# Forward Computational Modeling of Respiratory Airflow
## 2 Mathematical Structural and Spatial Geometric Models  
Prior to advancements in medical imaging technologies with high spatial resolution, mathematically based geometric models of the lungs were used to develop airway trees for CFD simulations. Based on initial designs of [PERSON] [6] and [PERSON] et al. [12; 20], geometric models have been created to account for asymmetric bifurcation angles, cross-sectional areas, diameter, and length of airway branches. One such model by [PERSON] et al. [15] is based on two simple principles that determine airway branch dimension and spatial dimensional arrangements for a dichotomous branching airway tree. This model assumes that the amount of fluid delivery through a branch is proportional to the volume of the region it supplies and that the terminal branches of the airway tree are homogeneously arranged within the organ [15]. The relationship between flow rate and airway branch diameter is established based on power law, assuming circular airway segments with constant diameters [15]. Both parameters are normalized by their respective values in the parent branch (Equation (1)). \[\dot{V}^{\prime}=(d^{\prime})^{n} \tag{1}\]  
where \(\dot{V}^{\prime}\) and \(d^{\prime}\) are the normalized flowrate and diameters, respectively, for any daughter branch, and \(n\) is the diameter exponent. Based on the principle of minimum energy loss for steady laminar flow in a rigid, circular conduit, \(n\) is empirically assigned a value of 3 [21]. According to [PERSON]'s law [22], the diameter relationship between the parent and daughter branches is given as  
\[d_{0}{}^{n}=d_{1}{}^{n}+d_{2}{}^{n}+\ldots\ldots+d_{i}{}^{n} \tag{2}\]  
where \(d_{0}\), \(d_{1}\), \(d_{2}\), and \(d_{i}\) are the diameters of the parent, first, second, and \(i\)th daughter branch respectively. Furthermore, a ratio \(f\) can be defined as the flow fraction (between the daughter and parent branch) in the first daughter branch, and the diameters of the daughter branches are directly related to the parent branch as  
\[d_{1}=d_{0}f^{\frac{1}{2}} \tag{3}\]  
\[d_{2}=d_{0}(1-f)^{\frac{1}{2}} \tag{4}\]Hence, if the dimensions of both daughter branches are unknown, and a desired flow is assumed in either branch, the dimensions of both branches can be obtained. [PERSON] (1973) further derived the relationship between the diameter \(d\) of the parent and daughter branches and the bifurcation angles as  
\[\cos\theta_{1}=\frac{{d_{0}}^{4}+{d_{1}}^{4}-{d_{2}}^{4}}{{2{d_{0}}}^{2}{d_{1}} ^{2}} \tag{5}\]  
\[\cos\theta_{2}=\frac{{d_{0}}^{4}+{d_{2}}^{4}-{d_{1}}^{4}}{{2{d_{0}}}^{2}{d_{2} }^{2}} \tag{6}\]  
where \(\theta_{1}\) and \(\theta_{2}\) are the bifurcation angles between parent branch and the first and second daughter branches respectively. When the flow fraction \(f\) for one of the daughter branches is known, the equations for the branching angles become:  
\[\cos\theta_{1}=\frac{\left[1+{f^{4/n}}-{(1-f)}^{4/n}\right]}{2{f^{2/n}}} \tag{7}\]  
\[\cos\theta_{2}=\frac{\left[1+{(1-f)}^{4/n}-{f}^{4/n}\right]}{2{\left(1-f \right)}^{2/n}} \tag{8}\]  
The algorithm for generating this geometric model is governed by basic and supplementary rules ([PERSON], 2008) which were used to successfully generate an airway tree (Figure 1) with approximately 54,000 branches utilizing the above equations. [PERSON] et al. (2013) generated a 3D geometric model of the lungs based on these principles, for an airway tree having 65,536 branches, and developed a CFD model of orally inhaled medication that could predict drug deposition characteristics and physiological parameters in the lungs, showing the utility of this geometric model. The airway tree is often assumed to be fractal in nature (i.e., having detailed self-similar structures at different scales), with a large degree of heterogeneity and lack of a characteristic scale. [PERSON] (1964) theorized the nature of the airway tree as a fractal-based bifurcating tree with a rectangular shape with a dimension of \(\sqrt{2}\times 1\)([PERSON], 1964; [PERSON], 1964).