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Modeling the adhesive strength of nanoparticles with neural networks

The use of nanoparticle for the early diagnosis, treatment and imaging of a number of disorders has raised in the last years due to the fact that they can be administered at the systemic level and can be transported by blood flow and reach any site within the macrovascular and microvascular circulation.

The aim of this study is to predict the vascular adhesion of nanoparticles from their wall shear rate and their diameter.


  1. Data set
  2. Neural network
  3. Training strategy
  4. Testing analysis
  5. Model deployment

1. Data set

The data set contains 58 instances, each of them with two input variables and one target. The first input (Shear_rate) contains information about the wall shear rate of the nanoparticle, the second input (Particle_diameter) has the value of the diameter of each particle. Finally, the target (Particles_adhering) is number of particles adhering per unit area to the collagen substrate. The task "Report data set" arranges all this information in a table as follows.

Variable description

Finally, for the analysis, we will use the task "Split instances" to divide the instances as follows: 60% for training and 40% for testing.

2. Neural network

In this section, we are going to design the model that we will train later to approximate the number of particles adhering per unit area.

The model will be a neural network composed by:

A graphical representation of the neural network, generated by the task "Report neural network" is depicted next. From the left to the right we have the two inputs, the scaling layer, the principal components layer, the hidden layer, the unscaling layer and the outputs. As we said before, the number of inputs and of principal components will be 2. The number of nuerons in the hidden layer, or complexity, is 3.

Neural network graph
Neural network graph.

Lastly, we will use the minimum-maximum method in the scaling and the unscaling layers.

3. Training strategy

The procedure used to carry out the learning process is called training (or learning) strategy. The training strategy is applied to the neural network in order to obtain the best possible performance. The type of training is determined by the way in which the adjustment of the parameters in the neural network takes place.

The following chart shows how the performance decreases with the iterations during the training process. The initial value is 8.9629, and the final value after 96 iterations is 0.049.

Training chart.

The next table shows the training results by the quasi-Newton method. They include some final states from the neural network, the loss index and the training algorithm.

Here the final parameters norm is not very big and the final loss and gradient norm are closed to zero. The total training time was around 1 second.

4. Testing analysis

Once we have trained the model, it is time to test its predictive capacity. This will be done by comparing the outputs from the neural network against the real target values for a set of data never seen before. The testing analysis will determine if the model is ready to be used in the production phase.

The next chart illustrate the linear regression for the variable Particles_adhering.

Regression plot.

The next table shows the values of the intercept, the slope and the correlation for the previous linear regression analysis.

Regression table
Regression table.

For a perfect fit, the values of the intercept, slope and correlation should be 0, 1 and 1, respectively. In this case, their values are close to the ideal ones, so the model shows a good performance.

5. Model deployment

The neural network is now ready to make predictions of good quality about the number of particles adhering for new values of wall share rate and diameters.

The task "Calculate directional output" allows us to study the behaviour of the output variable Particle_adhering as function of only one input.

The next picture shows the number of particles adhering as a function of the wall shear rate and for a reference point of the particle diameter of 3.38896.

Directional plot.

As we can see, for this value of the particle diameter, the number of particles adhering keeps more or less constant till the wall shear rate reaches the values around 70 and then it starts drecreasing.

On the other hand, the next image represents the number of particles adhering as a function of the particle diameter and for a reference point of the wall shear rate with value 73.4211.

Directional plot.

In this case, for this value of the wall shear rate, the number of particles adhering increases till the particle diameter reaches the value 5 and then it starts decreasing.

As well, we can use the mathematical expression of the neural network, which is listed next.

				(Particles_adhering) = (0.5*(scaled_Particles_adhering+1.0)*(74.75-13.22)+13.22);