Management of Blood Component Preparation

1
Management ofBlood Component Preparation

• Speaker Chun-Cheng Lin
• National Taiwan University
• Co-authors Chang-Sung Yu, Yin-Yih Chang

2
Outline

• Introduction to
• the blood component preparation problem (BCPP)
• A linear time algorithm for the BCPP
• Some variants of the BCPP
• Conclusion and future work

3
Introduction

• Transfusion therapy
• to transfuse the specific blood components
• needed to replace particular deficits
• for some medical purposes.
• Whole blood
• contain all blood elements
• a source for blood component production.
• Blood component preparation
• the indication for the use of unfractionated
  whole blood
• ? almost does not exist now
• separating specific cell components from the
  whole blood
• a lot of different methods (processes) or
  equipment
• ? different use, efficiency and quality

4
A Process of Separating Blood Components
Whole Blood
soft-spin centrifugation
Packed Red Blood Cells
Platelet-Rich Plasma
hard-spin centrifugation
washing
Washed Red Blood Cells
Platelet Concentrate
Fresh Plasma
freezing
Fresh Frozen Plasma
thawing centrifugation freezing
Frozen Plasma
Cryoprecipitate

• Implied value (ai) Consider both the revenue
  contributed by patients or insurance and the
  costs induced by blood of collection, testing,
  preparation, preservation, storage, processing
  time, etc.
• Demand limit (di).

5
Blood Component Preparation Problem

• Blood Component tree
• vertex vi a blood component
• with value ai and demand limit di
• the amount xi of vi is derived from
• the amount of its parent
• according to a given ratio ri
• Initial assignment of xi
• x1 Q other xi 0
• The Blood Component Preparation Problem (BCPP)
• Given an initial volume Q of the whole blood and
• an n-vertex blood component tee T
• (where demand limit di implied value ai),
• determine the assignments of xi
• so that (1) the total value is maximized
• (2) while the demand limit of each
  component is satisfied

6
A linear programming approach

• The BCPP problem can be solved by linear
  programming.

Comment There exist a lot of software tools for
the linear programming problem, users just need
to describe the BCPP as a linear program and then
use those tools to solve it without implement it.
7
Example

Q 100
8
Motivations

• Linear programming (LP) problem
• 2 drawbacks to solve the BCPP by LP
• The worst-case algorithm for the LP problem may
  not be executed efficiently (its time complexity
  is nonlinear)
• It may not be convenient for users to directly
  describe the constraints of the LP for a general
  derivatives tree.

9
Main result

• Main Theorem There exists a linear time
  algorithm for the BCPP in the size of vertices.
• Characteristic value ? (vi) of vi
• Compute the following formula in the bottom-up
  fashion
• e.g.,
• ? (v8) 2 ? (v9) 6
• ? (v6) max( 3, 0.7?2 0.3?6 ) 3.2

10
Our linear time algorithm (1/4)

• Step 1 ? vertex vi in top-down fashion of T,
• xi is assigned di, and then the remaining amount
  is forwarded to the next level
• if not enough to satisfy any demand limit, then
  return false.
• for convenience, we express that xi di yi.

3
10
100
0.8
0.2
4
4
8
5
x4
x2
40
50
x1
0.2
x5
0.5
1
0.3
x8
2
12
1
7
3
8
1
9
x9
x6
x3
x7
121.6
713.4
80
94
0.7
0.3
2
13
6
6
135.2
61.8
11
Our linear time algorithm (2/4)

• Step 2 ? vertex vi in bottom-up fashion of T,
• If vi is a leaf, then yiM yi
• o.w., yiM minvj?Child(vi) yjM / rj ,
• yim yj rj yiM for each vj ? Child(vi).
• (In fact, yiM is the maximal possible amount of
  vi flowed from its descendents and yim is the
  amount of every descendent of vj of vi after yiM
  is achieved)

y1m
0
10
y1M
y2m
y3m
0
2
8
4
y2M
y3M
y7m
y6m
y4m
0
11
2
0
1.6
13.4
6
4
y7M
y6M
y4M
y8m
y9m
1
0
5.2
1.8
y8M
y9M
12
Our linear time algorithm (3/4)

• Step 3
• Initially, all the leaves of T are marked.
• For each internal vertex vi in the bottom-up
  fashion of T, compute?? (vi). If ? (vi) ai,
  then vertex vi is marked.

v4, v5, v7, v8, v9 are leaves, and hence,
marked. ? (v6) max(3, 0.7?20.3?6) 3.2 gt 3
a6 ? v6 is unmarked. ? (v3) max(8, 1?1)
8 a3 ? v3 is marked. ? (v2) max(4,
0.2?20.3?10.5?3.2) 4 a2 ? v2 is marked. ?
(v1) max(3, 0.8?40.2?8) 4.8 gt 3 a1 ? v1 is
unmarked.
y1m
0
10
y1M
y2m
y3m
0
2
8
4
y2M
y3M
y7m
y6m
y4m
0
11
2
0
1.6
13.4
6
4
y7M
y6M
y4M
y8m
y9M
1
0
5.2
1.8
y8M
y9M
13
Our linear time algorithm (4/4)

• Step 4 Let U V. ? vi ? U in top-down fashion
  of T, if vi is unmarked then output xi di 0
  otherwise, do the following
• Output xi di yiM
• ?? vj in the top-down fashion of the subtree Tvi
  rooted at vi, if vj is marked, then output xj
  dj yjm otherwise, for evry children vk of vj,
  ykm ? ykm rk yjm, and then output xj dj 0
  (i.e., yjm 0)
• U?? U \ V(Tvi)

3
10
y1m
0
100
10
y1M
0.8
0.2
4
4
8
5
y2m
y3m
0
2
40
50
4
y2M
8
y3M
0.2
0.5
1
0
0.3
y6m
2
12
1
7
3
8
1
9
11
2
0
y7m
y4m
0
121.6
713.4
80
94
y6m
1.6
13.4
6
4
y4M
y7M
0.3
0.7
2.4
0.6
2
13
6
6
y8m
y9m
1
0
135.2
61.8
5.2
1.8
y8M
y9M
14
Observations of our outputs

• Observation 1.
• Observation 2. In the output of our algorithm,
• For every vertex vi in R1, ai ?lt ? (vi).
• For every vertex vi in R2, ai ? ? (vi).
• Observation 3. Any feasible solution of the BCPP
  is reachable from the output of our algorithm.

R1
above the band
on the band
R2
R3
below the band
Output of Step 1 of our algorithm
Output of our algorithm
15
Comparison of solutions
R1
above the band
on the band
R2
R3
below the band
Output of Step 1 of our algorithm
Output of our algorithm
R11

R12
R1

R13
band
R2
R3
Any feasible solution
16
Blood Component Preparation Problem

• Theorem. The BCPP can be solved in O(n) time.
• Proof. Skipped.
• Extension How to choose multiple standardized
  processes so that the total value is maximized?
• The preparation and preservation of blood
  components are considered within a time frame.
• In practice, we may execute more than one
  process simultaneously within a certain time
  frame.
• Fortunately, for the standardization of executing
  processes, the number of alternative processes is
  fixed constant.
• The extended problem can be solved roughly in
  polynomial time.

17
Modified Blood Component Preparation Problem

• The deriving operation may be nonreversible.
• ? require the volume of the components at
  higher levels ?is more.
• The Modified Blood Component Preparation Problem
  (BCPP)
• Given the initial volume Q of the whole blood
  and
• an n-vertex blood component tree T
• (every vertex has its demand limit),
• determine the assignments of xi
• so that (1) the volume of the components at
  higher levels
• is remained as more as possible
• (2) while the demand limit of every
  component is satisfied.
• Algorithm
• Steps 1 and 2 are the same as those of the
  previous.
• Step 3 Output x1 d1 y1M xi di yim, ?
  vi ? V \ v1.
• Corollary. The BCPP can be solved in O(n) time.

18
Conclusion and future work

• We have defined a new problem called the blood
  component preparation problem (BCPP), and
  proposed not only a linear programming solution
  but also a linear time algorithm for the BCPP.
• Some variants are also given in this work.
• A line of future work includes
• to investigate the tractability of the
  derivatives graph problem on some special cases
  of graphs
• to take more factors (e.g., time and inventory)
  into account in the BCPP
• to evaluate the effectiveness of the BCPP applied
  in practical environment
• to investigate the sensitivity analysis and the
  critical paths of the BCPP.

19
Thank you for your attention!