A Bayesian hierarchical model for maximizing the vascular adhesion of nanoparticles

Abstract

The complex vascular dynamics and wall deposition of systemically injected nanoparticles is regulated by their geometrical properties (size, shape) and biophysical parameters (ligand–receptor bond type and surface density, local shear rates). Although sophisticated computational models have been developed to capture the vascular behavior of nanoparticles, it is increasingly recognized that purely deterministic approaches, where the governing parameters are known a priori and conclusively describe behaviors based on physical characteristics, may be too restrictive to accurately reflect natural processes. Here, a novel computational framework is proposed by coupling the physics dictating the vascular adhesion of nanoparticles with a stochastic model. In particular, two governing parameters (i.e. the ligand–receptor bond length and the ligand surface density on the nanoparticle) are treated as two stochastic quantities, whose values are not fixed a priori but would rather range in defined intervals with a certain probability. This approach is used to predict the deposition of spherical nanoparticles with different radii, ranging from 750 to 6,000 nm, in a parallel plate flow chamber under different flow conditions, with a shear rate ranging from 50 to 90 $\text {s}^{-1}$. It is demonstrated that the resulting stochastic model can predict the experimental data more accurately than the original deterministic model. This approach allows one to increase the predictive power of mathematical models of any natural process by accounting for the experimental and intrinsic biological uncertainties.

Appendix 1: MCMC details

Since the posterior distribution of the parameters of interest cannot be computed in closed form, we require the use of MCMC methods for inference. Specifically, lacking closed form full conditional distributions, we employ Metropolis-Hastings steps for each of the three parameters of interest. Briefly, the Metropolis-Hastings algorithm can be used to obtain samples from any target probability distribution, say $p(x)$, provided it is possible to compute the value of a function, $q(x)$, proportional to $p(x)$. The sample values are produced iteratively, as part of a Markov Chain, with the distribution of the next sample being dependent only on the current sample value. At each iteration of the algorithm, candidate value is proposed. Then, with some probability, the candidate is either accepted or rejected. If accepted, the candidate value is used in the next iteration; otherwise, the candidate value is discarded, and the current value is retained for the next iteration. The probability of acceptance is determined by comparing the likelihoods of the current and candidate sample values with respect to the target distribution $p(x)$. For the purposes of this algorithm, we have simplified this probability by proposing candidates from the prior distributions.

At iteration $b$, to update $\beta _1^{(b)}$, the candidate, $\beta _1^*$ is generated from a Gaussian distribution, truncated below at 0, with mean $1\times 10^{-24}$ and standard deviation $5\times 10^{-24}$. The candidate is accepted probability given by the minimum of 1 and the following ratio

$$\begin{aligned} \frac{\prod _{s}\prod _{a}\text {Beta}\left( y_{sa}; P_a(S,a,\beta _1^*,\beta _2^{(b-1)})\phi ^{(b-1)},(1-P_a(S,a,\beta _1^*,\beta _2^{(b-1)}))\phi ^{(b-1)}\right) }{\prod _{s}\prod _{a}\text {Beta}\left( y_{sa}; P_a(S,a,\beta _1^{(b-1)},\beta _2^{(b-1)})\phi ^{(b-1)},(1-P_a(S,a,\beta _1^{(b-1)},\beta _2^{(b-1)}))\phi ^{(b-1)}\right) } \end{aligned}$$

For the update of $\beta _2^{(b)}$, the candidate, $\beta _2^*$ is generated from a Gaussian distribution, truncated below at 0, with mean $1\times 10^{13}$ and standard deviation $5\times 10^{13}$. The candidate is accepted probability given by the minimum of 1 and the following ratio

$$\begin{aligned} \frac{\prod _{s}\prod _{a}\text {Beta}\left( y_{sa}; P_a(S,a,\beta _1^{(b)},\beta _2^{*})\phi ^{(b-1)},(1-P_a(S,a,\beta _1^{(b)},\beta _2^{*}))\phi ^{(b-1)}\right) }{\prod _{s}\prod _{a}\text {Beta}\left( y_{sa}; P_a(S,a,\beta _1^{(b)},\beta _2^{(b-1)})\phi ^{(b-1)},(1-P_a(S,a,\beta _1^{(b)},\beta _2^{(b-1)}))\phi ^{(b-1)}\right) } \end{aligned}$$

Finally, the precision parameter, $\phi ^{(b)}$, is updated through a proposal $\phi ^*$ from a Gamma distribution with shape 2 and scale 0.05. The proposal is accepted with probability given by the minimum of 1 and the following ratio

$$\begin{aligned} \frac{\prod _{s}\prod _{a}\text {Beta}\left( y_{sa}; P_a(S,a,\beta _1^{(b)},\beta _2^{(b)})\phi ^{*},(1-P_a(S,a,\beta _1^{(b)},\beta _2^{(b)}))\phi ^{*}\right) }{\prod _{s}\prod _{a}\text {Beta}\left( y_{sa}; P_a(S,a,\beta _1^{(b)},\beta _2^{(b)})\phi ^{(b-1)},(1-P_a(S,a,\beta _1^{(b)},\beta _2^{(b)}))\phi ^{(b-1)}\right) } \end{aligned}$$