Predicting doubling times during brain organoid development
Motivation:
Let’s suppose we have a lab which uses brain organoids to investigate human brain development. By seeding appropriate extracellular matrices(ECMs) with thousands of human pluripotent stem cells(hPSCs) we may grow nonvascularized brain organoids. These tend to develop into spheroids for reasons that I try to explain below.
Now, if the brain organoid’s spherical surface applies diffusion constraints on the transport of oxygen and nutrients to all cells in the interior, we may ask how much time is required for the volume of a spherical brain organoid with radius to double. The value of this analysis is that if we can estimate the expected doubling time with reasonable confidence, we may predict the time to maturation.
Furthermore, I propose that predicting doubling times during brain organoid development as a fundamental challenge that could advance first principles approaches to understanding organoid development.
Caveat: Spherical Brain Organoids aren’t directly comparable to the human brain but they may be likened to the hydrogen atom for human brain development.
Assumptions:
In order to proceed with our analysis a number of assumptions are necessary. The following are considered sufficient:

An insignificant fraction of cells(< 5%) die before the spherical brain organoid has attained maximal volume, implying that the spherical organoid hasn’t grown too large.

During development, the distribution of each cell type converges to an equilibrium distribution where the distribution of each cell type(neurons,glia, oligodendrocytes) is unimodal and tightly concentrated around its mean. Furthermore, we assume that the equilibrium distribution is isotropic i.e. spatially homogeneous.

Spherical symmetry is maintained via efficient mechanisms for cell signalling that coordinate the entire resource allocation process.

The packing density of cells is invariant to slight perturbations of the spherical geometry and therefore if the brain organoid’s geometry is denoted by :
\begin{equation} \text{Mass}(\mathcal{B}) \approx k_1 \cdot \text{Vol}(\mathcal{B}) \approx k_2 \cdot N \end{equation}
where is the total number of cells and and are constants.

Half the organoid volume is exposed to air and the other half is embedded in ECM. Symmetry of this sort is necessary for our analytical arguments to be plausible.
A rational account for the spherical shape of brain organoids:
The isoperimetric inequality states that given a compact Euclidean manifold with fixed boundary , then satisfies the following inequality:
\begin{equation} \text{Vol}(\mathcal{B}) \leq \frac{1}{6 \sqrt{\pi}} \cdot \text{Vol}(\partial \mathcal{B})^{3/2} \end{equation}
where we have equality if and only if is a sphere.
Given the uniqueness of the sphere it’s reasonable to suppose that this shape isn’t an accident and that it’s probably advantageous to the brain organoid. Here I posit two possible advantages in terms of energy loss and cell signalling.

Minimisation of energy loss:
If heat is mainly lost by means of conduction via the boundary of the brain organoid then it would be advantageous to the brain organoid if this surface was minimal.

Efficient cell signalling:
If we assume that the cells in an embryoid body communicate by means of some complex network and that the packing density of cells is isotropic then it’s sufficient to minimise the average euclidean distance between cells. This minimisation process yields the sphere.
At this point a mathematical biologist might remark that brain organoids aren’t vascularized and therefore resource allocation must be diffusionconstrained. Surely a flat disklike morphology would be more appropriate? The error in this argument is that it fails to consider that resource allocation is at the service of coordinating the developmental process. Whatever is ideal for cell signalling shall constrain how resource allocation operates.
The expected number of doubling episodes during brain organoid development:
Before trying to estimate doubling times it might be instructive to analyse a related question. If is the mass of a spherical brain organoid, is the average density of a human brain and the vast majority of cell divisions are symmetric:
\begin{equation} M_{\mathcal{B}} \approx N_0 \cdot \overline{m_c} \cdot 2^D \end{equation}
\begin{equation} M_{\mathcal{B}} \approx \frac{4}{3} \pi r^3 \cdot \rho_{\text{brain}} \end{equation}
\begin{equation} \rho_{\text{brain}} \approx \frac{1400 g}{1260 \text{cm}^3} \approx \frac{1.1 \cdot 10^{3} g}{1 \text{mm}^3} \end{equation}
where is the average mass of a mature cell, is the average number of cell divisions and is the number of cells seeded per embryoid body.
By equating (2) and (3) we find that:
\begin{equation} D(N_0,r) = \frac{1}{\ln 2} \cdot \ln \big(\frac{4 r^3 \cdot \rho_{\text{brain}}}{N_0 \cdot \overline{m_c}}\big) \end{equation}
Now, if we make the reasonable assumption that the mass of a eukaryotic cell is bounded between one nanogram and a thousand nanograms we may infer that [2]:
\begin{equation} \overline{m_c} \approx 10^2 \text{ng} = 5 \cdot 10^{7} \text{grams} \end{equation}
so we have:
\begin{equation} D(N_0,r) \approx \ln(4r^3 \cdot \rho_{\text{brain}})  \ln (N_0) + 7\ln(10) \end{equation}
and if we use the bounds from [1]:
\begin{equation} 5000 \leq N_0 \leq 10000 \end{equation}
\begin{equation} 1.5 \text{mm} \leq r \leq 2.5 \text{mm} \end{equation}
we find that:
\begin{equation} 5.00 \leq D(N_0,r) \leq 5.83 \end{equation}
Estimating the doubling times during brain organoid development:
Disclaimer: In the analysis that follows we don’t make any assumptions on the proportion of cell divisions that are symmetric. This makes it more robust than the previous analysis on the expected number of doubling episodes during brain organoid development.
Given the formula for the volume of a sphere, if denotes the volume of a spheroid with radius where we have:
\begin{equation} V_{n+1} = 2 \cdot V_n \implies r_{n+1} = 2^{\frac{1}{3}} \cdot r_n \end{equation}
and given that brain organoids aren’t vascularized they must be diffusionconstrained. In this scenario, it’s reasonable to assume that:
\begin{equation} \text{growth rate} \sim \text{metabolic rate} \sim \frac{\text{vol}(\partial \mathcal{B})}{\text{vol}(\mathcal{B})} \approx \frac{4 \pi r^2}{\frac{4}{3} \pi r^3} = \frac{3}{r} \end{equation}
where we made the implicit assumption that during the elapsed time for doubling we have an approximate equality of the following averages:
\begin{equation} \langle \text{growth rate of cell population} \rangle \approx \langle \text{growth rate of organoid volume} \rangle \end{equation}
Now, given (13) if we denote the growth rate by we have:
\begin{equation} \frac{3k}{2^{\frac{1}{3}} \cdot r_n} \leq g_r \leq \frac{3k}{r_n} \end{equation}
where is an unknown constant.
It follows that if the volume of the brain organoid is currently the expected doubling time must be approximately:
\begin{equation} T_n \cdot g_r = V_{n+1} \end{equation}
\begin{equation} V_{n+1} = 2 \cdot V_n = \frac{8}{3} \pi r_n^3 \end{equation}
using (15) we find that the doubling time must be in the interval:
\begin{equation} \frac{8}{9k} \pi r_n^4 \leq T_n \leq \frac{8 \cdot 2^{\frac{1}{3}}}{9k} \pi r_n ^4 \end{equation}
and if our uncertainty over is expressed as a uniform distribution on this interval the expected doubling time is given by:
\begin{equation} \mathbb{E}[T_n] = \frac{4 \pi r_n^4}{9k} \cdot (1+2^{\frac{1}{3}}) \end{equation}
Discussion:
I must clarify that this theoretical analysis represents just the first attempt at a firstprinciples approach to predicting the time required for a brain organoid to double its volume. The main objective of this analysis was to advance concepts that are useful for analysing the development of brain organoids. This includes the metabolic activity of cells, their packing density, mechanisms for cell signalling and equilibrium distributions over cell types at the terminal phase of development.
To validate my formulas that predict the expected waiting time for a spherical brain organoid to double its volume we may use tools from data analysis. Specifically, we may use a combination of computer vision and nonlinear regression to infer a functional relationship between the doubling time and potentially relevant variables.
If the fourth derivative of the interpolated curve resulting from such an analysis is a positive constant then my theoretical analysis is broadly correct.
Acknowledgements: I would like to thank Bradly Alicea for constructive feedback on this theoretical analysis.
References:
 Yakoub AM, Sadek M. Development and Characterization of Human Cerebral Organoids: An Optimized Protocol. 2018.
 Haifei Zhang. Cell. http://softmatter.seas.harvard.edu/index.php/Cell. 2009.
 Modeling a neurodevelopmental disorder with human brain organoids: a new way to study conditions such as epilepsy and autism. https://neuroscience.berkeley.edu/modelingneurodevelopmentaldisorderhumanbrainorganoidsnewwaystudyconditionsepilepsyautism/. 17/09/2018.