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Numerical Modeling for Subcutaneous Injection Pressure

Felten, Sina Marie LU (2025) In Master’s Theses in Mathematical Sciences NUMM03 20251
Mathematics (Faculty of Engineering)
Centre for Mathematical Sciences
Mathematics (Faculty of Sciences)
Mathematical Statistics
Abstract
To simulate subcutaneous injection pressure, numerical solvers are developed and subsequently used to train a predictive, data-driven surrogate model that delivers fast and reliable outputs. Different approaches are examined for solving a nonlinear partial differential equation that describes the pressure during subcutaneous injections over time and in a semi-infinite spatial domain.
For the spatial discretization, the finite difference method (FDM), the finite element method (FEM) and the infinite element method (IEM) are applied, paired with the implicit Euler method to integrate in time. The FDM and FEM truncate the domain, while the IEM accounts for the entire semi-infinite domain.
All three approaches yield comparable numerical... (More)
To simulate subcutaneous injection pressure, numerical solvers are developed and subsequently used to train a predictive, data-driven surrogate model that delivers fast and reliable outputs. Different approaches are examined for solving a nonlinear partial differential equation that describes the pressure during subcutaneous injections over time and in a semi-infinite spatial domain.
For the spatial discretization, the finite difference method (FDM), the finite element method (FEM) and the infinite element method (IEM) are applied, paired with the implicit Euler method to integrate in time. The FDM and FEM truncate the domain, while the IEM accounts for the entire semi-infinite domain.
All three approaches yield comparable numerical results, though the FDM requires the finest mesh among them. To reduce high computational costs, a surrogate model based on Gaussian process regression is built, enabling statistical analysis via Monte Carlo simulation that requires a significantly larger number of samples from different system configurations. The surrogate model allows efficient sampling of parameters from their probability distributions, as many times as needed. Once trained, it can be used indefinitely, providing reliable outputs for tolerance analysis. This enables the testing of various combinations of drugs, tissues and injection devices, supported by the small error between the predicted and the true solutions. (Less)
Popular Abstract
Designing an injection device is a long journey and requires knowledge and tuning of several parameters. In this process, understanding the pressure behavior in human skin tissue during injections can support design choices and provide information on the pain experience of patients without conducting invasive experiences. The pressure in the tissue during injections is influenced by different parameters ranging from the injection device, over the drug's characteristics, to the skin properties. In this thesis the pressure is simulated numerically which is not only non-invasive, but it also gives the freedom to try many different drug, device and skin property combinations without manufacturing a device and building a laboratory test set-up.... (More)
Designing an injection device is a long journey and requires knowledge and tuning of several parameters. In this process, understanding the pressure behavior in human skin tissue during injections can support design choices and provide information on the pain experience of patients without conducting invasive experiences. The pressure in the tissue during injections is influenced by different parameters ranging from the injection device, over the drug's characteristics, to the skin properties. In this thesis the pressure is simulated numerically which is not only non-invasive, but it also gives the freedom to try many different drug, device and skin property combinations without manufacturing a device and building a laboratory test set-up.

The pressure during subcutaneous injections can be described by an equation that incorporates drug, device and skin properties as parameters. However, such equations are often difficult or even impossible to solve analytically. This thesis applies numerical techniques to approximate the solution, though these methods can still be computationally intensive. When designing a device, it is important to perform tolerance analysis to examine how variations in parameters affect the output, here the pressure in the tissue, which requires numerous simulation runs. In practice, running the approximation solvers thousands of times can be computationally infeasible. To reduce this heavy computational load, a model is developed that imitates the behavior of approximation solvers. This model allows the generation of outputs for desired drug, device and skin parameter combinations, even without having knowledge of the underlying pressure equations. As a result, the technique enables non-scientists and non-engineers to gain insights into how these parameters influence subcutaneous injection pressure. (Less)
Please use this url to cite or link to this publication:
author
Felten, Sina Marie LU
supervisor
organization
alternative title
Numerisk modellering av subkutant injektionstryck
course
NUMM03 20251
year
type
H2 - Master's Degree (Two Years)
subject
keywords
Subcutaneous injection, finite difference, finite element, infinite element, Gaussian process regression, surrogate modeling, Monte Carlo simulation
publication/series
Master’s Theses in Mathematical Sciences
report number
LUNFNA-3046-2025
ISSN
1404-6342
other publication id
2025:E53
language
English
id
9208347
date added to LUP
2025-08-06 15:53:18
date last changed
2025-08-06 15:53:18
@misc{9208347,
  abstract     = {{To simulate subcutaneous injection pressure, numerical solvers are developed and subsequently used to train a predictive, data-driven surrogate model that delivers fast and reliable outputs. Different approaches are examined for solving a nonlinear partial differential equation that describes the pressure during subcutaneous injections over time and in a semi-infinite spatial domain.
For the spatial discretization, the finite difference method (FDM), the finite element method (FEM) and the infinite element method (IEM) are applied, paired with the implicit Euler method to integrate in time. The FDM and FEM truncate the domain, while the IEM accounts for the entire semi-infinite domain.
All three approaches yield comparable numerical results, though the FDM requires the finest mesh among them. To reduce high computational costs, a surrogate model based on Gaussian process regression is built, enabling statistical analysis via Monte Carlo simulation that requires a significantly larger number of samples from different system configurations. The surrogate model allows efficient sampling of parameters from their probability distributions, as many times as needed. Once trained, it can be used indefinitely, providing reliable outputs for tolerance analysis. This enables the testing of various combinations of drugs, tissues and injection devices, supported by the small error between the predicted and the true solutions.}},
  author       = {{Felten, Sina Marie}},
  issn         = {{1404-6342}},
  language     = {{eng}},
  note         = {{Student Paper}},
  series       = {{Master’s Theses in Mathematical Sciences}},
  title        = {{Numerical Modeling for Subcutaneous Injection Pressure}},
  year         = {{2025}},
}