Modeling and computation of growth in soft biological matter

在不均匀生长的生物软组织中，由于机械不稳定性而产生的褶皱现象会引起各种各样的表面形态。应用范围从哮喘呼吸道的不受欢迎的折叠，到皮肤的皱纹，到大脑卷曲，它最大限度地增加了神经元的数量，缩短了它们之间的距离。在这里，我们主要使用大脑发育过程中皮质折叠的例子来研究生长诱导的稳定性丧失。一方面，大脑仍然是我们最不了解的器官。另一方面，脑生长既包括生长诱导的机械不稳定性，也包括机械诱导的生物生长，包括形态发生学生长的浅层灰质层和拉伸诱导的白质基质生长。使用力学的非线性场论和有限增长理论，在项目的第一阶段，我们建立了一个初步的脑增长计算模型，它提供了对增长诱导的初级和次级不稳定性的第一次洞察：外层的适度增长会产生规则的正弦皱纹图案；进一步的持续增长会导致与高级皱纹模式相关的次要不稳定性-表面分成越来越复杂的形态。在项目的第二阶段，我们将改进我们的模型，使其更符合实际的本构方程和生长规律。为此，我们将用脑组织的生物力学测试来补充我们的计算研究。我们将结合材料性能的时间和区域变化，并密切考虑相应的微观结构。这样的实验研究不仅可以对大脑发育进行真实的模拟，而且还可以彻底改变对脑组织的计算建模。因此，我们的项目相关工作还可以为脑瘤等疾病的医疗治疗和创伤性脑损伤等损伤的预防做出贡献。通过我们的计算模型，我们将探索健康和病理性大脑发育的主要方面。由于大脑结构与大脑功能密切相关，皮质发育畸形是癫痫和发育迟缓等精神疾病的常见原因。计算模型可以从细胞水平上的大量研究的破坏到器官水平上的形式和功能之间架起桥梁。它可以解释为什么会出现皮质畸形。了解大脑发育的潜在机制将加强对皮质畸形的早期诊断，并最终促进精神障碍的治疗和预防。

Wrinkling phenomena emerging from mechanical instabilities in inhomogeneously growing soft biological tissue can evoke a wide variety of surface morphologies. Applications range from undesired folding of asthmatic airways, via wrinkling of skin, to brain convolutions, which maximize the number of neurons and minimize their distance. Here, we study growth-induced loss of stability primarily using the example of cortical folding during brain development. On the one hand the brain still remains our least understood organ. On the other hand brain growth involves both growth-induced mechanical instabilities and mechanically-induced biological growth with a morphogenetically growing superficial gray matter layer and a stretch-induced growing white matter substrate. Using the nonlinear field theories of mechanics supplemented by the theory of finite growth, in the first phase of the project we have established a preliminary computational model of brain growth, which provides first insights into growth-induced primary and secondary instabilities: Moderate growth in the outer layer generates a regular pattern of sinusoidal wrinkles; further continuing growth induces secondary instabilities associated with advanced wrinkling modes - the surface bifurcates into increasingly complex morphologies. During the second phase of the project, we will refine our model regarding more realistic constitutive equations and growth laws. For this purpose, we will supplement our computational investigations with biomechanical testing of brain tissue. We will incorporate temporal and regional variations in material properties with close consideration of the corresponding microstructure. Such experimental studies will not only allow for realistic simulations of brain development but could also revolutionize computational modeling of brain tissue in general. Accordingly, our project related work could additionally contribute to the medical treatment of diseases such as brain tumors and the prevention of injuries such as traumatic brain injury. With our computational model we will explore major aspects of healthy and pathologic brain development. As brain structure closely correlates with brain function, malformations of cortical development are common causes of mental diseases such as epilepsy and developmental delay. The computational model can bridge the scales from largely studied disruptions on the cellular level towards form and function on the organ level. It can explain why cortical malformations emerge. Understanding the underlying mechanisms of brain development will enhance early diagnostics of cortical malformations and ultimately facilitate treatment and prevention of mental disorders.