A Visual Feedback Control Framework Oriented to Lung Branching Pattern Formation Simulation

Abstract

Mathematical modeling based on reaction-diffusion partial differential equations, is a well-known way to study the mechanism of lung branching pattern formation. However, model parameter identification is one of the key problems in the research, which is both labor-intensive and of low efficiency. To work out this problem, a visual feedback control framework is proposed in this paper. The framework based on automatic control theory is built, by means of pattern feature feedback, to identify model parameters. The results suggest that this framework is effective and feasible.

6.1 Introduction

In humans and most other vertebrates, the lung is known as the essential respiratory organ. Scientists from stem cell biology, molecular biology, biochemistry, theoretical modeling, and other disciplines have attempted to understand the mechanisms of lung pattern formation for years [1–3]. Yet it is still a long-standing work.

A well-established paradigm in the theoretical study of lung branching pattern formation is mathematical modeling based on reaction-diffusion theory proposed by Turing [4]. Following the work of Turing, Gierer and Meinhardt put forward an Activator-Inhibitor model [5], which has also been utilized for many other biological processes [6–9]. In order to investigate the mechanism of lung development in the modeling study, a suitable model with applicable parameters need to be chosen, which is capable of reproducing key characteristics of lung [10, 11]. Because different model parameter values lead to different simulation patterns, the determination of model parameters becomes quite important after the model is selected. However, such a model comprises many parameters, and most parameter values need to be estimated from a wide range. Hence, model parameter identification is a critical problem in the modeling study of lung pattern formation.

In biological research, researchers manually select the simulation approximate to the actual biological pattern by trying large quantities of parameter combinations [10–14]. In this way, the problem of model parameter identification is settled. However, several obvious disadvantages of this manual selection method, such as labor-intensive, low-efficiency, and lack of quantitative evaluation criteria can be seen.

Oriented to simulation of lung branching pattern formation based on reaction-diffusion partial differential equations, we provide a visual feedback control framework in this paper. Instead of manual work, this framework can automatically search out the simulation result that conforms to the target pattern and realize model parameter identification.

6.2 Visual Feedback Control Framework

The visual feedback control framework, as shown in Fig. 6.1, consists of three modules: visual simulation module, pattern feature extraction module, and model parameter identification module. The visual simulation module is equivalent to the actuator in this feedback control system, in which we present and output the numerical simulations of the lung branching model as images based on visualization; the pattern feature extraction module is equivalent to the sensor, in which we extract and quantify the features of lung pattern simulation to compare with the reference input; the model parameter identification module is equivalent to the controller to realize model parameter correcting.

Fig. 6.1

Visual feedback control framework

The operating procedures of this framework are listed in detail as follows:

Step 1::

Input the information about the target pattern and initialize the model parameters.

Step 2::

Numerically solve reaction-diffusion partial differential equations in the mathematical model with the parameter values, and represent the solutions by a two-dimensional image as the simulated lung pattern output.

Step 3::

Extract and quantitatively describe features of the simulated pattern by a series of image processing, then take the features as a feedback.

Step 4::

Adjust the model parameters by utilizing the pattern feature deviation between the target pattern and the simulation. If the deviation drops to zero or this framework reaches its maximum running time limit, then this framework would stop running and output the simulation result with the corresponding model parameters. Otherwise go to Step 2.

6.3 Methods of Visual Feedback Control Framework

6.3.1 Visual Simulation Module

In this framework, we use a reaction-diffusion model postulating 4 variables and 14 parameters [11]. The model is defined as:

$$ \begin{aligned} \frac{\partial A}{\partial t} =&\, \frac{{cA^{2} S}}{H} - \mu A + \rho_{A} Y + D_{A} \nabla^{2} A \\ \frac{\partial H}{\partial t} =&\, cA^{2} S - vH + \rho_{H} Y + D_{H} \nabla^{2} H \\ \frac{\partial S}{\partial t} =&\, c_{0} - \gamma S - \varepsilon YS + D_{S} \nabla^{2} S \\ \frac{\partial Y}{\partial t} =&\, dA - eY + \frac{{Y^{2} }}{{1 + fY^{2} }} \\ \end{aligned} $$

(6.1)

where $A$, $H$, and $S$, are respectively, the activator BMP-4, the inhibitor MGP, and the substrate chemical ALK1-TGF-$\beta$ complex, and $Y$ is a marker variable of cell differentiation ($Y = 1$ means the cell is differentiated).

In this model, it is assumed that differentiated cells $Y$ produce the activator $A$ at rate $\rho_{A}$ and the inhibitor $H$ at rate $\rho_{H}$, with taking up substrate $S$ at rate $\varepsilon$. The activator $A$ is in a process degraded at rate $\mu$ and diffusing at rate $D_{A}$ requiring $S$ and inhibited by $H$. Similarly, the inhibitor $H$ is also produced by $A$, degraded at rate $v$, and diffuses at rate $D_{H}$. Substrate $S$, is supplied at rate $c_{0}$, degrades at rate $\gamma$, and diffuses at rate $D_{S}$. Lastly, $Y$ also can degrade at rate $e$ and increase in the presence of $A$.

Here, we set model parameters $d$, $\rho_{H}$ and $\varepsilon$ in the equations above, the adjustable parameters need to be identified according to the lung branching pattern features selected to quantitatively characterize. More detail on the selection of the identified parameters will be reported in the following section (see Sect. 6.3.2).

Initial conditions of model are given by setting $A = 0.001$, $H = 0.01$, $S = 1$, $Y = 0$, uniformly distributed in space. The constant parameter values used are: $c = 0.002$, $\mu = 0.16$, $\rho_{A} = 0.03$, $D_{A} = 0.02$, $v = 0.04$, $D_{H} = 0.3$, $c_{0} = 0.02$, $\gamma = 0.02$, $D_{S} = 0.06$, $e = 0.1$, $f = 10$.

We perform numerical simulation of model on a 200 × 200 grid, with no-flux boundary conditions, using two-step Runge-Kutta methods. We directly save the solution of the variable Y as the two-dimensional simulation of lung branching pattern. The simulation is stopped and outputted when the growth of lung branching pattern reaches a certain degree. In order to improve the simulation efficiency, we present a parallel computing technique based on Graphics Processing Unit (GPU) and Compute Unified Device Architecture (CUDA) programming.

6.3.2 Pattern Feature Extraction Module

In this paper, based on the growth characteristics, the lung branching pattern is roughly divided into four categories: alternating side branching pattern, zygomorphic side branching pattern, tip branching pattern, and hybrid branching pattern (see Fig. 6.2). Alternating side branching pattern and zygomorphic side branching pattern are formed by monopodial branching in which lateral branches grow on both sides from a long straight main stem, and secondary lateral branches grow out in the same way taking lateral branches as main stem, and so on. The distinction between the two patterns is the symmetry of the side branching growth. Tip branching pattern and hybrid branching pattern are formed by dichotomous branching in which the main stem bifurcates into two equally sized branches from its apex, and likewise, each branch can split into two daughter branches. Hybrid branching pattern can be considered as a kind of tip branching pattern with side branching growing out. In view of this, for hybrid branching pattern, we further distinguish whether side branches grow on either or both the main stem and dichotomous branches. Figure 6.2d shows the simulation picture of hybrid branching pattern with side branches growing on both main stem and dichotomous branches.

Fig. 6.2

Four different lung branching patterns with labeling corresponding extracted features: a alternating side branching pattern; b zygomorphic side branching pattern; c tip branching pattern; d hybrid branching pattern

Besides pattern category information, we use the width of main stem and the spacing between two adjacent branches as the extracted pattern features, by considering the growth morphology of lung branching pattern and biological concerns on the study of lung development. These extracted pattern features are used to describe lung branching pattern. Various branching patterns may have different definitions of the spacing between two adjacent branches with different measure methods (see Fig. 6.2). In Fig. 6.2, w and s are given as the width of main stem and the spacing between two adjacent branches. In addition, further analysis indicates that the change in the values of parameters $d,\,\rho_{H}$ and $\varepsilon$ in the model influences a change in the pattern features and pattern category information. Thus, $d,\,\rho_{H}$ and $\varepsilon$ are chosen to be the identified parameters.

To sum up, pattern feature extraction, as shown in Fig. 6.3, proceeds along two stages: pattern discrimination and feature extraction. We set an unknown branching pattern as the input object of pattern feature extraction. In the first stage, we judge whether the unknown pattern is formed by monopodial branching or dichotomous branching. If its branch growth is entirely monopodial, we further confirm whether the pattern belongs to alternating side branching pattern or zygomorphic side branching pattern based on the symmetry of the side branching growth. If its branch growth is dichotomous, then we judge whether the pattern has side branches or not. If so, the pattern belongs to hybrid branching pattern, and we should further identify which kind of hybrid branching the pattern belongs to by determining the growth of these side branches. Otherwise, the pattern belongs to tip branching pattern. In the second stage, by knowing the category of the pattern, we extract and quantify the corresponding features by a series of image processing methods such as skeleton extraction and pixel scan.

Fig. 6.3

The flowchart of pattern feature extraction

6.3.3 Model Parameter Identification Module

In this framework, we optimize model parameters by using differential evolution (DE) algorithm, which is an efficient heuristic approach for global optimization in high-dimensional space based on a stochastic and parallel direct search method [15].

We define the cost function as:

$$ F = \delta_{x,y} + \sum\limits_{i}^{n} {u_{i} \left( {a_{i} - \bar{a}_{i} } \right)^{2} } $$

(6.2)

where the Kronecker delta is $\delta_{x,y} = \left\{ {0, x = y;\infty , x \ne y} \right\}$, with $x$ and $y$ representing the current simulation and the target pattern, n is given as the number of the quantitatively described pattern features, and $a_{i}$ is the ith quantitatively described feature value of the current simulation, with $\bar{a}_{i}$ and $u_{i}$, respectively, corresponding to the standard feature value and the feature weight value.

The initial extent of the adjustable parameters is: $d \in \left[ {0.0050,\,0.0200} \right]$, $\rho_{H} \in \left[ {0.000005,\,0.000150} \right]$, $\varepsilon \in \left[ {0.030,\,1.200} \right]$.

6.4 Results

6.4.1 Simulation of Lung Branching Pattern Set as the Target Pattern

We set different categories of simulation patterns as the target pattern to test this framework. The feature information about each target pattern with the known model parameters, is extracted as the reference input of the visual feedback control framework. Four examples are presented in this section, respectively, for setting alternating side branching pattern, zygomorphic side branching pattern, tip branching pattern, and hybrid branching pattern as the target pattern. These target patterns are shown in the first row of Fig. 6.4, while the corresponding result patterns, obtained through framework processing, are shown in the second row of Fig. 6.4. Their pattern feature information and model parameter values are reported in Table 6.1.

Fig. 6.4

The target patterns and the corresponding result patterns of the framework, in the case of different categories of simulation patterns set as the target pattern. The first row shows the target patterns, while the second row shows the corresponding result patterns

Table 6.1 The feature information and model parameter values of target patterns and corresponding result patterns obtained through framework processing, in the case of different categories of simulation patterns as the target pattern

In the example of alternating side branching pattern as the target pattern, with their shared pattern category and feature information, the result pattern is similar to the target pattern from the point of view of the growth morphology, and the difference between their model parameter values is tiny (see Fig. 6.4a1, a2, and Table 6.1, rows 3 and 4). In the examples of zygomorphic side branching pattern and tip branching pattern as the target patterns, besides their corresponding target patterns and result patterns have pattern category and feature information in common, both result patterns are almost the same as their suitable target patterns, and there is little difference between the model parameters of their target patterns and result patterns (see Fig. 6.4b1, b2, c1, c2, and Table 6.1, rows 5, 6, 7, and 8). For instance, in hybrid branching pattern as the target pattern, the result pattern and target pattern belong to one branching pattern category, while there is still a feature distinction between the target pattern and the result pattern; however, the result pattern is very close to the target pattern; furthermore, both model parameter values are near (see Fig. 6.4d1, d2, and Table 6.1, rows 9 and 10).

6.4.2 Actual Lung Branching Pattern Set as the Target Pattern

In order to test this framework further, different categories of actual lung branching patterns are chosen as the target pattern. Two examples are presented as follow, in which both the target patterns are acquired from the literature [11]. The first example sets the actual lung pattern that belongs to alternating side branching pattern as the target pattern in this framework (see Fig. 6.5a1), while the second example sets the actual lung pattern that belongs to tip branching pattern as the target pattern (see Fig. 6.5b1). The corresponding result patterns are obtained through visual feedback control framework processing (see Fig. 6.5a2, b2). The two examples demonstrate that the proposed framework can find the simulation pattern approximate to the actual lung pattern.

Fig. 6.5

The target patterns and the corresponding result patterns of the framework, in the case of actual lung branching patterns, are set as the target pattern. The first row shows the target patterns, while the second row shows the corresponding result patterns

6.5 Conclusion and Future Work

With the basic principle of feedback control system, the visual feedback control framework is proposed to solve the problem of model parameter identification in the modeling study of lung. The results of this framework show that the method presented in this paper is reasonably feasible and practically significant. Compared with the manual model parameter selection in biological research, this framework greatly reduces the staff workload, standardizes the selection criteria, and improves the overall efficiency. Furthermore, we believe that the design principle of the visual feedback control framework is not limited to the study of lung development as it can be easily generalized to other types of biological pattern formation.

In our ongoing work, we are improving and refining our algorithms of the proposed framework. We also plan to extend our framework to three-dimensional simulation of lung pattern formation. We will increase the number of the adjustable model parameters and the extracted features of lung branching pattern, which may lead to form some new growth patterns of the lung. These studies may make our framework more closely integrated with biological research and contribute to better understanding of the biophysical mechanisms of lung pattern formation.