DOI: 10.5061/dryad.866t1g21x
title: Data from: Exploiting nozzle geometry to predict resolution in extrusion-based bioprinting: mathematical modelling of a power-law fluid

description: Extrusion-based additive manufacturing is a popular technique used in the
 fabrication of three-dimensional constructs. Owing to the non-linear
 manner in which process parameters impact resolution and printability, the
 optimal combination remains platform- and material-specific. This study
 proposes a user-friendly, adaptable model to predict the diameter of a
 printed line of material through extrusion-based bioprinting. Exploiting
 the geometry of an arbitrary, axisymmetric nozzle, and assuming a
 power-law fluid, the model generated determines a relationship between the
 printed filament diameter and the pressure drop, nozzle travel speed,
 nozzle geometry, and material flow properties. Employing the model prior
 to printing enables engineers to restrict process parameter space and
 minimise the dependence on the current print-and-test methodology before
 an optimal combination of process parameters is determined. Two materials
 (a poly(vinyl alcohol)-based hydrogel and Nivea Crème), two temperature
 conditions, and three nozzle sizes were used for model validation,
 presenting good agreement with model predictions. When the shear-thinning
 property is included, R2 &gt; 0.97. This model provides context and
 direction for future optimisation-driven design research for this
 advancing manufacturing technology.
description: # Research data supporting "Exploiting nozzle geometry to predict
 resolution in extrusion-based bioprinting: mathematical modelling of a
 power-law fluid"
 [https://doi.org/10.5061/dryad.866t1g21x](https://doi.org/10.5061/dryad.866t1g21x) ## Description of the data and file structure Data is stored as CSV files (PVA_25G.csv, NC_25G.csv, NC_22G.csv, NC_20G.csv) and code stored as MATLAB Code (.m) (where its Optimisation, Statistics and Machine Learning, and Bioinformatics Toolboxes are used). Users should ensure all files (data and code) are in the same folder upon use. ### File naming Each CSV file contains print diameter measurements for one material and one nozzle geometry. The **materials** include: * A poly(vinyl alcohol)-based (**PVA**) hydrogel, or * Nivea Crème (**NC**) The **nozzle geometries** include: * **25G** (0.00025 m) * **22G** (0.00041 m), or * **20G** (0.00058 m) There are four data sets, PVA with a 25G conical nozzle and NC with either a 25G, 22G, or a 20G conical nozzle, where a particular range of air pressures have been used in each case. Each CSV contains the air pressures used (in kPa) across the top row and the print diameter measurements (in mm) for each air pressure in its corresponding column. For each set-up and material considered, nine lines of hydrogel were continuously printed onto the build platform via extrusion-based bioprinting, with each line measured at three points (start, middle, end) providing 27 data points. For the Nivea Crème experiments, this was repeated three times for each air pressure, providing 81 data points in total. During the study, a power-law fitting is compared with a Newtonian fitting (when n = 1). As such, two MATLAB scripts are provided, one for an arbitrary n and one for when n = 1. The CSV files are named according to: **MATERIAL_NOZZLEGEOMETRY.csv**. For example, NC_20G.csv indicates the file containing the print diameter measurements for Nivea Crème with a 20G conical nozzle. There are two MATLAB scripts, **Model_Fitting_Optimisation.m** and **Model_Fitting_Optimisation_N.m** for a power-law and Newtonian fitting, respectively.  ### File content By uncommenting out the section of code corresponding to the required material/nozzle geometry combination, the air pressures (in kPa) are called in from the corresponding CSV file and stored in **P** and the diameter measurements (in mm) stored in **data**. For air pressure **(1, x)** of **P**, its corresponding diameter measurements are stored in **(1:27, x)** and **(1:81, x)** of **data** for PVA and Nivea Crème, respectively. P and data are then converted to SI units, to produce **DeltaP** (Pa) and **data2** (m), respectively. Other variables: * **D_i**: nozzle inner diameter, taken to be 0.00402 (m) across all nozzle geometries * **D_o**: nozzle outer diameter, taken to be 0.00025, 0.00041, and 0.00058 (m) for 25G, 22G, and 20G nozzles * **U**: nozzle travel speed, taken to 0.005 (m/s), and * **L_c**: nozzle length, taken to be 0.032 (m) across all nozzle geometries An average diameter (and the standard deviation, **stdev**) for each pressure is calculated, and stored in **means**. Taking parameters, **n** and **K**, the flow behaviour and consistency indices, respectively, the fitting of the mathematical model (**d**) to the data (**means**) is optimised via **objective** (the minimum sum of squared residuals) and **initialGuess** (an appropriate initial guess for parameters [n, K] for either PVA or NC) by implementation of fminsearch. Once convergence is achieved, the optimised parameters, n and K, are stored in **bestParams**, and the corresponding fitted values in **fitted_model**. The coefficient of pressure drop (**deltaP**), determined from the model, is provided in **a**. Statistics such as R-squared, are calculated and printed and the experimental data and model fitting plotted. A Student's t-test is conducted and a false discovery rate (FDR) correction made in order to detect any statistical significance between the model prediction and the experimental data for each element of **deltaP**. The only difference in the Newtonian script is that the flow behaviour index, **n**, is defined as 1, and the **initialGuess** (and correspondingly **bestParams**) comprises of only one component, a suitable initial guess for the consistency index, **K** (the viscosity in a Newtonian regime). It is noted that this script can be used to set **n** to any value (such as the average optimal **n** values produced across material/nozzle geometry combinations from the power-law script), but the **initialGuess** will need to be changed accordingly.  ### Note * The initial guess for Nivea Crème was taken from Paxton *et al.* [*Biofabrication, 9:044107*, 2017] * With a lack of documentation on the rheology of the PVA hydrogel composition used and a Newtonian behaviour being assumed for non-Newtonian materials (in the Newtonian script case), **initialGuess** was chosen such that convergence was obtained