal depth, and gyrification index among five
regions.
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3.3. Regional growth models outperform classic unified growth models
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Figure 12. Symbolic regression growth model vs classic growth model.
3.4. Growth ratio values influence folding evolution more than growth trajectory
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Figure 13. Three distinct growth models for region 9.
Figure 14. Impact of growth trajectory on the folding patterns.
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, et al.
, et al.
3.5. Multi-region model provides more realistic folding results than single-regional model
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Figure 15. Brain folding patterns of a multi-region model.
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Figure 16. Multi-region model vs single-region model.
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4. Discussion
, et al.
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Conducting simulations on a single brain region allows for comparative analysis of distinct growth
theories and their effects on brain folding. However, this approach may lead to unrealistic folding
patterns due to the artificially imposed boundary conditions. Our findings indicate that a multi-
region computational model, which considers three adjacent regions simultaneously, offers a more
reliable result by producing more uniformly distributed folding patterns. In the future, a brain-wide
model encompassing all 18 parcellated regions is expected to yield more realistic folding
predictions, by integrating regional growth models derived through symbolic regression. Moreover,
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our study can be improved by addressing the following issues: First, we assumed uniform cortical
thickness across each brain region. Incorporating anatomically accurate cortical thickness,
accounting for both gray and white matter in the model construction process, would provide a
more convincing geometric model. Second, the tangential growth within the cortical layer was
assumed to be uniform. In reality, this growth varies spatially, as evident in differential growth
within the six-layered cortex.90 Future studies should consider adopting a spatially dependent
growth profile, as proposed by Tallinen, et al.8. Third, in current study, the tangential and radial
growth models were characterized based on different datasets, as shown in Figures 6 and 7. Future
studies that integrate surface area and cortical thickness measurements from the same dataset
would significantly enhance the integrity and rigor of the predicted growth model. Last but not
least, the brain tissue in our model was treated as an incompressible hyperelastic material described
by the neo-Hookean strain energy function. Incorporating a regional hyperelastic model with a
degree of compressibility, characterized though symbolic regression, could account for the
heterogeneity in stiffness and further enhance the reliability of our simulation results.
5. Conclusion
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Author Contributions
Conflicts of Interest
Data Availability
.
Acknowledgements
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Reference
Cerebral Cortex18
Cerebral Cortex31
Autism research15
Neuroepidemiology54
JAMA psychiatry77
Frontiers in cellular neuroscience13
Seminars in Cell & Developmental Biology140
Proceedings of the National Academy of
Sciences111
Cerebral Cortex22
Developmental cell52
Frontiers in Cell and Developmental Biology9
J Mech Behav Biomed Mater29
Journal of Neuroscience38
Nature385
Cerebral Cortex22
J Biomech Eng
132
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Cerebral Cortex24
Developmental neuropsychology24
Science189
Journal of
Biomechanics139
Physical Biology10
Journal of the Mechanics and Physics of Solids72
Brain Multiphysics2
Physical review letters121
Cerebral Cortex Communications2
Journal of Theoretical Biology
264
Extreme Mechanics Letters18
Cerebral Cortex
34
Cerebral Cortex19
Proceedings of the National
Academy of Sciences119
Proceedings
of the National Academy of Sciences116
International Journal of Solids and Structures132
Biomechanics and Modeling in Mechanobiology20
Scientific Reports6
Journal of biomechanics27
Nonlinear solid mechanics: a continuum approach for engineering science
Archives of Computational Methods in Engineering
30
International Journal for Numerical Methods in Engineering 125
arXiv preprint arXiv:2402.05238
Computer Methods in Applied Mechanics and
Engineering419
Construction and Building Materials280
arXiv preprint arXiv:2305.01582
Journal of
Magnetic Resonance Imaging53
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Dassault Systemes Simulia Corporation, Providence, RI, USA3
Cerebral Cortex33
Computer Methods in Biomechanics and Biomedical Engineering
Journal of the Mechanical Behavior
of Biomedical Materials76
Acta Mechanica Solida Sinica25
Scientific Reports
11
Human Brain Mapping43
Human Brain
Mapping41
Brain Multiphysics4
PloS one8
Neuroimage173
Magnetic resonance in medicine78
Developmental Cognitive Neuroscience63
PLOS ONE7
Cerebral Cortex34
Medical Image Analysis25
Human Brain Mapping40
Cerebral Cortex25
NeuroImage268
Journal of
Neuroscience29
Proceedings of the National Academy of Sciences
116
Nature Reviews Neuroscience19
Nature Physics
12
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Human Brain Mapping
39
Scientific Reports
13
Philosophical Transactions of the Royal Society B-
Biological Sciences373
J Neurosurg Pediatr21
Neuroimage68
PLoS computational biology16
Scientific Reports5
Brain
Multiphysics3
Nature Communications12
The European Physical Journal
Special Topics229
Annals of biomedical engineering43
Malaria Journal11
Journal of the neurological sciences
216
Journal of the Mechanical Behavior of Biomedical
Materials150
arXiv preprint arXiv:2310.10762
Computer Methods in Applied Mechanics and Engineering
405
Acta Biomaterialia160
Journal of the Mechanics and Physics of Solids112
Brain Multiphysics2
Extreme Mechanics Letters4
Human Brain Mapping43
Acta biomaterialia99
PLOS Computational Biology18
Journal
NeuroImage185
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Data Availability Statement
The original contributions presented in the study are included in the
article/supplemental material. Further inquiries can be directed to the corresponding
authors. The dHCP dataset is publicly available at the Developing Human Connectome
Project repository: http://www.developingconnectome.org. The BCP dataset is publicly
available in NIMH Data Archive: https://nda.nih.gov/edit_collection.html?id=2848.
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Open Access Article. Published on 02 January 2025. Downloaded on 1/3/2025 10:20:58 AM.
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DOI: 10.1039/D4SM01194E