Optimizing particle size for targeting diseased microvasculature: from experiments to artificial neural networks

Figures & data

Table 1 Size and zeta potential of polystyrene carboxylate fluorescent microspheres

Dnominal (μm)	0.75	1.00	2.00	4.50	6.00
Dmeasured (μm)	0.72	0.97	1.83	4.90	6.60
Zeta potential (mV)	−73.1 ± 2.5	−87.3 ± 3.1	−70.6 ± 4.4	−82.2 ± 7.0	−52.4 ± 0.4

Figure 1 (A) Parallel plate flow chamber apparatus, used for in vitro adhesion experiments, consisting of: 1, a syringe pump; 2, a syringe with particles suspended in solution; 3, inlet tubing; 4, parallel plate flow chamber; 5, outlet tubing; 6, microscope; 7, digital camera; and 8, computer. (B) Images depicting an epifluorescent microscope with setup of the system, and (C) the parallel plate flow chamber with tubing network installed over the stage of the microscope.

Figure 2 (A) Fully connected neural network with two hidden layers. (B) Typical scheme of a neuron and its activity: $g_r({\displaystyle \sum _i}{w}_{ri}{i}_i+b_r)$.

Figure 3 Number of particles (n_s) adhering per unit area in a parallel plate flow chamber, under different flow conditions (shear rate to the wall [S]), as a function of particle diameter d. Single marks represent the experimental data, as from Table S1.

Figure 4 (A) The n_s(d) relationship as predicted from ANN231 for particle diameters not tested in the experiments. (B) The n_s(d) relationship as predicted from ANN231 for shear rates (S) not tested in the experiments. (C) Root mean squared error of the test set of data for the ANN231.

Note: The red arrow depicts the minimum (2.6 × 10⁶ iterations).

Figure 5 (A) The n_s(d) relationship as predicted from ANN2321 for particle diameter not tested in the experiments. (B) The n_s(d) relationship as predicted from ANN2321 for shear rates S not tested in the experiments. (C) Root mean squared error of the test set of data for the ANN2321.

Note: The red arrow depicts the minimum (1.75 × 10⁶ iterations).

Table 2 Root mean squared error for learning and test sets of the two ANNs used

ANN	RMSE learning set	RMSE test set
231	0.04620	0.03678
2321	0.04623	0.03460

Abbreviations: ANN, artificial neural networks; RMSE, root mean squared error.

Figure 6 The optimal diameter d_opt as a function of S for the ANN231 and ANN2321.

Figure S1 Root mean squared error of the learning set of data for ANN231.

Figure S2 Root mean squared error of the learning set of data for ANN2321.

Table S1 Experimental data for the number of particles n_s adhering per unit area in a parallel plate flow chamber

S [sec^−1]	d [μm]	ns[#/mm^2]
50	0.720	25.60
50	0.720	16.56
50	0.720	22.13
50	0.968	25.79
50	0.968	29.16
50	0.968	27.40
50	1.829	40.01
50	1.829	38.57
50	1.829	41.04
50	4.899	64.49
50	4.899	52.76
50	4.899	54.23
50	6.596	61.55
50	6.596	55.70
50	6.596	74.75
75	0.720	17.78
75	0.720	21.50
75	0.720	17.78
75	0.968	17.15
75	0.968	19.35
75	0.968	18.61
75	1.829	17.30
75	1.829	20.81
75	1.829	20.08
75	1.829	23.45
75	1.829	29.90
75	1.829	28.21
75	4.899	48.85
75	4.899	60.42
75	4.899	42.43
75	4.899	55.28
75	4.899	54.64
75	6.596	47.32
75	6.596	39.24
75	6.596	35.77
75	6.596	40.40
75	6.596	46.74
75	6.596	50.78
90	0.720	14.92
90	0.720	15.31
90	0.720	14.07
90	0.968	13.22
90	0.968	14.65
90	0.968	14.92
90	1.829	13.68
90	1.829	17.59
90	1.829	13.68
90	4.899	28.57
90	4.899	29.71
90	4.899	32.00
90	4.899	29.71
90	4.899	30.28
90	6.596	27.18
90	6.596	29.23
90	6.596	18.98
90	6.596	24.62
90	6.596	22.05

Table S2 Values for mean and standard deviation of number of particles (n_s) adhering per unit area in parallel plate flow chamber experiments

Wall shear rate S [s^−1]	Particle diameter d [μm]	Mean adhesion μ[#/mm^2]	Standard deviation σ[#/mm^2]
50	0.72	21.43	4.56
50	0.968	27.45	1.69
50	1.829	39.87	1.24
50	4.899	57.16	6.39
50	6.596	64.00	9.76
75	0.72	19.02	2.15
75	0.968	18.37	1.12
75	1.829	23.29	4.90
75	4.899	52.32	6.88
75	6.596	43.38	5.75
90	0.72	14.77	0.63
90	0.968	14.26	0.91
90	1.829	14.98	2.26
90	4.899	30.05	1.25
90	6.596	24.41	4.06