Page 322 - IJB-10-2

International Journal of Bioprinting Continuous gradient TPMS bone scaffold

Therefore, when designing bone scaffolds, in addition to There are two main approaches to expressing the
ensuring favorable mechanical properties, it is also crucial porous structure of minimal surfaces. According to the
to consider whether the scaffold structure can maintain Enneper–Weierstrass parameter representation method,
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high permeability. TPMS can be accurately calculated :

In summary, although a high porosity can increase the x Ree( i w (1 2 ) R d

permeability of the continuous gradient TPMS structure, ( w ) () (I)
0
it also reduces its mechanical properties. Therefore, y Ree( i w (1 2 ) R d

when considering different application requirements, ( w ) ()
0
it is essential to strike a reasonable balance between w

mechanical properties and permeability. This study aims z Ree( i w ( ) 2 Rd
()

0
to optimize and design a continuous gradient TPMS bionic In this method, we use the following notation: i = ± 1,
bone structure for bone repair. In this study, we combined τ represents a complex variable, θ is the valve cover angle,
experimental and simulation approaches to investigate the Re denotes the real part of the complex variable, and R(τ)
parametric design, mechanical properties, permeability, is the Weierstrass function for different types of TPMS
and application of the continuous gradient TPMS bone- units. For example, the Weierstrass functions of G, D, and
like structure. The main purpose of this study is to obtain P surfaces can be expressed as :
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a bone scaffold structure that matches the performance of 1
human femur and to provide theoretical basis for further R() (II)
4
8
realization of human bone tissue repair. 14 1
However, it is worth noting that this method can only
2. Design, performance test, and generate a limited number of TPMS units. Currently, several
simulation analysis of continuous Weierstrass functions for minimal surfaces have been
gradient TPMS structure discovered. Just like other mathematical methods, TPMS
porous structures can be generated using Equation III :
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2.1. Parametric design and 3D printing of
continuous gradient TPMS structure ()r k A cos 2 (hr ) P k C (III)
k

The design method plays a crucial role in the quality of k1 k k
generated TPMS models. The geometric characteristics of Here, A represents the amplitude, λ is the periodic
k
k
the minimal surface structure, including parameters such as factor, and P is the phase function. Building upon this
k
porosity (or volume ratio), unit cell pore size, and thickness, foundation, the common TPMS units are presented in
are key factors that influence its performance. Therefore, Table 1. It can be observed from the TPMS implicit function
geometric design serves as the foundation for effectively expressions in Table 1 that ω and C are two important
controlling the application performance of TPMS structures parameters that influence the period and curvature of
in different fields. Unlike traditional foam lattice structures, TPMS. In the context of TPMS porous structures, the
TPMS enables the design of more intricate structural volume ratio of the two components is solely related to the
characteristics to mimic natural porous structures. curvature parameter C. Therefore, by assigning different
values to the periodic parameter ω and the curvature
Compared to other porous structures, TPMS exhibits parameter C, gradient or non-uniform TPMS porous
three significant characteristics. Firstly, TPMS represents structures can be generated.
an implicit surface, allowing for the complete expression of
the geometric structure through algebraic equations, which Table 1. Common mathematical expressions of TPMS structure
can be simplified as f(x, y, z) = C, where C is a constant.
Based on this, TPMS is considered an isosurface. Secondly, Unit name Mathematical expression 3D model
TPMS demonstrates periodicity in three independent Gyroid (G) f(x, y, z) = sin(ω x)cos(ω y) +
y
x
directions. The parameter function facilitates easy control sin(ω z)cos(ω x) +
x
z
over the distribution range and period of the model. Lastly, sin(ω y)cos(ω z) = C
y
z
TPMS is characterized as a minimal surface, meaning that
the mean curvature of TPMS is zero, resulting in a smooth Schwarz P(P) f(x, y, z) = cos(ω x) + cos(ω y) +
y
x
cos(ω z) = C
surface reminiscent of natural phenomena like soap z
bubbles and leaves. 31
Volume 10 Issue 2 (2024) 314 doi: 10.36922/ijb.2306