The use of nanoparticles for the early diagnosis, treatment and imaging of a number of disorders has raised in the last years.

Indeed, 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.

This is an approximation project, since the variable to be predicted is continuous (adhesive strength).

The basic goal here is to model the adhesive strength, as a function of the shear rate and the particle diameter.

The data set contains three concepts:

- Data source.
- Variables.
- Instances.

The file nanoparticle_adhesive_strength.csv contains 58 rows, each of them with 3 columns.

This data set has the following variables:

**shear_rate**: Wall shear rate of the nanoparticle. In 1/seconds, used as Input.**particle_diameter**: Diameter of each particle. In micrometers, used as Input.**particles_adhering**: Number of particles adhering per unit area to the collagen substrate. Used as Target.

For the analysis, we split the instances as follows: 60% for training and 40% for testing.

We can calculate the distributions of the variables. The following chart shows the histogram for the adhesion variable.

As we can see, the adhesion has a semi-normal distribution.

The inputs-targets correlations help us to understand which input variables might have a bigger influence on the target variable.

The particle diameter has a bigger correlation with the particles adhering than the shear rate. On the other hand, the particle diameter correlation is positive and the shear rate correlation is negative.

The second step is to set the model parameters. For approximation project type, it is composed by:

- Scaling layer.
- Perceptron layers.
- Unscaling layer.

Mean and standard deviation scaling method is set as the scaling method, while the minimum and maximum unscaling method is set as the unscaling method. The activation function chosen for this model is the hyperbolic tangent activation function and the linear activation function for the hidden layer and the output layer respectively.

A graphical representation of the neural network is depicted next. It contains a scaling layer, 2 perceptron layers and an unscaling layer. The number of inputs is 2, and the number of outputs is 1. The complexity, represented by the numbers of neurons in the hidden layer is 3.

The fourth step is to select an appropriate training strategy. It is composed of two things:

- A loss index.
- An optimization algorithm.

The loss index defines what the neural network will learn. It is composed by an error term and a regularization term.

The normalized squared error is used here as error term. The regularization term is the L2 regularization. We use a weak weight for this regularization term.

The optimization algorithm is in charge of searching for the neural network parameters that minimize the loss index. Here we chose the quasi-Newton method as optimization algorithm.

The following chart shows how the error decreases with the iterations during the training process.
The final values are **training error = 0.062 NSE** and **selection error = 0.098 NSE**, respectively.

The objective of model selection is to improve the generalization capabilities of the neural network or, in other words, to reduce the selection error.

Since the selection error that we have achieved so far is very small (0.098 NSE), we don't need to apply Order selection nor inputs selection here.

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 move to the production phase.

The next chart illustrates the linear regression analysis for the variable particles_adhering.

For a perfect fit, the values of the intercept, slope and correlation should be 0, 1 and 1, respectively.
In this case, we have **intercept = -2.57**, **slope = 1.11** and **correlation = 0.946**.
The achieved values are close to the ideal ones, so the model shows a good performance.

In the model deployment phase the neural network is used to make predictions about the number of particles adhering for new values of wall share rate and diameter.

We can plot Directional outputs to study the behavior of the output variable particle_adhering as function of single inputs. The following reference point is used:

**shear_rate**: 73.421 1/seconds.**particle_diameter**: 3.388 micrometers.

The next picture shows the number of particles adhering as a function of the wall shear rate around the reference point.

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 decreasing.

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.

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.

scaled_shear_rate = 2*(shear_rate-50)/(90-50)-1; scaled_particle_diameter = (particle_diameter-3.38896)/2.38932; y_1_1 = tanh(0.767924+ (scaled_shear_rate*-0.811769)+ (scaled_particle_diameter*2.21102)); y_1_2 = tanh(2.08552+ (scaled_shear_rate*-1.88506)+ (scaled_particle_diameter*-1.04971)); scaled_particles_adhering = (-0.772609+ (y_1_1*0.646181)+ (y_1_2*0.542708)); particles_adhering = 0.5*(scaled_particles_adhering+1.0)*(74.75-13.22)+13.22;

- "Optimizing particle size for targeting diseased microvasculature: from experiments to artificial neural networks", Daniela P Boso, Sei-Young Lee, Mauro Ferrari, Bernhard A Schrefler, Paolo Decuzzi. International Journal of Nanomedicine, 2011.