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Mathematical Modeling of Guided Neurite Extension in an Engineered Conduit with Multiple Concentration Gradients of Nerve Growth Factor (NGF)

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Abstract

Neurotrophic factors such as nerve growth factor (NGF) provide essential cues to navigate growing axon toward their targets. Concentration and concentration gradient of NGF are key parameters affecting the growth rate and direction of neurites and axons. However, the maximum distance for guided nerve growth under stimulation of a single concentration gradient is limited and is thus unfavorable in nerve regeneration. Since the sensitivity of PC12 cells to NGF signals is restorable even after brief removal of the factors, exposure to multiple concentration gradients of the factor can achieve longer distances and greater rates of guided growth. In this study, a mathematical model simulating nerve growth in a virtually constructed nerve conduit incorporating multiple NGF concentration gradients is established. Using a genetic algorithm, optimized initial profiles of NGF able to achieve 4.5 cm of guided growth with a significantly improved growth rate has been obtained. The model also predicts an inverse relationship between the diffusion coefficient of the factor and the neurite growth rate. This model provides a useful tool for evaluating various conduit designs before fabrication and evaluation.

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Acknowledgment

The authors would like to thank Dr David Wilmshurst, the University of Hong Kong’s Technical Writer, for editing this manuscript.

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Authors and Affiliations

  1. Department of Mechanical Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, China

    T. H. Z. Tse & J. Lam

  2. Medical Engineering, Department of Mechanical Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, China

    B. P. Chan & C. M. Chan

Authors
  1. T. H. Z. Tse
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  2. B. P. Chan
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  3. C. M. Chan
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  4. J. Lam
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Corresponding author

Correspondence to J. Lam.

Appendices

Appendix A

Figure 10 indicates the characteristics of the neurite outgrowth under optimal initial profiles of neurotrophic factor for gradient factor k = 30, 50, and 70 mL cm2 ng−1 h−1, respectively (refer to “Geometric Characteristics of Initial Profiles” section in Methodology for the meaning of optimal initial profile). It is found that neurite outgrowths share the same features under different gradient factors, i.e., rapid

Figure 10
figure 10

Regions A, B, and C represent rapid growth region, baseline growth region, and temporary stall region, respectively

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growth region, baseline growth region, and temporary stall region (refer to Figs. 5a–5d for detail).

Appendix B

Flow chart for optimizing initial profile by genetic algorithm is shown in Fig. 11.  

Figure 11
figure 11

Flow chart for optimizing initial profile by genetic algorithm

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Steps 1 to 6: Several initial profiles (individual solutions) are randomly generated to form initial population. The pre-defined fitness function are input in Steps 2 to 6 to calculate their fitness correspondingly. Fitness here is defined as the time consumed for neurite outgrowth for 5-cm conduit. A small fitness means a higher suitability of the initial profile for nerve reconnection.

Step 7: Compliance with criteria is checked. Generally speaking, criteria may include generation, time limit, fitness (suitability of the initial population), stall generation, and stall time limit. If the criteria are not fulfilled, the programme will continue to search for better initial profiles.

Step 8: The programme chooses some initial profile with better fitness from the new generation to prepare for the next new generation. A number of less fit profiles are also selected to maintain the diversity of each generation, preventing premature convergence on poor solutions.

Steps 9 to 10: The crossover and mutation mathematical processes (these terms occur in genetic algorithm 6, 7) are performed to obtain better initial profiles, after which a new population forms.

Steps 2 to 11 are repeated until the criteria are fulfilled. The programme is then terminated with a best initial profile in output.

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Tse, T.H.Z., Chan, B.P., Chan, C.M. et al. Mathematical Modeling of Guided Neurite Extension in an Engineered Conduit with Multiple Concentration Gradients of Nerve Growth Factor (NGF). Ann Biomed Eng 35, 1561–1572 (2007). https://doi.org/10.1007/s10439-007-9328-4

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