I am fitting a linear mixed effects model with a natural spline function for age in order to describe the nonlinear trajectory for a repeated outcome `y` (bone mineral content in grams) across time (`age` in years).

How can I solve the spline derivatives to get the velocity curve and estimate for each individual their peak velocity (grams/year) and age at peak velocity (years) from this model?

This is the example data

```
dat <- structure(list(id = c(1001L, 1001L, 1001L, 1001L, 1001L, 1002L, 
    1003L, 1004L, 1004L, 1004L, 1004L, 1004L, 1004L, 1004L, 1005L, 
    1005L, 1005L, 1005L, 1005L, 1006L, 1006L, 1006L, 1006L, 1006L, 
    1007L, 1007L, 1008L, 1008L, 1008L, 1008L, 1008L, 1009L, 1009L, 
    1009L, 1010L, 1010L, 1010L, 1011L, 1012L, 1012L, 1012L, 1013L, 
    1013L, 1014L, 1015L, 1015L, 1015L, 1016L, 1016L, 1016L, 1016L, 
    1016L, 1017L, 1017L, 1018L, 1020L, 1020L, 1021L, 1021L, 1021L, 
    1021L, 1022L, 1022L, 1023L, 1023L, 1023L, 1023L, 1023L, 1023L, 
    1023L, 1023L, 1023L, 1023L, 1024L, 1024L, 1024L, 1024L, 1024L, 
    1025L, 1025L, 1025L, 1026L, 1026L, 1026L, 1026L, 1027L, 1027L, 
    1028L, 1028L, 1028L, 1028L, 1028L, 1028L, 1028L, 1029L, 1029L, 
    1029L, 1029L, 1029L, 1029L, 1030L, 1030L, 1030L, 1030L, 1030L, 
    1030L, 1030L, 1030L, 1031L, 1031L, 1031L, 1031L, 1032L, 1032L, 
    1032L, 1032L, 1032L, 1033L, 1033L, 1033L, 1033L, 1034L, 1034L, 
    1034L, 1034L, 1034L, 1035L, 1035L, 1036L, 1037L, 1037L, 1037L, 
    1037L, 1039L, 1039L, 1040L, 1040L, 1040L, 1040L, 1040L, 1040L, 
    1041L, 1041L, 1041L, 1041L, 1041L, 1041L, 1042L, 1042L, 1042L, 
    1042L, 1042L, 1042L, 1042L, 1043L, 1043L, 1043L, 1043L, 1044L, 
    1044L, 1044L, 1045L, 1045L, 1045L, 1045L, 1045L, 1045L, 1047L, 
    1048L, 1048L, 1049L, 1049L, 1049L, 1049L, 1051L, 1051L, 1052L, 
    1052L, 1052L, 1052L, 1052L, 1053L, 1053L, 1053L, 1053L, 1053L, 
    1054L, 1054L, 1054L, 1054L, 1054L, 1054L, 1054L, 1054L, 1056L, 
    1056L, 1056L, 1056L, 1057L, 1057L, 1058L, 1058L, 1058L, 1058L, 
    1058L, 1060L, 1060L, 1060L, 1061L, 1061L, 1061L, 1061L, 1061L, 
    1062L, 1062L, 1062L, 1062L, 1062L, 1063L, 1063L, 1063L, 1064L, 
    1064L, 1064L, 1064L, 1065L, 1065L, 1066L, 1066L, 1066L, 1066L, 
    1066L, 1066L, 1067L, 1067L, 1067L, 1068L, 1068L, 1068L, 1068L, 
    1068L, 1068L, 1068L, 1069L, 1070L, 1070L, 1070L, 1071L, 1071L, 
    1071L, 1072L, 1072L, 1072L, 1072L, 1072L, 1073L, 1073L, 1073L, 
    1073L, 1074L, 1074L, 1074L, 1075L, 1075L, 1075L, 1075L, 1075L, 
    1075L, 1076L, 1076L, 1076L, 1077L, 1077L, 1077L, 1077L, 1077L, 
    1077L, 1078L, 1078L, 1078L, 1078L, 1078L, 1078L, 1078L, 1080L, 
    1080L, 1080L, 1080L, 1081L, 1081L, 1082L, 1082L, 1082L, 1083L, 
    1083L, 1084L, 1085L, 1085L, 1085L, 1085L, 1085L, 1085L, 1086L, 
    1086L, 1086L, 1087L, 1087L, 1087L, 1087L, 1087L, 1087L, 1087L, 
    1087L, 1088L, 1088L, 1088L, 1088L, 1089L, 1089L, 1089L, 1089L, 
    1089L, 1090L, 1090L, 1091L, 1091L, 1091L, 1091L, 1091L, 1092L, 
    1092L, 1092L, 1092L, 1092L, 1093L, 1093L, 1093L, 1093L, 1094L, 
    1094L, 1094L, 1094L, 1094L, 1095L, 1095L, 1095L, 1095L, 1096L, 
    1097L, 1097L, 1098L, 1098L, 1098L, 1098L, 1098L, 1099L, 1099L, 
    1099L, 1099L, 1099L, 1099L, 1099L, 1099L, 1100L, 1100L, 1100L, 
    1101L, 1101L, 1101L, 1101L, 1103L, 1103L, 1103L, 1103L, 1103L, 
    1103L, 1103L, 1104L, 1104L, 1104L, 1104L, 1105L, 1105L, 1105L, 
    1106L, 1106L, 1106L, 1106L, 1106L, 1106L, 1106L, 1106L, 1106L, 
    1107L, 1108L, 1110L, 1111L, 1112L, 1117L, 1123L), y = c(1934.047646, 
    1075.598345, 1956.214821, 2000.38538, 2000.38538, 732.315937, 
    3119.86, 624.951231, 791.2764892, 1884.530826, 624.951231, 1047.57, 
    1047.57, 791.2764892, 1238.306103, 1555.042976, 2547.870529, 
    2547.870529, 2467.385, 1181.635212, 1181.635212, 565.306282, 
    2016.027874, 2016.027874, 712.6134567, 635.2537841, 2167.362267, 
    2575.574188, 2167.362267, 2480.028259, 2575.574188, 2875.363243, 
    1180.139938, 2828.037147, 3017.119362, 2722.940933, 2167.92, 
    2409.652458, 2245.442558, 724.1520328, 635.6034756, 1649.08326, 
    966.8182507, 865.2717723, 1570.23, 916.1300105, 1180.999973, 
    2351.32885, 2418.851707, 2290.038887, 2224.060562, 2509.52, 1174.589081, 
    1540.219376, 2692.26, 1300.899734, 1100.650177, 1786.628242, 
    1705.842979, 543.8596134, 1786.628242, 2115.374241, 2331.46, 
    875.949604, 2241.945103, 2319.666939, 2316.220234, 719.7139549, 
    2042.803307, 719.7139549, 1132.977503, 875.949604, 2316.220234, 
    1737.18, 1351.629826, 1291.44593, 1291.44593, 1108.26586, 1028.979719, 
    1291.44593, 2068.934227, 2440.784416, 1036.72, 894.6663704, 2449.184731, 
    1109.9, 672.9310664, 2072.320354, 2114.215416, 2114.215416, 1805.422001, 
    2461.18, 2101.374248, 2105.879, 1600.086481, 2866.84, 1600.086481, 
    2807.311, 3055.569931, 1600.086481, 2602.287521, 2690.007614, 
    620.5975037, 2608.4, 2722.3, 2713.66185, 2608.4, 1590.002, 2198.211, 
    2488.097725, 2198.211, 2322.616348, 2627.1, 2418.328346, 2601.661034, 
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    2767.479, 1047.144354, 1047.144354, 1995.119, 1995.119, 707.6093158, 
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    2700.723, 2817.668579, 2700.723, 1349.90691, 1476.19994, 1552.95, 
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    1056.678543, 1571.936, 1210.89, 1210.89, 673.7005405, 687.7842464, 
    1016.86, 1217.866, 1493.791817, 2246.726913, 1054.821, 1054.821, 
    563.6580887, 1054.821, 1540.429863, 2209.006493, 1437.835186, 
    2191.308, 1412.128944, 2724.164597, 2791.705185, 2727.774208, 
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    1907.60466, 1770.658, 760.6319065, 912.8781966, 912.8781966, 
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    2100.048149, 758.3851737, 758.3851737, 2643.373329, 367.7795143, 
    866.0683727, 718.5049658, 866.0683727, 1906.694649, 2291.48, 
    2190.560314, 744.1710777, 1498.981777, 2460.912292, 590.1345787, 
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    2272.232933, 1801.729801, 2299.800095, 2272.232933, 2299.800095, 
    1895.828438, 1757.75, 1050.279345, 1757.75, 1326.09478, 1326.09478, 
    1633.119305, 1558, 1167.971405, 1828.16, 1788.571758, 2175.469, 
    1071.039494, 941.6030864, 2053.067215, 1461.02132, 1597.646778, 
    1885.321567, 2195.704372, 2195.704372, 1675.768558, 3157.550789, 
    1565.173126, 2195.704372, 3157.550789, 2404.836883, 2541.045593, 
    585.7223682, 2465.177761, 2678.462074, 500.3733997, 2465.177761, 
    781.342, 898.3551559, 2465.177761, 2465.177761, 1807.02, 1418.888027, 
    1797.36, 1807.02, 2200.06, 2218.369926, 2200.06, 1986.642735, 
    2088.292, 2069.139, 1507.901432, 2061.395798, 2075.164864, 2081.913219, 
    2081.913219, 483.8579493, 1857.88, 2578.772636, 1857.88, 1857.88, 
    1039.632153, 2288.28, 2288.28, 1831.349922, 2349.23, 933.1002788, 
    2626.298935, 1521.744, 933.1002788, 2626.298935, 1984.760715, 
    2450.333, 1732.339031, 1984.760715, 2731.9, 869.2320918, 1785.72, 
    1922.798, 3081.28, 1508.8, 2421.288597, 1922.798, 1268.074959, 
    1569.05, 1808.115, 1569.05, 1268.074959, 2165.724808, 2165.724808, 
    1808.115, 2084.149837, 2693.027184, 2464.489, 2607.653496, 1012.837271, 
    1012.837271, 2673.190872, 2635.290516, 2773.42, 2635.290516, 
    2654.772674, 2377.905655, 2679.014969, 2654.772674, 1226.40016, 
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    1873.817, 2274.930534, 2317.429165, 959.1709613, 1328.159428, 
    1328.159428, 1328.159428, 959.1709613, 1630.28, 1610.54982, 2507.05302, 
    750.467966, 750.467966, 821.2255058, 802.8240452, 2829.47879), 
        age = c(31.54004107, 11.95071869, 27.88501027, 27.88501027, 
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```

This is the code for fitting the model and plotting the mean predicted trajectory

```
    library(nlme)
    library(splines)
    library(tidyverse)

# LINEAR MIXED MODEL WITH NATURAL SPLINE FUNCTION FOR AGE
    nspline_model <- lme(y ~ ns(age, df = 4), data = dat, random = ~ age | id)

# PLOT MEAN FITTED NATURAL SPLINE TRAJECTORY
    dat$ns_pred <- fitted(nspline_model, level = 0)
    ggplot(data = dat, aes(x = age, y = ns_pred)) + geom_line() 

```

This is a possible solution using the `getPeakTrough` function and `smooth.spline` but i am not sure if it is correct - or how to get subject specific peak velocities rather than just the mean values.

```
    # POSSIBLE SOLUTION FOR THE NATURAL SPLINE MODEL???
    x <- dat$age
    y <- fitted(nspline_model, level = 0)
    y <- predict(smooth.spline(x, y, df = 4), x, deriv = 1)$y
    sitar::getPeakTrough(x, y, peak = T)
    # mean age at peak velocity = 11.82447 years; 
    # mean peak velocity 187.13404 grams/year

```