Blood-tissue+partition+coefficients

Approximating Blood-Tissue Partition Coefficient Measurements with Solubility Measurements
Researchers: Andrew SID Lang, William E Acree Jr., Caitlin Derricott, Emily Knight

Introduction
The Abraham model has been applied to blood-tissue partition coefficients [1]. For example, The blood-brain-barrier partition coefficient can be modeled using the equation:

Equation 1: log BB = 0.547 + 0.221 E -0.604 S -0.641 A -0.681 B + 0.635 V -1.216 Ic

where E, S, A, B, and V are the standard solute descriptors and Ic is taken to be one if the solute is a carboxylic acid and zero otherwise (because carboxylic acids undergo ionization. The pH of blood is 7.4 and carboxylic acids are ionized at this pH).

The above model gives us a way to approximate log BB for compounds with known Abraham descriptors [2]. If Abraham descriptors are not know, then we can either get them indirectly from models [3] - which is advantageous for virtual compound libraries, or for specific compounds, derive them from available solubility and partition measurements [4]. Sometimes measurements for a compound cannot be found in abundance. In the case where few measurements are known, a significant number of measurements have to be performed in order to approximate the descriptors with any degree of certainty.

Instead of deriving the solute descriptors through a series of measurements in order to approximate log BB, we propose a method to approximate log BB using only a couple (or a few) solubility measurements.

Procedure
We took all solvents with measured Abraham coefficients and normalized them by multiplying each coefficient by the average (mean) corresponding solute descriptor value ( Eave = 0.902, Save = 1.016, Aave = 0.144, Bave = 0.492, Vave = 1.330) for our set of compounds with known Abraham descriptors (not including those with nAcid > 0). We did this to account for the relative contributions that each coefficient makes towards the final partition value.

We then regressed the normalized coefficients for these solvents against the normalized coefficients in the Log BB equation using the following code in R: code mydata = read.csv(file="20141022Coefficients-NormalizedReadyForR.csv",head=TRUE,row.names="coefficient") library(leaps) attach(mydata) leaps<-regsubsets(normalized ~ 0 + .,nvmax=2,data=mydata,nbest=5,really.big=TRUE,intercept=FALSE) for (n in 1:10) { print(coef(leaps,n)) }
 * 1) summary(leaps)

[output] (Intercept)        X74 -0.09548583 0.16409286 (Intercept)         X50 -0.1057396  0.1664516 (Intercept)         X43 -0.0981550  0.1654905 (Intercept)         X42 -0.1011318  0.1665445 (Intercept)         X45 -0.1020970  0.1671912 (Intercept)          X40          X88 -0.001642993 0.683174473 -0.497378870 (Intercept)          X45          X53 0.001946871 0.448904998 -0.308778314 (Intercept)          X51          X71 0.002899478 0.370024092 -0.224092036 (Intercept)          X45          X68 0.001891418 0.455433153 -0.309829606 (Intercept)          X46          X71 0.004117552 0.360386611 -0.239561717 [output]

fit <- lm(normalized ~ 0 + X74,data=mydata) summary(fit)

[output] Call: lm(formula = normalized ~ 0 + X74, data = mydata)

Residuals: e        s         a         b         v -0.278023 -0.015804 -0.002069 -0.033820 -0.068669

Coefficients: Estimate Std. Error t value Pr(>|t|) X74 0.15165    0.02068   7.332  0.00184 ** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.1444 on 4 degrees of freedom Multiple R-squared: 0.9307,    Adjusted R-squared:  0.9134 F-statistic: 53.75 on 1 and 4 DF, p-value: 0.001842 [output]

fit <- lm(normalized ~ 0 + X40 + X88,data=mydata) summary(fit)

[output] Call: lm(formula = normalized ~ 0 + X40 + X88, data = mydata)

Residuals: e         s          a          b          v 0.0001244 -0.0026766 -0.0011036 -0.0002149 -0.0001633

Coefficients: Estimate Std. Error t value Pr(>|t|) X40 0.686187   0.002623   261.6 1.23e-07 *** X88 -0.500414  0.002444  -204.7 2.57e-07 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.00168 on 3 degrees of freedom Multiple R-squared:     1,     Adjusted R-squared:      1 F-statistic: 2.133e+05 on 2 and 3 DF, p-value: 1.865e-08 [output] code The results show that we can approximate log BB as follows (X74 = methylcyclohexane), see the (xlsx)

Equation 2: log BB = 0.510 + 0.152 log P_mcy -1.216 Ic

with a predicted adjR2 value of 0.91 and where log P_mcy = log P_(methylcyclohexane/water) = log S_methylcyclohexane - log S_water under most reasonable conditions (note carboxylic acids may dimerize in methylcyclohexane) [4].

To test our model, we compared log BB values calculated using equation (1) to those calculated using equation (2) and the following data provided by Bill Acree (RMSE 0.168):
 * name || E || S || A || B || V || logP_mcy || AD logBB (1) || ONS logBB (2) ||
 * water || 0.000 || 0.450 || 0.820 || 0.350 || 0.167 || -4.01 || -0.38 || -0.10 ||
 * ethanol || 0.250 || 0.420 || 0.370 || 0.480 || 0.449 || -1.78 || 0.07 || 0.24 ||
 * 1-propanol || 0.240 || 0.420 || 0.370 || 0.480 || 0.590 || -1.05 || 0.16 || 0.35 ||
 * acetone || 0.180 || 0.700 || 0.040 || 0.490 || 0.547 || -0.90 || 0.15 || 0.37 ||
 * t-butanol || 0.180 || 0.300 || 0.310 || 0.600 || 0.731 || -0.56 || 0.26 || 0.42 ||
 * 2-methyl-1-propanol || 0.220 || 0.390 || 0.370 || 0.480 || 0.731 || -0.41 || 0.26 || 0.45 ||
 * 1-butanol || 0.220 || 0.420 || 0.370 || 0.480 || 0.731 || -0.40 || 0.24 || 0.45 ||
 * neon || 0.000 || 0.000 || 0.000 || 0.000 || 0.085 || 0.60 || 0.60 || 0.60 ||
 * argon || 0.000 || 0.000 || 0.000 || 0.000 || 0.190 || 1.02 || 0.67 || 0.67 ||
 * nitrogen || 0.000 || 0.000 || 0.000 || 0.000 || 0.222 || 1.06 || 0.69 || 0.67 ||
 * krypton || 0.000 || 0.000 || 0.000 || 0.000 || 0.246 || 1.25 || 0.70 || 0.70 ||
 * methane || 0.000 || 0.000 || 0.000 || 0.000 || 0.250 || 1.34 || 0.71 || 0.71 ||
 * xenon || 0.000 || 0.000 || 0.000 || 0.000 || 0.329 || 1.61 || 0.76 || 0.75 ||
 * sulphur hexafluoride || -0.600 || -0.200 || 0.000 || 0.000 || 0.464 || 2.36 || 0.83 || 0.87 ||
 * benzene || 0.610 || 0.520 || 0.000 || 0.140 || 0.716 || 2.38 || 0.73 || 0.87 ||
 * toluene || 0.600 || 0.520 || 0.000 || 0.140 || 0.857 || 2.90 || 0.81 || 0.95 ||
 * ethylbenzene || 0.610 || 0.510 || 0.000 || 0.150 || 0.998 || 3.44 || 0.91 || 1.03 ||
 * cyclohexane || 0.310 || 0.100 || 0.000 || 0.000 || 0.845 || 4.07 || 1.09 || 1.13 ||
 * methylcyclohexane || 0.240 || 0.060 || 0.000 || 0.000 || 0.986 || 4.75 || 1.19 || 1.23 ||
 * styrene || 0.850 || 0.650 || 0.000 || 0.160 || 0.955 || 5.15 || 0.84 || 1.29 ||

We see good results as illustrated in the figure below.



We see log BB can be approximated fairly well just using logP_mcy values which in turn can be attained in most circumstances from simple solubility measurements in methylcyclohexane and water. We do note however that as the Abraham model for log BB improves our models will also need updating and it is feasible that in the future another solvent other than methylcyclohexane will be predicted as best for approxiamting logBB values.

Regressing against two solvents gave us the following eqution for log BB

log BB = 0.483 + 0.446 log P_(heptane/water) - 0.308 log P_(1-chlorobutane/water) -1.216 Ic

with a predicted adjR2 value of 0.99 and where log P_(heptane/water) = log S_heptane - log S_water and log P_(1-chlorobutane/water) = log S_1-chlorobutane - log S_water under most reasonable conditions (carboxylic acids may dimerize in these solvents).

Thus log BB can easily be approximated by measuring its solubility in two (or for a more accurate results three) solvents, namely methylcyclohexane and water (or pentane, carbon disulfide, and water). The results are so good for the two-solvent (X1, X2) approximation several other choices would give accurate results and solvents could thus be chosen for convenience sake. For example, it looks like that it is good to have one of the solvents be an alkane (see table below), so let's pick hexane and then see what solvents would be a good match for hexane (X41):

code leaps<-regsubsets(normalized - X41 ~ 0 + .,nvmax=1,data=mydata,nbest=10,really.big=TRUE)

for (n in 1:10) + { + print(coef(leaps,n)) + }

[output] (Intercept)        X56 0.04708982 -0.83915134 (Intercept)        X57 0.2135541 -1.3039898 (Intercept)          X4 0.08579438 -0.91845706 (Intercept)        X40 -0.0662102 -0.8323128 (Intercept)         X48 -0.08385477 -0.84289618 (Intercept)        X46 -0.06944242 -0.81177586 (Intercept)        X72 -0.08579303 -0.81583914 (Intercept)        X47 -0.08768122 -0.83694456 (Intercept)        X91 0.03366287 -1.11731138 (Intercept)        X59 0.1396774 -0.8748496 [output] code

We see hexane is a good pairing with: carbon tetrachloride, trifluoroethanol, isopropyl myristate, pentane, hexadecane, undecane, cyclohexane, dodecane, peanut oil, and dibutyl ether.

Results
The regression analysis was performed on other blood-tissue partition equations for in vivo processes, as log P, at 37 °C [5] without normalizing by the average descriptor values: Note: The skin permeation coefficients are for In vitro permeation with log Kp in cm s−1 all other coefficients are for in vivo processes.
 * Process || c || e || s || a || b || v || Ic ||
 * Blood–brain || 0.547 || 0.221 || −0.604 || −0.641 || −0.681 || 0.635 || −1.216 ||
 * Blood–muscle || 0.082 || −0.059 || 0.010 || −0.248 || 0.028 || 0.110 || −1.022 ||
 * Blood–liver || 0.292 || 0.000 || −0.296 || −0.334 || 0.181 || 0.337 || −0.597 ||
 * Blood–lung || 0.269 || 0.000 || −0.523 || −0.723 || 0.000 || 0.720 || −0.988 ||
 * Blood–kidney || 0.494 || −0.067 || −0.426 || −0.367 || 0.232 || 0.410 || −0.481 ||
 * Blood–heart || 0.132 || −0.039 || −0.394 || −0.376 || 0.009 || 0.527 || −0.572 ||
 * Blood–skin || −0.105 || −0.117 || 0.034 || 0.000 || −0.681 || 0.756 || −0.816 ||
 * Blood–fat || 0.077 || 0.249 || −0.215 || −0.902 || −1.523 || 1.234 || −1.013 ||
 * Water–skin || 0.523 || 0.101 || −0.076 || −0.022 || −1.951 || 1.652 || 0.000 ||
 * Skin permeation || −5.420 || −0.102 || −0.457 || −0.324 || −2.608 || 2.066 || 0.000 ||

In general, the blood-tissue partition coefficient is approximated using the following equation:

log P_blood/tissue = c_0 + c_1 X1 + c_2 X2 + Ic

where c_0 is the intercept and Ic is the carboxylic acid modifier; c1 is the coefficient multiplier for the log P value of compound X1, and c_2 is the coefficient multiplier for the log P value of compound X2 (c_2 = 0 for 1-variable approximations).


 * The top five results for log BB are as follows:**


 * blood-brain 1-variable || 0.547 ||
 * solvent || c || c0 || c1 || p || R2 ||
 * X74 || methylcyclohexane || 0.246 || 0.5054875 || 0.16875 || 0.00138 || 0.9401 ||
 * X50 || 1,9-decadiene || 0.104 || 0.52899032 || 0.17317 || 0.00251 || 0.9193 ||
 * X43 || octane || 0.231 || 0.51007003 || 0.15987 || 0.00224 || 0.9237 ||
 * X72 || cyclohexane || 0.159 || 0.52201315 || 0.15715 || 0.0028 || 0.9148 ||
 * X45 || decane || 0.186 || 0.51718048 || 0.16032 || 0.00282 || 0.9145 ||


 * Similarly, the top five results for the other processes are:**


 * blood-muscle || 0.082 ||
 * solvent || c || c0 || c1 || p || R2 ||
 * X57 || trifluoroethanol || 0.395 || 0.063356 || 0.0472 || 0.1955 || 0.3759 ||
 * X55 || chloroform || 0.191 || 0.07672649 || 0.02761 || 0.1735 || 0.4061 ||
 * X54 || dichloromethane || 0.319 || 0.0740888 || 0.0248 || 0.2132 || 0.3536 ||
 * X56 || carbon tetrachloride || 0.199 || 0.0777215 || 0.0215 || 0.2275 || 0.3364 ||
 * X65 || iodobenzene || -0.192 || 0.08615104 || 0.02162 || 0.2428 || 0.3191 ||


 * blood-liver || 0.292 ||
 * solvent || c || c0 || c1 || p || R2 ||
 * X57 || trifluoroethanol || 0.395 || 0.24957305 || 0.10741 || 0.1513 || 0.4394 ||
 * X74 || methylcyclohexane || 0.246 || 0.2809423 || 0.04495 || 0.2361 || 0.3266 ||
 * X50 || 1,9-decadiene || 0.104 || 0.28738136 || 0.04441 || 0.2644 || 0.296 ||
 * X55 || chloroform || 0.191 || 0.28278234 || 0.04826 || 0.2805 || 0.2799 ||
 * X49 || 2,2,4-trimethylpentane || 0.32 || 0.2793408 || 0.03956 || 0.2926 || 0.2682 ||


 * blood-lung || 0.269 ||
 * solvent || c || c0 || c1 || p || R2 ||
 * X57 || trifluoroethanol || 0.395 || 0.1651545 || 0.2629 || 0.04008 || 0.6918 ||
 * X74 || methylcyclohexane || 0.246 || 0.23886992 || 0.12248 || 0.05695 || 0.6372 ||
 * X50 || 1,9-decadiene || 0.104 || 0.2561404 || 0.12365 || 0.0693 || 0.6031 ||
 * X55 || chloroform || 0.191 || 0.24312523 || 0.13547 || 0.07863 || 0.5797 ||
 * X49 || 2,2,4-trimethylpentane || 0.32 || 0.2327024 || 0.11343 || 0.07877 || 0.5793 ||


 * blood-kidney || 0.494 ||
 * solvent || c || c0 || c1 || p || R2 ||
 * X57 || trifluoroethanol || 0.395 || 0.4429818 || 0.12916 || 0.1746 || 0.4046 ||
 * X74 || methylcyclohexane || 0.246 || 0.48103334 || 0.05271 || 0.2744 || 0.2859 ||
 * X50 || 1,9-decadiene || 0.104 || 0.48857016 || 0.05221 || 0.3008 || 0.2605 ||
 * X55 || chloroform || 0.191 || 0.48345107 || 0.05523 || 0.3316 || 0.2335 ||
 * X49 || 2,2,4-trimethylpentane || 0.32 || 0.4792736 || 0.04602 || 0.3345 || 0.2311 ||


 * blood-heart || 0.132 ||
 * solvent || c || c0 || c1 || p || R2 ||
 * X57 || trifluoroethanol || 0.395 || 0.062 || 0.17744 || 0.03284 || 0.7193 ||
 * X74 || methylcyclohexane || 0.246 || 0.113 || 0.07916 || 0.06759 || 0.6076 ||
 * X50 || 1,9-decadiene || 0.104 || 0.124 || 0.08062 || 0.07638 || 0.5851 ||
 * X49 || 2,2,4-trimethylpentane || 0.32 || 0.109 || 0.07273 || 0.09441 || 0.5437 ||
 * X43 || octane || 0.231 || 0.115 || 0.07153 || 0.09471 || 0.543 ||


 * blood-skin || -0.105 ||
 * solvent || c || c0 || c1 || p || R2 ||
 * X24 || ethanol/water(10:90)vol || -0.173 || 0.1918161 || 1.7157 || 0.0001669 || 0.979 ||
 * X12 || DMF || -0.305 || -0.05844175 || 0.15265 || 0.0001956 || 0.9773 ||
 * X23 || ethanol/water(20:80)vol || -0.252 || 0.09946272 || 0.81136 || 0.0003916 || 0.9679 ||
 * X22 || ethanol/water(30:70)vol || -0.269 || 0.03408914 || 0.51706 || 0.0006143 || 0.9598 ||
 * X18 || ethanol/water(70:30)vol || 0.063 || -0.11973255 || 0.23385 || 0.0005348 || 0.9625 ||


 * blood-fat || 0.077 ||
 * solvent || c || c0 || c1 || p || R2 ||
 * X88 || carbon disulfide || 0.047 || 0.064954605 || 0.256285 || 1.14E-06 || 0.9983 ||
 * X67 || ethylbenzene || 0.093 || 0.0493604 || 0.2972 || 1.57E-05 || 0.9935 ||
 * X70 || p-xylene || 0.166 || 0.0277727 || 0.29655 || 1.62E-05 || 0.9934 ||
 * X69 || o-xylene || 0.083 || 0.05226932 || 0.29796 || 1.97E-05 || 0.9928 ||
 * X91 || peanut oil || 0.574 || -0.12834276 || 0.35774 || 0.0008404 || 0.953 ||


 * water-skin || 0.523 ||
 * solvent || c || c0 || c1 || p || R2 ||
 * X75 || THF || 0.223 || 0.43751295 || 0.38335 || 2.79E-06 || 0.9973 ||
 * X81 || N-formylmorpholine || -0.032 || 0.53818944 || 0.47467 || 2.84E-05 || 0.9913 ||
 * X13 || dibutylformamide || 0.332 || 0.38916084 || 0.40313 || 1.56E-05 || 0.9936 ||
 * X85 || acetone || 0.313 || 0.39460114 || 0.41022 || 2.59E-05 || 0.9917 ||
 * X76 || 1,4-dioxane || 0.123 || 0.47371144 || 0.40072 || 2.24E-05 || 0.9923 ||


 * skin-permeation || -5.42 ||
 * solvent || c || c0 || c1 || p || R2 ||
 * X75 || THF || 0.223 || -5.53170293 || 0.50091 || 0.0001648 || 0.9791 ||
 * X60 || methyl tert-butyl ether || 0.341 || -5.58776518 || 0.49198 || 2.34E-05 || 0.9921 ||
 * X58 || diethyl ether || 0.35 || -5.595987 || 0.50282 || 3.54E-05 || 0.9903 ||
 * X22 || ethanol/water(30:70)vol || -0.269 || -4.9673268 || 1.6828 || 0.001522 || 0.937 ||
 * X21 || ethanol/water(40:60)vol || -0.221 || -5.1467114 || 1.2366 || 0.001265 || 0.9425 ||


 * 2-variable coefficients**


 * blood-brain 2-variables |||||||| 0.547 ||
 * solvent1 || c |||| solvent2 || c || c0 || c1 || p || R2 ||
 * X42 || heptane || 0.297 || X66 || toluene || 0.125 || 0.45117932 || 0.46044 || 0.000391 || 0.9947 ||
 * X43 || octane || 0.231 || X66 || toluene || 0.125 || 0.48459881 || 0.42974 || 7.92E-05 || 0.9982 ||
 * X44 || nonane || 0.24 || X66 || toluene || 0.125 || 0.47497705 || 0.48008 || 0.0004171 || 0.9944 ||
 * X45 || decane || 0.186 || X68 || m-xylene || 0.122 || 0.50038846 || 0.4592 || 1.79E-05 || 0.9993 ||
 * X43 || octane || 0.231 || X67 || ethylbenzene || 0.093 || 0.472768 || 0.44587 || 7.01E-05 || 0.9983 ||


 * blood-muscle |||||||| 0.082 ||
 * solvent1 || c |||| solvent2 || c || c0 || c1 || p || R2 ||
 * X49 || 2,2,4-trimethylpentane || 0.32 || X51 || 1-hexadecene || 0.116 || -0.0107512 || 0.4529 || 0.04075 || 0.8816 ||
 * X36 || 1-heptanol || 0.035 || X84 || ethylene glycol || -0.27 || -0.0020195 || 0.1854 || 0.1483 || 0.7198 ||
 * X30 || 1-butanol || 0.165 || X84 || ethylene glycol || -0.27 || -0.03736505 || 0.21589 || 0.1236 || 0.7518 ||
 * X55 || chloroform || 0.191 || X59 || dibutyl ether || 0.176 || 0.07475273 || 0.11273 || 0.006448 || 0.9654 ||
 * X49 || 2,2,4-trimethylpentane || 0.32 || X44 || nonane || 0.24 || 0.000392 || 0.9388 || 0.04187 || 0.8794 ||


 * blood-liver |||||||| 0.292 ||
 * solvent1 || c |||| solvent2 || c || c0 || c1 || p || R2 ||
 * X19 || ethanol/water(60:40)vol || -0.04 || X80 || N-methyl-2-piperidone || 0.056 || 0.336086 || 0.60886 || 0.001459 || 0.9871 ||
 * X18 || ethanol/water(70:30)vol || 0.063 || X11 || N-ethylformamide || 0.22 || 0.36579914 || 0.80642 || 0.01333 || 0.9438 ||
 * X17 || ethanol/water(80:20)vol || 0.172 || X80 || N-methyl-2-piperidone || 0.056 || 0.22836036 || 0.47659 || 0.005441 || 0.9691 ||
 * X39 || octadecanol || -0.096 || X79 || N-methylpyrrolidinone || 0.147 || 0.36228256 || 0.30679 || 0.02378 || 0.9173 ||
 * X16 || ethanol/water(90:10)vol || 0.243 || X80 || N-methyl-2-piperidone || 0.056 || 0.20549356 || 0.4286 || 0.007922 || 0.9603 ||


 * blood-lung |||||||| 0.269 ||
 * solvent1 || c |||| solvent2 || c || c0 || c1 || p || R2 ||
 * X32 || 3-methyl-1-butanol || 0.073 || X27 || 2-propanol || 0.099 || 0.3218737 || 2.0713 || 0.00211 || 0.9835 ||
 * X20 || ethanol/water(50:50)vol || -0.142 || X11 || N-ethylformamide || 0.22 || 0.7257252 || 1.7856 || 0.07541 || 0.8215 ||
 * X21 || ethanol/water(40:60)vol || -0.221 || X80 || N-methyl-2-piperidone || 0.056 || 0.68674853 || 1.72937 || 0.001563 || 0.9865 ||
 * X29 || 2-methyl-2-propanol || 0.211 || X6 || N-ethylacetamide || 0.284 || 0.3042222 || 0.9758 || 0.0003914 || 0.9946 ||
 * X22 || ethanol/water(30:70)vol || -0.269 || X80 || N-methyl-2-piperidone || 0.056 || 0.9410796 || 2.36584 || 0.004666 || 0.9721 ||


 * blood-kidney |||||||| 0.494 ||
 * solvent1 || c |||| solvent2 || c || c0 || c1 || p || R2 ||
 * X23 || ethanol/water(20:80)vol || -0.252 || X87 || DMSO || -0.194 || 0.92373252 || 2.03498 || 0.0001233 || 0.9975 ||
 * X23 || ethanol/water(20:80)vol || -0.252 || X9 || formamide || -0.171 || 1.0139219 || 2.5962 || 0.03288 || 0.8974 ||
 * X31 || 2-butanol || 0.127 || X90 || tributyl phosphate || 0.022 || 0.40783367 || 0.79943 || 0.001471 || 0.9871 ||
 * X22 || ethanol/water(30:70)vol || -0.269 || X87 || DMSO || -0.194 || 0.75434917 || 1.26831 || 0.001073 || 0.9895 ||
 * X21 || ethanol/water(40:60)vol || -0.221 || X87 || DMSO || -0.194 || 0.61679969 || 0.91581 || 0.001315 || 0.988 ||


 * blood-heart |||||||| 0.132 ||
 * solvent1 || c |||| solvent2 || c || c0 || c1 || p || R2 ||
 * X39 || octadecanol || -0.096 || X79 || N-methylpyrrolidinone || 0.147 || 0.21453465 || 0.38488 || 0.006347 || 0.9657 ||
 * X19 || ethanol/water(60:40)vol || -0.04 || X80 || N-methyl-2-piperidone || 0.056 || 0.1851036 || 0.76094 || 0.0002552 || 0.996 ||
 * X18 || ethanol/water(70:30)vol || 0.063 || X11 || N-ethylformamide || 0.22 || 0.2098766 || 0.9478 || 0.02701 || 0.91 ||
 * X53 || 1-chlorobutane || 0.222 || X71 || nitrobenzene || -0.196 || -0.0693916 || 0.4976 || 0.2066 || 0.6506 ||
 * X18 || ethanol/water(70:30)vol || 0.063 || X80 || N-methyl-2-piperidone || 0.056 || 0.11141671 || 0.67863 || 0.0005619 || 0.9932 ||


 * blood-skin |||||||| -0.105 ||
 * solvent1 || c |||| solvent2 || c || c0 || c1 || p || R2 ||
 * X12 || DMF || -0.305 || X79 || N-methylpyrrolidinone || 0.147 || 0.0072516 || 0.29355 || 8.51E-05 || 0.9981 ||
 * X23 || ethanol/water(20:80)vol || -0.252 || X84 || ethylene glycol || -0.27 || 0.22089576 || 1.54723 || 0.0002055 || 0.9965 ||
 * X18 || ethanol/water(70:30)vol || 0.063 || X84 || ethylene glycol || -0.27 || -0.20430483 || 0.46661 || 0.0005899 || 0.993 ||
 * X31 || 2-butanol || 0.127 || X33 || 1-pentanol || 0.15 || -0.1116483 || 0.8469 || 0.0002163 || 0.9964 ||
 * X20 || ethanol/water(50:50)vol || -0.142 || X84 || ethylene glycol || -0.27 || -0.09356472 || 0.64464 || 0.0003629 || 0.9949 ||


 * blood-fat |||||||| 0.077 ||
 * solvent1 || c |||| solvent2 || c || c0 || c1 || p || R2 ||
 * X30 || 1-butanol || 0.165 || X27 || 2-propanol || 0.099 || -0.14452933 || 2.90901 || 1.67E-06 || 0.9999 ||
 * X46 || undecane || 0.058 || X50 || 1,9-decadiene || 0.104 || 0.0795048 || 0.6 || 0.000768 || 0.9916 ||
 * X88 || carbon disulfide || 0.047 || X67 || ethylbenzene || 0.093 || 0.0715957 || 0.3647 || 4.58E-05 || 0.9987 ||
 * X88 || carbon disulfide || 0.047 || X57 || trifluoroethanol || 0.395 || 0.07875685 || 0.2703 || 3.27E-05 || 0.999 ||
 * X88 || carbon disulfide || 0.047 || X66 || toluene || 0.125 || 0.073668 || 0.3435 || 4.59E-05 || 0.9987 ||


 * water-skin |||||||| 0.523 ||
 * solvent1 || c |||| solvent2 || c || c0 || c1 || p || R2 ||
 * X27 || 2-propanol || 0.099 || X89 || sulfolane || 0 || 0.502735789 || 0.204689 || 8.08E-07 || 0.9999 ||
 * X83 || acetonitrile || 0.413 || X90 || tributyl phosphate || 0.022 || 0.45830486 || 0.141688 || 3.98E-08 || 1 ||
 * X6 || N-ethylacetamide || 0.284 || X86 || butanone || 0.246 || 0.417296244 || 0.147212 || 5.50E-07 || 0.9999 ||
 * X3 || butyl acetate || 0.248 || X48 || hexadecane || 0.087 || 0.407784123 || 0.504905 || 5.57E-08 || 1 ||
 * X5 || N-methylacetamide || 0.09 || X71 || nitrobenzene || -0.196 || 0.52935714 || 0.26483 || 1.61E-05 || 0.9994 ||


 * skin-permeation |||||||| -5.42 ||
 * solvent1 || c |||| solvent2 || c || c0 || c1 || p || R2 ||
 * X16 || ethanol/water(90:10)vol || 0.243 || X83 || acetonitrile || 0.413 || -5.6188514 || 0.3992 || 0.002202 || 0.9831 ||
 * X16 || ethanol/water(90:10)vol || 0.243 || X78 || cyclohexanone || 0.038 || -5.4961234 || 0.2646 || 0.003174 || 0.9784 ||
 * X1 || methyl acetate || 0.351 || X17 || ethanol/water(80:20)vol || 0.172 || -5.5907014 || 0.3698 || 0.00236 || 0.9823 ||
 * X29 || 2-methyl-2-propanol || 0.211 || X77 || propylene carbonate || 0.004 || -5.4934282 || 0.3434 || 0.001826 || 0.9851 ||
 * X22 || ethanol/water(30:70)vol || -0.269 || X71 || nitrobenzene || -0.196 || -5.0966903 || 1.0577 || 0.006749 || 0.9643 ||


 * Calculating the coefficients for methylcyclohexane and 1,9-decadiene for all processes gives:**

Partition Coefficients for Methylcyclohexane
 * Process || p || R2 || c_1 || 0_0 ||
 * Blood-brain || 0.0014 || 0.94 || 0.169 || 0.505 ||
 * Blood-muscle || 0.2401 || 0.32 || 0.021 || 0.077 ||
 * Blood-liver || 0.2361 || 0.33 || 0.04495 || 0.281 ||
 * Blood-lung || 0.05695 || 0.64 || 0.122 || 0.239 ||
 * Blood-kidney || 0.2744 || 0.29 || 0.053 || 0.481 ||
 * Blood-heart || 0.06759 || 0.61 || 0.079 || 0.113 ||
 * Blood-skin || 0.05174 || 0.65 || 0.111 || -0.132 ||
 * Blood-fat || 0.000858 || 0.95 || 0.285 || 0.007 ||
 * Water-skin || 0.03418 || 0.71 || 0.289 || 0.452 ||
 * Skin permeation || 0.01667 || 0.8 || 0.403 || -5.519 ||

Blood Brain EquationVariables for 1,9-decadiene (X50) 1,9- Decadiene was a decent blood-partition coefficient.
 * Process || p || R2 || c_1 || c_0 ||
 * Blood-brain || 0.0025 || 0.92 || 0.173 || 0.529 ||
 * Blood-muscle || 0.2697 || 0.29 || 0.021 || 0.080 ||
 * Blood-liver || 0.264 || 0.3 || 0.044 || 0.287 ||
 * Blood-lung || 0.0693 || 0.6 || 0.124 || 0.256 ||
 * Blood-kidney || 0.3008 || 0.26 || 0.0502 || 0.489 ||
 * Blood-heart || 0.07638 || 0.59 || 0.08 || 0.124 ||
 * Blood-skin || 0.03598 || 0.71 || 0.12 || -0.117 ||
 * Blood-fat || 0.0005379 || 0.96 || 0.297 || 0.046 ||
 * Water-skin || 0.02297 || 0.76 || 0.311 || 0.491 ||
 * Skin permeation || 0.01016 || 0.84 || 0.43 || -5.465 ||


 * Blood-Muscle || X90 || X289 || X108 || X2 || X277 ||
 * co || -0.019 || -0.033 || -0.020 || -0.018 || -0.018 ||
 * c1 || 0.026 || 0.044 || 0.023 || 0.023 || 0.022 ||
 * mult r2 || 0.38 || 0.37 || 0.3214 || 0.3159 || 0.3025 ||


 * Blood-Liver || X289 || X224 || X14 || X90 || X91 ||
 * co || -2.6E-02 || 0.017 || 0.013 || 0 || 0.015 ||
 * c1 || -0.099 || 0.046 || 0.045 || 0.047 || 0.041 ||
 * mult r2 || 0.408 || 0.320 || 0.291 || 0.269 || 0.264 ||


 * Blood-Lung || X289 || X224 || X14 || X90 || X91 ||
 * co || -0.048 || 0.001 || -0.003 || -0.018 || -0.002 ||
 * c1 || 0.246 || 0.121 || 0.122 || 0.131 || 0.114 ||
 * mult r2 || 0.681 || 0.615 || 0.583 || 0.560 || 0.5563 ||


 * Blood-Kidney || X289 || X224 || X14 || X90 || X91 ||
 * c0 || -0.048 || 0.001 || -0.003 || -0.018 || -0.002 ||
 * c1 || 0.12 || 0.052 || 0.052 || 0.053 || 0.046 ||
 * mult r2 || 0.381 || 0.269 || 0.245 || 0.218 || 0.214 ||


 * Blood-heart || X289 || X224 || X14 || X91 || X26 ||
 * c0 || -0.061 || 0.013 || 0.009 || 0.013 || 0.019 ||
 * c1 || 0.167 || 0.08 || 0.081 || 0.075 || 0.074 ||
 * mult r2 || 0.706 || 0.596 || 0.573 || 0.533 || 0.532 ||

Conclusion
Several blood-tissue processes have been approximated using just a few solvent/solvent partition coefficients or common solvent solubility measurements. Several blood-tissue coefficients (blood-skin, blood-fat, water-skin, and skin-permeation) had very good (R2 > 0.97) models using just 1-variable approximations and gained relatively little improvement when 2-variable were used.

In the case of blood-brain, we have a very good 1-varibale model with an R2-value of 0.91 that can be improved to 0.99 when a 2-variable model is used. This case is borderline and whether you use the 1-variable or a 2-variable model should depend upon circumstances.

For the other processes (blood-muscle, blood-liver, blood-lung, blood-kidney, and blood-heart) the R2-value for the 1-variable models ranged from 0.58 for blood-kidney to 0.82 for blood-lung. All of the R2 values for these processes increased to at least 0.94 when 2-variables were used. Thus it seems that 2-variable models are more appropriate in these case. However, in a pinch, the 1-variable models are attractive because they all share the same three solvents (methylcyclohexane, trifluroethanol, and 1,9-decadiene). That is, with just one logP_mcy measurement, 5 blood-tissue partition values can be estimated with reasonable accuracy. This is not surprising, as the actual processes are similar in nature. It is also not surprising that the blood-fat partition coefficient can be approximated using the peanut oil-water coefficient.

All the 2-variable models had very good R2 values and there should not be a need to step up to 3-variable models. The models do so well that we can get good approximations from many different logP combinations. This adds to the convenience of our method.