The very local interstellar medium (VLISM) neutral hydrogen and proton densities are still not precisely known even after 3 decades of deep space research and the existence of the EUV and other diagnostic data obtained by Pioneer 10/11, Voyager 1/2, and other spacecraft. The EUV data interpretation, in particular, has suffered because of inadequate neutral hydrogen-plasma models, difficulty of calculating the multiply scattered Lyman α glow, and calibration uncertainties. Recently, all these difficulties have been significantly reduced. In the present work we have used the latest state-of-the-art supersonic VLISM neutral hydrogen-plasma and Monte Carlo radiative transfer model, incorporating neutral density, temperature, and velocity variations, actual solar line shape, realistic redistribution function, Doppler and aberration effects. This work presents the methodology of the radiative transfer code and the first results of the comparison of the model predictions with the Pioneer 10 data. Monte Carlo radiative transfer calculations were carried out for five neutral hydrogen-plasma models and compared with Pioneer data. The first results are quite encouraging. We found that the VLISM ionization ratio is between 0.2 and 0.5 and that the VLISM neutral hydrogen density is <0.25 cm-3. The present calculation suggests that the Pioneer 10 photometer-derived intensities (Rayleighs) need to be increased by a factor of 2. If this model-derived calibration is used, then the difference between Pioneer 10 and Voyager 2 intensity values is reduced to ~2.2. The model, neutral hydrogen density = 0.15 cm-3 and proton density = 0.07 cm-3, is found to best fit the Pioneer 10 data.