Computational modeling of phagocyte transmigration for foreign body responses to subcutaneous biomaterial implants in mice
- Mingon Kang^{1},
- Liping Tang^{2} and
- Jean Gao^{1}Email author
https://doi.org/10.1186/s12859-016-0947-3
© Kang et al. 2016
Received: 29 October 2015
Accepted: 15 February 2016
Published: 29 February 2016
Abstract
Background
Computational modeling and simulation play an important role in analyzing the behavior of complex biological systems in response to the implantation of biomedical devices. Quantitative computational modeling discloses the nature of foreign body responses. Such understanding will shed insight on the cause of foreign body responses, which will lead to improved biomaterial design and will reduce foreign body reactions. One of the major obstacles in computational modeling is to build a mathematical model that represents the biological system and to quantitatively define the model parameters.
Results
In this paper, we considered quantitative inter connections and logical relationships among diverse proteins and cells, which have been reported in biological experiments and literature. Based on the established biological discovery, we have built a mathematical model while unveiling the key components that contribute to biomaterial-mediated inflammatory responses. For the parameter estimation of the mathematical model, we proposed a global optimization algorithm, called Discrete Selection Levenberg-Marquardt (DSLM). This is an extension of Levenberg-Marquardt (LM) algorithm which is a gradient-based local optimization algorithm. The proposed DSLM suggests a new approach for the selection of optimal parameters in the discrete space with fast computational convergence.
Conclusions
The computational modeling not only provides critical clues to recognize current knowledge of fibrosis development but also enables the prediction of yet-to-be observed biological phenomena.
Keywords
Computational modeling Foreign body responses Phagocyte transmigration Parameter estimationBackground
Medical implants, such as breast implants, encapsulated tissues/cells, neural electrodes, and eye implants, have experienced remarkable development and growth during the past decades. In the meantime, an increasing number of medical implant failures also has been reported. In earlier studies, it is well documented that excessive fibrotic responses are responsible for the failure of many medical implants [1–4]. Medical implants provoke unpredicted responses and reactions of the immune system, which are fibrotic capsule formations surrounding the medical device. To be specific, wound healing responses start with acute inflammatory responses, and then follow with fibrotic tissue reactions. The continuous inflammatory responses lead to overwhelming fibrotic reactions. Hence, comprehensive understanding of the mechanism governing the reactions can play an important role in successful implantation and development of the biomaterial while reducing its side effects and simultaneously improving the functionality of the implants.
Computational modeling and simulation have been highlighted in biomedical research for decades, since they not only discover the nature and their associations of the biological components but also provide quantitative predictions that go beyond the biological experiment. Therefore, in-depth understanding of the foreign body responses and its computational modeling will disclose the contributing components and help to predict the evolution, which would eventually lead to reduced failure rate of implantation. A number of research has been conducted for modeling and predicting foreign body fibrotic reactions. A hybrid model was proposed by combining a differential equation system and kinetics Monte Carlo algorithm to simulate and predict phagocytes responses at molecular level [5]. Chemical kinetics equations were adapted for building a predictive tool of foreign body fibrotic reactions [6, 7]. Partial differential equations were also utilized for macrophage spatial/temporal dynamics in foreign body reactions[8].
Tang et al. had conducted biological experiments with mice to recognize major components involved in foreign body reactions and to discover their responses and reactions [2]. To summarize the current biological understanding, the evolution of biomaterial-mediate inflammatory responses may be divided into six consecutive events: (1) phagocyte transmigration through the endothelial barrier, (2) chemotaxis toward the implants, (3) adherence to the biomaterial, (4) phagocyte activation, (5) fibrin deposition, and (6) fibroblast proliferation and collagen production [2]. Among these six procedures, we focused on comprehensive modeling of phagocyte transmigration. It is assumed that certain components such as mast cells, histamine, histamine receptors, and P/E selectins, are mainly involved in the event [9]. Histamine release has been reported to trigger inflammatory responses to biomaterial implants in human [10], and the event was reported by observing the concentration change of polymorphonuclear neutrophils (PMN) and monocytes/macrophages (M Φ), which are the most abundant type of phagocytes.
In this study, we conducted a mathematical modeling of phagocyte transmigration which is one of the processes involved in fibrosis formed around an implanted biomedical device. Furthermore, a new optimization approach, Discrete Selection Levenberg-Marquardt Algorithm (DSLM), based on our previous preliminary work [12] is developed for parameter estimation of the dynamic system for the accurate simulation of the biologic system. The new parameter estimation approach satisfies global non-linear optimization and converges in a feasible time. We will elucidate the biological experiment and elaborate the successive courses in the following sections.
Methods
Biological experiments for phagocyte transmigration
We have carried out biological experiments to build phagocytes transmigration models. We chose and tested the model system by using the experimental results obtained from mice, since a mouse model is commonly used to test the compatibility of medical implants and also to mimic foreign body reactions in human. For implantation, Swiss-Webster mice were anesthetized with isoflurane (Abbott Laboratories, North Chicago, IL), a 1.5 cm mid-abdominal or mid-dorsal longitudinal incision was made, and 0.6 cm diameter test disks and drug release pumps were implanted intraperitoneally (5 disks/mouse). The incision was closed with stainless steel wound clips. For studies in which the net recruitment of phagocytes was measured, control animals (operated but without implant placement) were always included. Immediately prior to examination, the animals were euthanized with C O _{2}. The implants were carefully removed from the peritoneal cavity or subcutaneous space and washed with sterile PBS. These samples were used for assay of inflammatory and fibrotic responses.
Inflammatory cells are recruited to the interface between biomaterial implants and skin tissue, which is the area of interest in this study. For measurements of the acute inflammatory responses, investigation was typically performed after sixteen hours. The sixteen hour time point is the earliest time point for maximal inflammatory cell recruitment.
Previous studies hypothesized that histamine might also play an important role in the recruitment of inflammatory cells to implants. To verify the idea, histamine receptors, antagonist pyrilamine (H1 receptor antagonist) and famotidine (H2 receptor antagonist), were injected to the peritoneal space. The combined treatment of H1 and H2 receptor antagonists dramatically decreased the number of phagocytes on implant surfaces as well as the number of PMN and M Φ recruited de novo to the peritoneal cavity (Fig. 3 b). The other intracellular substances via complex intracellular signaling may interfere with the activity of the inhibitory and stimulating actions, but the potential receptor interactions and signaling pathway have yet to be investigated. Therefore, in this study, we assume that histamine enhances phagocyte transmigration via both H1 and H2 receptors.
The half of the phagocytes in the knockout experiments (Fig. 3) can be suspected as a delayed accumulation. However, the knockout mice have normal inflammatory responses as shown in the previous work [13]. Moreover, the knockout mice triggered reduced long-term foreign body responses, fibrotic tissue formation, was shown in a recent publication [14].
Modeling for phagocyte transmigration
In this section, we build computational models of phagocyte transmigration by utilizing reported biological literature and investigating the phenomena of the observations.
Residual histamine
where k _{ e0} is the initial value of the source, k _{ k1} is the contraction rate of the source, k _{ w0} is the initial value of oscillation frequency, k _{ k2} is the contraction rate of oscillation frequency, and t _{1} is the starting point that the source is released.
where k _{ rhch } is the rate that residual histamine degrades.
Histamine
where k _{ rhch } is the histamine release rate which is the same as the residual histamine degrading rate as given in (3), and k _{ hs } is the rate that histamine degrades itself. The first term on the right hand side of (4) represents the increase of histamine released from mast cells. I _{ mc }(t) indicates the relative concentration level of mast cells for the knock-out experiment using mast cell deficient mice, where 0 ≤I _{ mc }(t)≤ 1. The second term on the right hand side of (4) represents the degradation of histamine itself.
H1/H2 Histamine receptor
where k _{ hchrt } and k _{ hchrb } are the upper/lower bounds of rate that histamine receptors combine with histamine, and k _{ hrs } is the rate that histamine receptors degrade themselves. In the same way as histamine modeling, the first term on the right hand side represents the propagation of histamine receptors combined with histamine, while the second term shows the degradation rate of histamine receptors. However, we adapted a hyperbolic form for the first term in (5), where it sets a maximally increasable bound with the change of histamine receptors since it would not increase unlimitedly.
P/E selectins
where k _{ hcsb } and k _{ hcst } are the upper/lower bounds of rate released by histamine in a hyperbolic form, and k _{ ss } is the selectins degradation rate.
Phagocytes
where k _{ pmnipb }, k _{ pmnipt }, k _{ mpipb } and k _{ mpipt } are the upper/lower bounds of rate for histamine receptors and selectins that increase permeability respectively, and k _{ pmnps } and k _{ mpps } are the contraction rate of the capillary permeability. I _{ pmnhr }(t) and I _{ mphr }(t) are the constants indicating block/non-block histamine receptors, and I _{ pmns }(t) and I _{ mps }(t) are the regulations that control block/non-block selectins.
where k _{ pmns } and k _{ mps } are the rates that PMN and M Φ degrade.
Results
Parameter estimation and simulation for phagocyte transmigration
Parameter estimation for phagocyte transmigration
The above section introduced a dynamic system which contains numerous parameters whose values reflect characteristics of the system. Thus, estimating parameters of the system is essential to discover the components’ behavior in the system and hence provides a successful quantitative modeling.
Estimated parameters by DSLM
Parameter | Description | Estimation |
---|---|---|
t _{1} | Starting time that the external source is released | 3.5000 |
k _{ e0} | An initial concentration of the external source | 11.2464 |
k _{ k1} | A self contraction of the external source | 0.1017 |
β | An initial concentration of the oscillation bound | 0.0134 |
k _{ w0} | An initial value of the oscillation frequency | 1.4037 |
k _{ k2} | A contraction rate of the oscillation frequency | 0.0937 |
k _{ rhch } | A rate that the residual histamine decayed | 0.3704 |
k _{ hs } | A rate that the histamine regulates itself | 0.0002 |
k _{ hchrb } | A rate that the histamine receptors are released | 0.9997 |
k _{ hchrt } | A upper bound rate that the histamine receptors are released | 0.0757 |
k _{ hrs } | A rate that the histamine receptors regulate themselves | 0.7232 |
k _{ hcsb } | A lower bound of rate that the selectins are released | 1.9997 |
k _{ hcst } | A upper bound of rate that the selectins are released | 0.2668 |
k _{ ss } | A rate that the selectins regulate themselves | 0.1024 |
k _{ pmnipb } | A lower bound of rate that increases the permeability for PMN | 0.2226 |
k _{ pmnipt } | A upper bound of rate that increases the permeability for PMN | 2.0000 |
k _{ pmnps } | A rate that the permeability of the capillary self degrades | 0.1003 |
k _{ mpipb } | A lower bound of rate that increases the permeability for M Φ | 0.0001 |
k _{ mpipt } | A upper bound of rate that increases the permeability for M Φ | 1.0108 |
k _{ mpps } | A rate that the permeability of the capillary self degrades | 0.1961 |
k _{ pmns } | A rate that the PMN self degrades | 0.0582 |
k _{ mps } | A rate that the M Φ self degrades | 0.0307 |
Simulation for phagocyte transmigration
Model parameter estimation
The parameters of the proposed mathematical equations can be estimated by nonlinear programming algorithm. There are a number of numerical optimization techniques such as Newton’s method, Broyden’s method, line search method, and trust-region method in nonlinear programming. Those methodologies are all local optimization techniques, where the object function is globally convex. However, the object function of the proposed models is unfortunately non-convex, which needs a global optimization technique. Therefore, in this study, we developed a global optimization algorithm, called Discrete Selection Levenberg-Marquardt (DSLM) for the parameter estimation. DSLM is a global optimization version extended from Levenberg-Marquardt (LM) algorithm, which is a gradient-based local optimization algorithm.
Least squares and Levenberg-Marquardt algorithm
where \(r_{i} = \tilde {y}_{i} - f(t_{i}, \textbf {x})\) and f(t _{ i },x) is the model function that has p parameters. Here, the goal is to find optimal x ^{∗} subject to minimization of the function F(x).
In order to solve the non-linear least squares problem, Isaac Newton proposed Newton’s method to find the minimum or maximum of a function, F(x), using the first (Jacobian) and second (Hessian)-order partial derivatives of the function [18]. Newton’s method can converge quickly if the initial guess is close to the optimum. However, it is often difficult or impossible to obtain the Hessian matrix of the object function. The steepest gradient method is a first-order optimization algorithm using gradient descent to find a local minimum or maximum [19]. It finds an optimum quickly even though the initial guess is far from the optimum and the system size is very large. However, as it goes close to optimum, it is often infeasible due to the constant step size. The Gauss-Newton algorithm remedies the shortcomings of both Newton method and the steepest gradient method. The Gauss-Newton method has an advantage that it does not need the second derivatives, and it has quadratic final convergence if initial guess is close to the optimum. Generally, if the function has small curve, it is expected to be super convergence. However, it may not produce a good performance if the curve of the first derivative of the function varies slowly, since it ignores the nonlinearity part of Hessian matrix.
where J _{ f }(x) is the first differential function of f(x), μ is a positive scalar called a Marquardt damping parameter, and I is an identity matrix. LM can be viewed as a blend of the Gauss-Newton method and the steepest descent method. LM shows similar performance as the Gauss-Newton method if μ is small, while it behaves like the steepest descent method when μ is large. If μ is zero, it will be exactly the Gauss-Newton method. Hence, LM is an adaptive algorithm to retain strength of the steepest gradient method, the Newton method, and the Gauss-Newton method. However, although LM is robust to find an optimal minimum, it has a limitation of local optimization. Therefore, it may fail to find the global minimum if it starts with an initial value belonging to other local curves, which is not suitable to solve the non-linear problems in many cases due to non-convexity.
Discrete Selection Levenberg-Marquardt (DSLM)
A naive global optimization is a NP-hard problem due to the curse of searching dimensions. Hence, most of global optimizations such as genetic algorithm and simulated annealing have adapted heuristic approaches [23–25]. DSLM seeks to search the global optimum for each dimension of x iteratively, instead of the entire p dimensions of x which causes of the infeasibility. This concept reduces the searching space to linear while in the mean time preserving the high performance as a local optimization does. DSLM mainly includes two procedures, selection and discretization.
Selection
Like other non-linear least squares algorithms, DSLM starts with initial values x ^{0} for iterative computation. To avoid the local optima, DSLM attempts to vary the initial guesses iteratively within the dimension of each parameter by fixing other parameters. Denote S _{ i } as a searching space corresponding to parameter x _{ i }. Global optimizers search the comprehensive spaces with complexity of S _{1}×S _{2}×⋯×S _{ p } for one iteration. On the other hand, the complexity of searching space in DSLM becomes S _{1}+S _{2}+⋯+S _{ p }, because it iteratively searches each parameter space. Then, it compares the scores of the functions calculated with an updated x by conducting the LM method and uses the output as the new initial values. Figure 7 b gives an example illustrating the selection of parameters in a 2-dimension space. The arrows show the direction of function scores by varying values on the horizontal axis with a fixed value on the vertical axis. The score of the function is calculated from the result to which the LM algorithm converges. Once the optimal value is calculated for one specific dimension, the parameter is updated as the new parameter. Then, the algorithm seeks the next optimal parameter with fixed parameters previously chosen as the optimum. The selection process iterates until all parameters converge.
Discretization
Discretization of each parameter space is the key part which will affect the performance and computational cost for DSLM algorithm. Given N number of discrete spaces of a parameter, the discrete spaces can be denoted as a vector, D=(D ^{1},…,D ^{ N }). However, the function score is not necessarily to be calculated for all discrete spaces. The LM method updates the initial \({x_{k}^{0}}\) to the new parameter, \({x}_{k}^{new}\) subject to converging toward its local optimum. Then, the DSLM algorithm marks the spaces between D ^{ k } and \(D^{k^{'}}\) where \({x_{k}^{0}}\) and \(x_{k}^{new}\) belong to, respectively. It further reduces the space to check. Unlike other competing algorithms using discretization, DSLM’s discretization does not affect the accuracy of the solution but only for preventing rechecking the space.
Pseudo-code
The following pseudo-codes, Algorithms 1 and 2, briefly illustrate the DSLM algorithm. DSLM starts with randomly chosen initial values x ^{0}={x _{1},…,x _{ p }}. It iterates updating x until x converges or the number of iterations is bigger than a maximum constant. DSLM ensures F(x ^{′})≤F(x), where F(x) is the function score with x. Lines 5−12 of Algorithm 2 illustrate the optimum selection for each parameter and marking the discrete spaces. After the selection of all parameters, it conducts the LM algorithm to obtain its comprehensive local optimum. The LM method was implemented as the way George, Sam and Ting proposed [26].
Discussion
Algorithm assessment
Benchmark functions
Name | Function (F(x)) | Dimension (d) |
---|---|---|
Rastrigin | \(10d + \sum _{i=1}^{d} [{{x_{i}^{2}}-10\cos (2\pi x_{i})}]\) | d={2,5,50} |
Michalewics | \(-\sum _{i=1}^{d} [\sin (x_{i})\sin ^{2m}(i{x_{i}^{2}} / \pi)]\) | d={2,5,10} |
Performance comparison (iteration numbers)
d=2 (mean ±std) | d=5 (mean ±std) | d=50 (mean ±std) | ||
---|---|---|---|---|
Rast. | GA | 51 ± 0 | 70.1 ± 14.41 | 174.9 ± 62.7 |
SA | 1744.4 ± 410.84 | 5003.5 ± 2126 | 41921 ± 6326.1 | |
DSLM | 5.2 ± 2.44 | 4.6 ± 1.64 | 22.5 ± 4.32 | |
Mich. | GA | 51 ± 0 | 58.3 ± 8.98 | 81.6 ± 29.6 |
SA | 1760.3 ± 765 | 3606.3 ± 880.7 | 8956.3 ± 2171.3 | |
DSLM | 3.5 ± 1.08 | 4.9 ± 1.6 | 7.4 ± 3.56 |
Conclusions
In this paper we presented a computational modeling framework for the study of biological system in response to biomedical implants. As an example of application, phagocyte transmigration has been studied to tackle the problem of fibrotic tissue formation surrounding the biomaterial implants, which causes implantation failure.
Foreign body reactions are complicated processes. Many cells and cellular products participate in the processes. To model such complex reactions, we focused on the key components identified in many previous works. Furthermore, the proposed model system is aimed at modeling short-term acute inflammatory responses at which there are many similarities between wound healing responses and foreign body reactions, while most of the chronic diseases, including foreign body reactions and fibrosis, are the long term outcomes. Based on the in-depth understanding of the successive reactions, mathematical equations are developed.
To complete the quantitative modeling, reverse engineering is conducted to estimate the parameters of the mathematical modeling equations. We developed a global optimization technique, DSLM, which overcomes the limitations of existing local optimization algorithms such as the LM algorithm and shows better performance than other global heuristic approaches. The proposed DSLM globally estimates the optimal parameters of the phagocyte transmigration modeling system, which discloses the inner nature of the dynamic system. To verify the global optimization capability of the modeling, the system was tested with the Rastrigin function and the Michalewics function. Nevertheless, DSLM is rather optimized to this study not as a general optimizer yet, since it needs more concrete analysis theoretically and empirically as a future work. With mathematical equations and estimated parameters by DSLM, we are able to further simulate and predict different aspects of phagocyte transmigration process under different conditions of system inputs.
Ethics statement
This research does not involve human subjects, human materials or human data. All animal data came from an early publication (Tang et al., PNAS, 95:8841-6, 1998). The animal experiments for that study were approved by the Albany Medical College’s Animal Care and Use Committee (IACUC) and in accordance with the National Institutes of Health guidelines for the use of laboratory animals.
Availability of supporting data
The generated transmigration data named Transmigration.xls and MATLAB source codes DSLM.zip are publicly accessible at: http://faculty.tamuc.edu/mkang/DSLM. The measurements of transmigration data can be found in Additional file 1. The source code as well as the synthetic data are provided in Additional file 2.
Declarations
Acknowledgements
This research was supported by NIH under grant R01EB00727.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver(http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Authors’ Affiliations
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