Optimising the fermentation throughput in biomanufacturing with bleed–feed

Figures & data

Figure 1. Long description. (a) A line graph displaying the biomass amount over time (when bleed-feed is not implemented). The x-axis represents the time and the y-axis denotes the biomass amount. The fermentation starts with a small amount of initial biomass. Next the biomass amount increases exponentially over time and then stays constant. Harvest time is indicated at the stationary phase. (b) A line graph displaying the biomass amount over time (when bleed–feed is implemented). The x-axis represents the time and the y-axis denotes the biomass amount. The biomass accumulates exponentially over time until bleed–feed is performed. The bleed–feed is performed instantaneously. Hence, the biomass amount immediately drops to a small value when bleed–feed is performed. The figure shows two examples to illustrate the biomass accumulation after bleed–feed, i.e. the starting biomass amount after bleed–feed is low in one example and high in the other. These two examples have the same cell growth rate. The example with a higher initial biomass reaches a certain biomass level sooner than the other one.

Biomass growth over time in current practice (a) and with bleed–feed (b). (a) Biomass growth without bleed–feed and (b) biomass growth with bleed–feed. (a) A line graph displaying the biomass amount over time (when bleed–feed is not implemented). The biomass amount increases exponentially over time and then stays constant. (b) A line graph displaying the biomass amount over time (when bleed–feed is implemented). The biomass accumulates exponentially over time until bleed–feed is performed. The biomass amount immediately drops to a small value when bleed–feed is performed. The figure shows two examples to illustrate the biomass accumulation after bleed-feed, i.e. the starting biomass amount after bleed–feed is low in one example and high in the other.

Table 1. Notation used in the renewal model.

$b_i$	Initial biomass amount for the ith cultivation
$\mathbb{E}\left[Y\right(t_b,b_2\left)\right]$	Expected yield obtained in a fermentation cycle when bleed–feed is performed at time $t_b$ and the second cultivation starts with $b_2$ units biomass
$f\left(m\right)$	Density of batches that enter the stationary phase at a biomass m
$F\left(m\right)$	Long-run fraction of batches that enter the stationary phase at a biomass of at most m
$g\left(t\right|b)$	Probability density function (pdf) of time to enter the stationary phase given starting amount of b units biomass
$G\left(t\right|b)$	Cumulative distribution function (cdf) of time to enter the stationary phase given starting amount of b units biomass
$m^{\prime }$	The critical biomass amount
$m(t,b)$	Biomass accumulation during the exponential growth phase by time t, when the fermentation starts with b units biomass
μ	Biomass growth rate
$\nu \left(m\right)$	Rate of entering stationary phase under m units of biomass
$R(t_b,b_2)$	Expected throughput when bleed–feed is performed at time $t_b$ and the second cultivation starts with $b_2$ units biomass
s	Bioreactor setup duration
$t_b$	Time to bleed–feed
$T_i$	Random time to enter the stationary phase at the ith cultivation
$t_i$	Realisation of the exponential growth phase's duration for the ith cultivation
$t_h$	Time to harvest a cultivation
$W(t,b)$	Random yield obtained from a cultivation by time t when the starting biomass amount is b

Figure 2. Plots of $\nu \left(m\right),G\left(t\right|b_1)$, $\mathbb{E}\left[W\right(t_h,b_2\left)\right]-b_2$ and $R(t_b,b_2^{\ast })$, for the industry base case (solid line), and a scenario with a higher risk of entering the stationary phase (dashed line).

Four line plots displaying (i) the rate function versus the biomass amount; (ii) the cumulative distribution function of entering the stationary phase versus time; (iii) the expected yield obtained from the second cultivation versus the starting biomass amount for the second cultivation; and (iv) the throughput function versus bleed–feed time. All four figures plot two specific cases using a solid and a dashed line, i.e. a base case which represents the industry case study (solid line), and the case when the risk of entering the stationary phase is higher (dashed line). The figures show that the high-risk case dominates the base case in the rate and cumulative distribution function plots whereas the base case dominates the high-risk case in the expected yield and throughput plots.

Figure 3. The rate function used in the base case (a) and the corresponding probability density function (b). (a) The rate function $\nu \left(t|b_1\right)$ and (b) the corresponding probability density function, $g\left(t|b_1\right)$.

(a) A line graph displaying the rate function (i.e. rate of entering the stationary phase) versus the biomass amount for the base case setting. The -axis represents the biomass amount and the -axis denotes the rate. The rate is equal to 0 until 9 g biomass, and then linearly increases until 20 g. The rate is equal to 1 when the biomass amount is greater than 20 g. (b) A bell-shaped graph displaying the probability density function versus time. The -axis represents the time and the -axis denotes the probability density function.

Table 2. Base case results and the room for improvement on current practice.

CP	NBP			BO			JBO
R	$t^{\prime }$	$R\left(t^{\prime }\right)$	%I	$t_b^{\ast }$	$R\left(t_b^{\ast }\right)$	%I	$t_b^{\ast }$	$b_2^{\ast }$	$R(t_b^{\ast },b_2^{\ast })$	%I
0.156	36.6	0.176	12%	37.3	0.177	13%	37.2	0.2	0.183	17%

Figure 4. Base case throughput $R\left(t_b\right)$ versus $t_b$ in BO strategy.

A line graph displaying the expected throughput as a function of the bleed–feed time (considering the base case under strategy BO). The expected throughput reaches its peak value of 0.177 at the bleed–feed time 37.3. This function has a convex shape before the peak and is decreasing after the peak.

Figure 5. Surface plot depicting base case throughput $R(t_b,b_2)$ versus $t_b$ and $b_2$ in JBO strategy, when $t_b$ and $b_2$ are optimised simultaneously.

A 3D surface plot displaying the expected throughput as a function of the bleed–feed time and the initial biomass amount for the second cultivation. This 3D plot is depicted for the strategy JBO. The expected throughput reaches its maximum value of 0.183 when the bleed–feed time is 37.2 and the initial biomass amount is 0.2 g.

Figure 6. Reducing two-dimensional JBO into two sequential one-dimensional optimisation problems: (a) $\mathbb{E}\left[W\right(t_h,b_2)-b_2$ versus $b_2$ and (b) $R(t_b,b_2^{\ast })$ versus $t_b$ for $b_2^{\ast }$.

(a) A line graph displaying the expected yield as a function of the starting biomass amount after bleed–feed (under the base case). The -axis represents the starting biomass amount for the second cultivation, and the -axis denotes the expected yield obtained from the second cultivation. The expected yield increases steeply until 0.2 g biomass, after which it decreases slowly. (b) A line graph displaying the expected throughput as a function of the bleed–feed time (under the base case). The expected throughput reaches its peak value of 0.183 at the bleed–feed time 37.2, and has a convex shape before the peak.

Table 3. Sensitivity on the biomass growth rate μ.

		NBP			BO			JBO
μ	CP R	$t^{\prime }$	$R\left(t^{\prime }\right)$	$\mathrm{\%}I$	$t_b^{\ast }$	$R\left(t_b^{\ast }\right)$	$\mathrm{\%}I$	$t_b^{\ast }$	$b_2^{\ast }$	$R(t_b^{\ast },b_2^{\ast })$	$\mathrm{\%}I$
0.04	0.147	54.9	0.147	0%	0	0.147	0%	0	–	0.147	0%
0.045	0.150	48.8	0.155	4%	49.5	0.156	4%	49.4	0.6	0.158	6%
0.05	0.152	43.9	0.163	7%	44.6	0.163	7%	44.6	0.4	0.168	10%
0.055	0.154	39.9	0.170	10%	40.6	0.170	11%	40.6	0.3	0.176	14%
0.06	0.156	36.6	0.176	12%	37.3	0.177	13%	37.2	0.2	0.183	17%
0.065	0.158	33.8	0.182	15%	34.5	0.183	15%	34.4	0.2	0.190	20%
0.07	0.160	31.4	0.187	17%	32.1	0.188	17%	32.0	0.1	0.196	22%
0.075	0.162	29.3	0.192	18%	30.0	0.194	19%	29.9	0.1	0.202	24%
0.08	0.164	27.5	0.197	20%	28.1	0.198	21%	28.1	0.1	0.207	26%

Figure 7. Throughput $R\left(t_b\right)$ versus bleed–feed time $t_b$ in BO strategy for slow growth ($\mu =0.04$).

A line graph displaying the expected throughput as a function of the bleed–feed time (for a slow growing batch under strategy BO). When the bleed–feed time is between 0 and 56, the expected throughput has a convex shape (i.e. it is first decreasing and then increasing), where the maximum throughput 0.147 is reached at time zero.

Table 4. Sensitivity on the critical biomass level $m^{\prime }$.

	CP	NBP			BO			JBO
$m^{\prime }$	R	$t^{\prime }$	$R\left(t^{\prime }\right)$	$\mathrm{\%}I$	$t_b^{\ast }$	$R\left(t_b^{\ast }\right)$	$\mathrm{\%}I$	$t_b^{\ast }$	$b_2^{\ast }$	$R(t_b^{\ast },b_2^{\ast })$	$\mathrm{\%}I$
1	0.040	0	0.040	0%	4.7	0.041	2%	1.4	0.1	0.051	28%
3	0.074	18.3	0.081	9%	20.5	0.082	10%	20.0	0.2	0.090	21%
5	0.104	26.8	0.115	11%	28.2	0.116	12%	28.0	0.2	0.124	19%
7	0.131	32.4	0.147	12%	33.4	0.148	13%	33.3	0.2	0.154	18%
9	0.156	36.6	0.176	12%	37.3	0.177	13%	37.2	0.2	0.183	17%
11	0.180	40.0	0.204	13%	40.4	0.205	13%	40.4	0.3	0.210	17%
13	0.203	42.7	0.230	13%	43.1	0.231	14%	43.1	0.3	0.236	17%
15	0.223	45.1	0.255	14%	45.3	0.255	14%	45.3	0.3	0.261	17%

Table 5. Sensitivity on maximum biomass that can be obtained H.

	CP	NP			BO			JBO
H	R	$t^{\prime }$	$R\left(t^{\prime }\right)$	$\mathrm{\%}I$	$t_b^{\ast }$	$R\left(t_b^{\ast }\right)$	$\mathrm{\%}I$	$t_b^{\ast }$	$b_2^{\ast }$	$R(t_b^{\ast },b_2^{\ast })$	$\mathrm{\%}I$
12	0.134	36.6	0.161	20%	36.8	0.161	20%	36.8	0.2	0.168	25%
15	0.144	36.6	0.167	16%	37.0	0.168	17%	37.0	0.2	0.175	21%
20	0.156	36.6	0.176	12%	37.3	0.177	13%	37.2	0.2	0.183	17%
25	0.167	36.6	0.183	10%	37.5	0.184	11%	37.5	0.3	0.190	14%
30	0.176	36.6	0.190	8%	37.7	0.191	8%	37.6	0.3	0.197	12%
35	0.185	36.6	0.195	6%	37.9	0.197	7%	37.8	0.3	0.203	10%
40	0.193	36.6	0.201	4%	38.1	0.203	5%	38.0	0.4	0.208	8%
45	0.201	36.6	0.206	3%	38.2	0.209	4%	38.1	0.4	0.213	6%
50	0.208	36.6	0.211	2%	38.4	0.214	3%	38.3	0.4	0.218	5%
55	0.215	36.6	0.216	0%	38.5	0.219	2%	38.4	0.4	0.223	4%

Table 6. Sensitivity on setup duration s.

	CP	BO			JBO
s	R	$t_b^{\ast }$	$R\left(t_b^{\ast }\right)$	$\mathrm{\%}I$	$t_b^{\ast }$	$b_2^{\ast }$	$R(t_b^{\ast },b_2^{\ast })$	$\mathrm{\%}I$
0	0.174	37.3	0.190	9%	37.2	0.2	0.197	13%
2	0.169	37.3	0.186	10%	37.2	0.2	0.193	14%
4	0.165	37.3	0.183	11%	37.2	0.2	0.190	15%
6	0.160	37.3	0.180	12%	37.2	0.2	0.186	16%
8	0.156	37.3	0.177	13%	37.2	0.2	0.183	17%
10	0.153	37.3	0.174	14%	37.2	0.2	0.180	18%
12	0.149	37.3	0.171	15%	37.2	0.2	0.177	19%
14	0.145	37.3	0.168	16%	37.2	0.2	0.174	20%