A Position- and Time-Dependent Pressure Profile to Model Viscoelastic Mechanical Behavior of the Brain Tissue due to Tumor Growth

Abstract

The irregular shape of the tumor and the viscoelastic mechanical behavior of the brain tissue cause a position- and time-dependent pressure profile at the interface of the tumor and brain during tumor growth, respectively. Calculating the resultant pressure profile through numerical approaches, such as finite element method (FEM), may contribute to the surgical and radiotherapeutic treatments designing for the brain tumor patients via precisely build statistical atlases of brain geometry. However, to date, FE modeling of the tumor growth in the brain has been limited to a) application of a constant or a time-independent pressure boundary, b) application of time-independent material properties for the brain tissue, e.g., elastic and hyperelastic, and c) application of a quasi-spherical shape for the tumor FE model. This study proposed a computational framework to calculate the resultant position- and time-dependent pressure profile on the brain tissue due to tumor growth. A FE patch of the brain tissue was constructed and an inverse dynamic FE-optimization algorithm was used to calculate its viscoelastic mechanical properties under compressive uniaxial loading. To calculate the simulation time that results in a stable (independent to shear relaxation modulus) mechanical response, several FE simulations at various displacement rates were performed. Finally, two patient-specific post-tumor resection FE models were input to the FE-optimization algorithm to calculate the optimized 3^rd-order position-dependent and normal distribution time-dependent pressure profile parameters. The optimized viscoelastic material properties, the most suitable simulation time, and the optimized 3^rd-order position- and -time-dependent pressure profiles were calculated.

1. Introduction

Primary brain tumors is a major public health concern (Eichberg et al., 2019). Annually, more than ~20,000 patients are diagnosed with a high-grade glioma, with ~100,000 to 200,000 new cases of brain metastases each year (Jemal et al., 2008). Construction of accurate brain tumor atlases requires a correct 3D registration of the brain tumor images to a stereotactic space (Clatz et al., 2005; Mohamed and Davatzikos, 2005). Although brain tumor atlases would have practical applications for a more comprehensive tumor resection planning and therapeutic approaches via the variable parameters in the tumor, such as grade, size, and progression (Cuadra et al., 2002; Dawant et al., 2002; Kyriacou et al., 1999; Mohamed et al., 2001), to date, there are still some challenges in the tumor vicinity due to the tumor irregular shape and progression complications (Cuadra et al., 2004; Dawant et al., 2002). The contribution of biological factors as well as the biomechanical forces may alter the tumor microenvironment, and in turn the progression and response to the treatment in the tumor may get affected (Jain et al., 2014).

Growth of a tumor inside a stiffer fixed-volume solid matrix of the skull causes a generation and accumulation of forces due to mechanical interactions at the interface of the tumor and the brain following by the mass-effect (Gerard et al., 2017; Mokri, 2001; Orringer et al., 2012) that is a main reason of neurologic injury (Ropper, 1986). These mechanical forces could hinder inducing apoptosis, cancer cell proliferation, and augment the invasive and metastatic potential of cancer cells influencing tumor progression (Cheng et al., 2009; Demou, 2010; Helmlinger et al., 1997; Janet et al., 2012; Kaufman et al., 2005; Stylianopoulos et al., 2012). Moreover, they can invoke up to ~24 mm brain shift during tumor resection (Bayer et al., 2017; Chen et al., 2011; Gerard et al., 2017; Miga et al., 2016; Nimsky et al., 2000).

Finite element method (FEM) is a numerical approach that allows calculating the resultant forces and mass-effects due to tumor growth under various loading and boundary conditions as well as material properties. Various 2D (Kyriacou et al., 1999; Mohamed et al., 2001; Wasserman and Acharya, 1996) and 3D tumor growth (Angeli and Stylianopoulos, 2016; Dumpuri et al., 2007; Hawkins-Daarud et al., 2012; Hogea et al., 2007; Hogea et al., 2006a; Hogea et al., 2006b; Lloyd et al., 2007; Mohamed and Davatzikos, 2005; Nimsky et al., 2004) as well as mass-effect (Dumpuri et al., 2010; Fan et al., 2011; Sun et al., 2014) FE models have been developed to calculate the mechanical forces and mass-effects due to tumor growth. These models have all been limited to whether using a constant outward pressure profile in the tumor boundary to control the growth or using a simplified quasi-spherical tumor shape. However, very recently our group showed that a constant pressure boundary unable to capture the realistic irregular shape of a tumor so the application of a variable pressure boundary as a function of tumor shape is highly preferred (Abdolkarimzadeh et al., 2021). Yu et al., (Yu et al., 2022) employed hyperelastic material model to estimate the brain deformations in patients after tumor removal. However, it is known that soft biological tissues in general (Dehoff, 1978; Estermann et al., 2022; Holzapfel, 2017; Humphrey, 2003; Karimi et al., 2017a; Maccabi et al., 2018) and the brain tissue specifically are highly viscoelastic (Bilston et al., 1997; Budday et al., 2017; Finan et al., 2017; MacManus et al., 2017; Qiu et al., 2020; Sundaresh et al., 2021), so their mechanical response depend on the rate of the applied load. In our prior study (Abdolkarimzadeh et al., 2021), we used an elastic material model to address the mechanical properties of the brain tissue. However, the results revealed a considerable difference when it comes to the nodal comparison between the pre- and post-surgery tumor shapes. The error may relate to the simple elastic material model of the brain tissue, while it has been well documented that the brain tissue is highly viscoelastic (Bilston et al., 1997; Budday et al., 2017; Finan et al., 2017; MacManus et al., 2017; Qiu et al., 2020; Sundaresh et al., 2021). Although the application of a time-independent material model could be able to result in local accuracy when it comes to local comparison, it is difficult to capture the mechanical response of the entire brain throughout the tumor growth cycle. To date, there is a gap in our knowledge about the role of the brain tissue viscoelasticity on the resultant pressure profile in the brain tissue due to the tumor growth. In this study, we hypothesized that the applied pressure boundary on the brain tissue due to the tumor growth is highly position- and time-dependent that stems from the viscoelastic mechanical behavior of the brain tissue (Bilston et al., 1997; Budday et al., 2017; Finan et al., 2017; MacManus et al., 2017; Qiu et al., 2020; Sundaresh et al., 2021) as well as the irregular shape of the tumor (Aghalari et al., 2021; Hussain et al., 2017; Hussain et al., 2018), respectively. Since the strength of the bulk tumor mass-effect and the final tumor size (Zacharaki et al., 2008) in addition to the tumor progression and response to treatment have been reported to be regulated by the resultant pressure parameters during the tumor growth (Angeli and Stylianopoulos, 2016).

This study is therefore aimed to calculate the position- and time-dependent pressure boundary on a viscoelastic brain tissue due to the tumor growth that may contribute to a precise estimation of mass-effect due to tumor resection surgery using a fully coupled inverse dynamic FE-optimization algorithm. Since we have a time-dependent pressure boundary that is being updated at each load increment (time-step), a simple elastic material model unable to consider that while viscoelastic material model understands a time-dependent pressure. In addition, a simple elastic material model results in the same stresses and strains in the tissue regardless of the rate of the applied load but only is sensitive to the magnitude of the load. However, in the case of tumor growth, the brain tissue is constantly being deformed so the applied pressure on the tissue is constantly being updated. Therefore, we required to employ a time- and position-dependent pressure profile to address the tumor growth. To do that, a FE patch of the brain tissue was constructed and subjected to a compressive load to calculate the viscoelastic parameters of the brain tissue using a FE-optimization algorithm (Abdolkarimzadeh et al., 2021; Rahmati et al., 2021) based on the experimental human brain data (Karimi et al., 2017b). Thereafter, a set of FE simulations were performed using the same patch model to calculate the most suitable simulation time that results in stability in the viscoelastic parameters of the brain tissue (independency to the shear relaxation modulus). The calculated viscoelastic parameters and the simulation time were then assigned to volume-meshed post-surgery FE models of the human brain (Abdolkarimzadeh et al., 2021) to calculate the optimized 3^rd-order position-dependent and normal distribution time-dependent pressure function based on the minimum distance between the surface-nodes of the post-surgery brain cavity and pre-surgery tumor. To model the tumor growth, the ideal method is to apply load on tumor in the first stages of its growth aiming to reach the final shape of the tumor before tumor resection. However, the data corresponding to the first stages of tumor are rare to access since tumor symptoms appear in the final stages of tumor growth and patients usually make an appointment to visit a doctor. Therefore, to an inverse FE approach was used in which the post-resection model was loaded to reach pre-resection tumor model.

2. Materials and Methods

2.1. Brain Patch Finite Element Model – Viscoelastic Material Properties

It has been well documented that the brain tissue is viscoelastic (Donnelly and Medige, 1997; Fallenstein et al., 1969; Forte et al., 2017) and its mechanical behavior can be addressed through Prony series (Forte et al., 2017; Miller and Chinzei, 1997; Prange and Margulies, 2002; Rashid et al., 2012). A viscoelastic material model has both viscous and elastic properties (Herrmann, 1968). The behavior of a viscoelastic material can be outlined via short- and long-time shear moduli as follows:

$$G\left(t\right)=G_{\infty }+(G_0-G_{\infty })e^{-\beta t}$$ (1)

where G₀ and G_∞ are the short-time and long-time shear modulus, respectively, and β is the decay constant. A Jaumann rate formulation is also used as follows:

$${\sigma }_{ij}^{\text{'}}=2{\int }_0^{t}G\left(t-\tau \right)D_{ij}^{\prime }(\tau )d\tau$$ (2)

where the prime denotes the deviatoric part of the stress rate, ${\sigma }_{ij}^{\prime }$, and the strain rate, D_ij.

To calculate the viscoelastic mechanical properties of the brain tissue, a brain patch FE model was constructed, subjected to a compressive strain, and the resultant stress was compared to the prior experimental brain compressive data by Karimi et al., (Karimi et al., 2017b). The FE model of the brain patch has the length, width, and thickness of 30 mm × 30 mm × 20 mm that match the experimental study (Karimi et al., 2017b) (Figure 5 of that paper) as shown in Fig. 1a. Thereafter, the patch model was subjected to a uniaxial compression in the X direction, where the displacement boundary condition (~40% strain) was applied to the FE model to mimic the uniaxial mechanical testing of human brain (Karimi et al., 2017b). The optimization algorithm of Fminsearch-Unconstrained nonlinear minimization was coupled with the solver of LS-DYNA (Livermore Software Technology Corporation, CA, US) to calculate the viscoelastic parameters of the patch FE model (Abdolkarimzadeh et al., 2021; Rahmati et al., 2021). Fminsearch finds the minimum of a scalar function of several parameters, starting at an initial estimation. An initial guess, herein G₀=528 kPa, G_∞=168 kPa, and β=700 1/s (Galford and McElhaney, 1970), with the upper and lower boundaries of 34< G₀ <528 kPa, 5.5< G_∞ <168 kPa, and 30< β <1000 1/s (Turquier et al., 1996; Zhang et al., 2001), were set for the patch FE model and was started to run with the cost function of mean squared error (Schmidt et al., 2009). The mean squared error was the sum of the squared differences between the experimental data (Karimi et al., 2017b) and optimization value. The resultant stress-strain in the gauge at the center of the FE patch model (Fig. 1a) was calculated and plotted versus the experimental data (Karimi et al., 2017b) (Figure 5 of that paper) as displayed in Fig. 1b. The final optimized viscoelastic material properties were assigned to the brain tissue FE models as listed in Table 1.

Fig. 5.

Comparative contour maps of deformation in the post-resection brain cavity versus the pre-resection tumor surface nodes in (a) patient #11 and (b) patient #12.

Fig. 1.

(a) FE model of the brain patch, including the strain gauge and boundary conditions. (b) Stress-strain curves representing the experimental uniaxial compressive testing of human brain patches, compared to identical FE model.

Table 1.

The average von Mises stress at different simulation times.

Time (μs)	Average von Mises Stress (kPa)
720	44.3
7200	21.4
10000	19.2
20000	16.3
30000	15.3
40000	14.8
50000	14.4
60000	14.4
70000	14.2
72e3	14.2
80000	14.0
90000	14.0
100000	14.0
720000	13.5
7200000	13.4
72000000	13.3
720000000	13.4

FE simulations based on the modeling needs can be done at various time scales (end time) and step times. Although timing would not be a primary concern for a time-independent material model, such as elastic and hyperelastic, it would have a considerable influence in time-dependent material model, such as viscoelastic. In this step, a set of FE simulations were performed using the same patch FE model with optimized viscoelastic parameters extracted from the previously described FE-optimization runs. The patch FE model was subjected to the same loading and boundary conditions as explained before but different simulation times to calculate the stable time that results in the independency of the mechanical response to the shear relaxation modulus of the brain tissue. The FE simulations as well as the corresponding stresses were also reported in Table 2.

Table 2.

The identified material parameters of the human brain tissue and skull (Razaghi et al., 2019; Razaghi et al., 2021).

Parameters	Short-time shear modulus G0 (kPa)	Long-time shear modulus G∞ (kPa)	Decay constant β (1/s)	Bulk modulus κ (kPa)	Density ρ (kg/m3)
Brain tissue	505	110	828	5625	1140
Parameters	Shear Modulus μ (MPa)	Yield stress σy (MPa)	Plastic hardening modulus	Bulk modulus κ (MPa)	Density ρ (kg/m3)
Skull	3470	41.80	0.0462	7120	1456

2.2. Image acquisition, registration, and segmentation

Pre- and post-resection MRI images of two brain tumor patients (patients #11 & #12) were employed from an online resources at the Montreal Neurological Institute in 2010 (group #3) (Mercier et al., 2012). The segmentation and FE model reconstruction were fully explained in our recent paper (Abdolkarimzadeh et al., 2021). Briefly, the MRI images were first converted into a stack of *JPG images, following by a relative and absolute rotation and/or transformation to fully match the pre- and post-resection images together through affine transformation custom Matlab script. This was a very important stage of the work since the patient may not be able to hold the head at the same spot before and after the tumor resection, so an accurate rotation and/or transformation would be a very critical step for the following FE simulations. An expert neuroradiologist also helped us in addition to the semi-automatic Matlab code to correct the possible mismatches that our code cannot fix since even an insignificant mismatch herein would influence the mass-effect calculation and interpretation of the numerical outcomes.

The brain and the skull were segmented using Mimics (Materialise Inc., Belgium) and the surface meshes were thereafter volume meshed with 10-noded tetrahedral elements (Karimi et al., 2021a; Karimi et al., 2021b; Karimi et al., 2021c), as the 10-noded tetrahedral elements have been resulted the same precision as 20-noded hexahedral elements at considerably a lower computational cost (Karimi et al., 2021b). The FE models of brains after tumor removal from the isometric and top views are presented in Fig. 2. The FE models of the brain #11 and #12 were made of 395,274/550,554 and 144,138/216,166 elements/nodes, respectively. Mesh quality assessment (Ansys, Canonsburg, PA, US) and mesh density analysis (Ansys/LST, Canonsburg, PA, US) were performed to first check the quality of the mesh and then estimate the errors because of element numbers (data are not reported here) (Karimi et al., 2021b). The skull is not represented herein since it would hinder us showing the resultant cavity in the brain due to tumor removal as well as the anatomical location and size of the tumor.

Fig. 2.

The post-resection FE models of (a) patient #11 and (b) patient #12 from the isometric and top views.

A custom Matlab script helped to define the load surface, equivalence the nodes, and write the final LS-Dyna *k file. A 10-core Intel® Xeon® CPU W-2155@3.30 GHz computer with 256GB RAM was used to run the simulations in explicit-dynamic LS-DYNA (Ansys/LST, Canonsburg, PA, US). The simulations were performed in one-step for 80 ms with time steps of 1 ms (80 time steps). Cerebrospinal fluid pressure (CSFP) was applied on the outer surface of the brain from 0–5 ms with the magnitude of −10 mmHg (−1.33 kPa) (Eklund et al., 2016; Turner et al., 1996) that is determined based on the human intracranial pressure in the supine body position. Thereafter, from 5–80 ms the pressure was set to +10 mmHg. The MRI images show the brain under applied CSFP, therefore, a negative CSFP would help to precondition the tissue and thereafter pressure it back to the same load boundary.

2.3. Inverse Dynamic FE-optimization algorithm

A fully coupled FE-optimization algorithm (Abdolkarimzadeh et al., 2021; Rahmati et al., 2021) was employed to optimize the pressure profile at the interface of the brain tissue and tumor. Flow-chart of the applied algorithm is presented in Fig. 3. The position- and time-dependent pressure profiles were the aims of the optimization algorithm while the distance between the brain cavity surface-nodes and the pre-resection tumor surface-nodes was the cost function. When it comes to the FEM, two approaches can be used in terms of the stress calculation, including forward and inverse (Moulton et al., 1995). In the forward approach, the loading and boundary conditions, and the material properties are the known variables while the strain in an element or displacement at the nodes are the unknown variables and need to be calculated using FE solver. However, in the inverse FEM, the loading and boundary conditions as well as the material properties are the unknown parameters while the strain in the elements or nodal displacements are the known parameters. Herein, in the brain FE model, the pressure profile at the interface of the brain tissue and tumor is unknown while we know the nodal coordinates before and after tumor removal that provide us with the nodal displacement. An inverse approach would need more computational hours due to various initial guesses while a forward approach with known parameters would need less hours for the same problem. To solve such an inverse FE model, herein we benefitted from a coupled FE-optimization algorithm (Abdolkarimzadeh et al., 2021; Rahmati et al., 2021). Fminsearch with different initial guesses of P₀, P₁, P₂, P₃ and σ for patients #11 and #12 with the upper and lower boundaries of 0 < P₀ < 60,0 < P₁ < 60,0 < P₂ < 60,₀ < P₃ < 60, and 1 < σ < 10 were set for the pressure profile and was started to run with the cost function of mean squared error. To do that, an initial guess was used to define a 3^rd order polynomial function for position-dependent pressure profile and a two-variable (mean and standard deviation) normal distribution function for time-dependent pressure profile. The pressure herein in the optimized one to result in a minimized cost function that is the difference between the nodal coordinates of the reconstructed MRI data and computational data (Fig. 3). The optimized position- and time-dependent pressure profiles include an optimized 3^rd-order pressure function and probability density function of normal distribution function that were defined as follows:

$$\overrightarrow{P}(x,y,z)={\overrightarrow{P}}_0+{\overrightarrow{P}}_1\overrightarrow{d}+{\overrightarrow{P}}_2{\overrightarrow{d}}^2+{\overrightarrow{P}}_3{\overrightarrow{d}}^3$$ (3)

$$P(t)=\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{{\left(t-\mu \right)}^2}{2{\sigma }^2}}$$ (4)

where P₀, P₁, P₂, and P₃ are the pressure parameters, $\overrightarrow{d}$ is the vector of the distance that denotes the normal distance between the center of each element and a surface node on the tumor in Eq. (3), and μ and σ are mean and standard deviation in Eq. (4), respectively. Since the value of pressure increases during the tumor growth, μ should be determined as the end time of the simulation. The other parameters, i.e., σ is computed as an optimization parameter. The final optimized position-time-dependent pressure profile is defined as follows:

$$P\left(x,y,z,t\right)=P\left(x,y,z\right).P\left(t\right)$$ (5)

Fig. 3.

Flow-chart of the fully coupled inverse FE-optimization algorithm.

The change in time-dependent pressure pattern as a function of σ for a desired time is shown in Fig. 4.

Fig. 4.

The normalized pressure as a function of time for various σ values.

The optimization algorithm of Fminsearch-Unconstrained nonlinear minimization was coupled with the solver of LS-DYNA to calculate the pressure profile parameters (Abdolkarimzadeh et al., 2021). Fminsearch finds the minimum of a scalar function of several parameters, starting at an initial estimate. The coupled FE-optimization algorithm run in Matlab environment using LS-DYNA solver for more than ~90 iterations and the finally converged pressure parameters are listed in Table 2. The cost function was used as follows:

$$CostFunction=\sum _{i=1}^n\left(\mathrm{m}\mathrm{i}\mathrm{n}\left(‖p_i^{BC}-p^T‖\right)\right)$$ (6)

where $p_i^{BC}$ is the position vector corresponding to the i^th node of the brain cavity, p^T is the position matrix of all tumor surface-nodes and n is the number of brain cavity surface-nodes. It is worth mentioning that we cannot certify that the optimized pressure profile parameters result in “absolute” minimum for a non-convex cost function. However, herein, the best “possible” minimum was calculated since the solver run more than ~90 iterations for different initial conditions to result in the least possible error so we can say the solution is “consistent” and not a “relative” minimum.

3. Results

The effective stresses at various simulation times in the FE model of the brain patch are listed in Table 1. The optimized pressure profile parameters for both FE cases are also calculated and summarized in Table 2. The contour maps of the tumor surface nodes in the pre- and post-resection before and after loading for both FE cases of #11 and #12 are presented in Fig. 5. The resultant pressure contours in the post-resection tumor surface (brain cavity) after loading for both cases of #11 and #12 are shown in Fig. 5.

The contour maps of displacement in the brain tissues of patients #11 and #12 at various simulation times are shown in Fig. 6. The contour maps of pressure and von Mises stress in the brain tissues of patients #11 and #12 are shown in Figs. 7 & 8.

Fig. 6.

Contour maps of pressure in the post-resection brain cavity in (a) patient #11 and (b) patient #12.

Fig. 7.

The contour maps of displacement in the brain tissue of (a) patient #11 and (b) patient #12 at various simulation times.

Fig. 8.

The contour maps of von Mises stress in the brain tissue of (a) patient #11 and (b) patient #12 at various simulation times.

4. Discussions

Accurate planning and navigation is required to maximize tumor resection is going to be as planned while make sure the nearby healthy tissues will not be damaged. That would require a pressure profile that not only could capture the irregular shape of the tumor during the growth in the brain but also the time-dependent viscoelastic behavior of the brain tissue. However, the currently available approaches whether have failed near the brain due to large deformations and irregular growth pattern of the tumor or have failed to address the time-dependency of the brain biomechanics. In this study, a fully coupled dynamic FE-optimization algorithm (Abdolkarimzadeh et al., 2021; Rahmati et al., 2021) to calculate the position-and time-dependent pressure profile in the brain tumor in post-tumor resection patients.

To calculate the viscoelastic mechanical properties of the brain tissue, a brain patch FE model was constructed and subjected to a compressive uniaxial loading (Fig. 1a). A fully coupled FE-optimization algorithm was employed to calculate the stress-strain in the gauge at the center of the patch FE model and compare it to the experimental data (Karimi et al., 2017b) (Fig. 1b). The calculated viscoelastic mechanical properties were then assigned to the post-resection FE models of the human brain (Fig. 2). A positive pressure was applied to the post-resection brain cavity through a fully coupled inverse FE-optimization algorithm (Fig. 3) to calculate the position- and time-dependent pressure profile at the interface of the brain tissue and the tumor that mimics the tumor growth. The pressure profile proposed in this study benefitted from a time-dependent pressure parameter, σ, which defines the action time that pressure comes into effect on the surface elements of the brain cavity (Fig. 4).

The results revealed a very good fit between the pre-resection surface nodes and that of the post-resection tumor surface after loading (Fig. 5) that implies the accuracy of the proposed position- and time-dependent pressure profile in simulating the tumor growth. The contour maps of the pressure in the brain cavity due to the tumor growth are shown higher magnitudes at the free surfaces where there is a contact with the skull (Fig. 6).

The deformation in the brain tissue was 0.8–1.1 cm (Fig. 7) that is in a good agreement with that of 0.23–1.45 cm (Angeli and Stylianopoulos, 2016). The areas close to the tumor showed the highest deformation while distal regions showed almost no deformation that implies that the biomechanical effects of the tumor growth are mostly local. The mass-effect of 16–44 mm was observed in the regions around the brain cavity (Fig. 7) that is within the range of brain shift due to tumor resection (~ 24 mm) (Bayer et al., 2017; Chen et al., 2011; Gerard et al., 2017; Miga et al., 2016; Nimsky et al., 2000). The accurate prediction of mass-effect depends on the applied pressure boundary since the mass-effect has a pivotal role in defining the premise for justifying surgery of the cerebrum (Bullock et al., 2006; Gonda et al., 2013; McKenna et al., 2012) as well as neurologic diseases, i.e., tumor (Gamburg et al., 2000), stroke (McKenna et al., 2012; Zazulia et al., 1999), and trauma (Bullock et al., 2006; Kim and Gean, 2011).

The brain tissues showed the maximum von Mises stress of 140 kPa (Fig. 8) that is in good agreement with 120 kPa reported by Angeli et al., (Angeli and Stylianopoulos, 2016). In our prior study, the brain tissue was modeled as a homogenous isotropic hyperelastic material (Abdolkarimzadeh et al., 2021) that the tissue mechanical response was entirely time-independent. In this study, the brain tissue was simulated as a time-dependent viscoelastic material model that allows us to have a better understanding of the resultant pressure profile on the brain tissue due to tumor growth. In our prior study, the application of a time-independent hyperelastic model caused an irregular deformation in the brain when the CSFP was applied on the outer surface of the brain FE model (Abdolkarimzadeh et al., 2021). In this study, however, because of the viscoelastic material model, the brain showed stable response and reached a realistic physiological equilibrium before and during the tumor growth. Brain tumors can take anywhere from few months to years to form (Islim et al., 2019; Sallemi et al., 2015; Yamada et al., 2021). Although, to date, based on the current computational resources it is still not plausible to conduct such long FE simulations to predict the tumor growth, it is possible to use time-dependent material models, such as viscoelastic, to “partially” capture the time-dependent mechanical response of the brain tissue due to the tumor growth over a course of reasonable time-scale. The application of time-independent elastic and hyperelastic material models not only cannot address the time-dependent mechanical behavior of the brain tissue but also would result in considerable error in the pressure profile and mass-effect calculations.

A position- and time-dependent pressure boundary would have implications for modeling of structures with unidentified volumetric pressure, i.e., Schlemm’s canal (Maepea and Bill, 1992; Stamer et al., 2015) that the intraocular pressure may vary both as a function of the shape of the canal as well as the aqueous outflow pulse (Johnstone et al., 2021). Understanding of the position- and time-dependent pressure profile in the Schlemm’s canal may contribute in calculating an accurate magnitude of the shear stress applies on the endothelial cells of the Schlemm’s canal (Madekurozwa et al., 2021; McDonnell et al., 2020; Reina-Torres et al., 2021).

The mass-effect plays a key role in defining the premise for justifying surgery of the cerebrum (Bullock et al., 2006; Gonda et al., 2013; McKenna et al., 2012). In neurologic diseases, ranging from trauma (Bullock et al., 2006; Kim and Gean, 2011), stroke (McKenna et al., 2012; Zazulia et al., 1999), to tumor (Gamburg et al., 2000), increased mass-effect is consistently associated with poor prognosis. Our results revealed that considering a variable pressure boundary causes a more realistic pressure contour around the tumor region, so it would result in a more accurate prediction of mass-effect in ventricles. Our results herein showed the mass-effect of ~5–11 mm in the regions around the brain cavity (Fig. 7), which is lower than that of brain shift reported during tumor resection ~24 mm (Bayer et al., 2017; Chen et al., 2011; Gerard et al., 2017; Miga et al., 2016; Nimsky et al., 2000). This difference could be related to the stiffness of the tissues at the vicinity of the tumor that affect the mass-effect. In addition, herein we only modeled two cases while more cases would be required if a wider range of mass-effect estimation is the goal.

This study is limited by the following considerations. First, the quality of the MRI data plays a key role in FE models reconstruction and the following FE simulations. Therefore, even an insignificant mismatch between the pre- and post-resection images would result in error in mass-effect and pressure profile calculation. Herein, the affine transformation was used to register and correct the images in addition to an expert neurologist to minimize any mistakes. Second, the brain tissue in this study was considered to behave as homogenous isotropic viscoelastic, however, it has been well documented that the brain tissue is highly anisotropic having dissimilar material properties for the gray and white matters (Antonovaite et al., 2021; Giordano and Kleiven, 2014; Mijailovic et al., 2021; Velardi et al., 2006). Third, mechanical properties of the human brain tissue in vivo have been poorly understood (Howells et al., 2012; Raboel et al., 2012; Tain and Alperin, 2008); hence, the mechanical properties of a cadaveric human brain tissue were adopted for this study (Karimi et al., 2017b). Fourth, to date, there is no experimentally validated gold standard to compare the computed mass-effect of the tumor versus the experimental data for the pressures and deformations. In this study, since the pre- and post-resection images were provided for the same patient so we could compare the images, calculate the nodal displacement between the two sets of images, and use that as a reference. Finally, herein we used the post-resection data to predict the tumor growth using a position- and time-dependent pressure profile. However, the tumor growth is a forward process as it initiates in the brain and starts to grow by the passage of time. Although this is the real case, for such simulations, we would require several images during the tumor growth and accessing to those data would not be possible unless an accurate follow up. Therefore, in this study, we used an inverse approach and hypothesized that the growth of the tumor in the brain can be inversely simulated.

In summary, herein we proposed a position- and time-dependent pressure profile to address the viscoelastic mechanical behavior of the brain tissue due to tumor growth. The brain tissue has been well shown to have a time-dependent mechanical response under the applied loadings. Therefore, a time-independent pressure profile cannot address the mechanical behavior of the brain tissue and would result in an inaccurate prediction of the brain tumor mass-effect as well as tissue deformation. In this study, the viscoelastic mechanical parameters of the brain tissue were calculated using an inverse FE-optimization algorithm based on the experimental compressive data of the human brain tissue. Several FE simulations were then performed using a patch FE model to calculate the simulation time that results in independency of the tissue mechanical response (shear relaxation modulus) to the time scale of the entire simulation. Finally, the viscoelastic mechanical properties as well as the simulation time were incorporated into the FE model of the post resection human brain with the tumor cavity to inversely calculate the 3^rd order position-dependent and normal distribution time-dependent pressure profile at the interface of the brain-tumor. The results revealed that the new proposed pressure profile can well capture the position- and time-dependent mechanical behavior of the brain tissue due to irregular growth of the tumor size and can be considered as a new computational framework for tumor resection surgical planning. The proposed method is computationally inexpensive, robust, accurate, and allows the surgeons to predict the resultant mass-effect before tumor resection, so eventually contributes to better surgical outcomes.

Table 3.

The pressure parameters for patients #11 and #12.

	P0 (kPa)	P1 (kPa)	P2 (kPa)	P3 (kPa)	σ
Patient #11
Pressure parameters	14.1	23	10	0	2
Patient #12
Pressure parameters	13.1	44	1	0	2.5