Growth and Morphogenesis

I have always been intrigued by geometrical problems, especially those arising in the life sciences. This started already in the final year of High School when my maths teacher showed me a book called On Growth and Form by D'Arcy Thompson. Well, that was the beginning of the story! Here is the rest..

Morphogenesis of the gastro-intestinal system

The intestine is one of the most structurally complex and fascinating organ in our body. A closer look at its inner surface reveals a variety of geometrical patterns. These are known to form since the early stages of embryo development. However, the mechanisms underlying their formation are unclear. 

Together with my supervisor Pasquale Ciarletta, we proposed a new model which aims at explaining the origins of these patterns. 
The idea is actually very simple: you can imagine the intestine as a two-layered cylinder made of two different elastic materials, the outer of which much stiffer than the inner one. During the embryonic development both grow, whereby the inner tends to push against the outer and viceversa depending on who grows "faster". This mechanism causes an accumulation of mechanical stresses in the tissues which eventually ends up being very unstable. At some point—in order to release these stresses—the inner surface of the cylinder encounters a dramatic change of shape. Basically it goes from an initial circular shape to a final buckled one. Depending on where you look at the intestine, you find different patterns and we showed that this variety mainly depends on the mechanical properties of the two embryonic layers and on their initial thickness.

Observed gastro-intestinal patterns and the results from numerical simulations of the embryonic growth.

The geometrical model of the intestine with the two growing embryonic layers and the mathematical description of differential growth.

Related publications:

My PhD thesis

V. Balbi, E. Kuhl, P. Ciarletta. Morphoelastic control of gastro-intestinal organogenesis: theoretical predictions and numerical insights.  J Mech Phys Solids, 2015, DOI

P. Ciarletta, V. Balbi, E. Kuhl. Pattern selection in growing tubular tissues. Phys Rev Lett, 2014, DOI.

V. Balbi, P. Ciarletta. Morpho-elasticity of intestinal villi. J Roy Soc Interface, 2013, DOI.

V. Balbi and P. Ciarletta. Mathematical modeling of morphogenesis in living materials (Chapter 4) in L. Preziosi et al. (eds) Mathematical Models and Methods for Living Systems. Lecture Notes in Mathematics, vol 2167. Springer, Cham

Viscoelasticity

After my PhD, I joined the University of Manchester as a postdoctoral researcher and worked with William J Parnell and Tom Shearer. We worked on another amazing side of the mathematics of soft tissues: the constitutive modelling, i.e. the mechanical law that a material obeys.
If I applied a force to a block made of a soft tissue, what would the resultant deformation be? The constitutive law tells you the answer. We are interested in soft tissues with a family of fibres (transversely isotropic) and we are currently working on a constitutive theory based on the quasi-linear viscoelastic theory.

Related publications:
​
V. Balbi, T. Shearer, W.J. Parnell. A modified formulation of quasi-linear viscoelasticity for transversely isotropic materials under finite deformation. Proc Roy Soc A, 2018, DOI.

M. Righi & V. Balbi. Foundations of viscoelasticity and application to soft tissue mechanics. In Modeling Biomaterials (pp. 71-103). Cham: Springer International Publishing, 2021, pdf here.

V. Balbi, T. Shearer, W.J. Parnell. Tensor decomposition for modified quasi-linear viscoelastic models: towards a fully nonlinear theory. Mathematics and Mechanics of Solids, 2023, DOI.

BRain Mechanics

From 2017 to 2019, I joined the School of Maths at NUI Galway as Marie Curie research fellow. Part of my Marie Curie project focussed on determining the mechanical properties of the brain. Following an impact to the skull, brain matter can experience large stretches, possibly resulting in Diffuse Axonal Injury, which is the second leading cause of death from traumatic brain injury. Moreover, the brain matter experiences a mix of deformation modes in an incident. A particularly interesting problem, both experimentally and theoretically is the torsion. In collaboration with Michel Destrade (NUI Galway), Antonia Trotta and Aisling Ní Annaidh (College of Engineering, University College Dublin) we measured and modelled the elastic response of the brain in torsion. Our results suggest that during a rotational head impact the brain develop not only high shear forces but also vertical forces. These in turn lead to higher levels of damage in the cerebral tissue. Our results could potentially help improving the design of helmets and head-guards in contact sports.

Sketch of a porcine brain and map of the cutting points where we took the tissue samples for testing.

Distribution of the vertical stresses in the brain. The colour map is obtained from Finite Elements simulations of a rotational impact in Abaqus.

Experimental setup: a cylindrical sample of brain tissue is glued to the two parallel plates of a rotational rheometer. By rotating the upper plate, the sample is then twisted. During the twisting the sample pushes up against the upper plate, indicating that vertical stresses are generated.

Related publications:
​
V. Balbi, A. Trotta, M. Destrade, A. Ní Annaidh. Poynting effect of brain matter in torsion,  Soft Matter, 2019, DOI.

Other brain related publications:
​
V. Balbi, M. Destrade, A. Goriely. The mechanics of human brain organoids, Physical Review E, 2020, DOI.