# Overview

Bioprinting, a form of additive manufacturing, offers a greater design freedom in tissue engineering. By enabling customised geometries using a patient's own cells, it holds potential to revolutionise healthcare.

In this context, **print resolution** is critical-particularly when fabricating tissue scaffolds or constructs that replicate the mechanical properties of native tissues.

This repository includes two main components:

* A **dataset** and accompanying **MATLAB scripts** used to validate the leading-order asymptotic model (equations (71)-(74) of the main article) against experimental data
* A **graphical user interface (GUI)** implemented in Python, enabling users to run simulations interactively, visualise filament centre-line geometries, and compute the *window of printability*—the range of air pressures expected to achieve optimal resolution in pneumatic extrusion-based bioprinting

---

## Repository Structure

### Model_Validation.zip

Contains:

* `NC_20G.csv`-experimental dataset
* `power_law_shooting.m`-MATLAB shooting method solver
* `power_law_shooting_sweep.m`-full parameter-space sweep producing CSV outputs for post-processing
* `experimental_data_power_law.m`-processes dataset and generates figures
* Supporting functions:
  `alpha_experimental.m`, `ode_system_power_law.m`, `shooting_residual.m`

### EBB_Window_of_Printability_Generator.zip

Contains:

* `EBB_Window_of_Printability_Generator.py`-GUI
* `Underlying_Code.py`-`Simulation` function
* Icons and splash screen assets
* `example_input.txt`-template parameter file

---

## Dataset Description

The dataset contains filament diameter and length measurements printed using Nivea Crème with:

* a **20G nozzle**
* a **6 mm/s nozzle moving speed**, and
* air pressures ranging from **55–60 kPa**

Each condition was repeated three times. Filaments were imaged with a **Dino-Lite Digital Microscope** and analysed using **ImageJ**.

---

## Underlying Model

The mathematical model assumes:

* a slender extruded filament
* power-law rheology for shear-thinning behaviour, and
* steady, axisymmetric flow

Asymptotic reduction of the incompressible Navier–Stokes equations yields a system of **three first-order ODEs** governing:

* radius, R^(0)^(s)
* its derivative dR^(0)^/ds, and
* centre-line orientation, θ^(0)^(s)

It captures a viscous-dominated regime and satisfies the integral

```math
∫₀¹ cos(θ⁽⁰⁾(s)) ds = H / L,
```

which forces the centre-line to reach the build platform at Z(s)=0.

Model assumptions include:

* Small aspect ratio, εⁿ = R~o~/L, with n the flow behaviour index, R~o~ the nozzle outer radius, and L some typical length of the filament
* Incompressible fluid
* Power-law constitutive law
* Negligible surface tension
* Moderately shear-thinning materials (e.g. 0.39 < n < 0.68)
* Leading-order parameter grouping α~PL~ = Re~PL~/Fr^2^ (gravitational and viscous forces)
* Leading-order parameter grouping εⁿCa~PL~ (viscous drag and surface tension effects)
* Filament is approximately axisymmetric about its centre-line
* Plug flow velocity profile within the filament

---

## Extrusion Speed to Air Pressure

To link theory with experimental operation, an **empirical expression** derived in previous work (see [Tansell](https://doi.org/10.1098/rsos.250504) *[et al.](https://doi.org/10.1098/rsos.250504)*[, 2025](https://doi.org/10.1098/rsos.250504)) relates **theoretical extrusion speed**-derived from the 3-ODE model-to **air pressure required by a pneumatic bioprinter**.

---

# MATLAB Code for Model Validation

The MATLAB scripts compute the experimental non‑dimensional parameter α~PL~, solve the ODE system using a **shooting method**, and identify the **velocity ratio** that best matches experimental filament geometries.

## Main Scripts

### power_law_shooting.m

* Iterates on β = y~2~(s=1) until the boundary condition, y~1~(s=0) = 1, is satisfied, using `fzero`
* After convergence, evaluates the integral to obtain the model-predicted nozzle offset to filament length ratio, H/L
* Compares predicted and experimental H/L to identify a velocity ratio corresponding to a best match
* Predicts the deposited radius R(s=L) = R~o~ sqrt(velocity_ratio_suggestion)
* Computes percentage error and recommended extrusion speed, ū~E~

### power_law_shooting_sweep.m

For convenience and reproducibility, this MATLAB script performs a full sweep of dimensionless parameter space of the leading-order asymptotic model.

* Solves the governing 3-ODE system using a **shooting method** for all combinations of:
  * velocity ratio, ū~E~/U~N~, and 
  * α~PL~
* Computes the resulting H/L
* Saves the complete solution for each parameter set to a **CSV file**

For each combination of ū~E~/U~N~ and α~PL~, the script outputs a separate CSV file named:
``<velocity_ratio>_<alpha_PL>.csv``

Each CSV contains:
* The computed H/L
* The solution profile over the arc-length parameter, s:
  * s
  * radius, R^(0)^(s)
  * its derivative dR^(0)^/ds, and
  * centre-line orientation, θ^(0)^(s)

These files may be loaded directly by the user to recreate figures appearing in the manuscript.

### experimental_data_power_law.m

* Computes mean and standard deviation of measured filament lengths and radii
* Fits quadratic (`poly2`) curves for:
  * filament length vs pressure drop
  * radius vs pressure drop
* Generates plots with error bars appearing in manuscript

### Supporting Functions:

* `alpha_experimental.m`-computes experimental α~PL~ and H/L
* `ode_system_power_law.m`-the 3-ODE system
* `shooting_residual.m`-used by `fzero`

---

# MATLAB Requirements

* MATLAB R2020b or later (recommended)
* Optimization Toolbox (for `fzero`)
* Curve Fitting Toolbox (for `fit`, optional but required to reproduce the manuscript plots)

---

# Python GUI: Window of Printability Generator

The GUI provides a user-friendly interface for running the model without writing/amending code. It is ideal for rapid parameter exploration, to visualise filament centre-lines, and compute minimum/maximum pressure drops-the **window of printability**.

## 1. Purpose of the GUI

The GUI enables user to:

* Enter printing and material parameters
* Import data files
* Run simulations interactively
* Visualise centre-line filament geometries
* Compute minimum and maximum air pressures that achieve optimal resolution

## 2. Files Included

* `EBB_Window_of_Printability_Generator.py`-GUI (Tkinter)
* `Underlying_Code.py`-`Simulation` function
* Icons and GUI assets:
  * `app_icon.png`, `app_icon.ico`, `app_icon.icns`
  * `splash_screen.png`
  * `error_icon.png`, `warning_icon.png`
* `example_input.txt`-demonstration input file

## 3. How the GUI Works

Pressing **Submit**, the GUI

1. Validates user inputs
2. Passes values to `Simulation`
3. Solves the 3‑ODE system (shooting method)
4. Computes extrusion speed to air pressure relationship
5. Returns:
   * A `Matplotlib` figure of centre‑line geometries
   * Minimum/maximum pressure drop
6. Enables export options

## 4. Installation

### Recommended (All OS) - Anaconda

1. Install [Anaconda](https://www.anaconda.com/products/distribution)
2. Create a new environment (recommended)

   ```bash
   conda create -n ebb python=3.10
   conda activate ebb
   ```
3. Install required packages

   ```bash
   pip install matplotlib numpy scipy pillow
   ```

   > These packages support the GUI and the underlying simulation code.
4. Run the GUI

   ```bash
   python EBB_Window_of_Printability_Generator.py
   ```

   > The GUI will open.
   >
   > From here, you can enter parameters, import `.txt` input files, run simulations, view centre-line geometries and windows of printability, and export results.

### Windows (without Anaconda)

1. Install Python (3.8+) from [python.org](https://www.python.org/downloads/)

   > **Important:**
   > Tick **"Add Python to PATH"** during installation.
   >
   > Python.org Windows installers include **Tkinter**, so no extra configuration is needed.
2. Install libraries

   ```bash
   pip install matplotlib numpy scipy pillow
   ```
3. Run the GUI

   ```bash
   python EBB_Window_of_Printability_Generator.py
   ```

### macOS (without Anaconda)

1. Install Python (3.8+) from [python.org](https://www.python.org/downloads/)

   > This version includes: **Tkinter** and correct Tcl/Tk version for stable GUI rendering.
2. Install libraries

   ```bash
   pip3 install matplotlib numpy scipy pillow
   ```

1) Run the GUI

   ```bash
   python3 EBB_Window_of_Printability_Generator.py
   ```

### Linux

Use Anaconda-Linux Python distributions vary widely in Tkinter support.

## 5. Running the GUI

1. Launch the GUI
2. Enter values manually or load a `.txt` file (see `example_input.txt` for a ready-made import file)
3. Press **Submit**
4. View centre-line geometry and minimum/maximum pressure drops
5. Export results if required

## 6. Input Parameters

* Flow behaviour index, n
* Consistency index, K (Pa·sⁿ)
* Density, ρ (kg/m³)
* Nozzle base inner radius, R~i~ (μm)
* Nozzle tip inner radius, R~o~ (μm)
* Nozzle length, L~N~ (mm)
* Nozzle travel speed, U~N~ (mm/s)
* Nozzle offset, H (mm)

Tooltips include definitions, units, and typical ranges.

## 7. Outputs

* Centre-line geometries (mm) comprising two curves: minimum pressure drop (solid line) and maximum pressure drop (dashed line)
* Minimum and maximum pressure drop values (kPa)-the **window of printability**

## 8. Features

* Custom splash screen and icons handled via Pillow
* Import `.txt` files with parameter names and values (**Import Data** button)
* Tooltips comprising definitions, units, and typical/suitable ranges of parameters
* Clear all inputs, results, and hide export button with confirmation popup (**Clear All** button)
* Terminate ongoing simulation and reset GUI state with confirmation popup (**Cancel** button)
* Progress bar updates in three phases: solving, matching, and plotting
* Visualisation of centre-line geometries
* Calculation of pressure drop range (the window of printability)
* Export options (**Export Results** button):
  * Choose to export figure, pressure drop data, or both via checkboxes
  * Supported formats: `.png`, `.pdf`, `.svg` for figures; `.txt`, `.csv` for data
* Input validation with custom error popups for missing/invalid inputs
* Out-of-range inputs trigger a confirmation popup before proceeding

## 9. Import & Export Functionality

### Import

* File must be `.txt` format
* Each line should follow `Label: Value` format
* Labels should match GUI labels or normalised names (punctuation, hyphens, and single letter tokens removed from `Label`)
* Unmatched labels trigger a popup with guidance

### Export

* Figure: `.png`, `.pdf`, `.svg`
* Data: `.txt`, `.csv`
* **Export Results** button appears only after simulation completes and disappears when the cancel button is pressed or a new set of parameters are submitted
* Export options:
  * Choose to export figure, pressure drop data, or both via checkboxes

## 10. Error Handling

The GUI checks for:

* Invalid or missing inputs
* Out-of-range parameters
* Unrecognised imported labels
* Failed ODE solutions
* Extremely small pressure drops bounds (< 10⁻² kPa)

Each triggers a context-specific popup message.

## 11. Known Limitations

* Axisymmetric nozzle geometry (specifically conical and straight nozzle geometries)
* Power-law fluids only
* Surface tension effects neglected
* Restricted to steady, catenary-like solutions
* Not suitable for highly viscoplastic materials at very low shear stresses

## 12. Future Extensions

Potential enhancements include:

* Herschel–Bulkley (yield stress) rheology
* Other axisymmetric nozzle types
* Surface-tension-corrected boundary conditions

---

# Citations

This project makes use of the following open-source libraries:

* NumPy: Harris et al., *Array programming with NumPy*, **Nature**, 2020. DOI: [https://numpy.org](https://numpy.org)
* SciPy: Virtanen et al., *SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python*, **Nature Method**s, 2020. DOI: [https://scipy.org](https://scipy.org)
* Matplotlib: Hunter, *Matplotlib: A 2D Graphics Environment*, **Computing in Science & Engineering**, 2007. DOI: [https://matplotlib.org](https://matplotlib.org)
* Pillow: Clark et al., *Pillow (PIL Fork) Documentation*, **Python Software Foundation**, 2010-present. DOI: [https://python-pillow.org](https://python-pillow.org)

Flow behaviour and consistency indices for NIVEA Crème were taken from:

* N. Paxton, W. Smolan, T. Böck, F. Melchels, J. Groll, and T. Jungst. *Proposal to Assess Printability of Bioinks for Extrusion-Based Bioprinting and Evaluation of Rheological Properties Governing Bioprintability*. **Biofabrication**, 9(4):044107, 2017. DOI: [10.1088/1758-5090/aa8dd8](https://doi.org/10.1088/1758-5090/aa8dd8).

Default value of 'Nozzle Base Inner Diameter', if unknown, is assumed from:

* J.C. Gómez-Blanco, E. Mancha-Sanchez, A.C. Marcos, M. Matamoros, A. Díaz-Parralejo, and J.B. Pagador. *Bioink Temperature Influence on Shear Stress, Pressure and Velocity Using Computational Simulation*. **Processes**, 8(7): 865, 2020. DOI: [10.3390/pr8070865](https://doi.org/10.3390/pr8070865)

Original empirical expression considering an arbitrary, axisymmetric nozzle geometry:

* A.V. Tansell, N. Mahmoodi, J.P. Crolla, R.J. Dyson, G.J. Luo, L.E.J. Thomas-Seale. *Exploiting Nozzle Geometry to Predict Resolution in Extrusion-Based Bioprinting: Mathematical Modelling of a Power-Law Fluid*. **R. Soc. Open Sci.**, 12: 250504, 2025. DOI: [10.1098/rsos.250504](https://doi.org/10.1098/rsos.250504).