Mathematica

Monte Carlo simulations were performed in Mathematica (Wolfram Research, Inc., Mathematica, Version 11.1, Champaign, IL (2017)) in order to determine the optimal time binning between the 5 time samplings tested. The mean of the metabolite-corrected arterial TAC from the fastest initial temporal sampling (11x10sec-6x15sec-5x20sec-3x300sec) extracted from the patients with interpolation to 1-second frames was used for the modeled arterial TAC (C_IDIF(t)). This modeled arterial TAC was applied for every investigated time sampling. The modeled tumoral TAC C(t) was obtained as follows:
$\text{C}\left(\text{t}\right)=\text{VB}{\text{C}}_{\text{IDIF}}\left(\text{t}\right)+\left(1-\text{VB}\right)\text{K}1{\text{e}}^{-\text{k}2\text{t}}\otimes {\text{C}}_{\text{IDIF}}\left(\text{t}\right).$
K1, K2 and VB were average values extracted for the 33 lesions using the 5 different time samplings according to the selected model.
For each time sampling, 1024 realizations of independent distributed Poisson noise (Added noise) were added to the modeled TAC as follows:
$\text{Added}\text{Noise}=\text{c}\left(\text{RandomInteger}\left[\text{PoissonDistribution}\left[\text{C}\left(\text{t}\right)\right]\right]-\text{C}\left(\text{t}\right)\right)/\text{Sqrt}\left(\text{dt}\right),$
where c is the scaling factor and dt is the frame duration.
Each realization was fitted to the model providing an estimation of the kinetic parameters. The mean and standard deviation of the estimated K1 values were computed from all the realizations and compared to a target K1 value as the objective, which is the average of the K1 values extracted from the 33 lesions.

All analysis, graphs and maps were performed or created in Mathematica (Wolfram Research, Inc., Mathematica, Version 14.0, Champaign, IL (2024)).

To investigate the optimal time sampling between the seven proposed time samplings, Monte Carlo simulations were performed in Mathematica (Wolfram Research, Inc., Mathematica, Version 11.1, Champaign, IL, USA (2017)). A modeled arterial TAC (C_IDIF(t)) was obtained from the mean of the arterial TAC from the fastest initial temporal sampling (8 × 3″–8 × 12″–6 × 30″) extracted from the patients with interpolation to 1-s frames. This modeled arterial TAC was applied for every investigated time sampling. Average K1 and average k2 parameters extracted from the lesions with the seven different time samplings according to the selected model provided a modeled tumoral TAC C(t) as follows: $$\mathrm{C}\left(\mathrm{t}\right)=\mathrm{VB}{\mathrm{C}}_{\text{IDIF}}\left(\mathrm{t}\right)+\left(1-\mathrm{VB}\right)\text{K1}{\mathrm{e}}^{-\mathrm{k}2\mathrm{t}}\otimes {\mathrm{C}}_{\text{IDIF}}\left(\mathrm{t}\right)$$
For each time sampling, 1000 realizations of independent distributed Poisson noise (added noise) were added to the modeled TAC as follows: $$\text{Added noise}=c\left(\text{RandomInteger}\left[\text{PoissonDistribution}\left[\mathrm{C}\left(\mathrm{t}\right)\right]\right]-\mathrm{C}\left(\mathrm{t}\right)\right)/\text{Sqrt}\left(\mathrm{dt}\right),$$
where c is the scaling factor and dt is the frame duration.
Each realization was fitted to the model providing an estimation of the kinetic parameters. The mean and standard deviation of the estimated K1 values were computed from all the realizations and compared to the target K1 value. The target K1 value was the average of K1 values extracted for all of the lesions from all of the time samplings with the selected model.

Desmos Graphing Calculator v1.9 [https://www.desmos.com/calculator (accessed on 5 September 2023)], which is a free multivariate open access software, was used to generate illustrations so that readers could duplicate the graphical results and explore their analytic goals at no expense. Mathematica [Wolfram, Mathematica/">https://www.wolfram.com/Mathematica/ (accessed on 5 September 2023), ver. 13.3] was used to confirm the (x, y) coordinates of graphical intersections and other analytical results.

Unless indicated otherwise, the number of replicates was three for each simulation, and six for each experiment. For comparative statistics, an unpaired, two‐tailed, Student's t‐test was performed in Wolfram Mathematica (version 12.4). To fit the data to the proposed function, the nonlinearmodelfit function of the Wolfram Mathematica (version 12.4) was applied.

LNP pKa was determined using the TNS binding assay. The TNS reagent was prepared as a 300 μM stock solution in DMSO. Following Zhang et al.^{43 (link)}, LNPs were diluted to 75 μM ionizable lipid, TNS to 6 µM in a total volume of ~93 µL of buffered solutions containing 20 mM boric Acid, 10 mM imidazole, 10 mM sodium acetate, 10 mM glycylglycine, 25 mM NaCl, and where the pH ranged from 3 to 10. The Cytation 5 Cell Imaging Multi-Mode Reader (Biotek) was used to read Fluorescence (Ex321/Em445). The pH was measured in each well after TNS addition. Mathematica (Wolfram Research) was used to fit the fluorescence data to the Henderson–Hasselbalch equation $\mathrm{RFU}={\mathrm{RFU}}_{\max }-\left({\mathrm{RFU}}_{\max }-{\mathrm{RFU}}_{\min }\right)/\left(1+{10}^{\mathrm{p}K\mathrm{a}-\mathrm{pH}}\right)$ to provide the pKa. We also tested two alternative TNS binding assay protocols (Supplementary Figs. 1 and 2 and Supplementary Tables 1 and 2) Sabnis et al.^{44 (link)}, where LNPs were diluted to 24 µM and TNS to 6.3 µM, and Jayaraman et al.^{24 (link)}, where LNPs were diluted to 40 µM and TNS to 1 µM.

For each problem instance, the integer sequences bounded by d_min and d_max and satisfying equations (2) and (3) were found with the algorithm outlined below.Algorithm

Solutions(N, D, A, d_min, d_max).

The output was further filtered according to eq. (4). The function IntegerPartitions is the one from Mathematica (Wolfram Research, Champaign, IL) version 7 and onwards, and it implements a multiply restricted integer partitioning algorithm that in this case finds all ordered sequences of exactly N integers adding up to A using only the elements appearing in the list s defined in the lines 1–4 above. The vast majority of time is taken by the call to IntegerPartitions. For the two real networks which had millions of solutions, IntegerPartitions was run in block mode where a subset of partitions was found with each call until all partititions had been found. While it is conceivable that more direct and efficient methods might exist to find the solutions, such discussion is outside the scope of this work.