Exploring the role of the outer subventricular zone during cortical folding through a physics-based model

  1. Mohammad Saeed Zarzor
  2. Ingmar Blumcke
  3. Silvia Budday

  • Curated by eLife

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    Through theoretical analysis, the authors argue that the proliferation of neurons in the outer subventricular zone, which is specific to humans, decreases the distance between neighboring sulci in the cerebral cortex and increases cell density in the ventricular zone. Though the exact mechanisms remain to be further elucidated, the compelling data and approach represent a valuable foundation for the study of cortical folding from the underpinning cellular level as well as the coupling role of mechanics and cellular biology. This study will be of particular interest to the large community of scientists studying the mechanisms of brain development and disorder and even possibly beyond.

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Abstract

The human brain has a highly complex structure both on the microscopic and on the macroscopic scales. Increasing evidence has suggested the role of mechanical forces for cortical folding – a classical hallmark of the human brain. However, the link between cellular processes at the microscale and mechanical forces at the macroscale remains insufficiently understood. Recent findings suggest that an additional proliferating zone, the outer subventricular zone (OSVZ), is decisive for the particular size and complexity of the human cortex. To better understand how the OSVZ affects cortical folding, we establish a multifield computational model that couples cell proliferation in different zones and migration at the cell scale with growth and cortical folding at the organ scale by combining an advection-diffusion model with the theory of finite growth. We validate our model based on data from histologically stained sections of the human fetal brain and predict 3D pattern formation. Finally, we address open questions regarding the role of the OSVZ for the formation of cortical folds. The presented framework not only improves our understanding of human brain development, but could eventually help diagnose and treat neuronal disorders arising from disruptions in cellular development and associated malformations of cortical development.

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  1. Version published to 10.7554/elife.82925 on eLife
  2. eLife

    Author Response

    Reviewer #1 (Public Review):

    After giving a very accessible introduction to cellular processes during brain development, the authors present the computational model used in this study. It combines the kinematics of cell proliferation with the mechanic of brain tissue growth and is essentially equal to their model presented in Zarzor et al (2021), but extended for the outer subventricular zone (OSVZ), see for example Figs. 2 in the present manuscript and in Zarzor et al (2021). This zone, which is specific to humans, provides a second zone of cell proliferation. The division rate in the OSVZ is smaller and at most equal to that in the ventricular zone.

    The authors present two main findings: The distance between sulci in the cortex is decreased whereas the cell density in the ventricular zone is increased in presence of the OSVZ. Furthermore, the "folding evolution", which is the ratio between the outer perimeter at time t and the initial perimeter increases in presence of the OSVZ. The strongest effect is seen, when division rates in both proliferating zones are equal. The authors compare the cases of varying and constant cortical stiffness, which they had also done in Zarzor et al (2021). Finally, they consider the feedback of cortical folding on OSVZ thickness.

    The computational model provides a sound description of how cell proliferation and migration combined with tissue mechanics yield cortical folding patterns. However, only a few parameter values are varied in a limited range. Also, it remains unclear to me, how important the specific functional dependencies of, for example, the cell division rate on the radial coordinate are. This point seems of particular importance because the effect of the presence of the OSVZ on the folding patterns seems rather minute, see Fig. 5. The authors do not propose experiments that could be used to test their description and results. Finally, the analysis is restricted to 2 dimensions.

    Thank you very much for the valuable suggestions. We agree that we are only able to show limited parameter studies in the manuscript. Therefore, we have now implemented a user interface that can be downloaded from Github (https://github.com/SaeedZarzor/BFSimulator) and will allow interested readers to directly change the parameter values and run the simulations.

    To better emphasize the effect of the presence of the OSVZ on the folding patterns, we have edited the corresponding section and figure in the revised manuscript to include a quantification of the distance between sulci:

    “In general, the distance between neighboring sulci decreases with increasing Gosvz, as marked in Figure 7. For the displayed cases, the distance decreases from d = 8.796 mm for Gosvz = 0 to d = 8.67 mm for Gosvz = 10 and finally d = 8.2 mm for Gosvz = 20. Interestingly, the cortical thickness and effective stiffness ratio at the first instability point (denoted by w in Figure 5) are the same for all these cases. Therefore, we attribute the observed differences to the faster increase in the cell density and thus cortical growth, cortical stiffness and the effective stiffness after the instability has been initiated.”

    In addition, we have added a new figure to show that the observed trends also hold true for 3D simulations:

    “Figure 8 demonstrates that the observed trends also hold true when extending the model to 3D. For the case of varying stiffness with a stiffness ratio of 3, a growth ratio of 3, and an initial division rate in the ventricular zone Gvz = 600, the folding complexity increases with increasing initial division rate in the OSVZ Gosvz.”

    Reviewer #2 (Public Review):

    Weaknesses

    • To account for the complexity of biological phenomena, the model relies on a large number of ad hoc choices whose consequences are difficult to predict.

    We fully agree that there are quite a number of model assumptions that we have to make. Still, we have achieved great agreement with the data from fetal brain sections, which in our opinion justified the assumptions made.

    To better explain the choice of parameters, we have now included the following paragraph in the manuscript: “The mechanical and diffusion parameters are adapted from the literature Budday et al. (2020); de Rooij and Kuhl (2018), while the geometry parameters are estimated based on histologically stained human brain sections and magnetic resonance images. For instance, to determine the MST factor, we measured the relative distance between the ISVZ and OSVZ in histologically stained images. The final value adopted is the result of dividing the measured distance by the expected time. When determining the growth problem parameters, numerical stability and algorithm convergence were major criteria.”

    • The physical model description is highly technical and out of reach for a non-specialist.

    Thank you for making this point! We have now adapted the model description to better emphasize the main features of the model and the feedback mechanisms between the mechanical growth problem and the cell density problem:

    “...is the Cauchy stress tensor formulated in terms of the elastic deformation tensor, as only the elastic deformation induces stresses. The Cauchy stress describes the three dimensional stress state in the spatial (grown and deformed) configuration and is computed by deriving the strain energy function…”

    “Through Equation 6, the cell density problem controls the effective stiffness ratio between cortex and subcortex (as the cortical stiffness changes while the subcortical stiffness remains constant) and thus also the emerging cortical folding pattern Budday et al. 2014; Zarzor et al. 2021.”

    “Through Equation 8, the amount of growth is directly related to the cell density - the higher the cell density, the more growth.”

    “The vector n represents the normalized orientation of radial glial cell fibers in the spatial configuration and controls the migration direction of neurons. As the brain grows and folds, the fiber direction changes. Through this feedback mechanism, the mechanical growth problem affects how neurons migrate and the cell density evolves locally.”

    “By applying Equation 16 for the VZ, we ensure that the division rate decreases from its initial value G_vz to a smaller value as the maximum stretch value s in the domain increases, i.e., with increasing gestational age. This constitutes an additional feedback mechanism between the mechanical growth problem and the cell density problem: As the maximum stretch and thus the deformation increases due to constrained cortical growth, the division rate in the VZ decreases, resulting in less newborn cells” and “G^s_osvz is the division rate in the OSVZ that decreases with increasing maximum stretch s in the domain”

    • The description of neurogenesis shows three zones of cell proliferation, each inhabited by a specific cell type. Despite its realism, the proposed model does not take into account the ISVZ where the intermediate progenitors operate.

    Indeed, in our model we have focused on two original sources of the cells which are radial glial cells and ORGCs. As we know so far, the intermediate progenitor cells are produced from those two cell types, so they are indirectly included in the model as a resulting cell density.

    • The experiment of comparing several regimes derived from the relative importance of proliferation in the VZ and OSVZ is not very clear. It leads to the observation of the evolution of cell density maxima over time, which seems insufficient to conclude the importance of the OSVZ for folding. One wonders whether the key parameter that leads to folding is the rate of OSVZ proliferation or simply the total quantity of neurons generated by the two or even the three zones.

    Thank you for this remark. We fully agree with the Reviewer that a key factor is the total quantity of neurons generated. However, the major question we intend to address here is where these neurons originate from and how the different proliferating zones interact. In other words, we do not question the existence of the OSVZ, but we are trying to build a computational model that can mimic all relevant cellular processes during brain development - to then study their individual effect on cortical folding. Therefore, we do not argue that the OSVZ is necessary for folding, but that it plays a crucial role in the speed of generating these folds and their complexity in the Conclusion section:

    “Our results show that the existence of the OSVZ particularly triggers the emergence of secondary mechanical instabilities leading to more complex folding patterns. Furthermore, the proliferation of outer radial glial cells (ORGCs) reduces the time required to induce the mechanical instability and thus cortical folding.”

    • The experiment on the heterogeneity of proliferation in the OSVZ is a bit frustrating. I would like to see a set-up corresponding to the mosaics found in ferrets and closely associated with folding patterns.

    This is a valuable point, thank you! We have now added new results showing a more distinct regional variation of the OSVZ and have adapted our conclusions regarding this point:

    “Also in the ferret brain, where a region close in structure to the primate's OSVZ was found, this region shows a unique mosaic-like structure Fietz et al. (2010b); Reillo and Borrell (2012). In this section, we aim to assess the effect of regional proliferation variations in the OSVZ on the emerging cortical folding pattern. We discuss two different heterogeneous patterns here, but have included more variations online through our user interface on GitHub, as described in the Data availability section. In the first case, the OSVZ division rate gradually decreases along the circumferential direction. In the second case, the division rate varies in a more random pattern. Figures 13 and 14 show how cortical folds develop in both cases for the varying cortical stiffness case, a division rate in the VZ of G_vz = 120 and an initial division rate in the OSVZ of G_osvz = 20. As expected, the evolving folding patterns slightly differ. In both cases, the first folds appear, where the cell proliferation rate is highest. Expectedly, those regions also show a higher cell density in the cortex than regions nearby. However, both cases lead to final patterns with similar distances between sulci and folding complexity (one period doubling pattern). In addition, gyri and sulci are distributed equally -- regardless of the division rate. Therefore, we may conclude that inhomogeneous cell proliferation in the OSVZ controls the location of first gyri and sulci but does not necessarily affect the distance between sulci (also referred to as folding wavelength) and the overall complexity of the emerging folding pattern. This agrees well with our previous finding that the characteristic wavelength of folding remains relatively stable for inhomogeneous cortical growth patterns Budday and Steinmann (2018). The simulation results are also consistent with the previously found remarkable surface expansion above the regions with higher proliferation in the OSVZ Llinares-Benadero and Borrell (2019).”

    “Finally, our simulations reveal that inhomogeneous cell proliferation patterns in the OSVZ can control the location of first gyri and sulci but do not necessarily affect the distance between sulci and the overall complexity of the emerging folding pattern.”

    Furthermore, in our code, we have added a user interface with multiple options for different OSVZ regional variations. The link to the code with the user interface shown below is now updated in the Data availability section.

    • It would be interesting to elaborate a little on the possibility of extending the model in 3D, which seems imperative to evaluate the nature of the folding pattern generated. Comparing them to reality is an essential step in gauging the credibility of the model. For instance, it would be interesting to test to which extent the model can father the type of variability observed in the general population (Mangin et al.). It will also be particularly interesting to work on the inverse model between the real folding patterns and the heterogeneous proliferation maps that can generate them.

    We fully agree with the Reviewer. Unfortunately, to the best of the Author’s knowledge, there is currently no data set providing both the 3D evolution of the folding pattern and the corresponding distribution of the cell density. Therefore, the validation of 3D results is difficult. Promisingly, our model achieved good agreement with data from histologically stained fetal brain sections regarding the local gyrification index, final cortical thickness, and cell density distribution, as presented in Zarzor, et al (2021). We have indeed initiated the collection of additional data, ideally for the 3D validation. However, this will take some time and is out of the scope of the current work. It is also a great suggestion to compare our 3D simulation results with the variability found in the general population. Indeed, we plan to do such work in the future but consider this out of the scope of the current work, which focuses more on the OSVZ.

    To still show that our model can be extended to 3D, we have now included the following results: “Figure 8 demonstrates that the observed trends also hold true when extending the model to 3D. For the case of varying stiffness with a stiffness ratio of 3, a growth ratio of 3, and an initial division rate in the ventricular zone G_vz = 600, the folding complexity increases with increasing initial division rate in the OSVZ G_osvz.”

    Reviewer #3 (Public Review):

    Zarzor et al. developed a new multifield computational model, which couples cell proliferation and migration at the cellular level with biological growth at the organ level, to study the effect of OSVZ on cortical folding. Their approach complements the classical experimental approach in answering open questions in brain development. Their simulation results found the existence of OSVZ triggers the emergence of secondary mechanical instabilities that leads to more complex folding patterns. Also, they found that mechanical forces not only fold the cortex but also deepen subcortical zones as a result of cortical folding. Their physics-based computational modeling approach offered a novel way to predictively assess the links between cellular mechanisms and cortical folding during early human brain development, further shedding light on identifying the potential controlling parameters for reverse brain study.

    Strengths:

    The newly developed physics-based computational model has several advantages compared to previous existing computational brain models. First, it breaks the traditional double-layer computational brain model, gray matter layer and white matter layer, by introducing the outer subventricular zone. Second, it develops multiscale computational modeling by bringing the cellular level features, cell diffusion, and migration, into the macroscale biological growth model. Third, it could provide a cause-effect analysis of cortical folding and axonal fiber development. Finally, their approach could complement, but not substitute, sophisticated experimental approaches to answer some open questions in brain science.

    Weaknesses:

    The cellular diffusion and migration seem determined and controlled by a single variable, cell density, which is one-way coupled with the deformation gradient of the brain model. However, cell migration and diffusion should be potentially coupled with stress and vice versa. Also, the current computational model can be improved by extending it to a 3D model. Finally, they can further improve the study of regional proliferation variation by introducing fully-randomized heterogenous cell density and growth in their model.

    Thank you. We apologize for the lack of clarity in the original submission. There are indeed more coupling mechanisms, which we have now better emphasized when introducing the model:

    “Through Equation 6, the cell density problem controls the effective stiffness ratio between cortex and subcortex and thus also the emerging cortical folding pattern Budday et al. 2014; Zarzor et al. 2021.”

    “Through Equation 8, the amount of growth is directly related to the cell density - the higher the cell density, the more growth.”

    “The vector n represents the normalized orientation of radial glial cell fibers in the spatial configuration and controls the migration direction of neurons. As the brain grows and folds, the fiber direction changes. Through this feedback mechanism, the mechanical growth problem affects how neurons migrate and the cell density evolves locally.”

    “By applying Equation 16 for the VZ, we ensure that the division rate decreases from its initial value Gvz to a smaller value as the maximum stretch value s in the domain increases, i.e., with increasing gestational age. This constitutes an additional feedback mechanism between the mechanical growth problem and the cell density problem: As the maximum stretch and thus the deformation increases due to constrained cortical growth, the division rate in the VZ decreases, resulting in less newborn cells” and “Gosvzs is the division rate in the OSVZ that again decreases with increasing maximum stretch s in the domain”

    In addition, we have added a new figure to show that the observed trends also hold true for 3D simulations:

    “Figure 8 demonstrates that the observed trends also hold true when extending the model to 3D. For the case of varying stiffness with a stiffness ratio of 3, a growth ratio of 3, and an initial division rate in the ventricular zone Gvz = 600, the folding complexity increases with increasing initial division rate in the OSVZ Gosvz.”

    Finally, we have added new results showing a more distinct regional variation of the OSVZ. Furthermore, in our code, we have added a user interface with multiple options for different OSVZ regional variations. The link to the code with user interface is available in the paper:

    “Also in the ferret brain, where a region close in structure to the primate's OSVZ was found, this region shows a unique mosaic-like structure Fietz et al. (2010b); Reillo and Borrell (2012). In this section, we aim to assess the effect of regional proliferation variations in the OSVZ on the emerging cortical folding pattern. We discuss two different heterogeneous patterns here, but have included more variations online through our user interface on GitHub, as described in the Data availability section. In the first case, the OSVZ division rate gradually decreases along the circumferential direction. In the second case, the division rate varies in a more random pattern. Figures 13 and 14 show how cortical folds develop in both cases for the varying cortical stiffness case, a division rate in the VZ of G_vz = 120 and an initial division rate in the OSVZ of G_osvz = 20. As expected, the evolving folding patterns slightly differ. In both cases, the first folds appear, where the cell proliferation rate is highest. Expectedly, those regions also show a higher cell density in the cortex than regions nearby. However, both cases lead to final patterns with similar distances between sulci and folding complexity (one period doubling pattern). In addition, gyri and sulci are distributed equally -- regardless of the division rate. Therefore, we may conclude that inhomogeneous cell proliferation in the OSVZ controls the location of first gyri and sulci but does not necessarily affect the distance between sulci (also referred to as folding wavelength) and the overall complexity of the emerging folding pattern. This agrees well with our previous finding that the characteristic wavelength of folding remains relatively stable for inhomogeneous cortical growth patterns Budday and Steinmann (2018). The simulation results are also consistent with the previously found remarkable surface expansion above the regions with higher proliferation in the OSVZ Llinares-Benadero and Borrell (2019).”

  3. eLife

    eLife assessment

    Through theoretical analysis, the authors argue that the proliferation of neurons in the outer subventricular zone, which is specific to humans, decreases the distance between neighboring sulci in the cerebral cortex and increases cell density in the ventricular zone. Though the exact mechanisms remain to be further elucidated, the compelling data and approach represent a valuable foundation for the study of cortical folding from the underpinning cellular level as well as the coupling role of mechanics and cellular biology. This study will be of particular interest to the large community of scientists studying the mechanisms of brain development and disorder and even possibly beyond.

  4. eLife

    Reviewer #1 (Public Review):

    After giving a very accessible introduction to cellular processes during brain development, the authors present the computational model used in this study. It combines the kinematics of cell proliferation with the mechanic of brain tissue growth and is essentially equal to their model presented in Zarzor et al (2021), but extended for the outer subventricular zone (OSVZ), see for example Figs. 2 in the present manuscript and in Zarzor et al (2021). This zone, which is specific to humans, provides a second zone of cell proliferation. The division rate in the OSVZ is smaller and at most equal to that in the ventricular zone.

    The authors present two main findings: The distance between sulci in the cortex is decreased whereas the cell density in the ventricular zone is increased in presence of the OSVZ. Furthermore, the "folding evolution", which is the ratio between the outer perimeter at time t and the initial perimeter increases in presence of the OSVZ. The strongest effect is seen, when division rates in both proliferating zones are equal. The authors compare the cases of varying and constant cortical stiffness, which they had also done in Zarzor et al (2021). Finally, they consider the feedback of cortical folding on OSVZ thickness.

    The computational model provides a sound description of how cell proliferation and migration combined with tissue mechanics yield cortical folding patterns. However, only a few parameter values are varied in a limited range. Also, it remains unclear to me, how important the specific functional dependencies of, for example, the cell division rate on the radial coordinate are. This point seems of particular importance because the effect of the presence of the OSVZ on the folding patterns seems rather minute, see Fig. 5. The authors do not propose experiments that could be used to test their description and results. Finally, the analysis is restricted to 2 dimensions.

  5. eLife

    Reviewer #2 (Public Review):

    Gyrencephaly has been linked to the split of the subventricular zone (SVZ) and the formation of an outer subventricular zone (OSVZ) during neurogenesis. This paper proposes a convincing multizone computational model of neurogenesis allowing exploration of the role of this OSVZ in the folding dynamics. This model is a bridge between knowledge of cell proliferation and migration and the physics of growth.

    Strengths
    • The computational model described in this paper is probably the most ambitious to date. It succeeds in translating the complexity of microscopic biological phenomena that describe cell proliferation and migration into physical phenomena from continuum mechanics. It is truly a tour de force.
    • The description of neurogenesis is particularly clear, within the reach of a naive reader despite its complexity. The figure illustrating the chronology of the phenomena at work is a success.
    • The paper builds on impressive efforts to estimate from real human brain sections some of the complex parameters of the model such as the density of cells at different stages of migration.
    • The physical model is able to show ripples in the deep zones of proliferation that seem induced by the folding of the cortex. This observation is consistent with feedback from folding on the organization of the migration, as these ripples are not part of the model. I do not know to what extent these ripples have been demonstrated in reality.
    • The model shows that significant proliferation in the OSVZ leads to a doubling of the frequency of folding, a phenomenon observed in reality in large brains, which gives rise to allometric laws between folding and brain size (see Toro et al., Germanaud et al.)
    • The paper includes an experiment based on heterogeneous proliferation in the OSVZ, which is difficult to model in more classical models such as Tallinen's one. This is a particularly interesting possibility for modelling spatial heterogeneity in the expression of genes that modulate neurogenesis (see Llinares-Benadero et al.).

    Weaknesses
    • To account for the complexity of biological phenomena, the model relies on a large number of ad hoc choices whose consequences are difficult to predict.
    • The physical model description is highly technical and out of reach for a non-specialist.
    • The description of neurogenesis shows three zones of cell proliferation, each inhabited by a specific cell type. Despite its realism, the proposed model does not take into account the ISVZ where the intermediate progenitors operate.
    • The experiment of comparing several regimes derived from the relative importance of proliferation in the VZ and OSVZ is not very clear. It leads to the observation of the evolution of cell density maxima over time, which seems insufficient to conclude the importance of the OSVZ for folding. One wonders whether the key parameter that leads to folding is the rate of OSVZ proliferation or simply the total quantity of neurons generated by the two or even the three zones.
    • The experiment on the heterogeneity of proliferation in the OSVZ is a bit frustrating. I would like to see a set-up corresponding to the mosaics found in ferrets and closely associated with folding patterns.
    • It would be interesting to elaborate a little on the possibility of extending the model in 3D, which seems imperative to evaluate the nature of the folding pattern generated. Comparing them to reality is an essential step in gauging the credibility of the model. For instance, it would be interesting to test to which extent the model can father the type of variability observed in the general population (Mangin et al.). It will also be particularly interesting to work on the inverse model between the real folding patterns and the heterogeneous proliferation maps that can generate them.

    Conclusion

    The computational model of neurogenesis described in this paper is the most sophisticated model proposed to date. It is a convincing step towards a model that could one day simulate perturbations of neurogenesis that may give rise to the gyration abnormalities observed in certain developmental pathologies. A better understanding of the genesis of these anomalies could contribute to their use as a signature of hidden deleterious events occuring during neurogenesis.

    References

    Toro, R., Perron, M., Pike, B., Richer, L., Veillette, S., Pausova, Z., & Paus, T. (2008). Brain size and folding of the human cerebral cortex. Cerebral cortex, 18(10), 2352-2357.
    Germanaud, D., Lefèvre, J., Toro, R., Fischer, C., Dubois, J., Hertz-Pannier, L., & Mangin, J. F. (2012). Larger is twistier: spectral analysis of gyrification (SPANGY) applied to adult brain size polymorphism. NeuroImage, 63(3), 1257-1272.
    Tallinen, T., Chung, J. Y., Rousseau, F., Girard, N., Lefèvre, J., & Mahadevan, L. (2016). On the growth and form of cortical convolutions. Nature Physics, 12(6), 588-593.
    Llinares-Benadero, C., & Borrell, V. (2019). Deconstructing cortical folding: genetic, cellular and mechanical determinants. Nature Reviews Neuroscience, 20(3), 161-176.
    Mangin, J. F., Le Guen, Y., Labra, N., Grigis, A., Frouin, V., Guevara, M., ... & Sun, Z. Y. (2019). "Plis de passage" deserve a role in models of the cortical folding process. Brain topography, 32(6), 1035-1048.

  6. eLife

    Reviewer #3 (Public Review):

    Zarzor et al. developed a new multifield computational model, which couples cell proliferation and migration at the cellular level with biological growth at the organ level, to study the effect of OSVZ on cortical folding. Their approach complements the classical experimental approach in answering open questions in brain development. Their simulation results found the existence of OSVZ triggers the emergence of secondary mechanical instabilities that leads to more complex folding patterns. Also, they found that mechanical forces not only fold the cortex but also deepen subcortical zones as a result of cortical folding. Their physics-based computational modeling approach offered a novel way to predictively assess the links between cellular mechanisms and cortical folding during early human brain development, further shedding light on identifying the potential controlling parameters for reverse brain study.

    Strengths:
    The newly developed physics-based computational model has several advantages compared to previous existing computational brain models. First, it breaks the traditional double-layer computational brain model, gray matter layer and white matter layer, by introducing the outer subventricular zone. Second, it develops multiscale computational modeling by bringing the cellular level features, cell diffusion, and migration, into the macroscale biological growth model. Third, it could provide a cause-effect analysis of cortical folding and axonal fiber development. Finally, their approach could complement, but not substitute, sophisticated experimental approaches to answer some open questions in brain science.

    Weaknesses:
    The cellular diffusion and migration seem determined and controlled by a single variable, cell density, which is one-way coupled with the deformation gradient of the brain model. However, cell migration and diffusion should be potentially coupled with stress and vice versa. Also, the current computational model can be improved by extending it to a 3D model. Finally, they can further improve the study of regional proliferation variation by introducing fully-randomized heterogenous cell density and growth in their model.

  7. Version published to 10.1101/2022.09.25.509401v1 on bioRxiv