Introduction 1 1 2 4 Fig. 1 Scheme of the liver cell bioreactor with the perfusion circuit, the two inflow streams and the outflow stream. Measured data were acquired from the waste  5 7 8 6 Materials and methods Cells and bioreactors 9 n 10 V 2 V 1 10 −1 1 F A t F B t 1 F A t F A −1 F A −1 t A F B t F B F B −1 t B 4 F 0 t 1 \documentclass[12pt]{minimal}
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$$ \begin{aligned}{} F_{A} (t) & = \left\{ {\begin{array}{*{20}c} {{F_{{A{\text{1}}}} }} & {{for}} & {{t < t_{A} }} \\ {{F_{{A{\text{2}}}} }} & {{for}} & {{t \ge t_{A} }} \\ \end{array} ,} \right. \\ F_{B} (t) & = \left\{ {\begin{array}{*{20}c} {{F_{{B{\text{1}}}} }} & {{for}} & {{t < t_{B} }} \\ {{F_{{B{\text{2}}}} }} & {{for}} & {{t \ge t_{B} }} \\ \end{array} ,} \right. \\ F_{0} (t) & = F_{A} (t) + F_{B} (t). \\ \end{aligned} $$\end{document} F A t c Ai 1 F B t c B −1 c Bi i c i t c i c Ai c i t Table 1 c i c 0 i c i ) c i ) c Ai F A t 1 s i Name c i c 0 I c i c i c Ai −1 s i LEU c 1 c 0 1 2258 1 HIS c 2 c 0 2 940 3 ARG c 3 c 0 3 2604 4 VAL c 4 c 0 4 987 1 TRP c 5 c 0 5 464 2 PHE c 6 c 0 6 1300 1 ILE c 7 c 0 7 500 1 ALA c 8 c 0 8 1825 1 TYR c 9 c 0 9 2215 1 LYS c 10 c 0 10 327 2 MET c 11 c 0 11 259 1 SER c 12 c 0 12 861 1 GLY c 13 c 0 13 1957 1 THR c 14 c 0 14 859 1 ASP c 15 c 0 15 284 1 ASN c 16 c 0 16 168 2 GLU c 17 c 0 17 265 1 GLN c 18 c 0 18 687 2 NH3 c 19 c 0 19 41 1 UREA c 20 c 0 20 0 2 PROT c 21 c 0 21 0 – Data c i,j,k i j k K K t k t k t k t k c i,j,k 1 i 1 i i L j j j j Differential equation system 2.1 2.10 1 Cells and bioreactors 2.1 i F A t F B t V 1 c Ai i 1 F A t F B t 1 2.1 F 0 t c i c i p 0 2.2 2.9 c i V 2 2 2.1 \documentclass[12pt]{minimal}
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$$ \frac{{{\text{d}}c_{{0,i}} }} {{{\text{d}}t}} = F_{A} (t)/V_{1} \cdot c_{{Ai}} + F_{B} (t)/V_{1} \cdot c_{{Bi}} - F_{0} (t)/V_{1} \cdot c_{{0,i}} - p_{0} /V_{1} \cdot (c_{{0,i}} - c_{i} )\quad {\text{for}}\;i = 1,...,21 $$\end{document} 2.2 \documentclass[12pt]{minimal}
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$$ \frac{{{\text{d}}c_{i} }} {{{\text{d}}t}} = p_{0} /V_{2} \cdot (c_{{0,i}} - c_{i} ) - p_{i} \cdot c_{i} ,\quad {\text{for}}\;i = 1,...,14 $$\end{document} 2.3 \documentclass[12pt]{minimal}
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$$ \frac{{{\text{d}}c_{{15}} }} {{{\text{d}}t}} = p_{0} /V_{2} \cdot (c_{{0,15}} - c_{{15}} ) + p_{{16}} \cdot c_{{16}} + p_{{18}} \cdot c_{{17}} - (p_{{15}} \cdot c_{{19}} + p_{{17}} + p_{{19}} \cdot c_{{19}} /(p_{{20}} + c_{{15}} )) \cdot c_{{15}} , $$\end{document} 2.4 \documentclass[12pt]{minimal}
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$$ \frac{{{\text{d}}c_{{16}} }} {{{\text{d}}t}} = p_{0} /V_{2} \cdot (c_{{0,16}} - c_{{16}} ) + p_{{15}} \cdot c_{{15}} \cdot c_{{19}} - p_{{16}} \cdot c_{{16}} , $$\end{document} 2.5 \documentclass[12pt]{minimal}
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$$ \begin{aligned}{} \frac{{{\text{d}}c_{{17}} }} {{{\text{d}}t}} & = p_{0} /V_{2} \cdot (c_{{0,17}} - c_{{17}} ) + {\sum\limits_{i = 1}^{10} {p_{i} \cdot s_{i} \cdot c_{i} } } + p_{{17}} \cdot c_{{15}} + p_{{22}} \cdot c_{{18}} \\ & \quad - (p_{{18}} + p_{{21}} \cdot c_{{19}} + p_{{23}} ) \cdot c_{{17}} , \\ \end{aligned} $$\end{document} 2.6 \documentclass[12pt]{minimal}
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$$ \frac{{{\text{d}}c_{{18}} }} {{{\text{d}}t}} = p_{0} /V_{2} \cdot (c_{{0,18}} - c_{{18}} ) + p_{{21}} \cdot c_{{17}} \cdot c_{{19}} - p_{{22}} \cdot c_{{18}} , $$\end{document} 2.7 \documentclass[12pt]{minimal}
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$$ \begin{aligned}{} \frac{{{\text{d}}c_{{19}} }} {{{\text{d}}t}} & = p_{0} /V_{2} \cdot (c_{{0,19}} - c_{{19}} ) + {\sum\limits_{i = 11}^{14} {p_{i} \cdot s_{i} \cdot c_{i} } } + p_{{23}} \cdot (1 - g(t)) \cdot c_{{17}} + p_{{16}} \cdot c_{{16}} + p_{{22}} \cdot c_{{18}} \\ & \quad - (p_{{19}} \cdot c_{{15}} /(p_{{20}} + c_{{15}} ) + p_{{15}} \cdot c_{{15}} + p_{{21}} \cdot c_{{17}} ) \cdot c_{{19}} , \\ \end{aligned} $$\end{document} 2.8 \documentclass[12pt]{minimal}
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$$ \frac{{{\text{d}}c_{{20}} }} {{{\text{d}}t}} = p_{0} /V_{2} \cdot (c_{{0,20}} - c_{{20}} ) + p_{{19}} \cdot c_{{15}} /(p_{{20}} + c_{{15}} ) \cdot c_{{19}} , $$\end{document} 2.9 \documentclass[12pt]{minimal}
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$$ \frac{{{\text{d}}c_{{21}} }} {{{\text{d}}t}} = p_{0} /V_{2} \cdot (c_{{0,21}} - c_{{21}} ) + p_{{23}} \cdot g(t) \cdot c_{{17}} $$\end{document} 2.10 \documentclass[12pt]{minimal}
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$$ g{\text{(}}t{\text{)}} = {\text{0}}\quad {\text{for}}\;t < {\text{3}}d\;{\text{else}}\;g(t) = p_{{24}} $$\end{document} 2 2.1 2.10 3 6 6 2.2 p 1 p 14 2.2 2 2.2 2.3 2.6 6 F B t Cells and bioreactors 2 2 2 2 2.8 3 2.10 c 21 2.9 t Fig. 2 circles i c i arrows m p m p 0 AAg AAn’  Fig. 3 n n  Table 2 p m p mlow p mhigh AST GOT GS GLDH p m 3 m Interpretation/enzymes p m p mlow p mhigh Unit PC_1 PC_2 0 Diffusion 49628 [34681, 132980] −1 0.1882 0.3530 1 Aminotransferases (Transamination) LEU 1.816 [0.000, 4.629] −1 0.2679 −0.0187 2 HIS 16.29 [3.42, 620.16] −1 0.0251 −0.2948 3 ARG 100.00 [88.75, 2655] −1 −0.1430 0.2253 4 VAL 1.147 [0.428, 23.112] −1 0.2506 −0.0617 5 TRP 15.99 [7.42, 833.88] −1 0.2022 −0.1063 6 PHE 4.148 [1.740, 9.520] −1 0.2595 −0.0364 7 ILE 3.723 [0.000, 6.213] −1 0.2491 −0.0268 8 ALA 1.573 [0.000, 6.213] −1 0.2514 0.0135 9 TYR 1.135 [0.559, 3.450] −1 0.1866 −0.2730 10 LYS 0.192 [0.000, 2.164] −1 0.2358 0.0053 11 Other specific reactions MET 34.53 [7.27, 284.76] −1 0.2587 0.1171 12 SER 10.80 [3.34, 102.71] −1 0.2698 0.0399 13 GLY 10.76 [2.69, 251.25] −1 0.2674 0.0849 14 THR 9.326 [3.16, 420.70] −1 0.2543 −0.0546 15 Asparaginase 0.0009 [0.0000, 0.002] −1 −1 0.1825 0.4175 16 3.593 [1.215, 8.505] −1 0.2254 0.2971 17 AST 50.08 [39.25, 59.37] −1 −0.0650 0.0341 18 GOT 56.73 [50.35, 71.85] −1 0.1819 0.1456 19 m 225.76 [187.7, 378.9] −1 −1 0.2000 −0.3185 20 5.00 [0.9375, 6.17] −1 −0.1023 0.1813 21 GS 0.0033 [0.001, 0.007] −1 −1 0.1198 −0.1839 22 0.0205 [0.001, 0.039] −1 0.1017 −0.3917 23 GLDH, Protein Synthesis 40.68 [24.12, 50.38] −1 0.1121 0.0330 24 0.5475 [0.465, 0.813] – −0.0057 −0.1114 c i,j,k 3 j. 3 c i t 2.1 2.10 1 c i,j,k 3 \documentclass[12pt]{minimal}
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$$ {\text{mse}} = \frac{1} {{I \cdot K}}{\sum\limits_{i = 1}^I {\frac{1} {{{\mathop {(\max }\limits_k }c_{{i,j,k}} )^{2} }}} }{\sum\limits_{k = 1}^K {{\left( {{c_{{i,j,k}} - {\int\limits_{t_{k} - 24h}^{t_{k} } {c_{{0,i}} (t)} }F_{0} (t){\text{d}}t} \mathord{\left/ {\vphantom {{c_{{i,j,k}} - {\int\limits_{t_{k} - 24h}^{t_{k} } {c_{{0,i}} (t)} }F_{0} (t){\text{d}}t} {{\int\limits_{t_{k} - 24h}^{t_{k} } {F_{0} (t)} }{\text{d}}t}}} \right. \kern-\nulldelimiterspace} {{\int\limits_{t_{k} - 24h}^{t_{k} } {F_{0} (t)} }{\text{d}}t}} \right)}^{2} } } $$\end{document} Results and discussions 3 4 p 0 p 24 2 2 3 Fig. 4 dots 3 n c i t thin lines c i t 3 thick lines  3 p m 5 7 4 6 4 p 0 3 p 1 p 24 p 0 11 12 Table 3 p m m H1 H2 H3 H4 H5 H6 H7 0 93762 48599 154190 60636 318770 94939 81010 1 2.4364 2.1102 2.7550 3.7578 5.8576 2.9302 0 2 224.4944 5.1335 36.9058 50.8039 38.0362 2.9139 21.9417 3 342.2608 285.5303 79.9927 97.7853 55.3615 89.3504 2789 4 1.5042 1.9442 1.3383 2.2465 3.2348 2.2254 0 5 8.3918 12.4793 19.5866 88.6396 44.3098 19.8193 4.8666 6 2.8625 3.7943 8.6391 10.5712 11.0843 4.0756 0.8345 7 4.8314 6.4938 4.0010 6.0186 10.1770 6.6223 0.0102 8 3.5440 0.9192 1.7771 2.0325 5.8235 2.7174 0 9 2.1389 1.5629 2.4463 1.5746 1.2908 0.8548 0 10 0 0.0 2.2151 1.5213 1.6998 0.4708 0 11 10.8005 24.3783 83.0280 53.8904 116.5868 25.3813 12.9908 12 10.0767 9.4709 21.1218 16.3735 31.1820 10.2536 3.3815 13 10.0573 5.1397 27.4213 22.6315 45.1116 10.2223 3.2532 14 11.6650 7.2170 8.5069 22.0442 25.2093 12.4954 3.0131 15 0.0008 0.0007 0.0018 0.0007 0.0129 0.0008 0.0007 16 5.1580 3.4742 3.8554 7.1251 30.9168 3.7947 1.2083 17 40.4685 59.4058 37.9076 63.5569 60.3473 43.3760 69.3468 18 69.9174 57.3297 72.5782 70.4225 109.7867 52.5201 63.2809 19 291.5727 192.9329 341.5933 334.0218 238.6410 263.7827 111.2798 20 5.5669 8.4229 3.2504 4.8261 3.8251 3.7808 8.6594 21 0.0014 0.0 0.0114 0.0026 0.0006 0.0052 0.0034 22 0.0513 0.0 0.0031 0.0280 0.0004 0.0157 0.0000 23 52.5970 35.6001 51.8982 49.6467 67.2338 39.3603 26.0489 24 0.5707 0.6833 0.1963 0.5933 0.6132 0.6823 0.6408 mse 0.0162 0.0143 0.0202 0.0115 0.0135 0.0145 0.0321 Fig. 5 Measured data of the 20 variables of bioreactor run H1 with the simulated kinetics of the model fitted to the data of H1  Fig. 6 5  4 7 p 1 4 8 7 8 Table 4 t B 1 F B t p 1 p 2 p 22 Run No. t B mse [-] p 1  −1 p 2 −1 p 22 −1 H1-H7 (averaged) 3 0.0099 1.816 16.29 20.5 H1 3 0.0162 2.4364 224.4944 51.2761 H2 3 0.0143 2.1102 5.1335 0 H3 3 0.0202 2.7550 36.9058 3.1020 H4 3 0.0115 3.7578 50.8039 28.0101 H5 3 0.0135 5.8576 38.0362 0.4057 H6 3 0.0145 2.9302 2.9139 15.7285 H7 3 0.0321 0 21.9417 0.0186 M8 3 0.0147 1.0367 8.7742 0.0497 M9 5 0.0330 0 2.8611 0.0381 M10 5 0.0199 0.0008 11.2947 0.0578 M11 3 0.0313 0.0010 32.3658 0.2347 M12 3 0.0259 0.9387 168.2299 0.0722 M13 2–4* 0.1559 0.0003 11.6439 0.1391 M14 3 0.0492 0.0003 10.9787 0.1615 L15 0 0.2747 0 39.9225 0.0572 L16 6 0.0803 0 2.5935 0.0268 L17 3 0.0864 0 137.2721 0.0131 L18 3 0.1321 0 1.2613 0.0560 Fig. 7 5  Fig. 8 p 0 p 24  8 13 p 1 p 14 p 2 p 3 p 9 2 4 p 1 p 1 p t p m p 1 8 r p 11 p 13 p 4 p 7 p 1 r p 4 p 7 p 1 p 4 p 7 8 p 2 p 22 p 2 p 22 p 2 p 22 2 5 p 2 p 22 14 p 21 3 15 Conclusion The kinetics of 18 amino acids and the related nitrogen-containing compounds NH3 and UREA in a primary human liver cell bioreactor were analyzed and modeled using a differential equation system. The model focuses on the kinetics of GLU and ASP as well as on the formation and elimination of NH3 and the synthesis of UREA. It describes the degradation of amino acids by transamination, oxidative deamination and other specific reactions. In addition, the activities of selected enzymes such as AST, GOT, GS, and GLDH as well as, in a more aggregated form, the activities of urea cycle enzymes are included. The differential equation system does not represent a fully mechanistic but rather a phenomenological model since essential metabolic activities had to be neglected because they cannot be identified based on the measured data. The differential equation system allows the analysis of a number of representative liver cell functions in terms of their kinetic behavior. The identification of the model parameters by fitting the model responses to the measured data was used to generate hypotheses about the causes of specific differences between the bioreactor runs. The model fits were found to be very satisfactory for eight high and medium performance runs. The model is however inadequate for low performance runs and with respect to the LYS kinetics also for high performance runs. This is probably caused by the neglection of proteases activities in the model that appear to be relevant for low and medium performance runs. Both, protein synthesis and degradation could not be modeled in detail due to the lack of representative protein measurements. The applied model based analysis of data obtained from the bioreactor system can be used to quantitatively evaluate the functional state of liver cell cultures under high performance conditions intended for clinical application in extracorporeal liver support systems. The approach can also be used to study the effect of several exogenous factors, e.g., of hormones or drugs, on hepatocyte metabolism in vitro. The model based analysis methods applied here therefore provide suitable tools for in silico studies supplementing in vitro studies of hepatocyte functions in a systems biological way.